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ENGINEERING JOURNAL / FIRST QUARTER / 2004/3
I
n 1990,AISC and ASCE jointly commissioned a task group to develop a design philosophy for predicting thestrength of bracing connections. The design methods inplace prior to that time were only applicable to connectionsmade to the webs of columns,as a proper methodology forbrace-to-column flange connections had not been agreedupon. The overall goal was to create a complete design phi-losophy that could be used to accurately predict the strengthof bracing connections and ensure ductile structuralresponse. The task group devised a study of five bracing connectiondesign methods. Among the five design methods evaluatedwere the Uniform Force Method,developed by Thornton(1991),and a modified version of a design method devel-oped by Richard (1986). The study included a comparisonof the predicted response of each model to the actualresponse found by testing in both isolated,Bjorhovde andChakrabarti (1985),and complete frame action cases,Gross(1990),for each bracing connection. After extensive studyand comparison the committee recommended the use of theUniform Force Method as the most accurate predictor of thetrue structural response of bracing connections. Although it was found that the Uniform Force Methodwas the most accurate predictor of true connection per-formance,the virtues of each design method were consid-ered for application to an overall design philosophy.Richard’s design method was based on empirical resultsderived from 54 finite element models. Richard’s studiesshowed that frame action significantly affects the gusset-to-frame fastener force distributions and that the gusset con-nection force distributions primarily depend on the plateaspect ratio and the brace angle. Richard’s data suggestedthat uneven stress distributions are produced across a gussetedge in a bracing connection,the peak stress of such a dis-tribution being approximately 1.4 times the average pre-dicted stress value at ultimate load (see Figure 1). Asexplained by Thornton (1984),the Uniform Force Methodcaptures the effect of frame action,but obviously themethod does not capture the uneven stress distribution.In response to Richard’s findings,the ability of a gussetconnection to redistribute forces and achieve a uniform dis-tribution was considered. In the case of a gusset welded tothe web of a W-shape member,the flexibility of the webwas assumed to be sufficient to redistribute an uneven forcedistribution. However,in the case of a bracing connectionwelded to the flange of a member,which has significantlymore rigidity than a connection to the web,the potentialinability of the system to accommodate force redistributionwas considered. The development of a peak stress inducedat some point across the welded connection might cause theweld to fail at the point where the stress is concentrated,causing an unzipping of the weld and a progressive failureof the welded connection. In this case,the uniform forcedistribution used in the Uniform Force Method might beviolated,resulting in failure below the theoretically pre-dicted strength of the connection. An example of this scenario can be seen in the 3rd Edi-tion
Load and Resistance Factor Design Manual of Steel
Rationale Behind and Proper Application of the Ductility Factor for Bracing Connections Subjected to Shear and Transverse Loading
CHRISTOPHER M. HEWITT andWILLIAM A. THORNTON
Christopher M.Hewitt is staff engineer, American Institute ofSteel Construction, Inc., Chicago, IL.William A.Thornton is president, Cives Engineering Corpo-ration, Roswell, GA.
Fig. 1. Summary of finite element model results from Richard’s work displaying the ratio of peak vs. average yield stress.
4/ ENGINEERING JOURNAL / FIRST QUARTER / 2004
Construction
(
LRFD Manual
),Chapter 13,Example 13.1(AISC,2001). Because of the proximity of the brace clawangle to the gusset to beam connection,a peak stress can beexpected at point A in Figure 2.To maintain ductility in the connection for the case of awelded joint subjected to both shear and transverse loading,such as the attachment of a gusset to the flange of a mem-ber,the
LRFD Manual
states,on page 13-28:From Richard (1986) it is recommended that the designfactored force be increased by 40 percent to ensure ade-quate force redistribution in the weld group and the valid-ity of the Uniform Force Method. Thus,(Note that if a moment existed on this interface the con-nection would be designed for the larger of the peak stress and 1.4 times the average stress.)This 40 percent increase in the design force for thewelded joint was adopted to account for uneven distribu-tions in a directly welded gusset edge connection to a mem-ber flange,predicted by a ratio of peak versus average stressin the joint,and has been the basis for the recommendationsof the
Manual
. By providing over strength in the connec-tion,this factor maintains the validity of the design assump-tions in the Uniform Force Method. Looking back atRichard’s test results,the upper bound value of a statistical90 percent confidence interval of the graphed data pointssuggests a value of 1.25 as an appropriate design value,assuming a normal distribution. The 90 percent confidenceinterval upper bound of 1.25 should replace the value of 1.4currently recommended in the
Manual
,particularly since aresistance factor is already applied in the design procedureto account for variations in material properties. Ultimately,the actual weld size need not exceed that required todevelop the strength of the thickness of plate used.
PROPER APPLICATION OF THE DUCTILITY FACTOR
The ductility factor should be applied to ensure adequatestrength across the weld under any loading condition,andits application should not be limited to connections to resistseismic loads. The most critical element to the proper appli-cation of this factor is a calculation of the “average stress”that is compatible with Richard’s work. In order to complywith the conditions of Richard’s analysis,the “averagestress”is taken as the average scalar value of the combinedaxial,shear,and bending stress across the connection. Fig-ure 3 shows typical distributions of shear and bendingstresses. The bending stress distribution is taken as rectan-gular because the forces considered in Figure 1 are at theultimate capacities of the 54 connections considered inRichard’s studies.The “average stress”considering a rectangular stress dis-tribution can be calculated in the following manner:
A
Fig. 2. Welded bracing connection from 3rd Edition
LRFD Manual
.
Forces between gusset and beam (shown acting on gusset)Stress due to V
b
Stress due to M
b
(rectangular)Stress due to H
b
Fig. 3. Gusset-to-beam connection stress distribution.
Let for 0/2
b b
f f x l
= = →
for /20
b b
f f x l
= − = − →
1.41.392
ubreq
P Dl
=
t
= thickness
ba
V f tl
=
bv
H f tl
=
2
4
bb
M f tl
=
ENGINEERING JOURNAL / FIRST QUARTER / 2004/5
From this integration,we arrive at the following calculationfor the average stress across the connection:Likewise,the peak stress value is then taken as:The connection design stress is then taken as the greaterof
f
peak
or 1.25
f
avg
,and the weld is sized accordingly. Bycompleting this calculation,the designer ensures that adesign peak stress larger than the weld strength is notinduced across the joint and ductility (in other words,loadredistribution capability) is maintained under any loadingcondition.Figures 4 and 5 show a real application of the appropri-ate use of this factor.In this example,
∆
V
b
is taken as 132 kips,to reduce thebeam shear to:221 + 369
−
132 = 448 kipswhich is just under the shear capacity of the beam. This isdone in order to avoid the need for a doubler plate for shearand is covered as special case 2 in the
LRFD Manual
,page13-36. Note from Figure 4 that a doubler was still required,but this is for axial force block shear and is not required tobe welded in the k-area.On the gusset to beam connections interface,the result-ant forces are:
H
b
= 643 kips shear
V
b
− ∆
V
b
=
369
−
132 = 237 kips axial
M
b
= 38.5
×
132 = 5094 kip-in. momentThe length of the connected edge is 36.5 in. (the set back from the column face is 1 in.). The gusset is 1¾ in. thick and is made of A992 (Grade50) steel. The gusset stresses are:Note that the thickness of the gusset plate was deter-mined by the limit state of Whitmore buckling. Therequired weld size for the gusset to beam flange is calcu-lated as follows:
0 /22 22 2)/2 0
1( ( )
l avg a b v a b vl
f f f f dx f f f dxl
−
= − + + + +
∫ ∫
2 22 2
1()()2
avg a b v a b v
f f f f f f f
+
= − + + +
22
( )
a b v peak
f f f f
= + +
64310.1ksi27 ksi o.k.1.7536.5
v
f
= = <×
×
×
Fig. 4. Sample bracing connection.
β=16.5α=∆
Fig. 5. Free body diagram of bracing connection.
2373.71 ksi1.7536.5
a
f
= =×
2
509048.73 ksi1.7536.5
b
f
×= =×
3.718.7312.445 ksi o.k.
a b
f f
+ = + = <
2 2
(3.71 8.73) 10.1 16.0 ksi
peak
f
= + + =
6/ ENGINEERING JOURNAL / FIRST QUARTER / 2004
REFERENCES
American Institute of Steel Construction (AISC) (2001),
Manual of Steel Construction,Load and Resistance Fac-tor Design
,3rd Edition,AISC,Chicago,IL.American Institute of Steel Construction (AISC) (1992),
Manual of Steel Construction,Volume II,ASD 9th Edi-tion,LRFD 1st Edition Connections
,3rd Printing,AISC,Chicago,IL.Bjorhorde,R. and Chakrabarti,S.K. (1985),“Tests of FullSize Gusset Plate Connections,”
Journal of Structural Engineering
,ASCE,Vol. 111,No. 3,March,pp. 667-684.Gross,J.L. (1990),“Experimental Study of Gusseted Con-nections,”
Engineering Journal
,AISC,3rd Quarter.,Vol.27,pp. 89-97.Richard,R.M. (1986),“Analysis of Large Bracing Connec-tion Designs for Heavy Construction,”
ProceedingsAISC National Engineering Conference
,Nashville,June,pp. 31-1through 31-24.Thornton,W.A. (1991),“On the Analysis and Design of Bracing Connections,”
Proceedings AISC National SteelConstruction Conference
,Washington,D.C.,June,pp.26-1 through 26-33. (Also available online atwww.cives.com)Thornton,W.A. (1984),“Bracing Connections for HeavyConstruction,”
Engineering Journal
,AISC,3rd Quarter,Chicago,IL.The stress for weld design is:
f
weld
= max (
f
peak
,1.25
f
avg
)
f
weld
= max (16.0,1.25
×
13.7) = 17.1 ksiand the weld size required is (in
1
/
16
in.)The required weld size is thus
11
/
16
in.
CONCLUSIONS AND RECOMMENDATIONS
A re-evaluation of the statistical data used to derive this fac-tor suggests that the 40 percent increase recommended inthe
LRFD Manual of Steel Construction
is overly conserva-tive. It is recommended that this factor be revised to 25 per-cent,applied consistently with the procedure outlined hereand in Richard’s work. A 25 percent stress increase is suf-ficient to ensure that the weld has sufficient strength toresist a peak stress in the weld with statistical confidencecomparable to that used in the design of other connectiontypes.
2 2 2 2
1(3.718.73)10.1(3.718.73)10.1213.7 ksi
avg
f
= − + + + + =
117.11.7510.721.39221.392
weld
t D f
× = × = = ×

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