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steeltubeinstitute.org/hss/hss-information/aisc-360-16

Limit State Table NOTES

Overall width of rectangular HSS main member, measured 90 degrees to the plane of the connection, in Overall width of rectangular HSS branch member, measured 90 degrees to the plane of the connection, in Overall height of rectangular HSS main member, measured in the plane of the connection, in Overall height of rectangular HSS branch member, measured in the plane of the connection, in Gap between toes of branch members in a gapped K-connection, neglecting the welds, in Hb / sin q

Projected lap length of the overlapping branch on the chord, in Design wall thickness of HSS main member, in

Design wall thickness of HSS branch member, in

Width ratio; the ratio of the branch member to chord diameter = Db/D for round HSS ratio of overall branch width to chord width = Bb/B for rectangular HSS

Load length parameter, applicable only to rectangular HSS; the ratio of the length of contact of the branch with the chord in the plane of the connection to the chord width = lb /B

Acute angle between the branch and the chord (degrees)

B =

Bb =

H =

Hb =

g =

lb =

lp =

t or tdes =

tb =

b =

h =

q =

definitions of parameters

2

The Tables provide the connection available strength per limit state. Variables are provided in referenced formulas where substitutions are required to apply connections to HSS members. Limits of applicability to equations shown are noted in this Table where applicable. Note that limits of applicability in AISC 360-10 do not carry over to AISC 360-16 unless specifically noted. The connection available strengths in AISC 360-10 and 360-16 Chapter K assume a main HSS member with ample end distance on both sides of an in-framing connection. When a connection is near the end of an HSS member, there may not be adequate length to develop the traditional yield line pattern. Therefore, although the end distance limit state is not explicit in the AISC 360-10 Specifiction, the EOR shall consider the effects of a modified yield line. Refer to AISC 360-16 Commentary pp. 16.1-463 thru 464 for commonly accepted alternatives for connections near an HSS member end, such as the use of a cap plate installed at the end of the HSS member. If alternate measures are not taken, a reduction on connection capacity shall be applied for limit states pertaining to the HSS chord element. In the absence of any connection end criteria, reduce the HSS connection available strength by 50% if the connection is located less than [B* sqrt (1-b)] from a rectangular HSS member end, or less than [D*(1.25 - b/2)] from a round HSS member end. For additional guidance related to the application of provisions in AISC 360-16 Specification J10 specific to HSS sections, refer to AISC 360-16 Commentary pp. 16.1-450. This Commentary section explains how the Specification J10 limit states specific to concentrated forces applied normal to a member can be applied to HSS sections. See derivations on pages 3-6 of this PDF to provide additional guidance on limit state.

See STI’s article HSS Limit States in Cap Plate Connections on page 7 for detailed discussion on local limit states of wall yielding and crippling in rectangular HSS cap plate connections. For most manufactured rectangular HSS, HSS chord shear yielding per Row 2 of the Truss Limit State Table is found to govern over shear yielding and shear buckling in the projected gap region of gapped K-connections or in the projected gap region between inclined branches of cross-connections. Shear yielding and shear buckling can be checked using Specification Section G4. For round HSS gapped K-connections and cross-connections, chord shear yielding and shear buckling at the gap is included in the empirically derived formulas given in Table K3.1; however, it can also be checked using Specification Section G5. Refer to the Axial and Truss Tables where noted: The term (Fyt/Fybtb) is deleted from eq. (K1-1) because chord punching shear is independent of the properties of the plate or branch. See STI’s article Stepped HSS T- and Cross-Connections Under Branch In-Plane and Out-of-Plane Bending on page 14 for additional guidance on limit state noted.

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Limit State Table NOTES

limit state table notes

3

Limit State Table Notes #5: Derivations

Moment Table: Derivation for Local Yielding of HSS Sidewalls In-Plane Moment for Rectangular HSS (Refer to Moment Limit State Table Cell F3)

Refer to AISC 360-10 Eqn K3-7: Mn = 0.5F*y t (Hb + 5t)2 [AISC 360-10 Eqn K3-7] Refer to AISC 360-16 Specification Section J10.2 to derive available strength to match AISC 360-10 Eqn K3-7, based on local yielding of rectangular HSS Sidewalls under in-plane moment.

Rn = Fywtw(5k + lb) For lend > H [AISC 360-16 Section J10.2] lb = Hb

tw = tdes of HSS chord = t k = t Consider a conservative 1:1 slope for dispersion in lieu of k = 1.5t. F*y = Fy for HSS T-Connections = 0.8Fy for HSS Cross-Connections 2 Sidewalls

φ = 1.00, Ω = 1.50 Per Cidect Design Guide 3, pp. 62, moment derived from 2 stress blocks can be represented as:

Rn = F*y t [½ lb + 2.5k] * (2 sidewalls) = F*y t (lb + 5k) Substituting:

Rn = F*y t[Hb + 5t] for lend > H

Rn = F*y t[Hb + 2.5t] for lend < H

Moment Arm =½ lb + 2.5k = ½ (lb + 5k)

Therefore, Mn = Rn x moment arm = ½ F*y t (lb + 5k)2

Substituting for lb and k:

Mn = 0.5F*y t [Hb + 5t]2 for lend > H

Mn = 0.5F*y t [Hb + 2.5t]2 for lend < H


LIMIT STATE TABle notes

Moment table derivations

4

1


Moment Table: Derivation for Local Yielding of Branches Due to Uneven Load Distribution

In-Plane Moment Conn for Rectangular HSS (Refer to Moment Limit State Table Cell F6)

Refer to AISC 360-10 Eqn K3-8:

Mn = Fyb[Zb – (1-!!"#!!

)BbHbtb] [AISC 360-10 Eqn K3-8]

φ = 0.95 Ω = 1.58

beoi = (!"!!

)( !!!!!"!!

)Bb ≤ Bb [AISC 360-10 Eqn K2-13]

Refer to AISC 360-16 Specification Section F7 to derive available strength to match AISC 360-10 Eqn K3-8, based on local yielding of rectangular HSS branches due to uneven load distribution under in-plane moment. Mn = FyZeff [AISC 360-16 Eqn F7-1]

The steps to determine Zeff are as follows: 1. Find effective width of the two HSS branch walls transverse to the chord HSS 2. Subtract plastic modulus of non-effective wall area from total HSS plastic modulus (Zb)

Be = (!"!!

)( !!!!!"!!

)Bb ≤ Bb (same as beoi) [AISC 360-10 Eqn K1-1]

= Effective width of branch

! Show Be as a fraction of Bb ! (!!!!

)Bb

Non-effective width = Bnon-effective = (1 - !!!!

)Bb

Znon-effective = 2Bnon-effectivetb (!!!!!)!

= 2(1 - !!!!

) Bbtb (!!!!!)!

= (1 - !!!!

) Bbtb (Hb-tb)

Approximation of Znon-effective = (1 - !!!!

) BbHbtb

(The approximation ignores the HSS thickness in the last term only which results in a conservative value that underestimates Zeff) Zeff = Zb – Znon-effective

2

= Zb - (1 - !!!!

) BbHbtb

Mn = FybZeff [AISC 360-16 Eqn F7-1]

Mn = Fyb[Zb – (1-!!!!

) BbHbtb]

Since the branch member is primarily an axial member rather than a flexural member, the following resistance/safety factors are applied for this limit state:

φ = 0.95 Ω = 1.58



5


Moment Table: Derivation for Local Yielding of HSS Sidewalls

Out-of-Plane Moment for Rectangular HSS (Refer to Moment Limit State Table Cell H3)

Refer to AISC 360-10 Eqn K3-10: Mn = F*y (B – t) (Hb + 5t) [AISC 360-10 Eqn K3-10] Refer to AISC 360-16 Specification Section J10.2 to derive available strength to match AISC 360-10 Eqn K3-10, based on local yielding of rectangular HSS Sidewalls under out-of-plane moment. Rn = Fywtw (5k + lb) For lend > H [AISC 360-16 Section J10.2]

tw = tdes of HSS chord = t k = t Consider a conservative 1:1 slope for dispersion in lieu of k = 1.5t.

lb = Hb / sinθ For θ = 90, lb = Hb

F*y = Fy for HSS T-connections = 0.8Fy for HSS Cross-Connections

Substituting to determine available strength: Rn = F*y t [5t + Hb] = F*y t (Hb + 5t) for lend > H Rn = F*y t [5t + Hb] = F*y t (Hb + 2. 5t) for lend < H φ = 1.00, Ω = 1.50

Determine available moment capacity: Mn = Rn x moment arm

Moment arm = B – t

Mn = F*y t (Hb + 5t) (B – t) for lend > H Mn = F*y t (Hb + 2.5t) (B – t) for lend < H



6

1


Moment Table: Derivation for Local Yielding of Branches Due to Uneven Load Distribution

Out-of-Plane Moment Conn for Rectangular HSS (Refer to Moment Limit State Table Cell H6)

Refer to AISC 360-10 Eqn K3-11:

Mn = Fyb[Zb – 0.5(1-!!"#!!

)2Bb2tb] [AISC 360-10 Eqn K3-11]

φ = 0.95 Ω = 1.58

beoi = (!"!!

)( !!!!!"!!

)Bb ≤ Bb [AISC 360-10 Eqn K2-13]

Refer to AISC 360-16 Specification Section F7 to derive available strength to match AISC 360-10 Eqn K3-11, based on local yielding of rectangular HSS branches due to uneven load distribution under out-of-plane moment. Mn = FyZeff [AISC 360-16 Eqn F7-1]

The steps to determine Zeff are as follows: 1. Find effective width of the two HSS branch walls transverse to the HSS chord 2. Subtract plastic modulus of non-effective wall area from total HSS plastic modulus (Zb)

Be = (!"!!

)( !!!!!"!!

)Bb ≤ Bb (same as beoi in 360-10 above) [AISC 360-16 Eqn K1-1]

= Effective width of branch

! Show Be as a fraction of Bb ! (!!!!

)Bb

Non-effective width = (1 - !!!!

)Bb

Znon-effective = !!![(1 - !!

!!)Bb]2*(2 walls)

Simplify and rearrange ! Znon-effective = ½(1 - !!!!

)2Bb2 tb

Mn = FyZeff [AISC 360-16 Eqn F7-1] = Fy(Zb – Znon-effective)

Mn = Fy[Zb – ½ (1-!!!!

)2Bb2tb]



2

Since the branch member is primarily an axial member rather than a flexural member, the following resistance/safety factors are applied for this limit state:

φ = 0.95 Ω = 1.58

7

HSS Limit States in Cap Plate Connections Cathleen Jacinto, PE, SE

FORSE Consulting Technical Consultant to the Steel Tube Institute

Introduction

A commonly used hollow structural section (HSS) connection type consists of ‘line loads’ applied to the end of an HSS member via a cap plate, or via the flange of a WT. One example of this connection type is a wide flange (WF) beam framing over an HSS column, in which the bottom flange of the WF beam is bolted to a cap plate which is welded to the top of an HSS column (Figure 1). In this case, WF beam shear reactions are imparted onto the HSS column below in local bearing. There may also be a moment transferred across the beam-to-column interface, resulting in tension and compression forces applied to the end of the capped HSS column. Alternately, a WT (or cap plate with a transverse stem plate) can be welded to the top of an HSS member to transfer concentrated loads (Figure 3).

Figure 1 - Wide Flange Beam to HSS Column With Cap Plate

There are several limit states required to be checked to confirm the adequacy of the cap plate, beam or WT, and the HSS column, but this article will focus on reviewing limit states specific to the HSS section. When a tension or compression force is applied to the end of a capped HSS axial member across nearly its full width, failure of the HSS walls is possible. Local yielding and local crippling of round or rectangular (including square) HSS sidewalls are potential limit states to consider. We will explore how to apply AISC 360 Specification Chapter J10 requirements to HSS wall limit state checks for HSS cap plate connections.

Shear Lag Design Considerations



Article 1

8

When a tensile or compressive force is transferred from a transverse web or plate to a cap plate, this force is dispersed over a width across the round or rectangular HSS walls below. This width is directly dependent on the thickness of the transverse web or plate and the thickness of the cap plate. A very thick cap plate will distribute this concentrated load more uniformly and across a larger width, activating more of the walls of the HSS column than a very thin cap plate. This is the shear lag effect; the potential for uneven stress distribution that results when the HSS cross section is not uniformly and fully loaded. The design method for checking the HSS walls in cap plate connections recognizes that shear lag will be present in the HSS if portions of the cross-section are not loaded. To understand how shear lag is considered in cap plate connections, we explore the load path through the connection elements. Per AISC 360 Specification , Section J10.2 Commentary for Local Yielding, a concentrated load from the transverse web or plate is spread over a stress zone with a slope of 2.5:1 in local yielding. This stress zone starts at the top of the cap plate and extends through the plate to the HSS walls below. Figure 2 illustrates the transverse web

bearing width, l b, oriented for load dispersion into the HSS wall of width, B. This can also be applied to the diameter, D, of a round HSS. The distribution slope relative to local yielding of

2.5:1 from each face of the transverse web produces a dispersed load width of (5t p + l b).

If (5tp + l b) is less than B, then only two HSS walls are engaged to resist the single concentrated load applied from the transverse web. And for these two effective walls, only a

partial width of (5t p + l b) can be considered to resist the applied load. Note, however, that if a moment is applied to the cap plate, then only two HSS walls are available to resist the moment force couple. In this case, one HSS wall resists the tension force and a second HSS wall resists the compressive force.

If (5tp + l b) is greater than B, then the full HSS section is effective in resisting the load. All four walls in a rectangular HSS walls or the full gross area of a round HSS can be considered. In this case, no shear lag is present. This is obviously the most efficient condition for resisting load in the HSS, but may require a very thick plate in order to accomplish

this. The next section will clarify how to more specifically apply shear lag considerations in checking the HSS section for local yielding and local crippling. HSS Column Cap Plate Connections – Limit State of Wall Local Yielding


Limit statetable notes

HSS Limit States in Cap Plate Connections Cathleen Jacinto, PE, SE

FORSE Consulting Technical Consultant to the Steel Tube Institute

Introduction A commonly used hollow structural section (HSS) connection type consists of ‘line loads’ applied to the end of an HSS member via a cap plate, or via the flange of a WT. One example of this connection type is a wide flange (WF) beam framing over an HSS column, in which the bottom flange of the WF beam is bolted to a cap plate which is welded to the top of an HSS column (Figure 1). In this case, WF beam shear reactions are imparted onto the HSS column below in local bearing. There may also be a moment transferred across the beam-to-column interface, resulting in tension and compression forces applied to the end of the capped HSS column. Alternately, a WT (or cap plate with a transverse stem plate) can be welded to the top of an HSS member to transfer concentrated loads (Figure 3).

Figure 1 - Wide Flange Beam to HSS Column with Cap Plate There are several limit states required to be checked to confirm the adequacy of the cap plate, beam or WT, and the HSS column, but this article will focus on reviewing limit states specific to the HSS section. When a tension or compression force is applied to the end of a capped HSS axial member across nearly its full width, failure of the HSS walls is possible. Local yielding and local crippling of round or rectangular (including square) HSS sidewalls are potential limit states to consider. We will explore how to apply AISC 360 Specification Chapter J10 requirements to HSS wall limit state checks for HSS cap plate connections.

Shear Lag Design Considerations

9

When a concentrated tensile or compressive force acts on the end of a round or rectangular HSS column with a cap plate, and the force is in the direction of the HSS axis, the walls of the HSS member could fail in sidewall yielding, with consideration of shear lag. The concentrated force may be due to local bearing or a moment connection.

Figure 3 - Cap Plate or WT Connection to HSS, Under Axial Load

The nominal strength in local yielding of round or rectangular HSS walls to resist tension or compression load can be determined by AISC 360 Specification Eq. (J10-2). In yielding, the load

is assumed to distribute from the top of cap plate down at a slope of 2.5:1, resulting in a dispersed load width of (5t p + l b).

Below is a summary of variables to assume in Specification Eq. (J10-2):

F yw = F y of HSS columnk = Cap plate thickness, tp

tw = HSS wall thickness, tdes

lb = Transverse plate or web thicknessd = B/2

Qf is omitted

Incorporating these variables into AISC 360-16 Specification Eq. (J10-2) results in the nominal strength in HSS wall local yielding, for one wall:

tRn = F yHSS des (5t )p + lb

.00, Ω .50Φ = 1 = 1

As discussed above, if (5t p + l b) < B, then only two HSS walls are effective due to shear lag. Therefore, the yielding nominal strength to resist a concentrated load, for two walls:

or for rectangular HSS (or for round HSS) F 5t( p + lb) < B < D :



When a tensile or compressive force is transferred from a transverse web or plate to a cap plate, this force is dispersed over a width across the round or rectangular HSS walls below. This width is directly dependent on the thickness of the transverse web or plate and the thickness of the cap plate. A very thick cap plate will distribute this concentrated load more uniformly and across a larger width, activating more of the walls of the HSS column than a very thin cap plate. This is the shear lag effect; the potential for uneven stress distribution that results when the HSS cross section is not uniformly and fully loaded. The design method for checking the HSS walls in cap plate connections recognizes that shear lag will be present in the HSS if portions of the cross-section are not loaded. To understand how shear lag is considered in cap plate connections, we explore the load path through the connection elements. Per AISC 360 Specification , Section J10.2 Commentary for Local Yielding, a concentrated load from the transverse web or plate is spread over a stress zone with a slope of 2.5:1 in local yielding. This stress zone starts at the top of the cap plate and extends through the plate to the HSS walls below. Figure 2 illustrates the transverse web

bearing width, l b, oriented for load dispersion into the HSS wall of width, B. This can also be applied to the diameter, D, of a round HSS. The distribution slope relative to local yielding of

2.5:1 from each face of the transverse web produces a dispersed load width of (5t p + l b).

If (5tp + l b) is less than B, then only two HSS walls are engaged to resist the single concentrated load applied from the transverse web. And for these two effective walls, only a

partial width of (5t p + l b) can be considered to resist the applied load. Note, however, that if a moment is applied to the cap plate, then only two HSS walls are available to resist the moment force couple. In this case, one HSS wall resists the tension force and a second HSS wall resists the compressive force.

If (5tp + l b) is greater than B, then the full HSS section is effective in resisting the load. All four walls in a rectangular HSS walls or the full gross area of a round HSS can be considered. In this case, no shear lag is present. This is obviously the most efficient condition for resisting load in the HSS, but may require a very thick plate in order to accomplish

this. The next section will clarify how to more specifically apply shear lag considerations in checking the HSS section for local yielding and local crippling. HSS Column Cap Plate Connections – Limit State of Wall Local Yielding

10

t ≤ F ARn = 2F yHSS des (5t )p + lb yHSS g

It should be noted that if the tensile or compressive force is due to an applied moment onto the cap plate, then only two HSS walls are available to resist the moment. Therefore, only one wall is effective in resisting the tensile force couple and one wall to resist compression. In this instance, the nominal strength for one wall per Eq. (1) should be used to resist each of the tension or compression coupled force.

If (5tp + l b) > B, then shear lag is not present, and the full HSS section is effective in resisting the concentrated load. Therefore, the nominal local yielding strength, for all HSS walls:

or for rectangular HSS (or for round HSS) F 5t( p + lb) > B > D :

ARn = F yHSS g

AISC 360-16 Specification Eq. (J10-2) applies to connection forces located greater than member depth, d, from the member end. This is analogous to wide-flange beams bearing on a seated

connection, and the full (5k + l b) width cannot disperse on both sides of the bearing plate at the beam end. In our connection condition, if the transverse web or plate is centered upon the HSS

cap plate, then the full dispersed (5k + l b) width can be achieved and Eq. (J10-2) can be used. However, if the transverse web is eccentric to the HSS column centerline and/or the full

dispersed (5k + l b) width per Figure 2 cannot be achieved, then it would be more appropriate to base the local yielding limit state check on Eq. (J10-3).

HSS Column Cap Plate Connections – Limit State of Wall Local Crippling

HSS Column Subject to Axial Compression

This limit state applies to the walls of a square or rectangular HSS member subject to a concentrated compressive force. The force is applied in the direction of the HSS axis, acting on the end of the HSS member via a cap plate. The concentrated load may be due to local bearing or a moment connection.

The capacity of the HSS sidewalls to resist crippling can be determined using AISC 360-16 Specification Eq. (J10-4). This equation assumes the compressive force is applied at a distance from the member end that is greater than or equal to d/2, while Eqs. (J10-5a) and (J10-5b) apply to forces at a distance to member end less than d/2. The Commentary for Section J10.3 clarifies that Eqs. (J10-5a) and (J10-5b) are intended for beam ends where the web is unsupported, such as at seated connections. Although our connection condition occurs at the end of the HSS member, the HSS walls are considered supported by the cap plate, and Eq. (J10-4) is applicable.



11

Similar to local yielding of the walls, the compression load is assumed to disperse from the base of the transverse web or plate through the cap plate over a specific width of the HSS walls. For buckling, a distribution slope can be assumed as 1:1 from each face of the transverse web. This is based upon a model in which the force is assumed to spread out at a 45 degree angle in both directions from the point of load for buckling and crippling, as shown in Figure 4. Therefore, the force will distribute across the full HSS width, B, and at a depth no greater than B/2. In terms of wide flange connections, the web local crippling phenomenon has been observed to occur in the wide flange web adjacent to the loaded flange. In our case, crippling would occur in the affected HSS wall(s) adjacent to the cap plate within a depth of B/2. Therefore, the web (or wall) depth is taken as half of the full width of the HSS, or B/2.

Figure 4 - Distribution slope model for local crippling/buckling

The coefficient, Qf, is omitted from Eq. (J10-4). Qf is intended for connections where forces are normal to the HSS wall compressive stress. In this cap plate connection, the compressive force is parallel to the compressive stress in the HSS wall.

Below is a summary of variables to assume in Specification Eq. (J10-4):

F yw = F y of HSS column tf = Cap plate thickness, tp tw = HSS wall thickness, tdes

lb = Transverse plate or web thicknessd = B/2

Qf is omitted

Incorporating these variables into AISC 360-16 Specification Eq. (J10-4) results in the nominal strength based on HSS wall local crippling, for one wall:

F or (5tp + lb) < B :


Eq. (4)


12

.8tRn = 0 2des 1{ + 6( B

lb )( tptdes )1.5}√EF yHSS

tptdes

Φ = 0.75, Ω = 2.00

If (5tb + l b) < B as discussed above, then two rectangular HSS walls are engaged to resist local crippling due to the axial compressive force. Therefore, Rn per Eq. (4) is doubled to account for two HSS walls resulting in Eq. (5). This equation is consistent with AISC 360-10 Specification Eq. (K1-15). If (5tb + l b) > B, all 4 HSS walls are engaged and local crippling will rarely govern over local yielding.

For the limit state of HSS wall local crippling, for two walls:

F or (5tp + lb) < B :

.6tRn = 1 2des 1{ + 6( B

lb )( tptdes )1.5}√EF y

tptdes

.75, Ω .00Φ = 0 = 2

Beam Over HSS Column Connection With Moment

Note that this limit state is also applicable for beam over rectangular HSS column joints in which moment is transferred across the beam-to-column interface.

Assuming that (5t b + l b) < B, then two HSS walls contribute to resisting the applied beam moment, resulting in one HSS sidewall to resist the compression couple force. Therefore, Eq. (4) for one wall shall be applied to determine the local crippling capacity of the HSS column.



13

AISC. 2010. “Specification for Structural Steel Buildings”, ANSI/AISC 360-10, American Institute of Steel Construction, Chicago, IL.

AISC 2011. “Steel Construction Manual, Fourteenth Edition”, American Institute of Steel Construction, Chicago, IL.

AISC. 2016. “Specification for Structural Steel Buildings”, ANSI/AISC 360-16, American Institute of Steel Construction, Chicago, IL.

AISC. 2017. “Steel Construction Manual, Fifteenth Edition”, American Institute of Steel Construction, Chicago, IL.

CIDECT 2009. “Design Guide 3: For Rectangular Hollow Section (RHS) Joints Under Predominantly Static Loading 2 nd Edition”, Comite International pour le Developpement et l’Etude de la Construction Tubulaire.

AISC 1997. “Hollow Structural Sections Connections Manual”, American Institute of Steel Construction, Chicago, IL.

References

AISC. 2010a. “Steel Design Guide Series 24: Hollow Structural Section Connections” , American Institute of Steel Construction, Chicago, IL.



14

Stepped HSS T- and Cross-Connections under Branch In-Plane and Out-of-Plane Bending

by Jeffrey A. Packer

Bahen/Tanenbaum Professor of Civil Engineering, University of Toronto, Ontario, Canada _____________________________________________________________________________________

Moment-resisting connections are sometimes required between square and rectangular HSS members that are welded directly to each other in a T- or Cross-connection configuration. If the branch-to-chord width ratio (β) is less than unity (i.e., the connection is “stepped”), the connection will generally not be fully rigid and will only be a partially restrained (PR) moment connection. As such, the connection will transfer moment, but the rotation between the connected members will not be negligible, and in the analysis of the structure the force-deformation response characteristics of the connection need to be included (AISC Specification 360-16 B3.4b(b)). Determination of the moment capacity of stepped HSS connections under branch bending is covered by AISC 360-16 K4.3, which refers to a small amount of information in Table K4.2 and to the broad coverage in Chapter J. This article thus provides more specific guidance, by examination of the potential failure modes or limit states. Limit States For both branch in-plane and out-of-plane bending, the limit states for stepped moment connections can be inferred from similar failure modes for T- and Cross-connections under branch axial loading. The common limit states (within the scope of AISC 360-16 Sections J10 and K1.2a) will be: (i) plastification of the chord connecting face; (ii) shear yielding (or “punching shear”) of the chord connecting face; (iii) local yielding of the branch in a T-connection (or branches in a Cross-connection) due to uneven load distribution in the branch(es). The connection moment capacity for each of these failure modes can be determined by using the Principle of Virtual Work, applied to assumed failure mechanisms or models. Solutions are derived below, for the separate cases of branch in-plane (ip) bending and branch out-of-plane (op) bending. Branch In-Plane Bending Limit State of Chord Plastification This can be analysed using a rectilinear yield-line mechanism in the chord connecting face, as shown in Figure 1. (This mechanism is also shown in a generic form in Fig. 9-5(b) of the AISC Manual (AISC, 2017)). In this case the external work done by the applied load is the moment, ipM , multiplied by the virtual

rotation, / / 2bH , where is a small virtual displacement. The internal work done, assuming rigid-

plastic behavior, can be taken as the work expended in plastic rotation of all the yield lines, which is

given by 2

4

yi i

F tl

, where il is the length of yield line i , i is the corresponding rotation of yield line i ,

and 2

4

yF t

is the plastic moment of resistance per unit length of the chord face. Equating the internal

and external work done, the small virtual displacement/rotation terms cancel out and an expression for the nominal in-plane moment capacity, n ipM , can be obtained. This then needs to be minimized with


limit statetable notes

Article 2

Stepped HSS T- and Cross-Connections Under Branch In-Plane and Out-of-Plane Bending

152

regard to the unknown length of the yield-line pattern shown in Figure 1, by differentiating the n ipM expression with respect to the angle . This yields the following minimum nominal moment

capacity:

21 2

sin 2 11

y bn ip f

F t HM Q

(1)

where is a non-dimensional bearing length ratio equal to /sin

bH B

and in Eq. (1) takes into account a

possible branch-to-chord member angle of inclination. fQ is an added factor, derived empirically from

connection tests, that takes account of the influence of chord connecting face compressive stress on the connection capacity, and is given by AISC 360-16 Eq. (K3-14). A resistance factor of φ = 1.0 (or Ω = 1.50) would apply to Eq. (1), consistent with other yield line equations in AISC 360-16, to produce a connection available flexural strength.

Figure 1: Yield-line mechanism for plastification of the chord connecting face, for in-plane bending

Limit State of Chord Shear Yielding (Punching Shear) Due to non-uniform loading around the cross-section of the branch member adjacent to the chord connecting face, only part of each transverse branch wall will be effective, shown by the shaded parts of the branch in Figure 2. Regardless of loading (axial, in-plane bending, or out-of-plane bending), the longitudinal branch walls will be fully effective (because they are adjacent and parallel to the stiff chord sidewalls), while the transverse branch walls will have varying degrees of effectiveness (because they are supported by the flexible chord face). The welds and chord face immediately adjacent to the highly loaded parts of the branch will also be highly loaded. Thus, if chord punching shear is to occur it will take place along the two U-shaped segments, each of length b epH B (see Figure 2). The amount of branch

rotation to punch through the chord thickness is such that t . Through-thickness punching will initiate in the four / 2epB zones, tapering to zero punching at the axis of rotation.



163

The external work done is / / 2ip ip bM M t H which can be equated to the internal work done by shear

forces multiplied by average displacements. Taking the shear yield stress as 0.6 yF one obtains:

0.6 2 0.6 2/ 2 2

ip y b y epb

t tM F t H F t B t

H

, or

0.6

sin 2sin

y b bn ip ep

F tH HM B

(2)

for the nominal moment capacity, and in Eq. (2) takes into account a possible branch-to-chord member angle of inclination. The punching shear effective width term, epB , is given by:

10

/ep b bB B B

B t

(3)

A resistance factor of φ = 1.0 (or Ω = 1.50) would apply to Eq. (2), consistent with AISC 360-16 Eq. (J4-3), to produce a connection available flexural strength, and similarly a chord stress influence function, fQ , is

not applied.

Figure 2: Failure model for punching shear of the chord connecting face, for in-plane bending

Limit State of Local Yielding of Branch(es) due to Uneven Load Distribution

As explained previously, the effective portions of the branch correspond to the shaded portions in Figure 2, except epB (the chord effective width for punching shear) in Figure 2 must be replaced by the

term eB (the branch member effective width). These are different because the branch member effective

width is influenced by the relative strengths of the branch and the chord on which it rests, whereas the chord punching shear effective width (Eq. (3)) is independent of the branch properties. AISC 360-16 Eq. (K1-1) gives the effective width of one branch transverse wall as:

10

/

ye b b

yb b

F tB B B

B t F t

(4)



174

Assuming a compact branch member, the branch nominal moment capacity may be derived by subtracting the moment contribution of the non-effective “flange” portions from the branch (in-plane) plastic moment capacity. Thus, for an inclined branch,

sin

b bn ip yb b yb b b e

H tM F Z F t B B

where Zb is the plastic section modulus of the branch about the axis of bending. Re-arranging terms and simplifying the moment arm to (Hb/sinθb) one obtains:

1sin

e b b bn ip yb b

b

B B H tM F Z

B

(5)

A resistance factor of φ = 0.95 (or Ω = 1.58) can be applied to Eq. (5), consistent with the same limit state for an axially loaded branch in AISC 360-16 Table K3.2, to produce a connection available flexural strength. The chord stress influence function, fQ , is not included because this is a branch failure mode.

Branch Out-of-Plane Bending Limit State of Chord Plastification This can be analysed using a rectilinear yield-line mechanism in the chord connecting face, as shown in Figure 3. (This mechanism is also shown in a generic form in Fig. 9-5(c) of the AISC Manual (AISC, 2017)). In this case the external work done by the applied load is the moment, opM , multiplied by the virtual

rotation, / / 2bB , where is a small virtual displacement. The internal work done can be calculated

in an analogous manner to the case of in-plane bending, and hence the nominal out-of-plane moment capacity, n opM , can be obtained. After optimization of length c in Figure 3, the following minimum

nominal moment capacity can be determined:

2 0.5 1 2 1

1 1

b bn op y f

H BBM F t Q

(6)

Figure 3: Yield-line mechanism for plastification of the chord connecting face, for out-of-plane bending



185

The chord stress influence factor, fQ , given by AISC 360-16 Eq. (K3-14) is again included. A resistance

factor of φ = 1.0 (or Ω = 1.50) would apply to Eq. (6), consistent with other yield line equations in AISC 360-16 and the in-plane bending case, to produce a connection available flexural strength. Limit State of Chord Shear Yielding (Punching Shear) As explained previously, if chord punching shear is to occur it will take place along the two U-shaped segments, each of length b epH B (see Figure 2), even under branch out-of-plane bending. Again, the

branch can be imagined to just punch through the chord thickness at the furthest point from the axis of rotation. Thus, through-thickness punching will initiate along the two bH lines, tapering to zero punching

at the axis of rotation. The external work done is / / 2op op bM M t B which can be equated to the

internal work done by shear forces multiplied by average displacements. Taking the shear yield stress as 0.6 yF one obtains:

0.6 2 0.6 2 1/ 2 2

epop y b y ep

b b

BtM F t H t F t B t

B B

, or

0.6 12

epn op y b b ep

b

BM F tB H B

B

(7)

for the nominal moment capacity. The punching shear effective width term, epB , is given by Eq. (3). A

resistance factor of φ = 1.0 (or Ω = 1.50) would apply to Eq. (7), consistent with AISC 360-16 Eq. (J4-3), to produce a connection available flexural strength, and the chord stress function, fQ , is not applied.

Limit State of Local Yielding of Branch(es) due to Uneven Load Distribution The effective portions of the branch again correspond to the shaded portions in Figure 2, even under branch out-of-plane bending, but epB (the chord effective width for punching shear) in Figure 2 must be

replaced by the term eB (the branch member effective width) given by Eq. (4).

Assuming a compact branch member, the branch nominal moment capacity may be derived by subtracting the moment contribution of the non-effective portions of the branch transverse walls from the branch (out-of-plane) plastic moment capacity. Thus,

22

4

yb bn op yb b b e

F tM F Z B B

where Zb is the plastic section modulus of the branch about the axis of bending. Re-arranging, gives:

220.5 1 e

n op yb b b bb

BM F Z B t

B

(8)

A resistance factor of φ = 0.95 (or Ω = 1.58) can be applied to Eq. (8), with no fQ factor, as for this limit

state under in-plane bending, to produce a connection available flexural strength. Limit State of Chord Distortional Failure



196

This additional limit state is cited as a potential failure mode for stepped rectangular HSS connections whenever a torque is applied to the chord member. This failure mode involves rhomboidal distortion of the chord cross section, with the torsional capacity given by (AISC 360-16 Eq. (K4-7)):

2n op y bM F t H t BHt B H

(9)

AISC 360-16 Table K4.2 gives a resistance factor of φ = 1.00 (or Ω = 1.50) for Eq. (9), as it is associated with a yielding failure mode. References AISC. 2016. “Specification for Structural Steel Buildings”, ANSI/AISC 360-16, and Commentary, American Institute of Steel Construction, Chicago, IL. AISC. 2017. “Steel Construction Manual”, 15th edition, American Institute of Steel Construction, Chicago, IL.

March 2019



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