Effective Length Factor for theDesign of X-bracing SystemsADEL A. EL-TAYEM and SUBHASH C. GOEL

X-bracing systems made with single angles are quitecommon in steel structures. In current practice, the design ofX-bracing members may be performed in one of two ways.The first method is to ignore the strength of the compressiondiagonal in resisting the imposed loads. Conversely, thesecond method recognizes the contribution of thecompression diagonal. This method requires overall bucklingof the full diagonal in the out-of-plane direction as well asbuckling of one half diagonal about the weak principal axisto be investigated in calculating the effective slendernessratio. However, for all the hot-rolled single equal leg angles(listed in Ref. 4), the radius of gyration about the weak axis,Z-axis, is greater than half of that in the out-of-planedirection. Accordingly, buckling of full diagonal in the out-of-plane direction governs the strength of single-anglecompression diagonals.

The purpose of this paper is to provide designers withsome recommendations regarding the effective length factorto be considered in the design of X-bracing systems. Therecommendations were drawn from experimental andtheoretical study of full-scale X-bracing specimens. For moredetails, refer to the original report by the authors.2

EXPERIMENTAL STUDY

Test Specimens

Five full-scale single-angle specimens and one double-angleX-bracing specimen made of A36 steel were included in thisstudy. The test setup is shown in Fig. 1, and the importantgeometrical properties of the test specimens are given inTable 1. All of the single-angle specimens were made fromequal-leg angles, whereas unequal-leg angles were used inthe double-angle specimen. End gusset plates were providedto connect the bracing members to the testing frame. Filletwelds were used for all the connections. The working stressdesign method per the AISC Specification,4 together withWhitmore's concept5 and Astaneh's recommendations,1 were

Adel A. El-Tayem formerly Graduate Student, Department of CivilEngineering, University of Michigan, Ann Arbor, Michigan.

Subhash C. Goel is Professor of Civil Engineering, University ofMichigan, Ann Arbor, Michigan.

Fig. 1. Test set-up

used to design and detail the test specimens. Typical detailsof a single-angle specimen are shown in Fig. 2. Quasi-staticcyclic loading was used for the tests with the small earlydeformation amplitudes used to help study the first bucklingload of the specimens.

Test Results

During the first cycle, and at small deformation levels, thecompression diagonal showed symmetrical lateraldeformations in the out-of-plane direction over its entirelength without any drop in load-carrying capacity. But, as thedisplacement level was increased, drastic changes inbehavior were noticed. These changes involved both thebuckling configuration and the axis of buckling. Only onehalf of the compression diagonal buckled about its weak axis(Fig. 3). It was after this that a drop in the load-carryingcapacity was observed in these specimens (Fig. 4). Thisbiased buckling can be explained by the interaction betweenthe two cross diagonals. The interconnection provides anelastic restraint against both lateral and rotationaldeformations of the compression diagonal at the point ofintersection. The restraint is sensitive to the force and sag inthe tension diagonal. Therefore, the tension member, at smalldisplacement level, was incapable of preventing the lateraldeformation of the full compression diagonal. The increase inthe force and sagging of the tension member

FIRST QUARTER / 1986 41

Table 1. Geometrical Properties of Test Specimens

Specimen Cross Section A in.2Lr

a

z

bt

b

tgc

AW0 L2 2 1.19 142.6 10.0 .438AW1 L4 4 1.94 88.6 16.0 .375AW2 L4 4 3/8 2.86 87.6 10.7 .625AW3 L2 2 5/16 1.46 143.0 8.0 .438AW4 L2 2 3/16 .902 141.4 13.3 .313

DAW1 2L 3 2 2.63 (62.0)d 10.0 .563aL = 71 in. (1.81m)b b

t stands for width-thickness ratio

ctg stands for thickness of gusset platedThe value in parenthesis stands for out-of-plane buckling

Fig. 2. Details of test Specimen AW1

associated with the increase in the displacement levelproduce a restraining interaction force large enough to causebiased buckling in one-half length of the compressiondiagonal.

The measured buckling loads at the instant of occurrenceof biased buckling for each of the diagonals, whichalternately sustained the compressive load, were measured bythe strain gages. The measured buckling loads in the firstcycle (normalized by Py) are plotted against the calculatednormalized buckling loads per the AISC formulas (safetyfactor removed) (Fig. 5). Results show that despite earlydisplay of lateral deformation over entire length, the buckledhalves of the system were capable of developing bucklingloads (in the first cycle) equivalent to those per the AISCformulas safety with the value of effective length being equalto about .85 times the half diagonal length. Fig. 3. Biased buckling of Specimen AW4

42 ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Fig. 4. Recorded hysteresis loops of Specimen AW4,first three cycles

Fig. 5. Normalized first buckling loads compared to AISC values

ANALYTICAL STUDY

Model

The biased buckling mode in which only one half of thecompression member deformed about its weak axis was theprevalent mode for all of the tested specimens. As a result,only one half of the compression member is represented by amodel shown in Fig. 6. This model incorporates the imposedelastic restraints against both rotational and translationaldeformations. End connections and the interaction at theintersection joint provide these restraints.

The differential equation of equilibrium in the biasedbuckling mode for this member can be written as follows:

EId ydx

Pd ydx

4

4

2

2 0= = (1)

Fig. 6. Model for buckled half to determine K factor

where,EI is the flexural rigidity of the cross section about bucklingaxis, P is the applied compressive load, x is the distancemeasured from the left end of the member and y is the lateraldisplacement transverse to the buckling axis of the crosssection. Introduction of the general boundary conditionsapplicable to this model and seeking the non-trivial solutionleads to the following equation:

[ ]1- h - h - h h + h + h +1 2 1 2 2 1 2F F FZ ( ) Sin[ ]2 - 2 + h h - 2 + 2 = 02Z ZF F( )1 2+ Cos (2)

where,k1, k2 and k3 are the stiffnesses of the attached springs, L ishalf diagonal length and the non-dimensional parameters h1,h2, F and Z are defined by the relations:

h , h , =11

22

2

3= = =

-EIk L

EIk L

P LEI

ZP

P k LF and .

Equation 2 is to be solved for discrete values of the loadparameter F. Consequently, the effective length factor K isdetermined by the following equation:

K =pF

(3)

Complete details of underlaying assumptions and derivationof Eq. 2 are given in the report.2

Special Cases

1. Pin connected at one end and fixed at the other

Solution:

For the above boundary conditions, the values of k1, k2 and k3are O, and , respectively. Consequently, h1 and h2 beco-

FIRST QUARTER / 1986 43

me equal to and Z equal to O. After substitution for theabove values, Eq. 2 becomes:

SinF + FCosF = 0

This equation has a solution at F = 4.488 and from Eq. 3 Kis found to be equal to 0.70.2. Member fixed at both ends but free to sway at one end

only

Solution:For such conditions the fixed end implies that k1 is equal to and, therefore, h1 is equal to 0. At the other end, restraintagainst rotation makes h2 equal to 0 while the free-to-swaycondition implies that k3 is equal to 0, thus, Z becomes equalto 1. Substitution for the values of h1, h2 and Z in to Eq. 2gives:

FSinF = 0

This equation has a solution at F = p, and from Eq. 3 K isfound to be equal to 1.

Test Specimens

To calculate the effective length factor K for each of thetested specimens, the values of k1, k2 and k3 need to bedetermined first. The rotational spring stiffness k2 depends onthe geometrical and mechanical properties of the end gussetplates, calculated as follows:

kEI

Lg

g2 = (4)

where,E is the modulus of elasticity, Ig is the moment of inertia ofgusset plate at Sect. a-a as shown in Fig. 7, and Lg is thedistance between the free and fixed edges of the gusset plate.The stiffnesses of the rotational and linear springs, k1 and k3,at the intersection joint are functions of the torsional andflexural properties of the two cross diagonals as well as themagnitude of the applied force. These stiffnesses aredetermined by using the conventional stiffness approach.Thus:

( )k EIL S rS SGJLm m

1 1 22

12= - +/ (5)

and

( )=

-=3

11

22133

/im

iSrSSLEIk (6)

where,i refers to the other three half diagonals, Lm is half length ofthe bracing angle, J is the torsion constant of the section andr is the degree of rotational fixity of end connections taken to

Fig. 7. Length and critical section of end gusset plate

be equal to

SEIL

SEIL

k

1

1 2+.

The values of S1 and S2 are adopted from Ref. 3 andexpressed as:

S

Cot

TanP

CothTanh

1

1

2 21

0

12

0=

-

-

-

F F

FF

F FF

F

/

/

if,

if, P

1 - 2

and

S

Cosec

TanP

CothTanh

2

2 2 1

0

20

=-

F F

FFF F

FF

- 1

1 -

1 - 2

/

/

if,

if, P

A short computer program was developed to solve for theeffective length factor K by an iterative procedure. Theoutput of this program is compared with the experimentalresults of Table 2, where the latter were determined by backsubstitution in the AISC buckling formulas. The comparisonshows that the experimental and theoretical values are ingood agreement.

44 ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Table 2. Theoretical and Experimental Values for Effective Length Factor, K

StiffnessSpecimen k1

(kip-in./rad.)k2

(kip-in./rad.)k3

(kip/in.)Ktheo. Kexp.

AW0 35.5 159.0 0.278 0.862 0.892AW1 797.0 107.8 1.3 0.838 0.836AW2 1405.1 516.0 2.048 0.806 0.873AW3 37.7 161.9 0.359 0.863 0.887AW4 34.5 72.9 0.216 0.875 0.846

DAW13620.0(3274.5)a 354.3(5. 105) 3.400(3.11) 0.799(.577) 0.805

aThe values in parentheses for specimen DAW1 are for in-plane buckling

CONCLUSIONS

The experimental and theoretical studies show the currentdesign procedure underestimates the contribution ofcompression diagonal in resisting applied loads. Based on theexperimental and theoretical results, the followingconclusions can be drawn:1. Design of X-bracing systems should be based on

exclusive consideration of one half diagonal only.2. For X-bracing systems made from single equal-leg

angles, an effective length of .85 times the half diagonallength is reasonable.

3. The proposed theoretical model can be used forestimating the effective length factor in any direction andfor any cross-sectional shapes.

ACKNOWLEDGMENTS

This investigation was sponsored by the American Iron andSteel Institute under Project 301B. The writers are thankfulto Walter H. Fleischer, Albert C. Kuentz and other membersof the task force for the encouragement they providedthroughout the research. The conclusions and opinionsexpressed in this paper are solely those of the writers and donot necessarily represent the views of the sponsor.

REFERENCES

1. Astaneh-Asl, A., S. C. Goel and R. D. Hanson CyclicBehavior of Double Angle Bracing Members with EndGusset Plates Report No. UMEE82R7, Department ofCivil Engineering, The University of Michigan, AnnArbor, Mich., August 1982.

2. El-Tayem, A.A. and S.C. Goel Cyclic Behavior of AngleX-Bracing with Welded Connections Report No. UMCE85-4, Department of Civil Engineering, The Universityof Michigan, Ann Arbor, Mich., April, 1985.

3. Maugh, L.C. Statically Indeterminate Structures J. Wileyand Sons, 1946, New York, N.Y.

4. American Institute of Steel Construction, Inc.Specification for Design, Fabrication and Erection ofStructural Steel for Buildings 1978, Chicago, Ill.

5. Whitmore, R.E. Experimental Investigation of Stresses inGusset Plates University of Tennessee EngineeringExperiment Station Bulletin, No. 16, May, 1952.

FIRST QUARTER / 1986 45

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