design of portal frames - notes

7/29/2019 Design of Portal Frames - Notes

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MSc Design of Steel Buildings Design of Portal Frames 1

MSc Design of Steel Buildings

Design of Portal Frames

1 INTRODUCTION

In most European countries, steel construction is used for the majority of non-domestic

single-storey buildings. This is due to the ability to design relatively light, long span, durable

structures in steel, which are easy to erect safely and quickly. The capacity to provide spans

up to 60 m, but more commonly around 30 m, using steel has proved very popular for

commercial and leisure buildings. The lightness and flexibility of this kind of steel structure

reduces the sizes and the costs of foundations and makes them less sensitive to the

geotechnical characteristics of the soil.

The brief for the design of the majority of single storey industrial buildings is essentially to

design a structure with a limited number of internal columns. In principle, the requirement is

for the construction of four walls and a roof for a single or multi-bay structure. The walls canbe formed of steel columns with cladding of profiled or plain sheet. The designer considers a

system of beams or frameworks (latticed or traditional) in structural steel to support the

cladding for the roof. Use is made of hot rolled hollow sections (circular, rectangular) and

traditional sections (I sections, angles, etc.) and also cold formed sections which, in many

cases, provide the most efficient and economic solution.

In the following, after mentioning the main components of common single storey steel

buildings, principles and detailed rules for designing traditional steel portal frames, which

represent the main structural systems in about 50% of single storey structures in the UK, will

be presented. Reference will be made to modern codes of practice, in particular to Eurocode 3

(EN 1993-1-1) and BS5950.

2 COMPONENTS IN SINGLE STOREY BUILDINGS

The skeleton of a typical single-storey building is shown in the Figure 1. It consists of

three major elements: cladding for both roof and walls, secondary steel members to support

the cladding and form framing for doors and windows and the main frame of the structure,

including all necessary bracing.


2/32

MSc Desi

Figure 1:

2.1

The

appearan

structure

has emerusual su

sheets a

There ar

2a), ii) d

composi

Single-s

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Figure 2

to 35 m

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span of

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uildings D

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om a subst

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3/32

MSc Desi

Figure 2

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ingle skin t

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4/32

MSc Desi

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Figure 3:

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Figure 4:

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Figure 5.

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ickness var

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stiff for por

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straint to th

variety of s

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carry all of

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ork (the pu

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ained mem

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tal frames

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the loads a

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5/32

MSc Desi

frame sp

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Figure 5:

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Figure 6:

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acings of 6

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Figure 6.

uildings D

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6/32

MSc Desi

be made

industria

a higher

generall

more. T

common

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2.3

Steel

building

configur

Figure 7beam), (c

external

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when th

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ain fram

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: Various tyPortal with

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es are wi

they com

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ane to be

crane loads

any case,

pports can

(a)

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(e)

sign of Port

sections. I

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also for sp

ely used

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are pitch p

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the past,

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their cold

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itch portal fwith crane, (

ortal frame

r the fully

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d purlins.

ormed cou

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ly 1,8 m int

the range

structural

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concept as

ame, (b) Ce) Two-span

either wit

rigid versi

orizontal di

es, but th

of steel. T

(

(

f)

purlin wa

s. Hot-rolle

his means

terparts, ty

ned portal

ervals. How

of standard

system in

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shown in Fi

rved portalportal frame,

pinned ba

on (Fig. 8b

splacements

additional

e frames a

)

)

common

purlins ha

that they a

ically 3 m

rames, whi

ever, they a

cold form

single stor

rm. Vario

ure 7.

rame (cellul(f) Portal wi

ses (Fig. 8a

), when it

. Less weig

cost of t

e construct

6

n

e

e

or

h

e

d

y

s

arth

),

is

t

e

d


7/32


from I-section (UKB, universal beam) rafters and columns with haunches at the connections

at the eaves as illustrated in Figures 9-10 . The haunch length is approximately 10% of the

span and can be formed from welded plate or more commonly a cutting from a rolled section.

The depth at the column face is typically slightly deeper than the rafter section. Portal frames

can be also built with tapered rather than haunched sections. Frames of this type are common

in the USA and are being used more frequently in Europe. The sections are fabricated from

plate on automated welding machines. The ability to vary web thickness, flange dimensionsand section depth results in high material efficiency. Deep slender sections are used to

maximise economy.

Figure 8: Arrangements for portal frames.

A single-span symmetrical portal frame (as shown in Figure 9) is typically of the

following proportions:

a span between 15 m and 50 m (25 m to 35 m is the most efficient). An eaves height (base to rafter centreline) of between 5 and 10 m (7,5 m is commonly

adopted). The eaves height is determined by the specified clear height between the top

of the floor and the underside of the haunch.

A roof pitch between 5and 10 (6 is commonly adopted). A frame spacing between 5 m and 8 m (the greater frame spacing being used in longer

span portal frames).

Members are hot rolled I sections rather than H sections or UKB (universal beam),because they must carry significant bending moments and provide in-plane stiffness.

Sections are generally S235 or S275. Because deflections may be critical, the use ofhigher strength steel is rarely justified.

Haunches are provided in the rafters at the eaves to enhance the bending resistance ofthe rafter and to facilitate a bolted connection to the column. Small haunches are

provided at the apex, to facilitate the bolted connection.


8/32

MSc Desi

Figure 9:

Figure 1

The

directionbe at bot

vertical

Figure 1

gn of Steel B

Single-span

: Details of

ortal fram

, stability ish ends of t

racing by a

: (a) horizon

uildings D

symmetrical

eam-to-colu

in-plane s

provided be building,

hot-rolled

tal bracings,

sign of Port

portal frame.

n connecti

ability is p

vertical bror in one b

ember at e

(b) longitudi

l Frames

n with haunc

ovided by

acing in they only (Fig

ves level.

al vertical b

hes.

rame conti

elevations.. 11). Each

acings.

uity. In th

The verticarame is co

(a)

(

longitudin

bracing mnected to t

)

8

al

ye


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3 DESIGN OF PORTAL FRAMES

When designing portal frames, different steps are usually considered: i) determine

possible loading conditions, ii) calculate factored design load combination(s), iii) estimate

the element cross sections, iv) analysed the frame for each loading condition and select the

sections and determine connections and v) check secondary modes of failure. In the

following, after mentioning the main loads to be considered when designing portal frames(Fig. 12), different analysis methods, characterized by different levels of accuracy and

complexity are reported. Then an effective design approach, which is based on the rigid-

plastic analysis, is described. This represents the current practice in the UK, which leads to

the lightest and hence the most economical form for a portal frame.

Simplified formulations for accounting for frame stability are considered in the last part of

the note. As plastic design methods result in relatively slender frames, checking frame

stability is a basic requirement; thus in-plane and out-of-plane stability of both the frame as a

whole and the individual members must be considered.

3.1 Loading

External Gravity Loads

The dominant gravity load is from snow. The general case is the application of a basic

uniform load, but with sloping roofs having multiple spans and parapets, the action of drifting

snow has to be considered. The design for main portal frames can be carried out using the

uniform load case, but the variable loads caused by drifting are to be applied to cladding and

purlins. For portal frames, the structure capacity is usually determined by the snow load case,

unless the eaves height is large in relation to span.

Figure 12: In-plane loads to be considered in portal frames design.

Wind Loads

With lightweight cladding and purlins and rails, wind loads are important. Cladding and

its fasteners are designed for the local pressure coefficient. Care must be taken to include the

total effect of both internal and external pressure coefficients.


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Internal Gravity Loads

Service loads for lighting, etc., are reasonably assumed to be globally 0,6kN/m2. As

service requirements have increased, it has become necessary to consider carefully the

provision to be made. Most purlin manufacturers can provide proprietary clips for hanging

limited point loads to give flexibility of layout. Where services and sprinklers are required, it

is normal to design the purlins for a global service load of 0,1 - 0,2kN/m2

with a reducedvalue for the main frames to take account of likely spread. Particular items of plant must be

treated individually.

Cranes

Where moving loads such as cranes are present, in addition to the gravity loads, the

effects of acceleration and deceleration have to be taken into account in the design. A quasi-

static approach is generally used in which the moving loads are enhanced and treated as static

loads in the design. The enhancement factors to be used, depend on the particular plant and

its acceleration and braking capacity. Manufacturers must be consulted where heavy, high

speed or multiple cranes are being used. To take into account dynamic effects due to cranes,

the maximum vertical loads and the horizontal forces are increased by specific factors. Therepeated movement of a crane gives rise to fatigue conditions. Fatigue effects are restricted to

the local areas of support, the crane beam itself, support bracket and the connection to main

columns. It is not normal to design the whole frame for fatigue as the stress levels due to

crane travel are relatively low.

Other Actions

In certain areas, the effects of earthquakes should be considered. In those countries

affected, there are maps which identify the seismic level of each zone and standards to

evaluate structural behaviour (Eurocode 8).

In common practice, it is not necessary to take into account differential settlement of less than

2,5cm. If differential settlement exceeds 2,5cm, its effects must be examined, both from thestructural and functional points of view. In less ductile structures, such as those constructed

with sections not in Class 1 or 2, it is always important to evaluate the sensitivity of the

structure in relation to possible differential settlement.

It is also general practice not to take into account the effects of temperature when the

maximum dimension of the building is less than 40 to 50 m, or when expansion joints have

been used which separate the structure into zones which do not exceed this dimension.

Elsewhere, it is important to evaluate the effects of variations in temperature. It is also

necessary to ensure that the characteristics of the finished structure, both the systems of

fastening and the seals in the envelope, are compatible with the inevitable deformations due

to change in climate.

3.2 Methods of analysis

According to current codes of practice, structures must generally be checked at two

different Limit States: at Ultimate Limit State (ULS) and at Serviceability Limit State (SLS).

In the case of portal frames, ULS is the most critical condition.


11/32


Figure 13: Curves representing the frame response determined using different approaches.

At ULS different methods can be employed for determining internal forces for each

individual structural member or critical loads for the structural system as a whole. These

techniques include first and second order approaches based on both elastic and plastic theory.

In particular eight different procedures can be employed: i) first order elastic analysis, ii)first-order rigid-plastic analysis, iii) elastic critical load analysis, iv) second-order elastic

analysis, v) second-order rigid-plastic analysis, vi) first-order, elastic-plastic theory and vii)

second-order, elastic-plastic analysis and viii) second-order, plastic zone analysis.

Conversely, when calculating deflections at working load levels for the purpose of checking

serviceability (SLS), it is usual to employ only linear elastic analysis.

First-Order Elastic Analysis

In first-order elastic analysis a linear relationship between the applied loading F and the

deformations () is considered. The internal force distribution in the frame is assumed to beunaffected by the displacements in the frame. Frame analysis can therefore be conducted

according to linear elastic principles. The frame responds according to line 1 in Figure 11.

First-Order Rigid-Plastic Analysis

Rigid-plastic analysis neglects the effects of elastic deflections and assumes that all

structural deformation takes place in discrete regions, called plastic hinges, where plasticity

has developed. When using first-order, rigid-plastic theory only the collapse condition is

addressed. This condition occurs when sufficient plastic hinges are assumed to have formed

to convert the structure into a mechanism. Thus the path by which this stage is reached, i.e.


12/32


the sequence of formation of the hinges and any intermediate distributions of internal forces,

are not considered. Curve 2 in Figure 4 gives the frame response according to this approach.

Due to the form of the analysis, no information is provided on the magnitude of the

deflections. The analysis gives only that all stiffness is lost at the collapse load and

deflections therefore (in theory) become uncontrolled.

Elastic Critical Load Analysis

Using elastic critical load analysis, it is possible to calculate the buckling loads for frames

under specific loading conditions. Depending upon the content and complexity of the frame,

several different buckling modes, each with its associated elastic vertical load, may be

possible. Once again the analysis provides no information on the magnitude of the

deflections; it simply identifies a particular load level. The curve 3 in Figure 4 gives the

representation of the critical load obtained by an elastic buckling analysis.

Second order elastic analysis

In second-order elastic analysis the effect of elastic deformations on the internal force

distribution is taken into account. The result is a transition from the linear analysis line 1 atlow loads to the elastic critical line 3 at large deflections. For frames the second-order effects

may be separated into 2 parts: i) reduction in the effective bending stiffnesses of individual

members due to compressive loading, ii) a destabilising effect due to the overturning moment

produced by the vertical loads acting through the horizontal deflections caused by the lateral

loads.

Second-order rigid-plastic analysis

If the deformations, that may develop as a result of the formation of the plastic collapse

mechanism are allowed for when formulating the equilibrium of the frame, then the result is

the developing mechanism curve of line 5 in Figure 4. This curve shows that equilibrium can

only be maintained with a reduction in the level of the applied loads.

First-order, elastic-plastic theory

If a linear elastic analysis is modified to allow for reductions in frame stiffness with the

progressive formation of plastic hinges at increasing levels of the applied load, then the

response curve of line 6 is obtained. This line exhibits progressive loss of stiffness as each

plastic hinge is formed and eventually merges with the rigid-plastic line 2.

Second-Order, Elastic-Plastic Analysis

When the analysis that traces the formation of plastic hinges also allows for the effects of

deformations in setting up the governing equations, then line 6 is modified somewhat intoline 7. Line 7 initially follows the first-order elastic line 1, but diverges from this line to

follow the second-order elastic line 4 as destabilising effects become more significant.

Formation of the first plastic hinge - which occurs at a slightly lower applied load than is the

case with the first-order, elasto-plastic analysis due to the larger deformation associated with

second-order analysis - further reduces the stiffness, causing line 7 to diverge from line 4.

This divergence becomes more pronounced as more plastic hinges form. The peak of this

curve corresponds to the failure load predicted by this type of analysis. At large deformations

line 7 tends to merge with the curve for the mechanism, line 5.


13/32


Second-Order, Plastic Zone Analysis

If the spread of plasticity both through the cross-section and along the member length is

taken into account, instead of assuming that it is concentrated into the desirable regions of the

plastic hinges, then the resulting type of analysis is usually termed plastic zone theory. It

provides an even closer representation of actual behaviour and leads to a curve similar to line

7. In principle, any of the above approaches to frame analysis may be adopted. In practice,some of the effects may be found to be of little real significance for certain classes of

structure, e.g. for many low-rise frames second-order effects are very small and may

reasonably be neglected. Certain cases may also arise where particular forms of response

should be avoided, e.g. for buildings containing heavy cranes which will cause repeated

loading, elastic design is normally employed. The more complex approaches will almost

certainly require the use of suitable computer software to implement the volume of

calculation. It is therefore important to select an approach which is compatible with both the

accuracy required and the level of importance of the project under consideration. In the case

single-storey pitch portal frames, it has been shown that the use of rigid-plastic analysis leads

to significant saving in material. Therefore rigid plastic analysis can be easily and effectively

used to determined the required capacity for rafter and stanchions. However, further checks

are required to consider the second-order effects.

3.3 Rigid-plastic analysis

The rigid-plastic analysis allows us to calculate the plastic collapse load (p in Fig. 13) ofstructures when it is related to indefinite development of deflection under constant load.

When the load at ULS is given, the rigid-plastic procedure can be used as a design method to

determine the required plastic bending capacity for the beams and columns.

The plastic collapse method does not assess deflections and does not deal with the stability of

individual members or the structure as a whole. Therefore, it should be used as a design

approach only when the strength is the overriding design criterion. Compared to the

traditional elastic approach, plastic collapse analysis can lead to a more effective use ofstructural materials and cost savings, is more easily applicable to a wide range of common

structures, and, unlike the elastic method, the results achieved are independent of initial

imperfections (lack of fit, settlements etc.). The fundamental concepts of the plastic collapse

approach were initially derived from observations of experimental tests (Fig. 14) and not

from the application of a sound mathematical theory as in the case of elastic methods.

Figure 14: Fix-ended beam test: stress concentration and plastic hinge concept (Kazinczy, 1914).


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Basic Hypotheses

When using plastic methods for designing and analysing framed structures, we rely on the

ability of the joints to transmit bending moments and on the resistance to bending of the

members. Consequently, assuming that the flexural capacity of joints is always greater than

that of the joined elements, the basic hypotheses for the plastic collapse theory refer to the

relation between bending moment and curvature for beams and columns, which is closely

associated with the behaviour of the component material. In this respect, the basic physical

property exploited in the plastic method is ductility, in the sense that the material is assumed

to be capable of deforming well into the plastic range without fracture and significant

strength degradation under constant or slowly increasing loads (Fig. 15). High ductility must

also characterize the behaviour of steel sections, which should experience large rotations

without degradation in strength.

Eurocode 3 introduces some restrictions on steel and cross sections to be used in

structures designed by means of plastic methods of analysis. These restrictions are needed in

order to guarantee that sections at least at the locations at which the plastic hinges may form,

have sufficient rotation capacity to permit all the plastic hinges to develop throughout the

structure. In particular, at plastic hinges, the code allows only for the use of structural steel

with an elongation at failure 15% and Class 1 member sections. Conversely, in the otherparts of the structure, section Class 2 may also be employed.

Plastic hinge concept

The plastic hinge concept can be explained by analysing the behaviour of a simply

supported beam loaded by a vertical force (Fig. 15). When the load is increased until reaching

the plastic moment Mp at midspan, plastic deformations extend over a region where the

bending moment exceeds the elastic moment My.

Figure 15: Simply supported beam at collapse: distribution of plastic deformations, (M-) relationand load-displacement curve (W-).

Because of the shape of the M- diagram, the curvature k remains very small outside theelasto-plastic region. Conversely, close to the point where the load is applied, the curvature is


15/32


very high. The beam therefore deforms very nearly as if it consisted of two rigid portions

connected by a hinge, a plastic hinge. The plastic hinge behaves like a real but rusty hinge,

needing (non-zero) momentp

M M= to rotate it.

Figure 16: Plastic collapse mechanism for a simply supported beam loaded at midspan.

The plastic limit analysis is aimed at determining the plastic collapse loading condition,

focusing on the evaluation of plastic mechanisms. Therefore, when using such approach in

designing and analysing structures, a rigid-plastic approximation for the (M-) relationshipcan be considered (Fig. 16). The elasto-plastic law (Fig. 15) is fundamental only when

analysing the nonlinear elasto-plastic structural response, for instance trough an incremental

elasto-plastic procedure.

Theorems of Plastic Collapse Analysis

When calculating the collapse load of structures characterized by more than one potential

mechanism, it becomes necessary to identify the actual collapse mechanism. The basic

theorems for plastic collapse1

(Greeberg, Prager, Horne) give us the principles governing the

plastic collapse in formal statements, which can be used to define the actual collapse

mechanism and then to calculate the true collapse load, or, alternatively, determine the

required plastic bending capacity.

Static or lower bound theorem

If, at any load factor , it is possible to find a statically admissible and safe BMD(satisfying equilibrium and yield conditions), then is either equal to or less than the loadfactorc at collapse.

Corollary 1: the collapse load of a structure cannot be decreased by increasing the strength of

any part.

1Horne,M.(1971)PlasticTheoryofStructures,MITPress,Cambridge,Massachusetts.


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Kinematic or upper bound theorem

If, for any assumed plastic mechanism (mechanism condition), the external work done by

the loads (at a positive load factor) is equal to the internal work at the plastic hinges, then is either equal or greater than the load factorc at collapse.

Corollary 2: the collapse load of a structure cannot be increased by decreasing the strength of

any part.

The importance of the second theorem is obvious, for it follows that if the values of external

loads (load factors) corresponding to all the possible collapse mechanisms are found, the

actual collapse load (load factor) will be the smallest of these values.

Uniqueness theorem

If, at any load factor , a BMD can be found which satisfies the three conditions ofequilibrium, mechanism, and yield, then that load factor is the collapse load factorc.

Corollary 3: the initial state of stress has no effect on the collapse load.

Corollary 4: if a structure is subjected to any programme of proportional or non-proportional

loading, collapse will occur at the first combination of loads for which a BMD satisfying the

conditions of equilibrium, mechanism, and yield can be found.

Summary of basic theorems for plastic collapse analysis:

Uniqueness

theorem c =

MECHANISM CONDITION

EQUILIBRIUM CONDITION

YIELD CONDITION

c - kinematic theor.

c

- static theorem

Static approach

Collapse in statically determinate structures occurs when a plastic hinge forms in the most

stressed section. Therefore the ultimate load assessment and plastic design can be carried out

using simple equilibrium considerations and the static theorem.

In the case of statically indeterminate structures, as the load increases beyond the elastic

limit, which is attained when the first hinge forms, plastic hinges appear in succession at

sections where the absolute value of bending moments has a local maximum equal to the

plastic bending capacity (plastic moment), until the structure turns into a mechanism at

collapse. Also in these situations, the ultimate load can be calculated using static methods,

which investigate the collapse state considering equilibrium and yield conditions. In this

respect, the equilibrium diagram method, which uses the graphical superposition offree andreactant bending moment diagrams, represents an effective strategy.


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Analysis (c unknown)by statics at collapse:

c pM WL =

Design (p unknown)by statics at collapse:

p cM WL=

Analysis (c unknown)by statics at collapse:

c p4M WL =

Design (p unknown)by statics at collapse:

p cM WL 4=

Figure 17: Collapse load for statically determinate structures calculated using the static approach.

Yield condition: MMp

Equilibrium condition: BMD is in equilibrium with external loads.

Mechanism condition: plastic hinges at critical sections.

By inspection:

A p

B p

A BC p

p

p c

M M

M M

M 2M2WLM M

3 33M2WL

2M W3 L

=

=

+ = + =

= =

Other examples:


18/32


A p

B p

2

A BC p

2p

p c 2

M M

M M

M MwLM M

8 2

16MwL2M w

8 L

=

=

+= + =

= =

A p

B

A BC p

p

p c

M M

M 0

M MWLM M

4 2

6MWL 3M W

4 2 L

=

=

+= + =

= =

Figure 18: Collapse load for simple statically indeterminate structures calculated using the static

approach.

The same procedure can be used for investigating portal frames. In this case different

collapse mechanisms, which define plastic hinge position, should be considered (Fig. 19).

After fixing the position of plastic hinges (collapse mechanism), the moment diagram at

collapse can be drawn and the yield condition can be checked.

Consider the frame in Figure 20, it is made statically determined so that free bending diagram

can be drawn. The reactant diagram may be constructed by considering the effect of

redundant actions destroyed by cutting the frame. Assuming that mechanism 1 is the true

collapse mechanisms, we can write the following simultaneous equations, which define

plastic hinges at B,C,D, and E:

At B: 2 pVl Sl

M Rh M4 2

+ =

At C:p0 M M =


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At D: 2 pVl Sl

M Rh M4 2

+ + =

At E: ( )1 1 2 pVl Sl

Hh M R h h M4 2

+ + + =

It gives: 1 2p p 1

1 2 1 2

h HhVl 1 VlM ; M=M ; S=0; R= Hh

h h 8 2 h h 8

= + + +

Figure 19: Basic collapse mechanisms for a pitch frame.


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Figure 20: Free and reactant diagrams.

The calculated Mp is a safe value only is the bending moment diagram at collapse

satisfies yield condition. It can be checked considering the value for bending moment at A:

( )A 1 2 pVl Sl

M M R h h M4 2

= + +

Substituting the values for S, R and M, we found the relation:1

0 Hh Vl 4 , which is

usually satisfied for frames of a wide range of span to height ration.

3.4 Graphical method for plastic design of portal frames

A simplified graphic procedure, based on the reactant and free moment distribution, can

be effectively used in the case of pitch frame, when the snow load determines the collapse

condition.

Figure 21: Pitch frames with pinned base columns.

For the governing load case we require a bending moment diagram that:


21/32


satisfies mechanism, equilibrium and yield condition (Uniqueness Theorem). In practicethis can be achieved by utilising hinges at: i) maximum moment near apex of rafter, ii)

maximum moment in stanchion at the bottom of the haunch (Fig. 21);

gives a haunch region that remains elastic. The haunch is usually a tapered section cutfrom the same UKB as the rafter, giving a maximum depth at the end plate connection of

just less than twice the rafter depth. The rafter end of the haunch must remain elastic andtherefore the moment at this point should remain less than Mpr/1.15 where Mpr is the

moment resistance of the rafter. 1/1.15 comes from the shape factor for a UKB;

gives a practical eaves bolted connection. This normally requires a limit on the moment atthe rafter-stanchion connection. Typically the notional moment at this point should be

around 1.5Mpr. In general the moment should be within the range: Mps/2.5 < Mpr< Mps,

where Mps is the moment resistance of the stanchion. Well proportioned designs will be

somewhere close to the middle of this range.

We can adopt a graphical technique to determine the moments for design of members

relying in the first instance on the application of a release to reduce the problem to a statically

determinate one. We release the horizontal reaction at one of the column bases, this meansthere are no moments in the stanchions and the free bending moment in the rafter is the same

as for a beam, thus giving the maximum free bending moment diagram: 2M wL 8= .

Then applying a horizontal reactant force to the roller gives a bending moment diagram in the

frame proportional to the height above the reactant force. This gives the reactant bending

moment diagram. Figure 22 illustrates the concept.

Figure 22: Free and reactant bending moment.

Thus the practical application of the graphical method goes as follows:


22/32


Draw the portal frame to scale and either back project the roof line to the horizontal toobtain the point A referred to in Figure 22. This point can also be determined using the

relationship:2 1 1 2L L h h= , where L1, h1 and h2 are defined in Figure 22.

On a separate diagram mark out the horizontal distances that you obtained from the firststep. Now draw a parabola as accurately as possible to represent the free bending moment

to a convenient vertical scale. The concept is illustrated in Figure 23.

Project a line from point A with a clear ruler such that the difference between the reactantmoment line and the free moment parabola satisfies the conditions of a haunched region

that remains elastic and a practical eaves bolted connection as discussed previously. it is

important to note that it is you who decides on the gradient of this line.

Figure 23: Graphical procedure to determine rafter and stanchions plastic bending capacity.


23/32


Once the reactant moment line is established, pick off values for rafter and stanchion

moment capacities and choose appropriate sections for both. From the figure you should also

be able to check the minimum haunch length to ensure that the haunch tip remains elastic

(Mpr/1.15). If using a haunch length of span/10 the check can still be applied to the tip of the

haunch. It can also be checked that the moment in the stanchion is within the range discussed.

Note that the position of the maximum moment near the apex varies as the slope of the

reactant line varies. In practice it can only occur at one of the purlins - where load istransferred to the rafter - generally the first or second from the apex. However the error is

insignificant and it is assumed to be capable of continuously variable location.

Approximate tabular method

Based on the assumptions introduced for the graphical design method, a number of design

charts for estimating member sizes have been produced. They can be used to determine

quickly the sizes of simple pinned based frame elements assuming that:

i) the depth of the rafter is approximately span/55 and the depth of the haunch below

the eaves intersection is 1.5 times the rafter depth.ii) The moment in the rafter at the tip of the haunch is 0.87Mp,r, so that it is assumed

that the haunch remains elastic.

Figure 24: Rise/span versus horizontal base force for various values of span/eaves height.


24/32


Figure 25: Rise/span versus required Mp of rafter for various values of span/eaves height.

The graphs in Figures 24,25,26, which can be used for the range of span/eaves height

between 2 and 5 and rise/span of 0 to 0.2 (where 0 is a flat roof), give: i) the horizontal force

at the base of the frame as a proportion of the total factored load wL, where w is the load per

unit length of rafter and L (=2L1) is the span of the frame (Fig. 24), ii) the required moment

capacity of the rafters as a proportion of wL2

and iii) the required moment capacity of the

stanchions as a proportion of wL2. In practical calculations, after determining the ratios

span/height (2L1/h1), rise/span (h2/2L1) and the total load (wL and wL2), tables in Figs 24, 25

and 26 can be used to calculate horizontal thrust at base, Mpr required for rafter and Mps

required for stanchion respectively.

Figure 26: Rise/span versus required Mp of stanchion for various values of span/eaves height.


25/32


Further considerations on rigid- plastic analysis

As it has been mentioned before, when designing pitched roof portal frames, attention is

mainly paid on the snow load, because it determines the frame sizes. Therefore the frame is

analysed under a uniformly distributed load, also neglecting the notional horizontal loads

which account for practical imperfections such as lack of verticality (Fig. 27). Such

simplification is admissible, as geometrical imperfections have a significant influence only onthe design of structures which are sensitive to second-order sway effects. Typical portal

frames are not particularly sensitive to such sway modes, and notional horizontal loads

usually have a less than 1% effect on the required plastic moment of resistance and can be

ignored in member sizing. The bending moment distribution considered in the analyses is

therefore symmetrical, thus leading to a symmetrical collapse mechanism (Fig. 28). This

mechanism is overcomplete and should not be analysed. However, if the constraint of

symmetry is applied, so that the apex moves down vertically without rotating, this

mechanism becomes 'complete' and can be considered in the calculations. Moreover even

though the position of the rafter hinges is unknown, in practical design it can be assumed at

the apex (see the graphical method in 3.4). In reality, as distributed load from the roof is

usually applied to the frame as a series of point loads through purlins, the rafter hinges

always form under the first or second purlin down from the apex (Fig. 28).

Sway imperfection Equivalent forces (notional horizontal loads)

Figure 27: Frame imperfection and notional horizontal loads (EC3).

Figure 28: Symmetrical collapse mechanism.

F2

F1 F1

F2

F1

F2

(F1+F2)/2 (F1+F2)/2


26/32


After determining required Mpr and Mps by using the graphical approach or the

approximate method with tables, actual sections for stanchions and rafters can be determined,

choosing among Class I sections. A potential reduction in plastic bending capacity caused by

axial forces should be considered, as well as the frame under wind loads should be analysed.

However, both checks are generally not relevant for common pitch frames. Finally the actual

plastic load factor p*, associated with the most critical load combination, which usually

includes factored dead and snow loads (w) can be determined (Fig. 29). It is p* >1, as itreflects the excess capacity due to the choice of discrete member sizes in some arbitrary way.

In the calculation, the actual plastic capacity of stanchions Mp,s* and rafters Mp,r* (in case

reduced because of axial force) are considered, as well as it is assumed that plastic hinges at

collapse form in the columns below the connections to the rafter first and then at the apex.

( )

( )

( )

p,s*

1

p,r* 1 2

p* 2

1

MR

h b

M R h h

w 2L 8

=

+ + =

Figure 29: Calculation of the load factorp associated with rigid-plastic analysis.

3.5 Second-order effects

First-order rigid-plastic analysis provides only an unsafe assessment of the actual capacity

of framed structures, as is does not account for second-order effects (Fig. 30). This is shown

in Figure 13, where the maximum capacity f of a structure, calculated using an accuratesecond order elasto-plastic procedure, is compared against the load factor at collapse p,which is determined by means of a rigid-plastic approach. It appears that the load factorffalls always well below p, thus preventing the use of simple plastic theory in practicalapplications. However, many common structures, as single-storey portal frames, are not

usually excessively slender and the strain hardening contribution, which ensures that plastic

hinges do not rotate at a constant moment but rather have a rising moment-rotation

relationship, is often sufficient to overcome the destabilising effect of axial compressiveloads or at least to ensure that the shortfall offbelow p is not excessive. However, in a safeassessment and design, there is a need for methods allowing for simple estimates of f, thusleaving the use of sophisticated nonlinear analysis procedures only for investigating unusual

structures or for slender structures, where the difference between fand p is significant.


27/32


where: h is the height from the column base to the inflexion point

is the sway relative to the column base of the inflexion point.

Figure 30: First and second order moments in a beam-column.

Merchant-Rankine approach

The Merchant-Rankine formula provides the most effective approximate method for

estimating the load factorfat failure. It is employed by current codes of plastic (Eurocode 3,BS5950) and represents an extension of the Rankine equation for the failure load of a column

under compression to frames:

f p cr

1 1 1= +

(1)

which leads to:

p

f

p cr1

=

+ (2)

where f is the actual load factor at collapse, p is the rigid-plastic load factor and cr is theelastic critical load factor.

Equation (1) was initially suggested on a purely empirical basis and only later Horne showed

that it has a theoretical justification provided that the critical buckling mode is similar to the

plastic collapse mode. While, in other situations, it is likely to be conservative. In the case of

single-story portal frames, which are relatively stocky structures, strain hardening in plastic

hinges and a small amount of restraint from the cladding can be sufficient to overcome the

destabilising effects of axial compressive loads. Recognising these factors, Wood suggested a

modification of eq. (2) to give:

p crf

p cr p

crf p

p

when 4 100.9

when 10

=

+

= >

(3)

P

H

x

M(h) =Hh +P

M(x) = Hx +P +P x / h

P

H

h

x

M(h) =Hh

M(x) =Hx


28/32


In the case of slender frames, whencr p 4.6 < according to BS5950, more accurate

nonlinear second order elasto-plastic analyses should be carried out.

Figure 31: Critical load and buckling mode for a pitch frame.

It is evident that any approach based on the Merchant-Rankine formula requires a good

estimate of the elastic critical load factorcr(Fig. 31). For multi-storey frames such value can

be calculated quite easily, using different alternative methods. Conversely the evaluation ofthe critical load for pitched roof portal frames is not simple and requires a specific treatment.In this case, the destabilising effect of the axial thrust in the rafter must be considered.

Davies developed an effective method, which is remarkably simple and give results thatare sufficiently accurate for all practical purposes. According to this approach, the criticalmode is assumed to be anti-symmetrical sway, with a corresponding deflection of the rafter(Fig. 31). Davies determined the critical load, solving the buckling problem in Figure 32,

where a small disturbing moment M at eave which gives rise to a rotation initiates framebuckling.

1

Figure 32: Elastic critical load calculation for a pitched roof frame.

An explicit expression forcr can be derived as a function of the axial load in the column P cand along the rafter Pf:


29/32


( )

( )

( )

c rcr 22

cr r r

r c

cr

r c r r

5E 10 R I Lwith R for fixed base frames

2RP h5P L I h

I I

5E 10 R for pinned base frames

L P h 0.3P L

+ = =

+

+ =

+

(4)

where Ic and Irare the second moment of era of columns and rafter respectively.

An estimate of the axial loads Pc and Pr, which is sufficiently accurate for practical design can

be obtained considering the notations in Figure 33 using the expressions:

c r

wL M wLP , H= and P H cos sin

2 h 2= = + (5)

Figure 33: Evaluation of the axial loads Pc and Prat collapse.

Typical portal frames, when well-designed, are characterised by values ofcrwhich are about5, so the consideration of second-order effects is extremely important. Being in the rage of

intermediate slenderness, the modified Merchant-Rankines formula (Eq. 3) can be

employed. This required that the load factor for plastic design should be increased

accordingly to the above equation forp and that all of the internal forces obtained by first-order analysis should be amplified in proportion.

3.6 Member stability

At ULS, it is essential that all members of the frame remain stable. In general, the

member stability checks should be carried out using final bending moment diagram for the

loading combination used to the to determine the member sizes. In structures designed using

elastic methods, this means ensuring that the frame elements are stable against both in-plane

and out-of-plane (lateral torsional buckling) failure. While, when using plastic design, also

different requirements need to be satisfied, to guarantee large rotational capacity at plastic

hinge position. The first requirement implies to use only Class I (plastic) sections at plastic


30/32


hinges location. This guarantees that the elements are sufficiently stocky for plastic hinges to

develop, without premature loss of strength due to local buckling. Moreover codes of practice

(e.g. Eurocode 3, BD5950) require to provided torsional restraints at all plastic hinge

positions, with adjacent lateral restraints within a specific distance Lm from the hinge

restraint. The Lm value can be calculated using the expression:

zm 1/ 222

pl, y yEd

2

1 t

38iL

W f1

57.4 756 C AI 235

= +

(6)

where Ed is the average compressive stress [N/mm2] due to axial load, fy is the design

strength [N/mm2], iz is the radius of gyration [mm] about the minor axis, I t is torsional

constant and C1 is a factor depending on the loading and end condition.

In a portal frame, purlins and sheeting rails are assumed to provide lateral restraint to the

flange to which they are connected. On their own, they do not provide either torsional

restraint to the section or lateral restraint to the remote flange if this flange is in compression.

Torsional restraints hold both flanges of a beam in position relative to each other (in the

lateral direction). The most usual way of achieving either of these forms of bracing is to use aknee brace connected to a purlin or sheeting rail as shown in Figure 34. Past research

indicates that each knee brace should be designed for a compression load equal to 2% of the

compression flange yield load and should have a stiffness that is given by a slenderness ratio

of at least 100.

Figure 34: Typical knee brace.

There are three different regions of the frame which require checking: the apex region, the

haunch zone and the columns. The apex region of the rafter contains a length of almost

uniform bending moment and this is the most difficult distribution to stabilise. However, the

most critical load condition places the outer flange in compression and this is laterally

restrained by the purlins. The requirements therefore depend critically on whether it has beenpossible to prove that this part of the frame is entirely elastic at the ultimate limit state. If this

part of the rafter contains a plastic hinge, then the purlin spacing is limited to Lm. If this part

of the rafter is elastic, and in any event for parts of the rafter remote from the plastic hinge,

then the standard requirements for elastic design apply. The purlin spacing is then restricted

to a value of LE (effective length depending upon the amount of end restraint and loading

condition). In the haunch region of the rafter, the purlins restrain the tension flange. If this

region contains a plastic hinge, then the purlin spacing in the vicinity of the hinge is again


31/32


limited by Lm. Provided that every length of purlin has at least two bolts in each purlin-rafter

connection; and the depth of the purlin section is not less than 0.25 times the depth of the

rafter, the bottom flange of the rafter can also be assumed to be restrained at the point of

contraflexure. If the design procedure suggested in previous sections has been followed, then

the haunch region of the rafter will generally be elastic. In this case the maximum purlin

spacing can be increased to that which results in a value of LE which satisfies standard lateral

torsional buckling checks.

Figure 35: Spacing for lateral and torsional restraints in columns and haunched rafter.


32/32

Under the dominant load condition, the column will also usually be restrained at intervals

along the tension flange by the sheeting rails. In a plastically designed frame, there will

almost always be a plastic hinge below the haunch. This will, of course, require a torsional

restraint. According to current codes, the distance to the next restraint should be L m.

However, this value should be modified to account for the actual bending moment profile,

which is not constant.

design of portal frames - notes

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