dimensioning of metallic conetctions ec 3

Dimensioning of Metallic Connections per EC3

Manual of Analysis and verification examples

Analysis reference and verification.

December 2007

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Contents

1. Basic Calculation principles...................................................... 1 1.1. Force distribution on connecting means 1.1.1. Transfer of torque coplanar to bolting on bolts 1.1.1.1. Free centre of rotation 1.1.1.2. Forced centre of rotation 1.2. Calculation of fillet weld strength 1.3. Impact of plate deformability in bolted connections 1.3.1. Failure of strut footing 1.3.1.1. Welding connection 1.3.1.2. Bolted connection 2. Connection of strut beam bolted on the beam body............. 28 2.1. Description of mechanical behavior 2.1.1. Transfer of stress through the connection means 2.1.2. Potential connection failure mechanisms 2.1.3. Additional check for hollow rectangular strut cross-section. 2.1.4. Classification of connection 2.2. Detailed calculation of connection 2.2.1. Clearance check 2.2.1.1. Bolt Clearances 2.2.1.2. Welding thickness check 2.2.1.3. Beam height adequacy check 2.2.2. Finding the maximum bolt shearing force 2.2.3. Bolting shearing strength 2.2.4. Strength under intrados compression 2.2.5. Tearing strength of connection beam segment

User Manual v3.0

Contents

2.2.6. Tearing strength of connection plate segment 2.2.7. Welding check 2.2.8. Strength against front failure of hollow strut footing 2.2.8.1 Connecting plate tensile strength 2.2.8.2. Check 3. Strut beam connection with double angular plates .............. 42 3.1. Description of mechanical behavior 3.1.1. Transfer of stress through the connection means 3.1.2. Critical connection failure mechanisms 3.1.3. Classification of connection 3.2. Detailed calculation of connection 3.2.1. Clearance check 3.2.1.1. Bolt clearance check 3.2.1.2. Angular plate height adequacy check 3.2.2. Shearing strength of a bolt 3.2.3. Strut bolt check under shearing 3.2.4. Strut bolt check under intrados compression 3.2.5. Beam bolt check under shearing 3.2.6. Beam bolt failure under intrados compression 3.2.7. Check of angular plates under shearing 3.2.8. Check of angular plates under bending 3.2.9. Beam section tearing strength 3.2.10. Tearing strength of an angular plate leg section on a beam 3.2.11. Tearing strength of an angular plate leg section on a prop 3.2.12 Case of welding on beam connection 4. Beam on Beam connection with double angular plates ....... 61 4.1. Description of mechanical behavior 4.1.1. Transfer of stress through the connection means 4.1.2. Critical connection failure mechanisms 4.1.3. Classification of connection 4.2. Detailed calculation of connection

User Manual v3.0

Contents

4.2.1. Clearance check 4.2.1.1. Bolt Clearances 4.2.1.2. Beam height adequacy check 4.2.2. Shearing strength of a bolt 4.2.3. Main beam bolt check under shearing 4.2.4. Main beam bolt check under intrados compression 4.2.5. Secondary beam bolt check under shearing 4.2.6. Secondary beam bolt check under intrados compression 4.2.7. Check of angular plates under shearing 4.2.8. Check of angular plates under bending 4.2.9. Secondary beam section tearing strength 4.2.10. Tearing strength of an angular plate leg section on a secondary beam 4.2.11. Tearing strength of an angular plate leg section on main beam 5. Continuity restoration of a beam subjected to bending and tensile stress with beam with body-foot plates............. 76 5.1. Description of mechanical behavior 5.1.1. Transfer of stress through the connection means 5.1.1.1. Transfer of bending moment 5.1.1.2. Transfer of pivotal force 5.1.1.3. Transfer of shearing force 5.1.2. Bolting stressing 5.1.3. Case of member with welded connections 5.1.3.1. Body plate 5.1.3.2. Footing plate 5.1.4. Plate stressing 5.1.5. Member cross-section stressing 5.1.6. Checks carried out 5.2. Detailed connection calculation 5.2.1. Clearance check 5.2.1.1. Bolt clearance check 5.2.1.2. Beam height adequacy check

User Manual v3.0

Contents

5.2.2. Beam cross-section checks at the restoration point 5.2.2.1. Complete cross-section check 5.2.2.2. Decreased cross-section check at the restoration point 5.2.2.2.1. Bending strength check 5.2.2.2.2. Shearing force strength check 5.2.3. Bolt check 5.2.3.1. Footing bolt check 5.2.3.1.1. Bolt strength under footing shearing 5.2.3.1.2. Bolt group strength under intrados compression 5.2.3.2. Body bolt check 5.2.3.2.1. Bolt strength under footing shearing 5.2.3.2.2. Bolt group strength under intrados compression 5.2.4. Footing and body plate check 5.2.4.1. Footing plate check under pivotal force 5.2.4.2. Body plate check under axial force. 6. Strut beam connection with front slabs ................................. 96 6.1. Description of mechanical behavior 6.1.1. General 6.1.2. The logic of semi-rigid connections and the necessity of ductile behavior 6.1.3. Classification of torque-transferring connections 6.1.4. Deformation capacity 6.1.5. Aiming at ductile behavior 6.1.6. The component method in general 6.1.6.1. Transfer of forces to the connection 6.1.6.2. Calculation of initial connection rigidity 6.1.6.3. Required rotating ability 6.1.7. The logic of equivalent short-T 6.1.7.1 Multiple bolt rows 6.1.8. Formation of short-Ts on front slab 6.1.9. Transfer of tensioning forces

User Manual v3.0

Contents

6.1.9.1. Feasibility of reinforcing strut footing 6.1.9.1.1. Reinforcement by transverse reinforcement plates 6.1.9.1.2. Reinforcing by backing plates 6.1.10. Transfer of compressive forces 6.1.11. Transfer of shearing forces 6.1.11.1. Bolt shearing forces 6.2. Detailed calculation of connection 6.2.1. Clearance check 6.2.1.1. Bolt clearance check 6.2.1.2. Check of strut body reinforcing plate clearances 6.2.1.3. Check of strut footing reinforcing plate clearances 6.2.1.4. Check of beam strut angle 6.2.1.5. Welding check 6.2.2. Initial calculations and assessment of general connection parameters 6.2.2.1. Values of the parameter of transformation β 6.2.2.2. Strut shearing surface 6.2.2.3. Calculation of effective width of the strut body under compression 6.2.2.4. Decreasing factor ω for the interaction of compression and shearing 6.2.3. Solving the connection 6.2.3.1. Finding connection component rigidity coefficients 6.2.3.1.1. Preliminary assessment of the number of tensioned bolt rows 6.2.3.1.2. Rigidity coefficients for the first tensioned bolt row. 6.2.3.1.3. Rigidity coefficients for the second tensioned bolt row. 6.2.3.1.4. Rigidity coefficients for the third tensioned bolt row.

User Manual v3.0

Contents

6.2.3.1.5. Rigidity coefficients for the forth tensioned bolt row. 6.2.3.1.6. Calculation of the equivalent rigidity coefficient keq 6.2.3.1.7. Other connection rigidity coefficients 6.2.3.2. Finding the strengths of the basic connection components 6.2.3.2.1. Strut body under shearing 6.2.3.2.2. Strut body under compression 6.2.3.2.3. Beam body and footing under compression 6.2.3.3. Repetitive process for finding the number of tensioned rows 6.2.3.4. Connection classification 6.2.3.4.1. Finding the initial rigidity SiM 6.2.3.4.2. Classification according to rigidity 6.2.3.4.3. Classification according to strength 6.2.3.5. Check of connection rotating ability 6.2.3.6. Check of shearing force 6.3. Flow diagrams 6.3.1. Finding the initial rigidity Sj,ini 6.3.2. Finding strength torque 7. Prop support connection....................................................... 163 7.1. Description of mechanical behavior 7.1.1. General 7.1.2. Elastic resistance of the prop base 7.1.3. Resistance of prop base with low quality mortar 7.1.4. Comparison of the calculations of concrete strength according to EC2 and EC3 7.1.5. Accumulation of stresses under the prop base 7.1.6. Effective length of the short-T plate base 7.1.7. Slip coefficient between steel and concrete

User Manual v3.0

Contents

7.1.8. Transfer of shearing forces through anchors 7.1.9. Transfer of shearing forces through friction and anchors 7.1.10. Bolt anchoring rules 7.1.10.1. Theoretical data 7.1.10.2. Check for bending of anchoring bodies 7.1.11. Impact of the biaxial nature of loading to the connection analysis procedure according to EC3 7.1.12. Assumptions in order to eliminate the difficulties arising due to the fact that EC3 does not provide instructions for biaxial loading 7.1.12.1. Strength of the tensioned side of the node 7.1.12.2. Strength of the compressed side of the node 7.1.12.3. Total connection strength 7.1.12.4. Necessity of employing interaction diagrams 7.1.12.5. Calculation of pivotal rigidity 7.2. Detailed connection calculation 7.2.1. Check of Distances 7.2.1.1. Check of anchor distances 7.2.1.2. Check of cement grout thickness 7.2.1.3. Check of weldings 7.2.2. Connection calculation 7.2.2.1. Finding the basic node parameters 7.2.2.2. Starting and ending points for neutral axis shift within the cross-section 7.2.2.3. Solving for the neutral axis at a specific point of the cross-section 7.2.2.3.1. Tensioned side of the node 7.2.2.3.1.1. Assessment of strength of each tensioned anchor 7.2.2.3.2. Compressed side of the node 7.2.2.3.3. Node strength 7.2.2.3.4. Pivotal rigidity of node 7.2.2.4. Check of anchoring on the foundation

Analysis reference & verification manual

1 Basic Calculation principals

1.1. Force distribution on connecting means

1.1.1. Transfer of coplanar to bolting torque on bolts

In this chapter, we shall discuss the method employed to transfer a coplanar to a torque bolting in order to find the acting shearing force at the most adverse bolt. One may discern two cases, depending on the “type” of the bolting rotation pole. Presuming that this pole is located in the Gravity centre of the bolt cluster, it is called Free Centre of Rotation. In the opposite case, where the centre of rotation would not coincide with the bolting centre, then such centre should called Forced Centre of Rotation. In the following sections, each case shall be discussed separately:


1.1.1.1. Free Centre of Rotation

In the event of slight deformations to bolted plates, a linear relation between the cutting force Ri, which is acting on bolt "i" and the displacement of “δi” may be assumed, resulting to calculating the force of each bolt proportionately to the distance from the rotation centre and the bend θ as shown in the following graph:

Fig. 1.1. Cluster of bolts loaded by coplanar torque.

Fig. 1.2. Determination of bolt forces


The algebraic relations of determining such forces are as follows:

Since bolting is undergoing true bending, then the equilibrium equations should be illustrated as follows:

From the two last relations it appears that the centre of rotation truly coincides with the gravity centre of the bolting. Finally, the external torque satisfies the relation:

where from it appears the shearing force to the most adverse bolt:

If an eccentric force would act on a bolt cluster, the following method could be employed: The force R results from the flexural torque M=eF; whereas F/n results from the eccentric force F (where the bolt number).


As shown, the eccentric force F may be replaced by a flexural torque M and a force of even direction on the centre of rotation. The bolt forces result from superposition due to bending and shear. The resultant force may be analyzed in components along the main axes:

In the previous sections, we assumed an elastic analysis for the distribution of forces on the bolts. It is reminded that the tension could be also distributed plastically, if the bolt shear strength was greater / smaller than the ductile strength against intrados compression, which would result to rendering the possibility of plastic intrados deformation and equalization of forces causing fatigue to each bolt.

Fig. 1.3 Determination of bolt forces on a free centre of rotation subject to eccentric force.


In such case, namely in a marginal failure condition, all bolts would be fatigued, their strength being in intrados compression and in direction vertical to the actinic distance between the bolt and the momentary centre of rotation.

The distance of the momentary centre of rotation from the Gravity Centre of the bolting is defined as x0. According to the plastic analysis, there results an algebraic system to be resolved, since the force equilibrium has been taken toward the axis x as well as that of torques about the centre of rotation A. From the solution of said system, the distance x0 as well as the strength shearing force VRd are taken. Since the cutting force did not have the direction of some main axis as in the previous case, then it would have been also required to set out the equilibrium equation to the other axis, in which case a 3x3 system would have resulted.

Fig. 1.4. Plastic force distribution on bolting due to torque


Fig. 1.5 Force distribution on bolts

Elastic distribution

Plastic distribution

Plastic deformations

Fig. 1.6. Relation between a force causing fatigue to a bolt and a growing deformation.

Elastic distribution

Plastic distribution


1.1.1.2. Forced centre of rotation In a front slab connection, as the one of the following figure, a

considerable rigidity difference may be observed between the tension and compression areas.

In the compression zone, the compressive force is transferred directly from the beam footing to the strut body. The deformations observed in this area are minor compared to those developed in the tensioned zone, where bending of the front slab, as well as of the strut footing is taking place. Due to such difference in rigidity, the point 1 of figure 1.7 is considered as the centre of rotation. For safety purposes, many a times it is considered that this also coincides with the location of the lower bolt row. The thicker and therefore more rigid the front slab becomes, the more the centre of rotation tends towards its lower end.

Assuming that the rigidity observed is equal to all tensioned bolt rows, the tensioning forces of each row are directly proportional to their distance from the centre of rotation.

In accordance with the foregoing, the following may apply:

Fig. 1.7. Forces of bolt rows in front slab connection, having forced centre of rotation (2 rows of bolts).

Forced centre of rotaton

Compressing forces


For bolts of the same size:

In fact, the rigidity of the tensioned bolt rows may differ considerably amongst them; for instance, the protruding section of the front slab of the upper beam footing is more flexible than the section located underneath the beam footing, where the body of the beam contributes greatly to rigidity increase. As a result, the row 2 of figure 1.8 would apparently receive greater load than the row 1.

Fig. 1.8. Impact of the front slab thickness on the force distribution on bolt rows.

Thick front slab Thin front slab


In case of thin front slabs, the differences in rigidity of the various bolt rows are more significant, whereas the distribution of forces deflects much more than a linear distribution.

As regards a front slab of ordinary dimensions, it could be reasonably accepted that the tensioning strength on the upper footing of the beam is distributed on rows 1 and 2. However, since the connection is loaded by the combination of flexural torque and tensioning strength FH, the forced rotation centre might not be applicable; and this is α condition it should be given closer attention. This depends on the size of the tensioning load FH.

In case the centre of rotation is forced (small FH) then such FH is transferred through the rigid point 1. The torque about this point is:

where (a) is the distance between the gravity centre axis of the beam and the point 1. Due to horizontal equilibrium, it is taken:

If , then D=0. For D<0, there is no forced centre of rotation. From

(1.15) and (1.16) it results that if , then there is a forced centre of

rotation; whereas if , then there is a free centre of rotation.

Fig. 1.9. Connection of a front slab under flexural torque and tensioning load.

Forced centre of rotation

Free centre of rotation


1.2. Calculation of fillet weld strength. In this section, we shall present the ground based on which the

strength of the fillet welds on the connections examined shall be calculated. As effective length l of a filler weld it shall be taken the length on which such fillet weld has its full thickness.

The strength of the designing may be calculated either by the Component Method or by the Simplified Method. In this section, reference shall be made to the first method, which is considered as being more accurate, since is has been employed in the calculation of the fillet weld strength in connection under examination. The particular method takes into consideration the tensions to the members to be connected (welded), as shown in Fig. 1.11.

Fig. 1.10. Fillet weld thickness


The fillet weld tensions are calculated as follows:

The alternative method proposed by Eurocode 3 gives:

Fig. 1.11. Tensions on members to be welded and tensions of fillet weld.


The second requirement for s. £ fu/gMw plays some part only if tension “t” is small, namely, when the resulting force deviates from the horizontal (see fig. 1.11).

If only σΖ, is existent, then for the fillet weld thickness it may be taken:

• For S235 steel

• For S355 steel

Since plastic analysis is employed for finding actions during

construction designing; and since the connection shall be implemented on a point where formation of plastic joint is anticipated, then the minimum welding thickness should be determined by the condition sz = fy. This would give:

• For S235 steel

• For S355 steel

This requirement is also used on statically undetermined

structures, elastically designed. It should be stressed out that, even in case of elastic designing, it is presumed that the members and the connections have adequate deformation capabilities, such as to be able to compromise with loads and tensions, which usually are not taken into consideration, i.e. loads attributed to temperature differences. This is one of the reasons for which the members being connected should creep, prior to their weld failing.

When the Eurocode relations apply with sz = fy, then the welding strength shall be at least equal to that of the member being connected. In other words, the failure shall be observed on the member and not on the welding. Accordingly, from the member creeping requirement, prior to the friable failure of the welding, it results that the welding designing may be based on the relation:


where the index “r” denotes the material of the member to be welded. For

instance, for steel S355 it may be taken: Since the value of the ratio fyr /fur may exceed the value 0.70, the following is required: If the deformation feature (ductility) is required, then the welding should be designed in a way to safely bear at least the 80% of the creep force of the weakest member to be connected.

From the foregoing, and referring to the example in Fig. 1.11, a lower limit of a double fillet welding thickness results:

• For S235 steel • For S355 steel

Finally, it is worth to be pointed out that fillet weld thicknesses,

which are calculated on the basis of the mean tension method of the Eurocode, are approximately 1.22 times more than thicknesses calculated on the basis of the component method, in which case the use of steel increases by 1.222 ? 1.5 times! 1.3. Determination of effective welding width between a plate and a non reinforced footing.

Supposing a strut footing on which a plate is welded. Since the

footing on which the plate is welded is not reinforced, then the load would tend to cause deformations, which would have a maximum value on the footing edges and a minimum value in the middle (due to body contribution). As a result, the welding sections that are close to the body would be loaded more, in which case it would be deemed necessary to take this distribution of deformation into account in the analysis. Therefore, only a section of both the fillet weld and the footing would be taken into account on calculating the strength of connection components.

For a double “T” cross-section, the effective width beff should be taken as follows:

however


where fy is the limit of member creep and fyp, is the limit of plate creep. In the case of a hollow cross-section, the following shall be

applicable, in correlation with the foregoing:

however

Fig. 1.12. Effective width of welding on a non reinforced footing.


21.3. Impact of the plate deformation ability on bolted connections. The examination of the impact of the plate deformation ability on a

bolted connection is effected on two basic simulations, the short “T” and the short “L”. The first applies to bilateral connections, such as to open cross-sections; and the latter to unilateral connections, such as closed cross-sections, angles, etc. It is apparent that the strength of the short "T" is twice as much as that of the short "L"; therefore, that is the one to be examined herein below.

Fig. 1.13. Short “T” and short “L” simulations


Supposing that the plate of Fig. 1.14 is welded to the footing of a double “T” cross-section and tensioned. The tensile strength of the connection, as regards the vertical member, shall be determined by the most critical failure mechanism of those manifested.

Correspondingly, the failure mechanisms for a front slab bolted on the strut footing shall be as follows:

Fig. 1.14. Failure mechanisms for plate welded on strut footing.

1. Plastic failure of the strut footing.

2. Creep – breakage of strut body.

3. Welding failure

Fig. 1.15. Failure mechanisms for plate welded on strut footing.

1. Plastic failure of the strut footing.

2. Creep – breakage of strut body.

3. Welding failure

4. Bolt failure

5. Front slab plastic failure (of short “T”)


It is apparent that the failure of such connection results from the analysis process, taking into consideration that the first (weaker and determining) failure mechanism is marginally effected. That is how the strength of each potential failure mechanism, which might occur, is calculated and the minimum is kept, according to which the connection strength result. EC3 provides adequate information on the node components and on said failure mechanisms for strut beam connections as well as on other connection types, such as strut bearing, etc. In the following sections, certain basic failure mechanisms on ordinary connections shall be discussed, and a preliminary approach in determining their strength shall be attempted. 1.3.1. Strut footing failure 1.3.1.1. Welding connection

Fig. 1.16 illustrates a welding connection on a footing of double “T”

cross-section.

Fig. 1.16. Plastic strength forces of a non-reinforced footing.


A part of the tensile power is transferred through rectilinear tensions without bending to the strut footing. The width to which such power applies is twc+2rc. The tensile power is given by the relation:

The remaining part of the power should be transferred through bending of the strut footing to the body. For that type of loading, a model may be adopted as shown in Fig. 1.17, based on a slab of bound movements on its three sides, loaded by a linear load at its centre. The application of plasticity theory maintains that the applied load is directly proportional to the plastic strength torque of the slab:

where: the coefficient “C” results experimentally, whereas a value of said coefficient equal to 14 is considered as safe.

Summarizing the above: the overall strength of a non-reinforced strut footing is given by the minimum value of the following, namely by the most critical mechanism out of the 2 mentioned above.

Fig. 1.17. Connection aspect


where beff is the effective width of the footing, as it has been determined in section 1.3. Because of the limited experiments pertaining to the assessment of coefficient “C”, the effective width, in connection with the determination of Ft2, is limited to 7tfc.

Accordingly:

In order for the connection to develop plastic deformations, ductility is required, which is ensured once the beam footing has crept prior to failing of the welding between footing - plate. Consequently, the designing strength of a non-reinforced footing should be at least equal to the 70% of the creeping strength of the beam footing:

Should this requirement not be satisfied, then the connection should be reinforced by reinforcement plates, as shown in Fig. 1.18:


1.3.1.2. Bolted connection Contrary to the previous condition of a welded connection, in case

of using front slab bolted on the strut footing, the overall tensile strength is transferred through the strut footing, resulting solely to flexural torques.

In order to appreciate the various forces developed in such connections, it would be useful to consider from the beginning a simpler model under tensile. Therefore, two short "Ts" are selected, which are connected by two bolts and loaded by tensile load Ft.

Fig. 1.18. Reinforcement of strut footing.


Fig. 1.19. Transfer of tensile strengths through bending and model for tensile strength transfer to bolted connections.


It is initially supposed that the force of each bolt is equal to 0.5Ft and that the footing has been designed to transfer such forces through bending. The necessary footing “tf” thickness should be calculated as follows:

Fig. 1.20. Form of short “T” failure.

Fig. 1.21. Critical failure mechanism π; bolt creep.


In this case, the following would be applicable:

In the foregoing equations, the bolt and strut footing dimensions have been calculated in a way that the tensile strength of bolts would determine the strength of the connection. In the event of failure, the footings would separate from each other.

In the event that stronger bolts are selected, then the extreme tensile strength would increase, obtaining values higher than those resulted in the previous case for Ft. Accordingly, footings would creep and the bolt deformation decrease at the same time. In case of failure, footings would not completely separate. This means that there is a contact surface between them resulting to developing contact forces. These forces would reasonably cause further flexural torque to the footings. When such forces are strong enough, then said torque would be equal to the plastification torque mpl. If this would be the case, four plastic joints would be formed in marginal condition.


Fig. 1.23. Critical failure mechanism is the slab-footing plastification.

Fig. 1.22. Critical failure mechanism is the footing plastification.


In this case, the following would be applicable:

where

Between the 2 types of failure described above, there is a third type, in which connection forces develop, the bolts however fail prior to footing plastification. In such case, both the bolts and the footing are considered as being critical.

Fig. 1.24. Critical failure mechanism is the slab – footing plastification and the bolt creep.


In relation to this case, the energy method for calculating the marginal tensile strength of the connection shall be illustrated. For the remaining 2 cases, respective relations may be set out towards finding the short “T” strength.

The equation of external and internal force power is stated as follows:

and for δ1, δ2

The previous cases are summarized in the following graph:

It is recommended that the connection is designed so that the critical point would be the failure mechanism (1), namely, critical would be the footing, the reason being that then the deformation ability would be adequately provided (more ductile mechanism). In the third failure type of Fig. 1.25,

Fig. 1.25. Relation between footing (slab) resistance and bolts for a Short “T”.


the deformation is provided to the connection solely by the lengthening of bolts, which is far less than the footing deformation of the previous mechanism. It should be noted that the deformation ability of mechanisms (1) and (2) might increase by selecting bolts threaded lengthwise.

In compliance with EC3, the tensioned zone of a non-reinforced footing should be considered as an aggregate of short “Ts” equal to the total of the effective lengths of each individual footing (Σ?eff). Adopting the plasticity theory, the effective length ?eff of a short “t” may be calculated as shown in Fig. 1.26.


Fig. 1.26. Effective length of an equivalent Short “t” for one and two rows of bolts.


2 Connection of beam – strut by a plate

bolted on the beam body.


2.1. Description of mechanical behavior

Fig. 2.1. Linear node simulation

Linear node simulation

Diagram of a beam shearing force in node area

Diagram of bending torques in node area


2.1.1. Transfer of tension through connection means

Said connection is required to transfer, as stated above, a shearing force from the beam to the strut through the connection means. Moreover, as shown in Fig. 2.4, the bolting is further stressed by a torque coplanar to it, which results from the linear simulation accepted as applicable. Specifically, the stress is transferred as follows:

• Shearing force and torque at the bolting centre of gravity by resolving a framing metallic carrier.

• These values create a force shearing the bolt threads. • The overall shearing and flexure are transferred through intrados

forces to the plate. • This transfers to the welding a torque vertical to its plane and a

shearing parallel to its axis (see fig. 2.3). • These values are transferred to the strut footing and are axially

received by it.

In the general case of bolting consisting of many columns and rows of bolts, the shearing force Ved is considered as being exercised at its gravity centre, as well as the torque of value e2Ved. Then, it is distributed to the bolts, following the calculation of the polar moment of inertia of the bolting as:

Fig. 2.3. Transfer of forces to strut

1 - Bolts (shear and intrados) 2 - Body of the beam 3 - Connection plate

4 W ldi

Fig. 2.3.Welding stress


where ‘η’ is the total number of bolts. In the previous sections, we assumed an elastic analysis for the distribution of forces on the bolts. It is reminded that the stress could be also distributed plastically, if the bolt shearing strength was greater / smaller than the ductile strength against intrados compression, which would result to rendering the possibility of plastic intrados deformation and equalization of forces stressing each bolt. In this case, the elastic distribution applied is acceptable towards safety; however, in certain cases, it may not be the best solution as regards economy. Accordingly, due to torque on the corner (most adverse) bolts, the horizontal force shall be

Fig. 2.4. Calculation of torque stressing the bolting and the welding


and the vertical

and the resultant active shearing on a (single-sheared) bolt shall be:

Fig. 2.5. Example of transfer of coplanar torque from the gravity centre of bolting to a bolt; and relation between the position and the force on a bolt.


2.1.2. Potential connection failure mechanisms Summarizing the above, one could point out the connection failure

mechanisms:

Fig. 2.6. Types of failure (ductile types are 1(B), 2 and 5)

1. Bolt failure: (a) under shearing. (b) under intrados compression

2. Yield of complete section

3. Breakage of decreased cross-section 4. Breakage of outer distance (tearing)

5. Plate yield 6. Welding breakage


2.1.3. Additional check for a hollow rectangular strut cross-section.

It should be pointed out that in the event of a hollow strut cross-section, a local phenomenon might appear relating to the aspect of its footing. This shall be dealt with by an additional check provided by EC3 relating to the axial strength of hollow cross-section footing.

2.1.4. Connection classification

Finally, is should be stated that the connection being examined is classified in class (a) of figure 2.8 due to its high flexibility.

Fig. 2.7. Local phenomenon for RHS cross-sections

Fig. 2.8. Connection types as regards rigidity

a. case of articulated connection b. case of rigid connection

c. case of semi-rigid connection


2.2. Detailed connection calculation

In this section, the particular connection shall be resolved in detail, so that the results may be compared to those of the previous calculation routine.

Accordingly, the characteristic example of the following connection shall be selected, following which all checks effected shall be presented. Suppose that the design shearing force is VEd=60 kN. It is reminded that all dimensions are in kN, mm, unless specified otherwise .

Fig. 2.9. Front view of connection (dimensions in mm)

Bolts Μ16/8.8 Steel type: S235

Fig. 2.10. Detail of connection plate (dimensions in mm)


2.2.1. Checking of clearances 2.2.1.1. Bolt clearances For the connection plate, it is:

2.2.1.2. Check of welding thickness

2.2.1.3. Check of beam height adequacy

Fig. 2.11. Distance of bolting centre of gravity from the left plate cheek.


Polar moment of inertia of bolting:

Torque in the bolting centre of gravity:

End bolt: xb=yb=40 mm

The end bolt shearing force (i.e. top, right) should be

where η = 4 bolts 2.3.2. Bolting shearing strength

Strength of a bolt:

where αν = 0.6 for quality 8.8. (the coefficient 0.85 comes in by virtue of the selected bolt quality for check under shearing)

Strength of bolting: 2.2.4. Strength under intrados compression

The strength of the group of bolts is given by the formula:

with ab = minad, fub/fu, 1, where coefficients k1 and ad are dependent on the examined row and location of a bolt (outer or inner), respectively.

According to the load transfer direction (1st row is the top one) it shall be:


• Row 1:

o Column 1:

o Column 2:

• Row 2:

o Column 1:

o Column 2:

therefore:

2.2.5. Beam section tearing strength Net tensioned surface:

Net sheared surface:


Since bolting is eccentrically stressed, then the tearing strength shall be given by the formula:

2.2.6. Tearing strength of connecting plate section

Net tensioned surface:

Net sheared surface: Since bolting is eccentrically stressed, then the tearing strength shall be given by the formula:

2.2.7. Welding check

The following stresses are calculated: - σ_ι_, regular stress, vertical to the weld - σII , regular stress, parallel to the welding axis - τ_ι_, shearing stress (at the neck level) vertical to welding axis - τII, shearing stress (at the neck level) vertical to welding axis

Torque stressing the welding:

Shearing force stressing the welding:

Fig. 2.12. Definitions of calculated stresses


Welding moment of inertia

Therefore

and Consequently, the welding is adequate. 2.2.8. Strength under front failure of hollow strut footing

This check would not have been required if the strut cross-section would have been of double-T. As mentioned in the mechanical behavior description section, said check shall be effected by considering the behavior as plastic, namely, by checking the axial strength of the strut footing to the tensile strength of the connection plate, a process favoring safety. 2.2.8.1. Connecting plate tensile strength

It is reminded that the tensile strength results from the minimum of the yield force for the entire surface and the decreased breaking force thereof due to existing holes. From fig. 2.13 we conclude that:

and The breaking force of the decreased cross-section shall be:


and the yield force of the complete cross-section:

(2.29)

in which case, Nt,Rd = 269.57 kN

2.2.8.2. Check

The above force shall be considered as action towards the check of the strut footing as regards front failure. According to EC3, the strength shall be:

where km=1 for tensioning.

Fig. 2.13. Decreased surfaces


3 Beam – strut connection

with double angular plates


3.1. Description of mechanical behavior

Fig. 3.1. Linear simulation of connection

Case of bolting on the beam body

Linear node simulation

Case of welding on the beam body

Diagram of a shearing force in node area (V)

Diagram of flexural holes in node area (M)


3.1.1. Transfer of stress through the connection means Said connection is required to transfer, as stated above, a

shearing force from the beam to the strut through the connection means shown in fig. 3.1. The bolting on the beam is further stressed by a torque coplanar to it, which results from the linear simulation accepted as applicable. Namely, the eccentricity of the joint positioning point generates a torque e1 VΕd, which shall be taken into account in the following sections. On the other hand, the bolts located at the strut footing are stressed solely by shearing forces. The mechanism transferring stress via the connection is described below.

In correspondence with figure 3.2, the path of forces into the connection means is as follows:

Fig. 3.2. Stress transfer to the connection means

Case a: Bolting on beam body

Case b: Welding on beam body


1. Shearing force and torque at the centre of gravity of the bolting on the beam, by solving a metallic carrier frame 2. ο (a) bolting on the beam: The previously mentioned forces

generate shearing loading on each bolt (double-sheared bolts).

ο (b) welding on the beam: The aforesaid forces stress the 3 fillet welds

3. The overall shearing force and bending are transferred to the leg of the angular plate on the beam.

4. Through shearing of the two angular plates (mainly and practically negligible bending due to their short length), the design shearing force is transferred to the legs of the angular plates on the strut body.

5. The design shearing force is transferred by intrados forces to the strut bolts, which are sheared (single-sheared).

6. Shearing is now received by the body or footing of the strut. In the general case of bolting consisting of many columns and

rows of bolts, the shearing force Ved is considered as being exercised at its gravity centre, as well as the torque of value e2Ved. Next, it is distributed to the bolts, following calculation of the polar moment of inertia of the bolting

Fig. 3.3. (a) Stress of welding on beam body. (b) Stress of bolting on beam body.


where ‘η’ is the total number of bolts. In the previous sections, we assumed an elastic analysis for the distribution of forces on the bolts. It is reminded that the stress could be also distributed plastically, if the bolt shear strength was greater/smaller than the ductile strength against intrados compression, which would result to rendering the possibility of plastic intrados deformation and equalization of forces stressing each bolt. In this case, the elastic distribution applied is acceptable towards safety; however, in certain cases, it may not be the best solution as regards economy. Accordingly, due to the torque applied on the corner (most adverse) bolts, the horizontal force shall be

and the vertical

and the resultant design shearing force on a (single-sheared) bolt shall be:

The angular plates shall be checked as regards bending due to the coplanar torque, which is also equal to

for each footing. The design shearing force shall be VEd / 2 for each plate.


Fig. 3.4. Calculation of torque stressing the bolting on the beam

Fig. 3.5. Cases of beam connection (a) on footing (b) on strut body


The angular plates shall be checked as regards bending due to the coplanar torque, which is also equal to

for each footing. Finally, a check shall be carried out on the angular plate

under shearing, with acting shearing force equal to on each one of them.

For a more detailed description of the transfer of coplanar bolting torque to individual bolts, the reader may consult the corresponding force distribution annex.

Assuming that instead of a bolting there is a welding on the beam, the torque on the centre of gravity thereof may be respectively calculated as stressing the three fillet welds. Finally, the resultant stress developed on the most adverse point of the welding shall be calculated, and this is

Fig. 3.6. Calculation of torque stressing the welding on a leg of an angular plate on a beam

centre of gravity of the welding


checked against the maximum permissible stress, in accordance with the normative frame that follows. 2.1.2. Critical connection failure mechanisms

Summarizing, the connection failure mechanisms are pointed out, for which the connection shall be checked:

• Strut bolt failure under shearing • Strut bolt failure under intrados compression

For bolting on the beam: ο Strut bolt failure under shearing ο Strut bolt failure under intrados compression ο Angular plate leg on beam failure under tearing ο Beam body failure under tearing

For welding on the beam: ο Fillet welding failure under maximum developed stresses

• Angular plate leg on strut failure under tearing • Strut body failure under tearing • Angular plate failure under shearing • Angular plate failure under bending

3.1.3. Connection classification The connection being examined is classified in class (a) of figure 3.7.

Fig. 3.7. Node classification according rigidity thereof.

a. case of articulated connection b. case of rigid connection

c. case of semi-rigid connection


3.2. Detailed connection calculation In this section, the particular connection shall be resolved in detail,

so that the results may be compared to those of the previous calculation routine.

Accordingly, the characteristic example of the following connection shall be selected, following which all checks effected shall be presented. Suppose that the design shearing force is VEd=110 kN. It is reminded that all dimensions are in kN, mm, unless specified otherwise.

Fig. 3.8. 1st front view of connection (dimensions in mm)

Bolts M12/4.6 Angular plates 1.90/10 Steel type: S235


Fig. 3.9. 2nd front view of connection (dimensions in mm)

Fig. 3.10. Detail of bolt positioning on angular plates

Angular plate leg on beam

Angular plate leg on strut


2.2.1. Checking of clearances

3.2.1.1. Checking bolt clearances

For the leg of the angular plate on the beam and strut we have:

3.2.1.2. Adequacy check of height of the angular plate

3.2.2. Shearing strength of a bolt

Shearing strength of a bolt

where αν = 0.6 for quality 4.6.

3.2.3. Strut bolt check under shearing

Due to the acceptance of the static simulation with the joint in the model, at the point of section of the gravity centre axis of the beam with the outer side of strut footing cheek, it is clear that said bolts would transfer solely a shearing force equal to the design shearing force VEd, without developing bending torque, at the same time, which would generate tensile forces on the bolts. Therefore, each of the 8 single-sheared bolts would be stressed by a design shearing force equal to


The bolting strength would be:

3.2.4. Strut bolt check under intrados compression The strength of the bolt cluster is given by the formula:

with ab = minad, fub/fu, 1, where coefficients k1 and ad are dependent on the examined row and location of a bolt (outer or inner), respectively.

According to the load transfer direction (1st row is the top one) it shall be:

• Row 1:

o Column 1:

o Column 2:

• Row 2:

o Column 1:


o Column 2:

Therefore, The strength of all 8 bolts on the main beam under intrados compression would be:

3.2.3. Beam bolt check under shearing

Bolting polar moment of inertia:

Torque coplanar to bolting:

Fig. 3.11. Coordinate system at the centre of gravity of bolting on a beam


The end (most adverse) bolt (xb=70mm and yb=130mm), is stressed by a shearing force:

where η = 4 bolts

and 3.2.4. Beam bolt check under intrados compression

The strength of the bolt cluster according to EC3 is given by the formulae:

• Row 1:

o Column 1:

o Column 2:

• Row 2:

o Column 1:

o Column 2:


Therefore

3.2.7. Check of angular plates under shearing In an angular plate a shearing force is acting equal to VEd/2, whereas the stressing torque is equal to xΚΒ VEd / 2 (the distance xΚΒ is defined in figure 3.11). In this particular example, it is:

and

the ductility criterion shall be applied

in order to verify whether the holes should be deducted from the shearing surface calculation,

in which case, the decrease of shearing surface due to the holes would not be necessary. Consequently,

the shearing strength of an angular plate is given by the formula:


3.2.8. Check of angular plates under bending As already mentioned, the torque stressing one angular plate is:

Since the above ductility criterion is satisfied, the subtraction of the holes surface for calculating the resistance torque on the angular plate cross-section is not required. Therefore:

whereas

3.2.9. Beam section tearing strength



Since bolting is eccentrically stressed, the tearing strength shall be given by the formula:

Fig. 3.12. Definition of surfaces Αην, Αηt


3.2.10. Tearing strength of an angular plate leg section on a beam



Since bolting is eccentrically stressed, then the tearing strength shall be given by the formula:

3.2.11. Tearing strength of an angular plate leg section on a strut



Since bolting is considered as not being eccentrically stressed, the tearing strength shall be given by the formula:

3.2.12. Case of welding on beam connection

Since the beam is welded instead of bolted, the following checks are carried out on the angular plates: Suppose the welding thickness is α=3 mm


The distance “e” from the outer side of the strut footing cheek to the gravity centre of the welding is equal to 66.18 mm. Next, the moments of inertia of the fillets shall be calculated:

The torque stressing the welding is:

Tensions of the fillet weldings along the main axii:

Resultant stress:

Fig. 3.13. Case of welding on the beam – angular plate connection


The correlation factor for steel S235 is defined as βw = 0.80; accordingly, the welding strength is:


4 Beam on beam connection with double angular plates


4.1 Description of mechanical behavior

Figure 4.1 Linear simulation of the connection and stress values

Linear simulation of node

Diagram of beam shearing force at the node area

Diagram of beam bending torque at the node area


4.1.1 Transfer of stress through the connection media

This connection is required to transfer, as mentioned above, a shearing force from the secondary beam to the main beam through the connection media shown in figure 4.2. The bolts on the secondary beam are also being stressed by a coplanar torque that results from the linear simulation, which has been accepted as valid. That is, the eccentricity of the point where the joint is being fixed creates a torque eVEd, which shall be taken into account later. On the other hand, the bolts on the body of the main beam are considered as bearing only a shearing force due to the small distance between the center of the joint, and this results in a practically zero torque, which would otherwise produce tensile forces on the bolts. In particular, the stress is borne as follows:

Figure 4.2 Bearing of stress at the connection


According to figures 4.2 and 4.3, the path of the forces on the connection media is as follows:

1. Shearing force and torque at the centre of gravity of the bolts on the secondary beam by solving a metallic frame bearing body

2. The above forces create a shearing load on each bolt (double-sheared bolts).

3. The overall shearing force and bending is transferred through intrados forces at the leg of the angular plate on the secondary beam

4. By shearing the two angular plates (mainly, and with practically insignificant bending due to their small length) the design shearing force is transferred on the legs of the angular plates on the main beam body

5. The design shearing force is transferred by intrados forces to the main beam bolts, which are sheared only once (single-sheared bolts)

6. The shearing force is then received by the body of the main beam

In the general case of bolting, which consists of many columns and rows of bolts, it is considered that the shearing force Ved is exerted at the center of gravity of the bolting area, as well as the torque e2Ved. Then it is distributed on the bolts after the calculation of the polar moment of inertia of the bolting area

where n is the total number of bolts. As mentioned above, an elasticity analysis has been assumed for the distribution of forces on the bolts. It should be reminded that the stress might also be plastically distributed,

Fig. 4.3 Stressing of bolts on the body of the secondary beam


provided that the shearing strength of the bolting area is greater than the higher ductile strength against intrados compression, which would enable plastic deformation of the covings and equalization of the stress forces on each bolt. For the case in question, the applied elastic distribution enhances safety and, in some cases, may be the best solution as regards cost-saving. Thus, the horizontal force due to the torque on the corner (most adverse) bolts shall be:

and the vertical force shall be

while the resultant shearing force that acts on a (single-sheared) bolt shall be:

Moreover, the angular plates shall be checked for bending due to their coplanar torque, which is equal to

for each plate. Finally, a shearing check shall be carried out for the angular plates with

Fig. 4.4. Calculation of torque stressing the bolting area on the secondary beam


acting shearing force on each one equal to . For a more detailed description of the coplanar bolting torque

transfer on individual bolts, the reader can refer to the corresponding force distribution annex. 4.1.2 Critical mechanisms of connection failure

Summarizing, we note the connection failure mechanisms for which the connection shall be checked:

• Main beam bolts shearing failure • Main beam bolts intrados compression failure • Secondary beam bolts shearing failure • Secondary beam bolts intrados compression failure • Failure of angular plate leg on the main beam due to tearing • Failure of angular plate leg on the secondary beam due to tearing • Main beam body tearing failure • Secondary beam body tearing failure • Angular plate shearing failure • Angular plate bending failure

4.1.3 Connection classification

Finally, we mention that the described connection is classified in category (4) of figure 4.5 due to its high flexibility.

Figure 4.5 Connection classification

a. articulate connection b. rigid connection

c. semi-rigid connection


4.2 Detailed connection calculation

In this section we shall solve the connection in question in detail, in order to enable comparison of the results with those of the previous calculation routine. So we select the typical connection example that follows and then we describe all checks to be carried out. Let the design shearing force be VEd=100 kΝ. It is reminded that all dimensions are in kN and mm, unless specified otherwise.

Figure 4.6 1'' connection section (dimensions in mm)

main (supporting) beam HEB300

secondary (supported) beam IPE 300

Bolts M12/4.6 Steel quality S235

equilateral angular plate 90/90/10, L = 160

Figure 4.7. 2nd connection section (dimensions in mm)


Convention: From now on, the main beam shall be indicated as ΚΔ, and the secondary beam as ΔΔ. 4.2.1 Check of Distances 4.2.1.1 Bolt distances

For the leg of the angular plate on the secondary beam we have:

Figure 4.8. Detail of bolt fixing on the angular plates

Leg of angular plate on the secondary beam

Leg of angular plate on the main beam


4.2.1.2 Check of beam height adequacy

where 30 + 40 + tf,ΚΔ = 89, the shearing of the upper footing of the secondary beam. 4.2.2 Shearing strength of a bolt

Shearing strength of a bolt:

where αν = 0.6 for quality 4.6. 4.2.3 Main beam bolts shearing check

Since we have assumed the static simulation with the joint on the model, at the intersection point of the two center of gravity axii of the connected beams, it is clear that these bolts transfer only a shearing force equal to the design shearing force Ved, and no bending torque is developed, which might otherwise produce tensile forces on the bolts. The more the main beam can swivel, the better this assumption approximates the actual behavior of the connection. Thus each one of the 8 single-sheared bolts is being stressed by a design shearing force equal to

The bolting area strength shall be:

4.2.4 Main beam bolts intrados compression check

The group of bolts strength is provided by the formula


With ab = min ad, fub/fu, 1, where coefficients k1 and ad depend on the checked bolt row and position (inner or outer) correspondingly.

Along the direction of load transfer (1st row is the top one) we have:

• Row 1:

o Column 1:

o Column 2:

• Row 2: o Column 1:

o Column 2:

Thus The intrados compression strength of all 8 bolts on the main beam is:


4.2.5 Secondary beam bolts shearing check

Polar moment of inertia of bolting area:

Torque, coplanar with the bolting area:

The edge (most adverse) bolt (xb=25 mm and yb=50 mm) is being stressed by a shearing force:

where n = 4 bolts

and 4.2.6 Secondary beam bolts intrados compression check

The group of bolts strength is provided by the formula

Figure 4.9. Coordinate system at the center of gravity of the bolting areaon the secondary beam


with ab = min ad, fub/fu, 1, where coefficients ka and ad depend on the checked bolt row and position (inner or outer) correspondingly.


• Row 1:

o Column 1:

o Column 2:

• Row 2:

o Column 1:

o Column 2:

Thus


4.2.7 Shearing check of angular plates An angular plate is subjected to a shearing force equal to VEd/2,

and the torque stressing it is equal to χΚΒ VEd / 2 (distance χΚΒ is defined in figure 4.9). In this particular example we have:

and

the ductility criterion shall be applied

to establish whether the holes must be excluded from the calculation of the shearing surface

so no decrement of the shearing surface due to the holes is required, and thus

The shearing strength of one angular plate is provided by the formula:

4.2.8 Bending check of angular plates

As mentioned above, the stressing torque on one angular plate is:

Since the above ductility criterion is met, no decrement of the hole

surfaces is required in order to calculate the resisting torque of the angular plate cross-section, thus

while


4.2.9 Secondary beam plate tearing strength

Net surface under tensile stress:

Net surface under shearing stress:

Since the bolting area is being stressed eccentrically, the tearing strength is provided by the formula:

Figure 4.10. Definition of c, w and k distances


5 Continuity restoration of a beam subjected

to bending and tensile stress with body-footing plates

5.1 Description of mechanical behavior

The connection in question is expected to restore the continuity of a member with double-T cross-section, by transferring, as mentioned above, the bending torque, and the shearing and pivotal force through the media.


As shown in figure 5.1, the connection can be considered as completely rigid; furthermore, this is the reason why the torque can be transferred through it. This is due to its high pivotal rigidity, to which the footing plates mainly contribute. If no such plates existed, the connection would be considered as articulate, since if there were only plates on the body the two members would pivot easily towards one another and the node would behave in a much more flexible manner.

Figure 5.1 Linear simulation of the connection and stress values at the restoration area

Diagram of bending torques at the node area [M]

Diagram of pivotal forces at the node area [N]

Diagram of shearing forces at the node area [V]


Later on, we shall describe the mechanism by which the stress values are transferred from the continuity restoration area to the connection media, the possible failure mechanisms, and the required checks.

The stress values that result by solving the metal frame are developed at the connection area of the two members. 5.1.1 Transfer of stress through the connection media

These forces must be transferred to the connection media (centers of gravity of bolting areas) and to the body footing plates, in order to produce the design actions of this connection. Initially, the quantities shown in figure 5.2 are transferred to the plates, and the plates transfer them through intrados forces to the bolts as shearing forces. These are then transferred to the overall cross-section of the member, beyond the connection area.

5.1.1.1 Transfer of bending torque

The bending torque is transferred to the body and footing plates in an elastic manner, according to their rigidity, that is, to their moment of inertia for bending stress. In particular, the torque received by the body plates shall be equal to:

Figure 5.2 Cross-section at the continuity restoration area and internal stress values


while the torque on the footing plates is:

This torque creates a pair of pivotal forces on the footing plates, and each force is equal to:

5.1.1.2 Transfer of pivotal force

The rigidity against pivotal stress is equivalent, as it is known, to the area of the member under stress, so according to the above we have:

Pivotal force on body plates: Pivotal force on footing plates:

Thus the total pivotal force (due to overlapping) on the footing plates shall be:

Figure 5.3. Moments of inertia of plates around axis y - y


5.1.1.3 Transfer of shearing force

The shearing force is considered as being received only by the body plates, since the shearing surface of the footing plates is practically negligible. 5.1.2 Stressing of bolts

1. Footing bolts: Their shearing stress is only equal to Nfinal,fl 2. Body bolts: They are subjected to shearing stress after

analyzing the coplanar torque into an equivalent bolt shearing force. This shall be equal to Mbolts,web=Mw-eVEd. Similarly, the pivotal force Mw and the shearing force VEd are distributed on each bolt and the resulting design force for the most adverse positioned bolt is calculated

where n is the number of bolts.

Figure 5.4. Plate surface areas


5.1.3 Case of member with welded connections 5.1.3.1 Body plate

Since, instead of bolting there is weld-fixing of one of the two members to be connected with the body plate, we correspondingly calculate the torque at the center of gravity of the fixing area (see fig. 5.6):

Note that we must first calculate distance e of the center of gravity of the welded area from the continuity restoration area. This is subjected to bending forces on the same plane, as well as to tensile and shearing forces. So we find the moment of inertia of the welded area towards the horizontal and vertical axii, as well as the acting torque:

Figure 5.5. Stressing of bolting areas on the body and footing

Single-sheared footing bolts (only shearing)

Double-sheared body bolts (shearing resulting from torque Mw, shearing force VEd and pivotal force Nw, after projecting them at the center of gravity of the bolting area).


Then we calculate the stresses at both directions and finally we calculate their resultant force fs, which is compared with the strength stress. 5.1.3.2 Footing plate

The weldings of the plate on the footing are being stressed only by the Nfinal,fl force, so we calculate the stresses on the welded area and we compare them with its strength stress (see fig. 5.6.).

5.1.4 Plate stressing

The body footing plates shall be checked as regards their tensile stressing by forces Nw and Nfinal,fl respectively. The tensile strength shall result as the minimum between the yield force for the total plate surface and the breaking force as reduced due to any holes.

Figure 5.6. Weldings (outer seams) on the left member

Body plate Footing plate


5.1.5 Member cross-section stress

Finally, the forces end up on the cross-section of the member to be connected. It must be ensured that this cross-section is able to safely bear such loads. Thus the cross-section shall be checked under bending, shearing, and tensioning simultaneously. The ductility criterion as regards whether the holes shall be subtracted in each check of this kind is as follows:

where Anet is the reduced surface of the part subjected only to tensioning, while A is the overall surface of the tensioned part of the cross-section or footing, depending on the case. In order to find these parts, the plastic neutral axis of the cross-section must be found. It muse be reminded that this axis does not necessarily pass from the center of gravity of the (twin symmetrical) cross-section, since there also exists a pivotal design force. So, if the ductility criteria are met, no reduction due to holes is required for the calculation of the strength torque of the cross-section. The same stands for the shearing check, where coefficient Anet corresponds to the overall shearing surface except the holes.

In particular, we must calculate both the tensioned surface of the cross-section as equilibrium point of the inner stresses with the externally applied pivotal force, and the reduced surface due to holes. If the above criterion is met for the tensioned part, we can then consider as negligible the reduction of the cross-section strength due to the holes in it. In the opposite case we must further reduce the plastic resistance torque of the cross-section due to the holes at the tensioned area of the body, considering that the category of the cross-section is 1 or 2, and the strength torque results as follows:

Regarding shearing, the shearing surface of the cross-section is calculated as

as well as the corresponding reduced ΑV,net due to the holes in the body.


We check whether the ductility criterion is met in order to deduce whether the shearing strength is affected by the holes in the body of the cross-section. Then we reduce the shearing surface and the design shearing strength results from the formula:

5.1.6 Checks to be carried out

Summarizing, we present the checks to be carried out for this particular connection:

• Bending check of cross-section • Shearing check of cross-section • Footing bolts shearing check • Footing bolts intrados compression check • Footing plate shearing check • Body plate tensioning check • Body bolts shearing check • Footing bolts intrados compression check

If there are weldings on the left member: • Footing welding check under maximum stress buildup • Body welding check under maximum stress buildup

5.2 Detailed connection calculation

In this section we shall solve the connection in question in detail, in order to enable comparison of the results with those of the previous calculation routine.

So we select the typical connection example that follows and then we describe all checks to be carried out. We suppose that we have a member with cross-section IPE 400, with design stress values at the restoration area NEd=30 kN, VEd=60 kN and MEd=150 kNm. The connection media and their qualities are shown in fig. 5.7. It must be reminded that all dimensions are in kN and mm, unless specified otherwise.


5.2.1 Check of Distances 5.2.1.1 Check of Bolt Distances

For the plate on the beam body we have:

Figure 5.7. Front view and section characteristic at the connection area.

Footing bolts: Μ20/4.6, body bolts: Μ16/4.6. Footing welding thickness 6 mm, body welding thickness 4 mm. Steel quality S235, dimensions in mm.


For the plate on the beam footing we have:

5.2.1.2 Check of beam height adequacy

where 30 + 40 + tf,KΔ = 89, the shearing of the upper footing of the secondary beam. 5.2.2 Beam cross-section checks at the restoration area

We shall calculate the strength values for the cross-section of the beam, and at the same time we shall check whether the ductility criterion is met.

to establish whether the holes must be excluded at the tensioned part of the cross-section. The cross-section is classified in category 1. 5.2.2.1 Check of the overall cross-section

Plastic resistance torque:


where 5.2.2.2 Check of reduced cross-section at the restoration area 5.2.2.2.1 Bending strength check

Footing surface:

Reduced footing surface:

so the ductility criterion is not being met, and we have to take into account the reduction of the cross-section strength due to the footing holes. Then we shall calculate the surface area of the tensioned part of the cross-section, due to the tensioning pivotal force and to the bending torque as shown in figure 5.8.

Figure 5.8. Shift of the center of gravity axis of the cross-section due to pivotal force

plastic neutral axis

tensioned rows of body bolts


Calculation of the tensioned cross-section surface area due to proper quantities N and M:

Assumption (1): the neutral axis intersects the cross-section body

where From the first equation we have:

and from the third one

so assumption (1) is valid. We check whether further reduction of the beam strength is required due to the holes in the body. The tensioned cross-section surface area shall be:

due to the tensioning pivotal force. So we must reduce the 2 footing holes and 2 from the 3 body holes.

The reduced cross-section surface area due to the holes is:

and again the ductility criterion is not being met, so we have to take into account the 2 body holes in order to reduce the plastic resistance torque of the cross-section. Thus we have to reduce all of the holes of the tensioned part (body and footing). Distance of footing holes from the neutral axis:

(as defined in fig. 5.8)

Distance of 1st tensioned row of body bolts from the neutral axis: hw1=147.18 mm


Distance of 2nd tensioned row of body bolts from the neutral axis: hw2=14.84 mm

The plastic resistance torque becomes:

5.2.2.2.2 Shearing strength check

The net sheared surface is:

The criterion

is met, so we consider that the body holes do not affect the shearing strength and the overall cross-section check, as described above, is valid. 5.2.3 Bolts Check 5.2.3.1 Check of Footing Bolts

The stress values at the continuity restoration area are distributed on the footing and body blades. The shearing force is considered as being fully received by the body blades, and the torque is distributed according to the blade rigidity. According to the above, we initially calculate the moments of inertia of the body and footing plates.

The torque shall be distributed as follows:


The pivotal force stressing each footing plate shall be:

and the pivotal force on each body plate shall be:

Due to bending, an additional pivotal force is applied on the feet, which is equal to:

The total design pivotal force on the footing bolts is then equal to:

5.2.3.1.1 Bolt strength under footing shearing



5.2.3.1.2 Strength of a group of bolts under intrados compression

The group of bolts strength is provided by the formula:

with ab = minad, fub/fu, 1, where coefficients k and ad depend on the checked bolt row and position (inner or outer) correspondingly.


• Row 1:

o Column 1:

o Column 2:

• Row 2:

o Column 1:

o Column 2:


• Row 3:

o Column 1:

o Column 2:

Thus

5.2.3.2 Check of Body Bolts

Torque acting at the center of gravity of the bolting area:

where x = 100 mm is the distance of the connection restoration area to the center of gravity of the bolting area.

5.2.3.2.1 Footing bolt shearing strength



5.2.3.2.2 Strength of a group of bolts under intrados compression

The group of bolts strength is provided by the formula:

With ab = min ad, fLb/fu, 1, where coefficients kt and ad depend on the checked bolt row and position (inner or outer) correspondingly.


• Row 1:

o Column 1:

o Column 2:

• Row 2:

o Column 1:

o Column 2:


• Row 3:

o Column 1:

o Column 2:

Thus

5.2.4 Check of footing and body plates 5.2.4.1 Check of footing plate under pivotal force

The tensile strength of the footing plate is calculated as the minimum between the yield force for the total surface and the breaking force as reduced due to any holes.

Reduced cross-section:

where t is the plate thickness, b is its width, and e2, pu, p2 are the bolt distances as specified in EK3.

Thus


5.2.4.2 Check of body plate under pivotal force

The tensile strength of the footing plate is calculated as the minimum between the yield force for the total surface and the breaking force as reduced due to any holes.

Reduced cross-section:

where t is the plate thickness, b is its width, and e2, plf, p2 are the bolt distances as specified in EK3.

Thus


6 Connections of beams –-

columns with end plate


6.1 Description of mechanical behavior 6.1.1 General

Torques are developed at nodes, on foundations (carriers) whose lateral stability is provided through the framing operation. Therefore, connections between beams - columns must be formed as torque connections, i.e. able to carry torques and the relevant shearing forces, from beams to columns. Development of connection torques is a function involving several factors such as the beam profile, column profile (open or closed cross-section profiles), connection type (bolted, welded or composite), method of construction (only factory - made or on - site welds), several geometrical or other restrictions and especially construction practices. The multitude of these factors is exemplary or the variety encountered in the formulation of such connections; however, their development is beyond the scope of the present paper. 6.1.2 The rationale of semi-rigid connections and the need for ductile behavior

The last decades saw the development of the concepts of semi-rigid connection and the partial - resistance connection; at earlier times, connections were designed either as articulated joints or as full-strength pilings. Due to the lack of theoretical and experimental research, it was often unclear whether connections designed as full-strength pilings, would also function in practice as designed by the engineer.

Due to the absence of suitable design models, connections were often designed much stronger and more rigid than needed, resulting in non-economical designs. This design would often contradict the overall design assumptions of the structure; it could therefore influence safety of the structure in question. Design of the initial nodes has been based on experimental observations. Test results as such were discussed at length and led to the development of simple Simulations for manual calculation purposes.

Test results were recently stored in databases, thus allowing prediction of node properties through analytical formulae. The most important properties are their bending torque strength (resistance) MjRd, deformation / rotational stiffness Sj and the deformation /rotational capacity φCd. These properties can be predicted with remarkable accuracy. The complexity of the problem due to the presence of defects, residual


stresses, friction etc., in some cases requires a behavior simulation through the use of detailed finite element simulations.

Hence, beam - column connections required to transfer torques, are often selected to be constructed as semi-rigid. This is usually dependent upon economic criteria of design, as mentioned. Formulation of continuous framing requires rigid or piled, full - strength connections. Contrary to that, in plain connections (e.g. shearing), are lower cost alternatives, but the resulting beams are of larger size. An intermediate solution (neither rigid connections, nor members of much larger cross sections), is implemented in semi-rigid connections.

In this case, the beam shall be dimensioned for Μ = MFREE - MCΟNN while linking plays a key role for semi-rigid frame design.

Figure 6.1 Quantity of support torque shall be determined upon design

Beam design


6.1.3 Classification of torque - transferring connections

Figure 6.2 Classification of connections according to strength, rigidity, and ductility

full strength

partial strength

where

Classification according to strength

Strength torque

Articulated

Classification according to rigidity

1. full strength, rigid, non-ductile 2. full strength, rigid, ductile 3. partial strength, rigid, ductile? 4. full strength, semi-rigid, ductile 5. partial strength, semi-rigid, ductile More ductile

Less ductile

Classification according to ductility


The key characteristics of a connection are ductility and classification as a partial connection. The first term pertains to the ability to develop a plastic joint and is fully dependent upon its rotational capability. The second term denotes that the strength torque of the connection is less than the adjoining beam. From the moment that beam strength torques are applied to connections, the latter shall be capable of making a sufficient rotation, so that the plastic zone may actually absorb energy. The rotation that the beam must withstand is then adjusted in regulatory terms and ranges between 0.02 and 0.04 rads. In practice, the strength torque of the connection selected, is within a range of 30-50% of the plastic strength torque of the beam. 6.1.4 Deformation capacity

To enable the connection to sufficiently rotate, some of its elements must initially exhibit controlled yield. Bent plates (flange, end plate etc.) and the body of the beam / column system under shearing are the prevailing node components selected to reach their ratio limit. In fact, the most important aspect is to avoid potential brittle failure of other joint components. This category includes welds as well as tensioned bolts. Thus, a component of the joint must be dimensioned as the «weakest»; this process is selected upon the design phase. Commonly, such a component is the bolted end plate, whether extending to the face or projecting on both opposite sides of the beam.

Figure 6.3. End plate types

End plate at the face of the beam

Projecting end plate


Check of rotational adequacy will be implemented by comparing rotational capability available with the necessary rotation of the beam at the joint position. The latter is related to the geometrical and stress conditions of the supported beam. For example, for a both-ends supported beam under distributed fixed load, the necessary rotation is provided from the following formula:

under accumulated load from :

etc. Rotational capability on the one hand depends upon configuration of the joint. Connections with a column of bolts often possess sufficient rotational capability. However, for larger shearing forces, where a second bolt column might be necessary, rotational adequacy must be controlled. Rotational adequacy must be also controlled on welded joints. Regarding bolted connections on end plates it is apparent that deformability must be the result of the deformability of the tensile zone, given the fact that deformability of the compressed part of the joint is low. To this end, deformability of the three T-stubs must be examined, as discussed in the introductory chapters. For a given deformability 6U, the resulting rotational capability is as follows:

where z is the distance between the compression and tension centers of gravity. As demonstrated in the reference chapter in deformability of the 3 T-stub failure mechanisms, sufficient rotational capability is often available in cases of critical end plate. For this purpose, the thickness of end plates as well as the height z of the joint shall be restricted.


Figure 6.4. Yield of end plate shall protect bolts and welds from undertaking additional loads

Figure 6.5. Impact of end plate thickness to the type of failure

Bolt strength (kN)

Critical end plate

Critical end plate and bolts

Critical bolts

Example for bolts Μ24 / 8.8

End plate thickness (mm)


6.1.5 Pursuit of ductile behavior

It is very common to adopt the rule of restricting thickness of the end plate to approximately 60% of bolt diameter (assuming that the quality of the latter is at minimum 8.8). Even thinner end plates would carry higher torques, with a risk of premature failure due to bolt breakage, before performing the necessary rotation of the node. End plates designed merely under the bending strength criterion, may have a thickness equal to or greater than bolt diameter. These are not considered to be ductile. Figure 6.5 e.g., would require an end plate 25 mm thick so that bolts Μ24 / 8.8 obtain their full plastic resistance. In addition to the above it must be specified that a thinner end plate will make a more ductile joint to behave in a less rigid manner in comparison to a thicker one.

Rigidity or stiffness is much more useful in portable frames in comparison to fixed ones. It contributes to the stability of the metallic frame regarding portable frames; specifically for fixed frames it only contributes to the restriction of beam shearing, while it restricts rotation of columns. Generally, upon selection of end plate thickness, rigidity and ductility of the joint may be regarded to be contradicting concepts, as the pursuit of one will diminish the other. Stiffness of end plate, intended to be the critical component of the joint, is proportional to its thickness in the power of two or three. Ductility under no circumstances may be ignored, thus larger and / or stronger bolts must be preferred, in fact restricting the thickness of end plate. An optimal combination would be Μ24 / 8.8 or 10.9 bolts on one end plate of 15 mm.

The above fully demonstrate the need to perform controls regarding deformations beyond structural adequacy of such joints. In regulatory terms, control of deformations shall be performed by applying rotational capacity checks according to EC3; however this is currently in a premature stage. A connection is considered capable of behaving as a plastic joint provided that any of the following conditions is met:

• Strength torque is restricted by shearing of the body of the pillar

• Strength torque is restricted by column flange in terms of bending; the failure mechanism is the first one (only flange is critical)

• End plate under bending would restrict strength torque and failure mechanism is the first (only end plate is critical)


Only in the special case of twin - side node arrangements (perimeter columns), it is possible that the first of the above mentioned conditions is met. In case a beam is located on both opposite sides of the column, active torques can be balanced, thus greatly reducing shearing of the column body; therefore, for plastic design the intent is that the 2 latter conditions apply.

Figure 6.6. The 3 rules specified by EC3 to allow sufficient rotational capability

Case 1: ductile joint when shearing strength of column body is critical Case 2: ductile joint when column flange is critical, commonly for thin flange Case 3: ductile joint when end plate is critical; the thickness of the plate as a function of bolt quality / size shall be determined in the design phase


6.1.6 The component method in general

This paragraph discusses certain general information on the node component method, as proposed by EC3 to analyze joints with end plates. Implementation of the component method will permit examination of several joint types in three main steps: classification of node components, estimation of the force - deformation diagram of each individual component during the initial rigidity phase, strength, force and deformation and eventually composition of these components to estimate overall behavior of the node. Component method principles are mainly based on the work by Zoetemeijer. It is possible that components are stressed by tension, compression and shearing.

The accuracy of this method is based on the precision of description of the main components by independent springs and the quality of the composition. The method allows very simple predictions for practical application on complex models, aiming at further extensions. It is assumed that component properties are independent, thus they can be easily simulated; however some components are not acting independently; they are affected by others. This can be overcome by applying simplifying assumptions, since a generic method based on an iterative calculation procedure would be complex for practical use.

Behavior of components can be generally described by a non - linear force - deformation plot. Behavior is often non - linear and includes several influences (strain hardening, contact between elements, membrane effects, 3-D stress distribution etc.), however it can be simplified to a bilinear or trilinear model. Model parameters are derived from component dimensions and material properties. The most important properties include design resistance FRd, rigidity coefficient k and deformation strength 6C. Joint properties are assured through the blending of properties of independent components. When a non - linear component behavior model is used, it is possible to employ a progressive blending process.

This approach allows the calculation of a full torque - rotation plot for the node. In actual applications, engineers usually need only the initial stiffness and bending torque resistance of the node. This can be easily assured by blending of the components which are described by a linear model.


6.1.6.1 Transfer of forces to the joint

To examine the behavior of the present joint, it shall be broken into several components, which shall be simulated by using springs. Connection of the individual springs in appropriate layout (series or parallel) will yield the resulting joint response. Breaking into components can be effected using appropriate subsystems, which sum up the response of several individual elements.

Distinguishing of subsystems is carried out depending on the stress they transfer. Thus, torque connections are broken into a tensioned and a compressed zone, whose response differs due to the difference in stress. Joints that only transfer axial forces can be examined in a similar manner, by examining force transfer areas. The following shall apply to torque connections:

Figure 6.7. Node rotation torque plot

Node resistance

Elastic limit

Initial stiffness

Experimental plot

Design plot

Deformability Rotation, rad

Figure 6.8. Joint broken down into subsystems


where: Fc, Rd , Ft,Rd the strength values of compressed and tensioned zones respectively, and z is the distance between the compression and tension centers of gravity.

Rotation capacity:

ή

depending on whether the zone under compression or tension is critical, where:

: deformability of compressed or tensioned zone : deformations, of compressed or tensioned zone - commonly elastic

- for a load level exhausting the resistance of the other zone

The following is a detailed description of the calculation of tensile compressive forces of the joint, and of lever arm z.

All torque connections are designed under the same rationale, irrespective of whether they have been selected to become rigid or full-strength. The design process shall ensure that all joint elements are capable of receiving the torque applied and a common shearing force. Axial action is largely negligible, as beams are considered to be part of a diaphragm which apparently does not carry loads along its axis.

All torque connections are designed under the same rationale irrespective of whether they are intended to be rigid or full-strength. Design process shall ensure that all joint components are able to carry the torque applied and a common shearing force. Axial action is largely negligible, as beams are considered to be part of a diaphragm which apparently does not carry loads along its axis.

For example, a typical beam - column joint is required to transfer torque by developing a pair of forces, a compressive force close to the lower beam flange and a tensile force at the upper section of the joint. In case it is applied to the axial direction, then the two above mentioned forces are equal. Therefore, top bolts must resist to the tensile force, while compression is assumed to be transferred through the contact of the end plate to the lower flange of the beam.


A large number of other node components, may limit the resistance of the joint. Each of them must be separately investigated and in case it is insufficient, it must be reinforced (using metal plates, depending on the case), as discussed thereafter. Regarding the joint examined, the following node components can be distinguished, both in topology terms and in terms of their loading (also ref. figure 6.9).

Figure 6.9. Node components


Loading type Point Required controls

a Bolt tensioning b End plate bending c Bending of column footing d Beam body tensioning e Column body tensioning f Welding between beam flange - plate

Tensioning

g Welding between beam body-plate Horizontal shearing h Column body under shearing

j Beam flange compression k Welding of beam flange L Column body under intrados compression

Compression

m Buckling of column body n Welding between beam body-plate p Bolt under shearing

Vertical shearing

q Bolting under intrados compression Table 6.1 Stressing of joint areas

Column body is subjected to accumulated loads in the tensile and

compressive zones; it can be controlled using empirical methods which determine an active area on the body. Buckling must be controlled - among others - in the compression zone.

Adjacent structural elements provide lateral assurance so that buckling length of the compressed part of the joint is equal to 0.7 times the body width.

Shearing of column body can become significant, especially under unilateral node layout or in a laterally non-shifting frame. On the contrary, if 2 beams converge from opposite sides of the column on a fixed frame, the quantities transferred out of two beams may be in equilibrium, thus the column body will not be subject to significant stress. Therefore, depending on the frame type, this element may be (or may not be) critical.


In most cases a plastic distribution over bolted nodes is assumed, however elastic or elastoplastic distribution can be also implemented. Regarding elastic distribution, forces over bolt rows are derived from tensile resistance of the upper row of bolts. Bolt forces must be checked to confirm that exceeding of resistance does not occur at any position.

When elastic distribution is assumed, a shearing force is transferred from all bolts; checks for shearing - tension contribution is then required. In this case, tensile force of the bolt shall include the increase due to the lever effect of bolts. During elastoplastic and plastic distribution, shearing force is determined from bolt rows that cannot convey bending torque. Alternatively, it may be assumed as combination of tension and shearing.

Tensile strength of a single row of bolts is calculated from column flange under bending Ft.fc.Rd, from the column body under tension Ft.wc.Rd, from edge plates under bending Ft.ep.Rd and from column body under tension Ft.wb.Rd. Overall resistance of components under shearing and compression is the minimum between column body frame under shearing Vwp,Rd/ β, of column body under compression Fc,wc.Rd and of beam flange under compression Fc,fb.Rd. Conversion coefficient β is used to calculate frame stiffness of column body, as this is subject to shearing stress on the left and right side of the node.

Regarding nodes with two or more bolt rows, resistance of the group of bolts of adjacent rows must be taken into account. When plastic distribution of forces on bolted nodes is assumed, resistance of a single row of bolts shall be calculated according to 1.9 Βt.Rd in order to ensure sufficient deformation capacity. When the ability to develop plastic articulation on nodes is required, strain hardening effects may lead to overloading of welds. In this case, welds shall be designed so as to undertake bending torques increased to 1,4 Mj.Rd for linked elements and 1,7 Mj.Rd for non-linked elements.


6.1.6.2 Calculation of initial joint stiffness

The component method may be used for calculating initial stiffness of a node Sj.ini. Stiffness can be calculated by adding elastic deformations of each of connection components. Using rigidity coefficient of the component (k), deformation δ is equal to:

where F is the force on component i and Ε is the modulus of elasticity.

Node rotation can be calculated from components' deformation taking into account the length of the arm between tension and compression zones:

Using formula 7.8, the node stiffness can be calculated as follows:

When the above mentioned formula is converted to the following expression:

the full torque-rotation graph can be calculated in straightforward manner. Regarding column - beam nodes and continuity of beams, a linear elastic behavior can be adopted, equal to 2/3 of the node's bending torque resistance. Factor μ that expresses the proportion of the initial stiffness to the string stiffness (shearing stiffness), is equal to one in the elastic phase and greater than one in the elastoplastic segment of the graph, i.e. equal to:


Form factor ψ is equal to 2.7 for welded nodes and bolted edge plates, and 3.1 for bolted flange plates.

The effective stiffness factor, keff of each bolt row under tensioning shall be calculated according to formula:

where k, are the stiffness factors of the main components that correspond to the particular row of bolts. These commonly are: the flexible column body under tensioning, the column flange under bending, the bolts under tensioning and the end plates under bending.

For joints of end plates with two or more bolt rows under tensioning, the main components from all bolt rows are represented by an equivalent rigidity coefficient, keq, determined according to the formula:

The equivalent lever arm shall be determined according to the formula:


6.1.6.3 Required rotational capability

The required rotational capability of nodes depends upon the structure type; it rarely exceeds 60 mrad. When the component method is employed, the end plate under bending and the frame of the column body under shearing represent ductile components. The bolt subject to tensile and shearing forces, as well as the welds, are common examples of brittle components. For this purpose, it is recommended to avoid designing bolts and welds that determine the strength of the joints. Prediction models allow a distinction between plastic nodes from brittle ones as well as hazardous solutions, not only regarding design assumptions, but also regarding behavior description.

6.1.7 The concept of the equivalent T-stub

Bending of a plate is a complex, three-dimensional problem in nature. To address this problem, Eurocode has introduced the "equivalent T-stub" concept.

Semi - empirical formulae provide the T-stub length which is aligned to the actual conditions of yield lines for a single or multiple rows of bolts.

The abovementioned 3 types of failure can be represented in a very simple manner in figure 6.11.

Figure 6.10. Rotation capacity, design resistance limit due to brittle failure of the second row of bolts at a beam - column joint

Torque

Rotation

Experimental plot

Bilinear model

Rotation capacity in plastic zone


Regarding the first mechanism, "double bending" is observed close to the bolts and at the joint with the T-stub body. The second mechanism combines bolt yield with a simple yield plot at the joining point between the flange and the body. Bolts are critical in mechanism 3, thus the failure pattern is only relevant to their yield. It is worth mentioning that only mechanism 3 utilizes the full strength of the bolts; regarding the other two mechanisms, part of their strength is utilized to counteract contact forces Q. In 1st failure pattern, only 70% of the bolts' tensile strength is available.

Figure 6.11. T-stub failure mechanisms

Thin plate, strong bolts

Bolt tensile strength

Thick plate, weak bolts


6.1.7.1 Multiple bolt rows

The process becomes more complex in cases of multiple bolt rows at close spacing, which interact in undertaking tensile forces. It is worth noting that a pair of rows will assume less than twice the force a single row would undertake. EC3 stipulates that outer bolt rows (above beam flange) are of key importance; due to their increased lever action, they will assume torques in a much more effective manner. This can be best understood if the calculation procedure is described. Resistance of the outer row is calculated as if the other bolt rows under tension are non - existent. The second row under tension shall assume a force resulting from the strength of rows 1 and 2 as these are assumed to be a bolt group, minus the minimum strength of row 1 or 2 assumed to be single rows etc., and assuming all possible bolt group combinations. The presence of reinforcement plates will restrict the number of rows whose forms of failure (curved yield) are combined.

At the end of this process, a column-type vector is determined, with all bolts force metrics. Clearly these can be restricted by a different component of the joint, dependent upon which component is considered to be critical. Commonly, column body is the critical component. Eventually, strength torque of the joint is calculated as S [Fti hi] where Fti is the force of the row of bolts i and hi its lever arm, taken from the centre of compression, which is initially assumed to be the bottom flange of the beam.

It must be stressed that this calculation formula is based on plastic distribution of bolts' forces; this fact requires that the node is capable of developing sufficient rotation, until joint T-stubs assume their full strength. As thickness values of the end plate and beam flange increase, the above mentioned control of rotation capacity becomes less meaningful, as bolts tend to become critical under such conditions and the footings of the respective T-stubs obtain much higher stiffness rates.

Furthermore, an alternative process based on triangular distribution of bolts' forces can be considered.

Then, the lever arm would determine the amount of force assumed by each row of bolts under tension, while footing thickness values for end plate columns would not influence the calculation in any way. The use of such a distribution in full strength joints is prohibited; therefore no further reference to it will be made.


6.1.8 T-stubs pattern on end plate

Bolts a1-a4 and b1-b4 near beam flange under tension, can be taken for bending torque resistance calculation, ref. figure 6.12. The bolts c2 and c3 could also be considered. However the bolts c1 and c4 can not be considered for the transfer of tension due to the limited stiffness of the end plate. These bolts together with the bolt row d could be useful as regards shearing.

Depending on the size of the end plate and bolt spacing, there are several possibilities for yield line patterns for bolts in rows a and b. The expected formation of yield lines at the extension of the joining plate is shown in figure 6.13. Bolts b1 and b4 are expected to present similar yield lines.

Figure 6.12 a) End plate with 4 bolts in row, b) separation into T-stubs , c) division of top bolt rows in separate T-stubs

Bolts under tensile force

Bolts only under shearing

Figure 6.13. Expected formation of yield lines at the extension of the connecting plate


6.1.9 Transfer of tensile forces

Many details are available regarding the calculation of tensile forces distribution on bolts; these are fully addressed in EC3, part 1.8. The basic tensile forces transfer mechanism to bolts has been referred to in an introductory chapter, taking into account the deformability of the flange or of the end plate depending on the case.

The strength of a row of bolts that may lead to failure conditions will be possibly restricted either by the bolts - carrying plate, or by strength of bolts, or by a combination thereof (a total of 3 forms of failure ). In case the end plate or the flange, respectively, is thin, then they shall be deformed due to bending. If they are thick, then the bolt will fail prior to yielding of the plate or the flange. In conditions where the thickness of the plate / flange obtains intermediate values, then the failure mechanism is complex and will result from a combination of the two independent factors mentioned above. Regarding the tensile force of the beam flange, this will spread in the body over a length equal to beff.


It is assumed that under the ultimate failure condition, when part of the body with a length of beff yields. Then tensile force Ft will become:

Respectively, for a welded joint, the same effective width beff, is used, both in the tensioned and the compressed zone, despite the experimental results that have shown that this is often longer in the zone under tensioning.

For bolted connections, the effective length of the T-stub of column body under tensile force is considered to be equal to the overall effective length of equivalent T-stubs. The column body can be reinforced either by flange reinforcing plates, or by plates welded on both sides of the column body (figure 6.15).

Figure 6.14. Effective length of column body zone under tensile force

Welded node End plate with 2 bolt rows*


6.1.9.1 Possibilities to reinforce column flange 6.1.9.1.1 Reinforcement with transverse reinforcement plates

In contrast to welded joints, bolted joints can be locally reinforced by plates located on the column, so that the force can be directly transferred by the beam flange to the column body, without any bending taking place.

Figure 6.15. Reinforcement using body plates

Figure 6.16 Reinforcement column flange


In case plates are used as in figure 6.16, the stiffness and strength of column flange will increase; a fact which is beneficial to the strength of the row of bolts adjacent to the rib. The strength of such rows can be calculated by introducing an equivalent T-stub with equivalent length of ?eff = α m1 where the value of coefficient α is taken from figure 6.17, formulated both according to the plasticity theory and the experimental results. This value will depend upon the geometry next to the reinforcing plate.

6.1.9.1.2 Reinforcement with plates parallel to a flange (backing plates)

In a corresponding manner, it is possible to reinforce column flange with plates bolted upon the former. Selection of length lbp of plates must be emphasized; it is recommended to be taken as equal to effective length ?eff

Figure 6.17 Coefficient α to calculate T-stub effective length of a row of bolts adjacent to column rib


of the equivalent T-stub of bolts. Thus, plastic torque Mpa is increased, not Mpb in figure 6.19, therefore such reinforcement is efficient only when failure mechanism 1 is critical (only end plate is critical).

Figure 6.18 Reinforcement column flange with backing plates

flange plate

Figure 6.19 Increase of Mpa with column flange plates


6.1.10 Transfer of compressive forces

The compressive force on a non-reinforced column body may cause local buckling. Following performance of experiments on column-beam joints, the following determination model has been adopted. The assumption made is that buckling starts when the average stress over a section whose effective width is equal to beff becomes equal to the yield stress. This model is essentially the same as the one used for the transfer of tensile forces to the column body. The ultimate compressive force results as follows:

The following are various variations of the geometry of the joint. It is noted that imaginary lines that determine the value of effective width, are inclined by 2.5:1 within the thickness of the flange of column and 1:1 within other elements of the joint.

Figure 6.20 Effective width of column body under compressive force


In case b, as shown in figure 6.21, it is assumed that the end plate under the beam flange will yield before buckling occurs in the column body. In this case, compressive force Fc is spread throughout the entire end plate

Figure 6.21 Effective length of the compressed part of the joint on column body


thickness.

The following must apply for the projection of end plate le :

In case the last formula is not met, then the lowest part of the end plate becomes plastic and the effective width is decreased to take the following value:

However, the lever arm between the compression and tension centers will increase, even marginally; this is considered an advantage (in other words, the center of compression does not precisely coincide with the lower beam flange).

In case of significant axial forces in the column, the critical load in the compressed zone will decrease. As long as regular tension sn is less than value 0,5 fy, their influence can be ignored. For even higher stresses sn, design strength of the compressed part of the joint shall be calculated

Figure 6.22 Behavior of compressed joining zone with projecting end plate


according to the formula:


where R is a reduction factor considered to be . The column can be also reinforced in the compressed zone with reinforcement plates between flanges or by plates welded on its body. 6.1.11 Transfer of shearing forces

On asymmetric joints, e.g. on perimeter columns, column body is also loaded with a shearing force, Fv. This may also happen on symmetric joints loaded in an asymmetric manner. For example, in the following joint, tensile force to the upper flange must be transferred through the shear panel formed; then it will counteract the respective compressive force of the lower flange of the beam.

Figure 6.23 Shearing stresses developed on a column with reinforced flanges (asymmetric node)


Assuming that geometry of the column body will not permit buckling due to the shearing force, the shearing strength of the panel will be equal to:

Column body could be reinforced by diagonal plates or plates

welded thereupon. When the first solution is selected, care must be taken so that bolt placement problems are avoided.

Care must be taken in cases as the following, where implementation of such reinforcements may entail undesirable results regarding the resulting stress condition. A similar example is shown in figure 6.25.

Figure 6.24 Reinforcement of column body against shearing


6.1.11.1 Bolt shearing forces

Generally, force distribution over bolts must be selected in an optimal manner. Thus it is possible to uniformly distribute shearing force over the bolts. In this case, the bolts must be checked according to the applicable rules for combined shearing / tension, i.e.:

In formula (6.23), tensile force Ft,Rd is the tensile resistance of each bolt, as resulting from solving the joint, thus formula (6.25) will result. Moreover, the shearing force is usually distributed over the compressed part of the node. If the load-bearing capacity of these bolts is sufficient, then the other bolts are merely tested as regards tensile strength. In addition, and in order to allow sufficient deformation capacity, it is important that bolts resistance to shearing is greater than their resistance under intrados compression, either on the plate or on the column flange.

Figure 6.25 Reinforcement of top node (stress condition in the panel will change)

Tensile force zone Compression zone

With reinforcement Without reinforcement


6.2 Detailed joint calculation

This section will provide the detailed solution of the particular joint, so that the results can be compared with the ones obtained from the previous calculation routine.

Thus, the following characteristic joint example is selected and thereafter all tests carried out are presented. Assume a column with ΗΕΒ300 section size and a ΙΡΕ 200 beam. Beam cross section will increase at the area of the joint, using a column of equivalent beam cross section, truncated at its upper flange. At the point of connection of the support beam flanges, a reinforcement of beam body is employed, to increase the resistance of the body against second order effects. Column flanges are also reinforced by reinforcing ribs at the compression center area (lowest flange of the beam), as well as in the area of the upper flange of the beam. A pair of ribs is used, to form a panel on column body; the latter is intended to receive the shearing force thereupon. Furthermore, reinforcement plates are also placed on both sides of column body to increase both its shearing resistance and its rotation stiffness as well. Finally, tensile force zone of the flange shall be reinforced by plates placed locally at the back of its flange (backing plates).

It is noted that the present example utilizes all possible reinforcements of a node of this type, to provide a comprehensive presentation of the Eurocode method and also to demonstrate the capabilities of the software developed with respect to handling a torque node between beam-column.

In the following, relevant drawings of the joint will be shown, to enable understanding of its geometry. It is reminded that all dimensions are in kN, mm, unless specified otherwise. The steel used is S235 grade for all

Figure 6.26 Example of shearing force distribution on bolt rows

Distribution of shearing force on non-tensioned bolts

Equivalent shearing force on each bolt


members and joining media. Furthermore, bolts are Μ20 / 8.8 and holes are regular (22 mm). All weldings are fillet welds with the following thickness values:

• Between beam flange and end plate: 5 mm • Between beam body and end plate: 3 mm • Between column body and column reinforcement plates: 6 mm • Between column body and beam flange: 3 mm

It is reminded that thickness of transverse ribs is at minimum equal to the flange of the member intended for reinforcement, therefore column ribs (reinforcements) are selected so as to have a thickness of 20 mm, and 10 mm for the beam.

Figure 6.27 Stress quantities acting on the node under examination


Figure 6.28 Joint front view

column flange ribs

local reinforcement plate in the tensioned zone 340/120/20

end plate 550/250/20

beam rib

section of ΙPE200

reinforcement plates on both sides of 800/220/15 body


Figure 6.29 Joint side view

Figure 6.30 Joint plan view


6.2.1 Spacing check 6.2.1.1 Bolt spacing check Regarding bolts on the end plate:

6.2.1.2 Spacing check of reinforcement plates column body

The following apply to reinforcement plates of the column body: According to paragraph 6.2.6.1 (9) (prEN 1993-1-8), the width of plates, bs, shall be sufficient so that the body reinforcement plate extends at least to the root of the mounting radius, thus for a malleable cross section, the check shall be performed in the following manner:

where run is the mounting radius of the column cross-section. Correspondingly, the maximum width of plates is equal to

Furthermore column body thickness must be less than or equal to the plate thickness, i.e.:


6.2.1.3 Spacing check of column flange reinforcement plates

Regarding the plates used for local reinforcement of the flange of the column (backing plates) the following shall apply:

Additionally :

In this particular case:

Figure 6.31 Determination of extreme dimensions for plates used for local reinforcement of the tensioned zone (backing plates)


6.2.1.4 Check of beam column angle

Regarding the beam column, specifically the angle between column beam flanges, γ, the following shall apply:

6.2.1.5 Check of weldings

Regarding weldings between joining media the following apply: Welding thickness between beam flange and end plate: a = 5mm<max a = 0.7x8.5 = 5.95mm Welding thickness between beam body and end plate: a = 3mm<max a = 0.7x5.6 = 3.92mm Welding thickness between column body plates and column body: α = 6 mm < max α = 0.7 χ 11 = 7.7 mm Welding thickness between beam-column and beam flange: α = 3 mm < max α = 0.7 x 5.6 = 3.92 mm 6.2.2 Initial calculations and estimation of general joining parameters 6.2.2.1 Values of transformation parameter Β

The following shall apply with respect to transformation parameter β:

6.2.2.2 Column shearing surface

It is calculated for a ductile column cross-section, according to the following formula as proposed in EC3, part 1.1, paragraph 6.2.6 (3):

According to paragraph 6.2.6.1 (6) (prEN 1993-1-8), the shearing surface may be increased by bstwc, due to the use of body reinforcement plates, where bs, twc, are the width and thickness of reinforcing plate and column body respectively.


Thus the shearing surface becomes:

6.2.2.3 Calculation of effective width of the column body under compression

According to paragraph 6.2.6.2 (1) (prEN 1993-1-8), for bolted connection to the end plate, the effective length of the column body under compression is:

where sp is the length resulting from a projection under 45° through the end plate (at least t,, and up to 2tp, provided that the section of the end plate close to the flange is sufficient). In particular, sp = 40 mm, hence: beffcwc = 293.96 mm, where tn is the thickness of the column flange, as it is assumed that the centre of compression of the joint is located in the column flange. 6.2.2.4 Reduction coefficient ω to account for interference between compression and shearing

According to table 6.3 it is derived that:

and

6.2.3 Solution of joint 6.2.3.1 Finding stiffness coefficients of joint elements


6.2.3.1.1 Pre-assessment of number of tensioned bolt rows

This will be arbitrary, taken as equal to the number of bolts located above the bottom flange of beam cross section, i.e. 4.

Thereafter, stiffness factors k1, k2, k3, k4, k5, k10, shall be calculated according to table 6.10 (prEN 1993-1-8). Of the above mentioned coefficients, all but the first two refer to bending of the end plate-flange or to tensioning of bolts and column body, thus they must be calculated separately for each row of bolts under tensioning conditions. On the other hand, stiffness factors k1, k2 referring to the column flange under shearing and under compression, can be calculated irrespectively of the rows of bolts. 6.2.3.1.2 Stiffness coefficients for the first row of bolts under tensioning

Stiffness coefficient ka (column body under tensioning):

According to table 6.11 (prEN 1993-1-8) parameter befftwc is taken to be the effective width of column body under tensioning from 6.2.6.3 (prEN 1993-1-8). For a node with one row of bolts under tensioning, beff,t,wc must be taken equal to less than the effective lengths ?eff (individually or as member of a group of rows of bolts) as given for the particular row of bolts in Table 6.4 (prEN 1993-1-8) (non-reinforced column flange) or in Table 6.5 (prEN 1993-1-8) (reinforced column flange). Instructions for calculating the effective length of the equivalent T-stub, are provided in table 6.4 (prEN 1993-1-8), since this is a row not adjacent to a column reinforcement structure (rib). Distances defined in the above mentioned table refer to the particular row:

m = 17.90 mm m2 = 36.61mm e = 80.00 mm e1 = 390.00 mm c = 45.00 mm

The above is an edge row of bolts, therefore effective length shall be derived from these formulae: An upper ceiling exists, thus effective lengths are ignored under the condition:


• The row of bolts is assumed to be individual ο For circular patterns:

ο For non - circular patterns:

• The row of bolts is assumed to be member of a group of rows of

bolts. ο For circular patterns:


Minimum effective length is equal to 112.47 mm; this constitutes the effective length of the first row of bolts for the column body under tensile force.

Thus, stiffness coefficient k3 shall be equal to:

where dc = hc 2tfc = 283.00 mm (6.44)

Stiffness coefficient k, (column flange under bending force):

Table 6.11 yields:

where length is equal to effective length calculated above, i.e. 112.47 mm. Distance m is defined in accordance with figure 6.8 (prEN 1993-1-8) and is equal to m = 17.90 mm. Thus, coefficient k4 will become:


Stiffness coefficient k5 (end plate under bending force, one row):

The effective length needed to calculate coefficient k5 will be derived from table 6.6 (prEN 1993-1-8). The distances defined in this table for this particular row are:

m = 38.81 mm mx = 34.34 mm e = 80.00 mm ex = 40.00 mm c = 45.00mm w = 90.00 mm

This constitutes a row of bolts within the beam flange under tensioning, therefore the effective length shall be derived from the following formulae:



• The row of bolts is assumed to be member of a group of rows of bolts.

It is not necessary to assume that the row belongs to a group. Minimum effective length is equal to 125.00 mm; this constitutes the effective length of the first row of bolts for the end plate under bending. Thus, stiffness coefficient κ5 shall be equal to:

Stiffness coefficient k10 (bolts under tensioning, single row of bolts):

For the calculation of this coefficient, value Lb (constituting the deformable bolt length, which is assumed to be equal to the retaining length - total thickness of metal and washer) is required, plus half of the sum of the thickness of bolt head and washer. Regarding geometry of the particular joint, it is derived that Lb = 61.95 mm, hence:


where As is the effective surface of a single bolt.

For the examined row (1st), effective stiffness coefficient keff,r, is calculated according to the formula:

where ί = 3,4, 5,10. (6.51) 6.2.3.1.3 Stiffness coefficients for the second row of bolts under tensioning

Stiffness coefficient k3 (column body under tensioning): Instructions for calculating the effective length of the equivalent T-

stub, are provided in table 6.5 (prEN 1993-1-8), as this is a row located adjacent to a column reinforcement structure. The distances specified in table 6.5 for this particular row, are as follows:


There is an upper ceiling, thus effective lengths under component e1 are ignored. Coefficient α from figure 6.11 shall be calculated after solving for values λ1 and λ2.

and

thus the resulting value of α shall be equal to 8.0.




• The row of bolts is assumed to be member of a group of rows of bolts ο For circular patterns:


Minimum effective length is equal to 97.02 mm, and this constitutes the effective length of the first row of bolts for the column body under tensioning.


where dc = hc 2tfC = 283.00 mm (6.58)

Stiffness coefficient k4 (column flange under bending):

Table 6.11 yields:

where length is equal to effective length as calculated above, i.e. equal to 97.02 mm. Distance m shall be derived in accordance with figure 6.8 (prEN 1993-1-8) and is equal to m = 17.90 mm. Thus, k4 coefficient will become equal to:


Stiffness coefficient k5 (end plate under bending, one row):

The effective length required to calculate k5 coefficient shall be derived from table 6.6 (prEN 1993-1-8). The distances defined in this table are as follows for this particular row:

m = 38.81 mm mx = 34.34 mm e = 80.00 mm ex = 40.00 mm c = 45.00 mm w = 90.00mm

This is a row of bolts outside the beam flange under tensioning, therefore the effective length shall be derived from these formulae:



• The row of bolts is assumed to be member of a group of rows of bolts.

No need to assume that this row is a member of a group

Minimum effective length shall be equal to 125.00 mm, and this constitutes the effective length of the first row of bolts for the end plate under bending.


Stiffness coefficient k10 (bolts under tensioning, a single row of bolts):

According to the above calculation it is derived that Lb = 61.95 mm, therefore:

where Α5 is the effective surface of a single bolt.


For the examined row (2nd), the effective stiffness coefficient keff,r, shall be calculated according to the formula:

where i = 3,4, 5,10. (6.65) 6.2.3.1.4 Stiffness coefficients for the third row of bolts under tensioning

Stiffness coefficient k3 (column body under tensioning):

Instructions for calculating the effective length of the equivalent T-stub, are provided in table 6.5 (prEN 1993-1-8), as this refers to a row located adjacent to a column reinforcement structure. The distances specified in table 6.5 (prEN 1993-1-8) for this particular row, are:


There is an upper ceiling, therefore the effective lengths under e1 component are ignored. Coefficient α from figure 6.11 (prEN 1993-1-8) shall be calculated after solving for λ1 and λ2 values.

and

and the resulting value of a α is equal to 8.0.






Minimum effective length is equal to 105.27 mm, where this constitutes the effective length of the first row of bolts for the column body under tensioning. Thus, stiffness coefficient k3 shall be equal to:

where

Stiffness coefficient k4 (column flange under bending): Table 6.11 yields:

where length £en is equal to the effective length as calculated above, i.e. 105.27 mm. Distance m shall be determined in accordance with figure 6.8 (prEN 1993-1-8) and is equal to m = 17.90 mm. Thus, coefficient k4 will become:

Stiffness coefficient k5 (end plate under bending force, one row):

The effective length required to calculate coefficient k5 shall be derived from table 6.6 (prEN 1993-1-8). The distances defined in this table for this particular row are as follows:


m = 38.81 mm mx = 34.34 mm e = 80.00 mm ex = 40.00 mm c = 45.00mm w = 90.00 mm

This is the first row of bolts underneath the beam flange subjected to tensioning; therefore the effective length shall be derived from the following formulae:

and



• The row of bolts is assumed to be member of a group of rows

of bolts ο For circular patterns:


Minimum effective length is equal to 217.66 mm and this constitutes the effective length of the first row of bolts for the end plate under bending. Thus, stiffness coefficient κ5 shall be equal to:


According to the above calculation it is derived that Lb = 61.95 mm, therefore:


where Α5 is the effective surface of a single bolt.

For the examined row (2nd), effective stiffness coefficient keff,r, shall be calculated according to the formula:

where i = 3,4, 5,10. (6.83) 6.2.3.1.5 Stiffness coefficients for the fourth row of bolts under tensioning

Stiffness coefficient k3 (column body under tensile force):

Instructions for calculating the effective length of the equivalent T-stub are provided in table 6.4 (prEN 1993-1-8), as this refers to a row not adjacent to a column reinforcement structure. The applicable distances as specified in table 6.4 (prEN 1993-1-8) for this particular row are:


There is an upper ceiling, thus effective lengths are ignored under the condition e1:






Minimum effective length is equal to 112.47 mm; this constitutes the effective length of the first row of bolts for the column body under tensioning.

Thus, stiffness coefficient κ3 shall be equal to:

where dc=hc 2tfC= 283.00 mm (6.88)

Stiffness coefficient kA (column flange under bending force):

Table 6.11 (prEN 1993-1-8) yields:

where length eff is equal to the effective length as calculated above, i.e. 112.47 mm. Distance m shall be specified in accordance with figure 6.8 (prEN 1993-1-8), being equal to m = 17.90 mm. Thus, coefficient κ4 will become:

Stiffness coefficient k5 (end plate under bending, one row):

The effective length required to calculate coefficient k5, shall be derived from table 6.6 (prEN 1993-1-8). The distances defined in this table for this particular row are:

m = 38.81 mm mx = 34.34 mm e = 80.00 mm ex = 40.00 mm c = 45.00 mm This constitutes an internal row of bolts close to the beam flange, therefore the effective length shall be derived from the following formulae:


and



• The row of bolts is assumed to be member of a group of rows of bolts ο For non - circular patterns:

ο For circular patterns:

Minimum effective length is equal to 230.73 mm; this constitutes the effective length of the first row of bolts for the end plate under bending. Thus, stiffness coefficient k5 shall be equal to:


According to the above calculation, it is derived that Lb = 61.95 mm, therefore:

where A5 is the effective surface of a single bolt. For the examined row (i.e. 2nd), effective stiffness coefficient keff,r, shall be calculated according to the formula:

where i = 3,4,5,10. (6.99)


6.2.3.1.6 Calculation of equivalent stiffness coefficient keq

The equivalent lever arm of joint zeq, is derived according to the formula specified in paragraph 6.3.3.1 (3) (prEN 1993-1-8):

where r is each tensioned row (r = 1, 2, 3, 4) and hr is the distance of each individual row from the centre of compression of the joint. It is concluded that zeq = 375.36 mm.

Thereafter, the equivalent stiffness coefficient is calculated as follows:

6.2.3.1.7 Miscellaneous stiffness factors of the joint

The only thing remaining to do is to calculate stiffness factors k1 and k2 of the joint, referring to column flange under shearing and compression respectively; nevertheless these will be needed at a later stage, for calculation of the rotation stiffness of the said node. Coefficient k1 is assumed to be infinite, as the node is assumed to be reinforced due to the existence of ribs on the column. In similar manner, coefficient k2 k1 is also assumed to be infinite, as the column of the node includes a reinforcing rib close to the compressible beam flange, as well as at the reinforcement plates of the column body. 6.2.3.2 Finding of strength values for the main components of the joint 6.2.3.2.1 Column body under shearing

Paragraph 6.2.6.1 (prEN 1993-1-8) shall apply. A condition is that the following inequality is valid:


Where . In fact

The design shearing strength of the non - reinforced body is provided by the following formula:

where Avc is the increased shearing surface of column cross section, due to the existence of body reinforcement plates. According to paragraph 6.2.6.1 (4) (prEN 1993-1-8), when transverse ribs are simultaneously used in the tensile and compressed zone, design plastic shearing resistance of the column body Vwp,Rd can be increased by Vwp,add,Rd as follows:

however

where: ds is the distance between the gravity center lines of ribs VΡl,Fc,Rd is the design plastic bending resistance of the column flange Vwp,add,Rd is the design plastic bending resistance of the rib

In particular:

hence


The resulting transformation coefficient β is equal to one, therefore reduction of the above mentioned shearing force resistance is not necessary. 6.2.3.2.2 Column body under compression

Coefficients ω and β used in chapter 6.2.6.2 have been already calculated. The design compression strength of the column shall be calculated according to the formula:

however

As reinforcement plates are used on the column body, column body thickness can be set to twice its value in the above mentioned formula. According to paragraph 6.2.6.2 (2) (prEN 1993-1-8), when maximum longitudinal compressive stress ocomEd exceeds 0.7fywc on the body (due to axial force and bending torque on the column) (next to the mounting radius for a ductile cross section, or on the root weld for e weld joint), its impact on the design compression strength of the column body must be taken into account by multiplying the value of FcwcRd supplied in formula 6.9 (prEN 1993-1-8) applying a reduction coefficient kwc, the latter is defined as follows:

• when

• when

For this particular case:

thus kwc = 1.


Regarding the reduction coefficient ρ for plate buckling we have: • for < 0,72: ρ = 1,0

• for > 0,72:

where η is its specific buckling capacity, as supplied by the following formula:

with for a column with ductile cross section

Compression resistance of the column can be increased due to the existence of a compressed rib on the column, by:

Therefore: 6.2.3.2.3 Body and beam flange under compression

The required resistance of beam flange shall be derived from chapter 6.2.6.7 (prEN 1993-1-8). Special attention must be given to the presence of the column as well, since the latter is located within the compressed area of the node.

Calculation of the strength of the column under body and flange compression, shall be performed initially. Shearing strength is :

thus reduction of strength torque of the column cross section - due to interference with the shearing force - is not necessary.

Thus, strength torque will become:


and the requested column strength is:

Regarding beam resistance, it shall be calculated in a similar manner to the calculation of column strength under compression, however

it shall be increased by , due to the existence of the rib on the beam. While bst and tst are the width and thickness of the reinforcing plate of the beam, respectively. Eventually, the resulting strength value shall be equal to 359.96 kΝ > 337.39 kN, therefore to enhance safety, the compressive strength for body and flange of the composite beam cross section due to the column, is considered to be 337.39 kN. 6.2.3.3 Iterative process for finding the number of rows under tensioning

This section will employ the iterative process presented in paragraph 6.3.2 (general flow diagram), to find the number of rows of bolts under tensioning. Partial bolt rows shall be regarded as individual or group members in order to find their critical tensile resistance load. The iterative process shall commence from the top row of bolts and shall stop when a row with negative strength is found. It is noted that (as repeatedly noted before), the plastic method for the determination of node strength values is implemented; this fact will detach this examination from the magnitude and direction of actions effected on the node.

Rows considered to be individual:

• 1st row of bolts: ο Column body under tensioning

According to paragraph 6.2.6.3 (3) (prEN 1993-1-8), on a bolted connection, the effective width beff,t,wc of the column body under tensioning must be taken as equal to the effective length of an equivalent T-stub corresponding to the column flange. The row in question is not adjacent to a rib, therefore the required effective lengths will be derived from table 6.4. Furthermore, and since it is assumed that an upper ceiling exists, the effective lengths shall be equal to:


Regarding distance values m, e, it follows that: m = 32 mm and e = 80 mm

According to table 6.2, we have:

Furthermore, the plastic torques developed on the equivalent T-stub of the flange are:

Due to the existence of local reinforcement of the column flange, the resistance force shall be:

where η = 40.75 mm

The required strength is provided by the following formula:

where

It is eventually found that o Column flange under bending


The examined row is not adjacent to a rib, therefore the effective length of the equivalent T-stub for bending of the column flange is provided by the following formulae:

where η = 40.75 mm

and

ο End plate under bending

The effective lengths for the end plate under bending and for a row above the flange under tensioning are provided by the formulae:


and

ο Beam body under tensioning

According to paragraph 6.2.6.8 (prEN 1993-1-8), the effective length is equal to the one mentioned above since it is considered that that the T-stub developed is equal to the one found in the case of the end plate under bending.

Strength is calculated as follows

Critical failure mechanism for 1st row of bolts assumed to be individual:

Beam body under tensioning with FT = 164.50 kΝ • 2nd row of bolts:

ο Column body under tensioning

The row is adjacent to a rib, therefore the required effective lengths may be inferred from table 6.5 (prEN 1993-1-8). The value of parameter α as inferred from figure 6.11 (prEN 1993-1-8), is equal to 7.71 for λ1 = 0.29 and λ2 = 0.39. The effective lengths are determined as follows:


According to table 6.2 it follows that:

Furthermore, the plastic torques developed on the equivalent T-stub of the flange are:

Due to the existence of a local reinforcement on the column flange, the following shall apply for the resistance force:

The required strength is provided by the following formula:

where

Eventually, it follows that

ο Column flange under bending

The effective length is equal to the value calculated above, = 251.28. Furthermore Fb = 141.12 kΝ, therefore

ο End plate under bending

Effective lengths for the end plate under bending and for a row of bolts above the flange under tensioning, are provided by formulae:


and

ο Beam body under tensioning

The effective length is equal to the above mentioned, =125.00 mm

The strength is calculated as follows:

Critical failure mechanism for 2nd row of bolts assumed to be individual: Beam body under tensioning with FT = 164.50 kN

In similar fashion, the strength value for the 3rd and fourth row of bolts can be derived. The calculated strength values, as well as the critical mechanisms are appropriately noted:

Critical failure mechanism for the 3rd row of bolts assumed to be individual: Column flange under bending with FT = 282.24 kN

Critical failure mechanism for the 4th row of bolts assumed to be individual: Column flange under bending with FT = 282.24 kN


Analysis will respectively carry on for all rows, assuming they are all individual as well as group members, in accordance with the general flow diagram shown in paragraph 6.3.2. Following completion of all iterations of the calculation process and assuming all bolts are group members, as stipulated by EC3, the following results will be derived, in conjunction with the final strength values for each row of bolts. One may observe the non-linearity of distribution of bolts' forces, a fact which is expected due to variation in rigidity between the tensioned and compressed areas.

Column body under shearing Column body under compression

Beam body & flange under compression

940,76 260,35 337,39

Table 6.2 Strength values of basic node components (in kΝ)

Rows considered to be individual Row i=l Column body under tensioning 529.43

Column flange under bending 282.24 End plate under bending 217.81

Beam body under tensioning 164.45 Fsolo = 164.45

Row i=2 Column body under tensioning 577.42 Column flange under bending 282.24

End plate under bending 217.81 Beam body under tensioning 164.45

Fsolo = 164.45 Row i=3 Column body under tensioning 577.42

Column flange under bending 282.24 End plate under bending 282.24

Beam body under tensioning 395.41 Fsolo = 282.24

Row i=4 Column body under tensioning 529.43 Column flange under bending 282.24

End plate under bending 282.24 Beam body under tensioning 395.41

Fsolo = 282.24


Rows considered to be group members

Row j = 1

Column body under tensioning over group 700.91 Column flange under bending over group 489.99

End plate under bending over group 489.99 Beam body under tensioning over group 493.62

Fg = 489.99

SumFsolo = 164.45 Fqroup = 325.54

Row i=2

Fup = 164.45 F(i) = 164.45

Row j = 1

Column body under tensioning over group 1Ε + 10 Column flange under bending over group 529.2

End plate under bending over group 529.2 Beam body under tensioning over group 1Ε + 10

Fg = 529.2


Row j = 2

Column body under tensioning over group 1Ε + 10 Column flange under bending over group 352.8

End plate under bending over group 352.8 Beam body under tensioning over group 1Ε + 10

Fg = 352.8


Row i=3

Fup = 328.9 F(i) = 8.39


Row i=4 Row j = 1 Column body under tensioning over group 1Ε + 10

Column flange under bending over group 705.6 End plate under bending over group 705.6

Beam body under tensioning over group 1Ε + 10 Fg = 705.6


Row j = 2 Column body under tensioning over group 1Ε + 10


Beam body under tensioning over group ΙΕ + 10 Fg = 529.2


Row j = 3 Column body under tensioning over group 1Ε + 10


Beam body under tensioning over group 1Ε + 10 Fg = 529.2


Fup = 611.14 F(i) = -273.75

Table 6.4 Strength values of rows considered as group members


The resulting force value for the fourth row was negative, therefore it is considered that only the first three rows are subjected to tensioning. The strength torque of the joint shall be derived as follows:

where η is the distance of the row from the centre of compression of the joint, which in this particular case is considered to be at the centre of the beam column flange. The strength torque has a value in excess of the action torque MEd = 70000 kNmm, therefore structural adequacy is achieved. 6.2.3.4 Classification of the joint 6.2.3.4.1 Finding initial stiffness Sj,ini

In accordance with paragraph 6.3.1 (prEN 1993-1-8) it follows that:

6.2.3.4.2 Classification regarding stiffness

According to paragraph 5.2.2.5 (prEN 1993-1-8), the following is the ultimate value of rotation stiffness Sjim of a node, in order to be regarded as rigid:

in fact, the above applies for kb = 25 (fixed frame), therefore the joint can be classified as rigid, in terms of stiffness. 6.2.3.4.3 Classification regarding strength

The bending strength of the beam cross section at the joint, taking into account the column, shall be:

Furthermore it follows that:

Hence, the joint is considered to be of full strength.


6.2.3.5 Check of rotation capability of joint

According to paragraph 6.4.2 (prEN 1993-1-8), the joint has insufficient rotational capability, since the critical mechanisms are as follows:

• Row 1: Beam body under tensioning • Row 2: Beam body under tensioning • Row 3: Body and beam flange under compression The abovementioned components of the joint should be stronger in

order to meet this criterion, while parts referring to bending of the flange and end plate could become weaker, so that these elements then become critical. Then, the type of failure would exhibit a more ductile behavior and rotational capability would be considered to be sufficient.

6.2.3.6 Check for shearing force This check is carried out pursuant to table 3.4 (prEN 1993-1-8).

The resulting strength value for each row of bolts is considered equal to the minimum of its strength under intrados compression and shearing force. The resulting strength values for each row are presented in table 6.5.

Row of bolts Strength under shearing

Strength under intrados compression

1 102.54 198.44 2 102.54 244.80 3 102.54 1627.36 4 102.54 1627.36 5 102.54 1627.36

Table 6.5 Strength of each row under shearing force

Apparently, the critical quantity for each row is the shearing strength of bolts, therefore the shearing strength of the entire joint will become:

Furthermore, it is necessary to examine the validity of the criterion of interference between tensioning and shearing for the bolts, i.e.:

This criterion is met for each row of bolts.


6.3 Flow diagrams 6.3.1 Finding initial stiffness Sj,ini

Flow diagram 6.1 Finding initial stiffness

Estimate of bolts under tensioning

Column body under tensioning K3J

For each row of bolts under tensioning

Column flange under bending K4,i

End plate under bending Κ 5,i

Bolts under tensioning Κ10,i

Column body under shearing ΚΙ

Column body under compression

Κ2 Initial stiffness


6.3.2 Finding of strength torque

Flow diagram 6.2 Finding strength torque

Column body under shearing Fcws

Column body under compression Fcwc

Body and beam flange under compression Fbfc

For each row of bolts i , starting from top

Column body under tensioning Fcwt

Column flange

under bending Fcfb

End plate under bending Fpb

Beam body

under tensioning Fbwt

For each row of bolts j above i

Column body under tensioning, of group i-j Fcwt,g

Column flange under bending, of group i-j Fcfb,g

End plate under bending,

of group i-j Fpb,g

Beam body under tensile, of group i-j Fbwt,g

Number of rows under tensioning nt = i-1

For each row of bolts j above i

For each row of bolts i from top up to nt

[hi: Distance of bolt row i from the middle under beam flange]

next iteration

NO YES


7 Prop support connection


7.1 Description of mechanical behavior 7.1.1 General

With the term 'prop base' we mean the lower part of the prop, the seating plate and the anchoring system (anchors). Usually the prop bases are designed without rigidity blades, but when they are expected to bear high bending torques, rigidity blades may also be used. The prop bases are seated either on a concrete foundation or on some other type of foundation (e.g. piles). Standard prEN 1993-1-8 includes rules for the calculation of their strength and rigidity. The process may be applied either to open or closed cross-section props. Moreover, it is possible to adopt design details that include seating plates reinforced with steel elements encased in the foundation concrete. The impact of the concrete base to the overall response of the prop base is not covered by standard prEN 1993-1-8.

The classic approach for designing articulated prop bases leads to seating plate thicknesses that ensure uniform distribution of stresses, so the seating plate may be simulated as a rigid element. The classic design of prop bases that bear torques is performed using elasticity analysis on the basis of the assumption that the cross-sections remain flat. By solving the equilibrium equations we can calculate the maximum stress on the concrete foundation (linear distribution of stresses) as well as the tensile stress at the anchoring system. Even if this methodology has been proven to be satisfactory for many years, it actually ignores the flexibility of the bending seating plate, anchoring system and concrete. The idea adopted in prEN 1993-1-8 transforms the flexible base to an active solid plate and permits development of stresses on the concrete equal to the accumulated compression resistance. To this end, we use plastic distribution of internal forces in the O.K.A.

To calculate the rigidity we can apply the connection components method as presented in a previous chapter. According to this method, we initially identify the important elements of the connection, which are called components, and then we define their strength and rigidity. These components are properly configured in order to provide the simulation of the overall connection.


The rules for calculating the resistance at the prop bases are provided in prEN 1993-1-8, Chapter 6.2.6, and the rules for calculating the rigidity are provided in Chapter 6.3.4. These chapters also include the characteristics of the prop base components, such as the components at the compressed side and at the bent plate (Chapter 6.3.2), at the prop footing and at the compressed body (Chapter 6.2.4.7), and at the tensioned side the tensioned anchoring system and the bending of the seating plate (Chapter 6.2.4.12). The method for receiving the horizontal shearing forces is provided in Chapter 6.2.1.2, while the limits for classifying the prop bases into categories are provided in Chapter 3.2.2.5.

Figure 7.1 Typical prop bases and selection of their componentsa) bolts inside the seating plate, b) bolts outside the seating plate

(possibly with rigidity elements)

Tensioned anchors and seating plate under bending

Prop footing and body under compression Prop footing and body under compression

Anchors under shearing

Reinforcing plate


7.1.2 Elastic resistance of the prop base

The prop bases operate by converting the flexible seating plate to an active solid plate. By limiting the plate deformations only within the elastic area, the simulation receives the uniform accumulated stress under the flexible plate. This provides a secondary assurance that the yield stress of the prop base is not exhausted. The calculation of the effective bearing area of the flexible seating plate is based on the effective width c

The elastic bending resistance per unit of length is calculated using the formula

Figure 7.2 Finite elements simulation for a prop base with short T cross-section, and concrete foundation under compression both before and after deformation, and main concrete stress

Figure 7.3 Simulations of prop bases


and the bending torque per unit of length, which acts on the seating plate and corresponds to a cantilevered beam of length c, is:

When these torques become equal, the bending torque resistance of the prop base reaches its limit, and the formula for the calculation of c is:

7.1.3 Resistance of prop base with low-quality mortar

The impact of the low quality mortar has been studied both by using experimental methods and numerically. It has been found that the thin layer of cement grout does not affect the concrete resistance. It is also expected that the thin layer of cement grout under 3-D compression between the concrete and seating plate shall behave like a liquid. Most of the mortars have high resistance in comparison to the concrete material. In such cases, the mortar may be ignored. In other cases, the resistance can be checked by assuming a distribution of regular stresses under the active plate at an angle of 45°. In case the mortar thickness is greater than 50 mm, the characteristic strength of the mortar shall be considered to be at least equal to the thickness of the foundation concrete [prEN 1993-1-8].

Figure 7.4 Distribution of stresses on the cement grout


7.1.4 Comparison of the calculations of concrete strength according to EC2 and EC3

Standards prEN 1992-1-1, Design of Concrete Structures, and prEN 1993-1-8, Design of Steel Structures, solve the same problem of concrete resistance when loaded by a steel plate. The limit of such compression is defined by the breaking of the concrete under compression. The relevant literature can be divided into to large categories. In the publications referring to research works related to the stress status of steel plates, most of which involve the use of prestressed anchors, and in the studies that focus on flexible plates loaded by the prop cross-section and bear the loads only through a part of the seating plate.

The experimental and analytical simulations include inter alia, ratios of concrete strength to the surface of the seating plate, the relevant depth of the concrete foundation, the identification of the seating plate installation area on the foundation, and the impact of reinforcement. The results of all such studies that relate to the breaking-through load and fully loaded plates provide quality information regarding the foundation of the prop base. The effects of failure are observed when an inverted stress distribution pyramid is created under the plate. The concrete ultimate state analysis application includes 3-D material behavior, plasticization and breaking. Experimental studies [Shelson, 1957; Hawkins, 1968, DeWoLf, 1978] led to the development of a suitable model for the stresses occurring at the prop bases, which has been adopted by the current standards. Moreover, it is required to perform a separate check of the concrete foundation under shearing, breaking-through and bending according to the final geometry of the prop base and all construction details. The impact of the flexible plate has been solved by introducing an equivalent solid seating plate [StockweLL, 1975], [Baniotopoulos, 1994]. This assumption corresponds to the actual non-uniform distribution of stresses where the max values follow the shape of the prop cross-section.

The realistic behavior of this model has been recently confirmed by experiments. 50 experiments in total have been carried out in order to investigate the concrete strength [DeWolf, 1978, Hawkins, 1968a]. In these experiments, the variants were the size of the concrete foundation, the size and the thickness of the seating plate. The test samples consisted of concrete cubes 150 to 330 mm, which were subjected to centric compression by a steel plate. Below we present the relation between the flexibility of the seating plate, which is expressed as the ratio of its thickness to the distance from its edge, and the corresponding bearing resistance. The bearing capacity of the test samples at failure varies from


1.4 to 2.5 times the capacity as calculated according to prEN 1993-1-8, with an average value of 1.75.

This simple and practical model has been modified and has been checked by experiments [BijLaard, 1982], [Murray, 1983]. It was also established [DeWolf, SarisLey, 1980], [Wald, 1995], that the stress rises as the eccentricity of the regular force increases. In the case where the distance from the edge of the plate to the edge of the foundation is constant and the eccentricity of the load increases, the contact surface decreases and at the same time the stress also increases. Moreover, in case of breaking of the concrete surface under the fixed edge, the problem must be addressed theoretically, and this defines the limits of the practical implementation of this method [Ivanyi, Baniotopoulos, 2000]. Figure 7.5 shows the impact of the concrete strength. From the comprehensive research program of Hawkins, 16 experiments with similar geometry and material properties were selected and are presented here [Hawkins, 1968a]; in these experiments the concrete strength was selected as the only parameter (the strengths used were 19.31 and 42 Mpa).


Figure 7.5 Relation between bearing resistance and prop base flexibility according to analytical formulae and experiments

Results of calculations Results of experiments

Figure 7.6 Relation between concrete strength - bearing capacity under ultimate load


7.1.5 Accumulation of stresses under the prop base

The mortar and concrete foundation resistance under compression has breaking under the seating plate as a limit. In the simulations used, the flexible seating plate has been replaced by an equivalent solid plate. The equivalent plate is defined by the cross-section of the prop increased by a strip with effective width c [prEN 1993-1-8]. The bearing resistance of the foundation corresponds to a 3-D concrete stressing status. In this case, the strength measured during the experiments was approximately 6.25 times higher than the resistance of the concrete under compression. The calculation of the bearing strength of the concrete fj is given in prEN 1993-1-8 as a function of the accumulation factor kj, with a max value of 5.0 for a square seating plate. The bearing resistance of the seating plate Fc,Rd is calculated by the following formulae:

The effective surface Aeff is shown in fig. 7.7. The mortar quality and thickness are introduced into the calculations through the node coefficient βj. For a βj=2/3, it is expected that the rated strength fck,g of the mortar shall not be less than 0.2 times the rated strength of the concrete foundation fck (fck,g < 0,2 fck) and the mortar thickness is tg< 0.2 min (a, b).


In the case of lower quality or larger thickness of the mortar layer, the latter must be checked separately. This check must be carried out in the same manner as the check for the concrete foundation strength.

Figure 7.7 Dimensions of concrete foundation and effective surface of the flexible seating plate

Only pivotal action

Effective surface around the cross-section

Effective surface around the compressed cross-section area

Combination of pivotal force and torque

Effective surface around the compressed cross-section area

Only bending torque

Pl. Neut. Axis

Pl. Neut. Axis


7.1.6 Effective length of the Short-T base plate

The effective length of a short-T under tensioning is affected by the failure mechanism of the short-T. Short-Ts at a prop base are similar to the short-Ts at the beam-prop connections, but their failure mechanism may be different. This is due to the large anchor lengths and the large thicknesses of the seating plates in comparison to the beam-prop connection plates. The failure mechanism usually results due to detachment of the short-T from the foundation, so no lever forces are produced. The resistance of the short-T when not in contact with the concrete is:

where is the plastic bending torque of the base per unit of length. The relation between failure mechanism 1, which corresponds to the beam-prop connection, and the short-T failure mechanism when in contact is shown in figure 7.9.

Figure 7.8 Short-T with no contact with the foundation


The limit between failure mechanisms with or without contact can be calculated using the analysis of the elastic deformations of the short-T, and it can be expressed in various manners, e.g. as the ultimate length of a bolt Lb,lim. For bolts longer than Lb,lim, neither contact effects nor lever actions are observed. That limit is:

where As is the surface area of the bolt cross-section and Lb is the free length of the anchor. For bolts encased in the concrete foundation, the Lb may be considered as equal to the length above the concrete surface Lbf, and the effective length of the encased section is estimated to be Lbe = 8 d, so Lb = Lbf + Lbe [Wald,1999].

Figure 7.9 Failure mechanism 1 for short-T at a prop base

Figure 7.10 Free length of anchors encased in the concrete foundation


The effective lengths of a short-T at a prop base are summarized below.

Figure 7.11 Dimensions of the prop base

Table 7.12 Effective lengths of short-T at a prop base

Bolts not used at the prop feet

Bolts in the prop feet

where lever forces appear where no lever forces appear


7.1.7 Slip coefficient between steel and concrete

The friction coefficient between the seating plate and the ground is provided in prEN 1993-1-8, paragraph 6.2.1.2. A value equal to Cf,d= 0.20 is used for cement-sand mortars, and a value Cf,d= 0.30 is used for special mortars. According to the CEB Guide [CEB, 1997], a value equal to 0.4 shall be used when a 3 mm thin layer of mortar is used. In such a case, a partial safety coefficient equal to gMf = 1.5 must be used. It must be noted that the friction coefficient acts, obviously, in favor of loading of the anchors due to the shearing force, so it must be considered as zero in favor of safety; thus we assume a smooth interface between the seating plate and the cement mortar. 7.1.8 Transfer of shearing forces through anchors

The horizontal shearing force can be received through friction, from the seating plate, the mortar and the foundation, through shearing and bending of the anchors, through a shearing element bonded under the seating plate or through direct contact of the seating plate with the foundation. In most cases, the shearing force is received through friction occurring between the seating plate and mortar. Friction is a function of the minimum compressive load and friction coefficient. Prestressing of the anchors leads to an increase of the resistance against the shearing force through friction. In cases where the horizontal shearing force cannot be received through friction due to the lack of a regular (vertical) force, provisions shall be made for the installation of elements suitable for receiving it (e.g. perforating nails).


The transfer of shearing forces through anchors has been widely implemented in the USA for many years [DeWoLf, Ricker 1990]. In case the holes, which are designed in a manner that ensures sufficient tolerances, are larger than the required size, they must be stabilized by a stabilization ring or by injecting resin in the anchor hole. Additionally, a conservative method is presented in [CEB, 1997]. According to this method, the anchors behave as cantilever beams with length equal to the mortar thickness increased by 0.5 d. Provided that the rotation of the anchor head is limited by the prop base, the opening is reduced to L/2.

In the model used for prediction of the design strength of the connection, as described in prEN1993-l-8, it is considered that the anchors shall be deformed whenever a tensioning force is applied on the anchor and a compressive force is applied on the mortar. That prediction is based on experimental and analytical work carried out by Bouwman [Bouwman et al., 1989]. The design methodology is simplified at the practical

Figure 7.13 Transfer of horizontal perforating forces at the prop base through a) friction from the seating plate, mortar and foundation

b) shearing and bending of the anchors c) shearing element bonded under the seating plate

d) direct contact of the seating plate with the foundation

Figure 7.14 Anchor subjected to bending according to CEB


implementations only by checking the shearing force and by limiting the ultimate conditions.

7.1.9 Transfer of shearing forces through friction and anchors

The resistance simulation used in prEN 1993-1-8, paragraph 6.2.1.2, is based on the assumption that the anchors loaded by a shearing force bend, so eventually tensioning is developed on them. The bending resistance of the anchors is relatively low, so plastic joints are created. This activates task-related hardening effects on the ductile material of the anchor. In order to transfer shearing only the anchors at the compressed area of the plate can be used. In this case, the shearing resistance consists of a friction resistance at the seating plate - concrete interface and of a reduced tensile resistance of the bolts, that is:

The resistance of the bolts against intrados compression at the seating plate and concrete must be checked separately.

Figure 7.15 Anchor behavior under shearing a) symbols, b) forces, c) force-deformation diagram

Tensile strength

Reduced tensile strength

Shearing and bending strength

Figure 7.16 Resistance to friction and tensioning on anchors

Resistance to friction


7.1.10 Bolt anchoring rules 7.1.10.1 Theoretical data

The rules of Table 3.2, prEN 1993-1-8, regarding the resistance of bolts against tensioning may be used for all steel qualities and for the anchors. The anchor force NEd must satisfy the formula:

where As is the net anchor cross-section, fub its ultimate strength, YMb the partial safety coefficient, and Bb the reduction factor for the reduced cross-section at the thread (the value Bb = 0.85 may be used).

The various types of anchors are shown in figure 7.17. In the case of high bending torques, it is possible to use anchors welded in groups as a mesh, but such a solution is costly. Models for calculation of the design resistance are proposed in the CEB Guide [CEB, 1997] and they are based on research carried out by Eligehausen [Eligehausen 1990]. In these models, the following failure mechanisms must be checked:

• Steel failure:

• Extraction:

• Concrete failure:

Figure 7.17 Types of anchors


• Failure due to concrete breaking:

Similar checks are required in the case of grouped bolts.

The calculation of the design resistance of an anchor includes the following parameters: The extraction resistance is:

where for non-cracked concrete: pk = 11fck and Ah is the bearing surface of its head; for circular and square heads:

The concrete resistance is:

where

is the rated resistance of an anchor. Fro non-cracked concrete we can use coefficient k1 = 11 (N/mm)0,5. The impact of the distribution geometry (interval) of anchors ρ and the distance form the edges e are included in the cone surface, see fig. 7.19.

Figure 7.18 Geometry of on-the-spot injected anchor with head


Consecutively the following apply:

As concrete cone width we can use the value pcrN = 3.0hef. The disruption to the distribution of concrete stresses may be taken into account by introducing coefficient:

Parameter Ψec,N is introduced in order to take into account the impact of a group of anchors, and is used for small encasing depths (hef < 100 mm). The resistance increases for anchors in non-cracked concrete through parameter:

Failure due to tearing for on-the-spot constructed anchors can be avoided, provided that the concrete is reinforced and the configuration of the anchors is dense. The distances of the anchors shall not exceed:

and the distance from the edge shall not be greater than:

Figure 7.19 Concrete cone at a) an individual anchor, b) a group of anchors, c) an individual anchor at the edge


The height of the concrete foundation shall not be less than:

where c0 is the required overlapping of the shoe reinforcement. 7.1.10.2 Check for bending of anchoring bodies

In case anchors with nuts or retention plate are being used, a check for bending is also required. From tables for square plates and pro-safety circular plates, the following applies:

where

with hρ equal to the thickness of the retention plate and b = 1. If the above check is not successful, the thickness of the retention plate or nut must be increased. 7.1.11 Impact of the biaxial nature of loading to the connection analysis procedure according to EC3

As is known, part 1.8 of EC3 does not provide details for dealing with the biaxial bending at metal element foundations; on the other hand it provides a summary of the fundamental principles for calculation of the connection of a prop to be founded, which is stressed by an axial force and a bending torque around its strong axis.

So, in a completely similar manner with the connection torque for a prop beam with front plate, we also propose the components method for the seating connection of a prop; on the basis of this method, as mentioned in chapter 7.2.6, the strength under torque is determined by the ultimate strength loads of each critical mechanism that may be realized.

Obviously, the critical mechanisms are directly related to the type of loading and it is difficult for any one of them to be reduced to generalized stress conditions using the instructions of the current version of EC. For example, suppose that we have a prop base stressed by a torque around its strong axis.


The basic components proposed are, in this case, related to the single-axis character of the connection. The components that are also related to tensioned elements, are characterized by an effective length of an equivalent short-T, which is considered to simulate the behavior of the connection. This length, in turn, relates to the arrangement of the bolts and takes into account any interactions between the rows of bolts for the realized failure type.

In the case of biaxial loading, with the neutral axis at random slope, intersecting the cross-section at random points, the term 'row of bolts' has no meaning; moreover, one of the mentioned components may be intersected by the plastic neutral axis, which makes non-valid certain checks, such as the check for 'prop body under tensioning' since, for example, the prop body may be tensioned partially, fully, or not at all.

The same stands for the mechanism checking the 'prop footing and body under compression'.

Thus we notice an inability of the EC to cover special cases, such as the biaxial stress case as discussed here, and this is attributed to the fact that EC is still in a premature stage.

Figure 7.23 Strength of basic connection components according to part 1.8 of EC3, for bending around the strong axis

tensioning compression

Basic connection components according to EC3: 1. Prop body under tensioning 2. Prop footing and body under compression 3. Seating plate with anchors under bending 4. Concrete under compression

Plastic neutral axis


7.1.12 Assumptions in order to eliminate the difficulties arising due

to the fact that EC3 does not provide instructions for biaxial loading

It is understandable that due to the above difficulties, certain assumptions must be adopted for analyzing the connection in order to adjust the provisions of EC3 in the case of bending around the two main axii of the prop. We must note that it is necessary to perform experimental procedures, and to carry out a detailed simulation of the node under the acting stress quantities in order to confirm such assumptions.

Initially, we must consider the node components relating to the geometry of the prop as non-critical, so that they do not contribute to the calculation of the strength torque of the node according to the components method. Using this assumption, no checks of the prop body under tensioning and footing body under compression are required. These concepts, as mentioned in the previous chapter, are valid mainly for bending around the strong axis.

Figure 7.24 Biaxial loading of prop under compression


As mentioned above, the basic components of the node, as proposed by EC3, are not valid in such a generalized loading.



From the above, we can deduct that the strength of the tensioned side of the node shall be determined only by the strength of the seating plate together with the anchors, against bending. Correspondingly, the strength of the compressed side of the node shall be determined only by the compression strength of the concrete. 7.1.12.1 Strength of the tensioned side of the node

In the case of random loading of the seating node, the neutral axis has random slope and it also intersects the cross-section at random points.

Figure 7.25 Case of loading of a prop cross-section under compression and biaxial bending




To deal with this general case, we create a number of short-Ts (with half the strength of the corresponding short-T) equal to the number of tensioned anchors. Each anchor that lies outside the footing of the prop cross-section is considered as creating a short-T with the seating plate, with width equal to its distance from the footing. Correspondingly, each tensioned anchor inside the feet creates a short-T with the body of the cross-section. The procedure to be followed is shown in fig. 7.25. Note also that the anchors outside the cross-section footing, which are also not inside the footing width, are not taken into account for the calculation of the strength torque. These are considered as receiving only a shearing force, and this is a pro-safety assumption.

Figure 7.26 Short-Ts formed at the tensioned area of the node

tensioning

compression



Thus the plastic forces of the anchors are independent of the loading and, consequently, of the position of the plastic neutral axis. In this phase of the calculation, the abovementioned assumption is of course involved, since we do not take into account any interactions between the anchor failure types, which normally are taken into account by considering the bolts as members of a group. Moreover, it is easy to calculate the torques around the strong and weak axii of the prop, which are created due to the tensile forces on the anchors.

When using this method for calculation of the anchor strength under tensioning together with the seating plate under bending, we also make another simplifying assumption regarding the formation of the short-Ts. These should be perpendicular to the plastic neutral axis, i.e. perpendicular to the direction of the resultant bending torque. This usually leads to more adverse results, since the bolts move away from the footing of the equivalent short-T, and thus their tensile strength decreases.

Figure 7.27 A more rational consideration of the short-Ts of the bolts,perpendicular to the direction of the resultant acting torque

tensioning

compression



Finally, in favor of safety, any reinforcing plates either at the strong or at the weak axis of the prop are not considered as reinforcement of the tensioned area, so they do not contribute to the tensile strength of the anchors. The reinforcing plates might otherwise be considered as reinforcing webbings of the tensioned zone, and in such a case the effective length of equivalent short-Ts would increase, resulting in a higher plastic strength. 7.1.12.2 Strength of the compressed side of the node

The strength of the compressed side of the node is determined only by the compression strength of the concrete. As effective we consider only the area under the compressed elements of the node and at a distance equal to c around them, while as compressed elements we can consider the feet and bodies, or their reinforcing plates, which are under compression. There, according to EC3, we consider as constant and equal to fr the distribution of stresses at the failure state. So, we have to calculate the compressed surface area in order to find the compression force, as well as the center of gravity of that area in order to determine the lever arm during the assessment of the strength torque.

The process involves strenuous calculations due to generality regarding the position and slope of the neutral axis. Initially, we form all candidate effective compressed areas, i.e. under each compressed body, footing, and reinforcing plate an orthogonal area is formed and projects by a distance equal to c at both sides of the element.


Figure 7.28 Effective foundation area around the compressed elements. The stress under this area is has constant distribution with value fj.

tensioning

compression


Figure 7.29 Effective area of a compressed element at distance c on both sides


Similarly for the compressed reinforcing plates.

After fully topologically defining the compressed area, we must calculate some inertial characteristics of this area in order to finally derive the plastic compression strength and the point where this force is applied. These shall be calculated in relation to the center of the prop, against which the intersection points of the compressed areas and the neutral axis will have already been calculated. The calculation is carried out as follows:

Figure 7.31 Compressed areas under a footing and body with double-T cross-section




where p = y, ζ and q = y, ζ and n is the number of corners of the examined effective area, and A, Αp, Apq, Appq are the moments of inertia of the first, second and third order towards ρ, q.

After completing the above procedure for all compressed effective areas and after increasing each inertial quantity by the value that corresponds to the current area, we derive as a sum the moments of inertia of the 0, 1st, 2nd, and 3rd order for the overall compressed zone. The important ones are the moment of zero order, which is equal to the surface area, and the moments of the 1st order towards y and z for identification of the position of the center of gravity. Indeed, towards a random coordinate system y, z, the position of the center of gravity of a closed flat surface section is provided by the formulae:

So we have determined the position of the resultant compressive force, and the center of gravity. The compressive force applied to the concrete shall be:

Figure 7.32 Definition of positive direction (counter-clockwise) for numbering of corners in an effective area


where fj is the characteristic compression strength of the node, which is also a function of the concrete quality and placement of the prop on the shoe. 7.1.12.3 Total connection strength

The total stress condition at the center of the prop shall be derived from the equilibrium of forces towards axis x, and torques around axis y and z. Finally, the quantities are derived as follows:

Axial strength (compressive force positive): where Fc and IFb, are the force at the compressed area and the total tensile strength respectively, exactly as they were calculated in the two previous paragraphs.

Strength torque around the strong axis:

Strength torque around the weak axis:

Figure 7.33 Coordinate system and position of the center of gravity of a compressed area



tensioned bolt i

compressed bolt, ignored during the calculation of the strength torque


7.1.12.4 Necessity of employing interaction diagrams

It must be noted that the method described above for the calculation for the connection strengths involves ultimate plastic strengths; both the anchors and the concrete shall be stressed by their max strength forces. Direct consequence of the above is the inability to find the exact stress status of a compressed and tensioned zone, since the values of the action torques and axial force do not affect the calculation process. The above strength calculation formulae have a compound form and a simple comparison of the actions with the corresponding strengths is not enough to ensure static adequacy. Instead, due to the interaction of these quantities (regular quantities), we must draw interaction diagrams for torques and axial forces; then, the adequacy of the connection shall be ensured provided that the loading lies within the diagram.

Figure 7.34 shows the process of successive placement of the neutral axis so that it passes through the whole cross-section. For each position we follow the described procedure in order to calculate the strengths, and derive a vector for the interaction diagram [NRd, MyRd, MzRd]T. The neutral axis is supposed to follow the elastic slope, i.e. the quotient of

Figure 7.34 Initial position, final position, and slope of the plastic neutral axis

tensioning

final position of plastic neutral axis

initial position of plastic neutral axis


the two bending torques acting on the cross-section. Nevertheless, after deriving the equilibrium equations for the deformed state, this slope may change, and this is a non-linear phenomenon. But due to the high sturdiness of the typically configured seating nodes, this slope is limited to minimum values (less than 45°), so it can be ignored. 7.1.12.5 Calculation of pivotal rigidity

In this paragraph we shall describe the basic instructions of EC3 for the calculation of the pivotal rigidity at prop seating connections, and we shall attempt to modify them in order to be applicable in this case of biaxial stress.

EC suggests the procedure shown in the table below for the calculation of pivotal rigidity Sr

Loading Lever arm z Pivotal rigidity Sj,ini

and and Left side under tensioning Right side under compression

where

και and Left side under tensioning Right side under tensioning

where


Loading Lever arm z Pivotal rigidity Sj,ini

και and Left side under compression Right side under tensioning

where

και and Left side under

compression Right side under compression

where

MEd > 0 is clockwise, NEd > 0 is tensioning, for μ. See 6.3.1(6)

Table 7.1 Calculation of pivotal rigidity at prop bases according to EC3

We must also note the following coefficients:

kT,l is the rigidity coefficient under tensioning of the left side of the node, and it must be taken as equal to the sum of rigidity coefficients k15 and k16 in relation to the left side of the node

kT,r is the rigidity coefficient under tensioning of the right side of the node, and it must be taken as equal to the sum of rigidity coefficients k15 and k16 in relation to the right side of the node

kC,l is the rigidity coefficient under compression of the left side of the node, and it must be taken as equal to the rigidity coefficient k13 in relation to the left side of the node

KC,r is the rigidity coefficient under compression of the right side of the node, and it must be taken as equal to the rigidity coefficient k13 in relation to the right side of the node


Finally, distances z are defined as follows:

a) Connection of prop base in the case of prevailing axial compressive force

b) Connection of prop base in the case of prevailing axial tensile force

c) Connection of prop base in the case of prevailing bending torque

d) Connection of prop base in the case of prevailing bending torque

Table 7.2 Definition of distances z

It is obvious that the previous formulae must be modified and generalized for the case of bending around the two main prop axis. The total connection rigidity shall be equal to the sum of the rigidities of the tensioned and compressed zone. So we must find the rigidities at these two zones, and the corresponding lever arms z. In favor of safety we assume that the components of the node related to the tensioned prop body and the compressed body and footing do not contribute to its pivotal rigidity. As a result, the node is considered more flexible, and this is a more adverse case.

As regards the tensioned zone, the rigidity shall be derived as a sum of coefficients k15 and k16, which, according to EC3, are defined as follows:


, when contact forces develop, (7.52)

, when no contact forces develop, (7.53)

, when contact forces develop, (7.54)

και , when no contact forces develop. (7.55)

So for each tensioned anchor we calculate coefficients k15 and k16, following the calculation of the corresponding effective length of the equivalent short-T ?eff per bolt, and the geometrical distance m. Then we add these two coefficients for all tensioned anchors. As regards the calculation of the lever arm for the tensioned zone, it is considered to be identical to the distance between the center of gravity of all tensioned anchors and the neutral axis, and this is an assumption in favor of safety, since usually the lever arm starts from the row of bolts that has the higher distance from the neutral axis.


For the compressed area of the node, the formula provided by EC3 is

and it is modified as follows where Aeff is the effective surface of the compressed cross-section. The lever arm of this zone is equal to the distance of the center of gravity of the compressed area from the neutral axis, as shown in fig. 7.35.

Figure 7.35 The lever arm of the tensioned zone is considered as equal to the distance of the center of gravity of the tensioned anchor polygon from the neutral axis

tensioning

compression


center of gravity of the tensioned anchor polygon


It must be noted that if only compression or only tensioning is applied at the cross-section, then we must ignore each corresponding rigidity coefficient from those mentioned above. Finally, in order to calculate the rigidity, we perform a calculation according to Table 7.2, but we use the values of zc, zT, k13, k14, κ15 that were calculated in this paragraph. 7.2 Detailed connection calculation

We select a prop with double-T cross-section, type ΗΕΒ 300, which is welded on the seating plate, whose dimensions are 700/400/25, using 6 mm thick outer seams at the footing and body. We also place reinforcing

Figure 7.36 The lever arm of the tensioned zone is considered as equal to the distance of the center of gravity of the effective compressed area from the neutral axis

tensioning

compression


center of gravity of the compressed area


plates on the strong axis of the prop; their dimensions are 700/450/15, and they are welded using 5 mm thick horizontal outer seams at the seating plate and 8 mm thick vertical outer seams at the prop footing. The anchors used are type Μ24/7.8, high coherence. The type of steel for all connection media is S235. The anchoring method and length shall be selected on the basis of calculations, so they are not defined beforehand. The shoe dimensions are 2500/2000/1500 and its concrete quality is C20/25. Finally, we use a 40 mm thick layer of cement grout under the whole surface of the seating plate, whose quality is the same as the shoe concrete. As presented in the following figures, we have deliberately selected a geometrical eccentricity both between the shoe and seating plate, and between the plate and prop in order to show the complete calculation path for a general geometry case. The stress quantities acting on the prop at the footing area are:

• Torque around its strong axis My,Ed= 250 kNm • Torque around its weak axis Mz,Ed= 80 kNm • Axial compressive load ΝEd= 1200 kN • Shearing force along axis z Vz,Ed = 280 kN • Shearing force along axis y Vy,Ed = 150 kN

It must be reminded that all dimensions are in millimeters, unless specified otherwise.


Figure 7.37 Plan view of foundation node and sign of positive bending torques

Figure 7.38 Side view of the connection


Figure 7.39 Front view of the connection

Figure 7.40 Detail of anchor placement on the seating plate


7.2.1 Check of Distances

7.2.1.1 Check of Anchor Distances

For the anchors on the seating plate we have:

7.2.1.2 Check of cement grout thickness

Regarding the cement grout thickness, according to paragraph 6.2.5 (7) of prEN 1993-1-8, we have:

Moreover, for the rated strengths of the cement grout and shoe concrete respectively, we have:

7.2.1.3 Check of Weldings

Regarding the welding thicknesses, the checks are as follows: • Between seating plate and prop body:

• Between seating plate and prop footing:

• Between prop footing and reinforcing plate:

• Between seating plate and reinforcing plate:


7.2.2 Connection calculation

7.2.2.1 Finding the basic node parameters

For the distances between the cheeks of the seating plate and shoe we have:

The corresponding effective surfaces of the shoe are:

The following applies:

so no decrement of surface A1,eff is required.

Coefficient bj is considered to be equal to 2/3 since the prerequisites of paragraph 6.2.5 (7) (prEN 1993-1-8) are met. Coefficient kj is calculated as follows:

Using formula 6.5 (prEN 1993-1-8) we get the value of coefficient c:

7.2.2.2 Starting and ending points for neutral axis shift within the cross-section

We initially suppose that we have a coordinate system whose center lies at the center of the seating plate.


The stress quantities shall become in this system:

The neutral axis shall have a slope that corresponds to the quotient of the above two action torques, i.e.:

and shall be shifted in order to pass through the whole seating plate, as shown in fig. 7.42.

Figure 7.41 Definition of coordinate system


The coordinates of points A and B are calculated to be equal to:

and

The neutral axis is shifted using a very small step, to ensure that the resulting interaction diagram is sufficiently precise. In order to present the calculation process, we shall consider an intermediate case for the neutral axis, for which the strengths against torques around the two axii and against an axial force shall be calculated. Thus we shall have a resulting vector that lies on the interaction surface. In the same manner we calculate the rest of its points.

Figure 7.42 The neutral axis shall be successively shifted in order to pass through the whole seating plate

shift direction of the neutral axis


7.2.2.3 Solving for the neutral axis at a specific point of the cross-section

Let the plastic neutral axis be at the position shown in fig. 7.43. Its line equation is: y = -3.125x 130

We observe that tensioning takes place a the left area of the node, so the anchors identified by numbers 1, 2, 3, and 4 in fig. 7.43 are tensioned and contribute to the node torque strength. 7.2.2.3.1 Tensioned side of the node

7.2.2.3.1.1 Assessment of strength of each tensioned anchor

The strength of each tensioned anchor shall be calculated from the formed short-Ts. As we have already mentioned, the anchors above the footing form a short-L to the prop footing, and the anchors inside the feet form a short-L to the body, as shown in figure 7.44.

tensioned area compressed area

Figure 7.43 The neutral axis intersects the body and footing of the prop, as well as the two reinforcing plates

neutral axis


The strengths shall be derived from table 6.6 (prEN 1993-1-8). For anchors 1 and 3, the procedure is as follows: The effective length of the short-T for a bolt within the footing of the prop is provided by:

From table 6.2, we shall calculate the plastic strength forces:

so contact forces may appear.

Figure 7.44 Formed short-Ts for tensioned anchors


The final tensile strength shall be equal to:

The rigidity coefficient κ15 of the checked anchor shall be:

and k16 shall be

For anchor 2 we have: The effective length of the short-T for a bolt outside the footing of the prop, provided that it is not an edge bolt, is:





The rigidity coefficient k15 of the checked anchor shall be:

and k16 shall be

Anchor 4 lies within the prop feet, so its effective length shall be calculated using the following formulae of table 6.6:

where m = 107.71 mm and α is calculated from table 6.11 for l1 = 0.57 and l2 = 0.28, and is found to be equal to 6.00.





The rigidity coefficient k15 of the checked anchor shall be:

and k16 shall be

7.2.2.3.2 Compressed side of the node

We follow the method described in paragraph 7.1.12.2 to calculate the surface area and the position of the center of gravity of the compressed side of the node. It must be reminded that in this case the stress distribution under ultimate failure condition is constant and equal to f,. The center of gravity is found to be at point C (140, 23) using the coordinate system of fig. 7.45. The compressed area is equal to Ac = 126792.61 mm2. Thus the compressive force on the concrete shall be:


7.2.2.3.3 Node strength

For this particular position and slope of the neutral axis we get the following stress values:

The axial strength is calculated as follows:

The torque around the strong axis of the prop due to the forces on anchors and the compressive force on the concrete shall be:

Correspondingly, for the strength torque around the weak axis:


Figure 7.45 Compressed effective area of the node

neutral axis


This triad constitutes a column vector on the interaction surface. Similarly, we shift the neutral axis so that it passes through the whole surface of the seating plate in order to get the corresponding values of the interaction diagram. The final picture shall be as follows:


Figure 7.46 Definition of coordinates xj, yi, xc, yc

neutral axis


It is established that loading lies within the diagram, so adequacy is ensured.

Figure 7.47 Interaction diagram for axial torque around the strong axis


Similarly, the diagram for the axial force and torque around the weak axis is:

Figure 7.48 Interaction diagram for axial torque around the weak axis


Loading lies again within the interaction diagram, so safety is also ensured as regards bending around the weak axis. 7.2.2.3.4 Pivotal rigidity of node

The calculation of the pivotal rigidity shall be carried out for the position of the neutral axis where the axial strength coincides with the compressive action. According to the assumptions described in paragraph 7.1.12.5, the lever arm of the tensioned area is equal to the distance of the center of gravity of the polygon of the tensioned bolts from the position of the neutral axis.

The resultant action torque shall be:


Figure 7.49 Definition of lever arms of tensioned and compressed area

neutral axis

center of gravity of the tensioned bolts polygon

center of compression


and the resultant strength torque shall be:

The relation

is met, so the connection lies within the elastic area and:

The rigidity of the tensioned area is calculated as the sum of rigidity coefficients k15 and k16 for all tensioned bolts. It is found to be equal to:

Similarly, for the rigidity of the compressed area, according to table 6.1, we have:

where Aeff is the compressed area for the above position of the neutral axis.

According to table 6.12 we have:

It must be noted that this rigidity relates to bending around the neutral axis and has nothing to do with the rigidity coefficients around axii y and z. The rotation around the neutral axis shall be:


This rotation may be reduced to separate rotations around the two main axii:

Where and λ is the direction coefficient of the neutral axis, i.e.:

7.2.2.4 Check of anchoring on the foundation

Anchoring shall be checked using the anchor that bears the highest tensioning force, which, according to previous calculations, has been found to be equal to Ft = 203.33 kΝ. It is supposed that high coherence anchors have been used. The coherence point is derived from ΕΚΩΣ2000, and it is found to be equal to 2.30 Μ Pa.

For straight-line anchoring:

For a typical value of reinforcement overlapping for shoes c = 100 mm, the above value for the anchoring length is sufficient since:

So we select straight-line anchoring. If the above condition was not met, it would be required to use another anchoring system.

dimensioning of metallic conetctions ec 3

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