distortional buckling of steel beams in cantilever-suspended span construction

Distortional buckling of steel beams in cantilever-suspended span construction

CHARLES ALBERT Canadian Institute of Steel Construction, Suite 300, 201 Consumers Road, Willowdale, Ont., Canada M2J 4G8

AND

HESHAM S. ESSA AND D. J. LAURIE KENNEDY Department of Civil Engineering, University of Alberta, Edmonton, Alta., Canada T6G 2G7

Received September 10, 1991

Revised manuscript accepted February 12, 1992

The behaviour of cantilever-suspended span systems is influenced by the type of loading and the presence of flange restraint. An experimental program consisting of 33 full-scale tests was undertaken to examine the stability of beams in a single overhang configuration. The results indicate that torsional flange restraint significantly enhances the buckling resistance and is particularly effective when combined with a web stiffener. Lateral bracing of the bottom flange at column supports is of considerable importance in maintaining stability. Simulating the proper boundary conditions is essential, since test specimens are very sensitive to unwanted restraints and can buckle in a higher energy mode.

A finite element model was developed that takes into account inelastic material behaviour, residual stresses and cross- sectional distortions. The predicted buckling capacities are in good agreement with experimental results.

Key words: cantilever-suspended span, steel beams, distortional buckling, flange restraint, inelastic behaviour.

Le comportement des poutres d'acier en porte-a-faux est influence par le type de chargement ainsi que les conditions de retenue des ailes. Une Ctude experimentale comportant 33 tests a pleine Cchelle a CtC menCe dans le but d'examiner la stabilitk de poutres continues avec une portCe en porte-A-faux. Les rksultats dkmontrent qu'une retenue en torsion amCliore la rksistance au dkversement de facon significative et est particulittrement efficace lorsque combinke a un rai- disseur soude a l'2me. Le support lateral de l'aile infkrieure A l'endroit des poteaux est d'une importance considerable dans le maintien de la stabilite. La simulation des conditions de retenue appropriees est essentielle puisque des retenues indesirables peuvent entrayner un mode de dkversement correspondant une charge critique supkrieure. .

Un modttle d'e1Cments finis a ete dCveloppC pour tenir compte du comportement inelastique, des contraintes rCsi- duelles et des distorsions de la section. La resistance au dkversement prkdite par le modttle indique une bonne correlation avec les rksultats experimentaux.

Mots clPs : poutres d'acier, porte-a-faux, dkversement avec distorsion, retenue des ailes, comportement inelastique. [Traduit par la redaction]

Can. J. Civ. Eng. 19, 767-780 (1992)

Introduction ,Cantilever span

Cantilever-suspended span construction is commonly used for low-rise commercial buildings. In this system, the beams of alternate bays cantilever over the top of columns and a simple span is suspended between the ends of the cantilevers, as shown in Fig. 1. The cantilever span length can be adjusted to balance the positive and negative bending moments in order to achieve an economical design. Members such as open-web steel joists with bottom chord extensions can provide lateral bracing to the top and bottom flanges and a properly welded joist shoe can apply torsional restraint to the top flange (Fig. 2). In some structural systems, due to an irregular grid spacing, joist locations do not coincide with columns and bracing may be omitted. Because the bottom flange of a cantilever is in compression at the columns, the absence of a bottom chord extension requires special attention to avoid compromising the overall stability.

The elastic stability of cantilevers has been investigated by Nethercot (1973) who examined various support conditions at the cantilever tip, with the load applied either on the top flange, at the shear centre, or on the bottom flange. Elias (1985) considered the effects of continuous lateral support on the top tension flange. While the behaviour of

FIG. 1 . Cantilever-suspended span construction.

Open-web steel joist

\Bottom chord extension 1;\ column

FIG. 2. Open-web steel joist, with bottom chord extension.

ordinary single-span cantilevers (with warping prevented at NOTE: Written discussion of this paper is welcomed and will be the root) is well understood, there is little available infor-

received by the Editor until February 28, 1993 (address inside mation on the buckling strength of continuous beams with front cover). overhangs. Prmed in Canada / Imprime au Canada

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768 CAN. J . CIV. ENG. VOL. 19, 1992

FIG. 3. Test setup, overall view.

r Pedestal (column) support r Loading frame

Cantilever span Main span

FIG. 4. Test setup.

The objective of this study was to establish a theoretical model capable of predicting the inelastic distortional behaviour of steel beams in the cantilever-suspended span system, reflecting actual construction details and to verify the model experimentally.

A finite element model was developed that takes into account inelastic material behaviour, residual stresses, and cross-sectional distortions. The experimental program consisted of 33 full-scale tests on beams of two different cross sections in a single overhang configuration.

Experimental program Test setup

The stability of cantilever steel beams was the focus of an experimental and analytical investigation at the Univer- sity of Alberta where a series of 33 full-scale tests was conducted. Figure 3 is a photograph of the overall test setup. As shown in Fig. 4, it consisted of five loading frames at 1.22 m spacing between two support pedestals representing columns, with an additional loading frame at the tip of a 1.22 m cantilever span. The single cantilever configuration with one span overhanging the support had a length of 8.53 m. The loads applied by the five frames between supports simulated the loads applied by joists, while the single frame at the cantilever tip simulated the load of a suspended span. A typical loading frame is shown in Figs. 5 and 6. A pair of loading rods fastened to the inner frame was pulled

vertically downward by a hydraulic jack located below the floor. The five interior jacks were all connected to the same oil manifold and therefore the interior loads were close to the same value. The load was applied to the test beam by the upper horizontal member of the inner frame. The set of four roller bearings enabled the inner frames to move smoothly and essentially without friction in the vertical direction within the outer frames and maintained the equilibrium of the inner frames as a test specimen underwent lateral buckling. Lateral movement of the test beam causes the inner frame to be loaded eccentrically, requiring a stabilizing couple that was formed by lateral forces at diagonally opposite rollers. Also, lateral restraint could be applied to the test beam by the reactions to the rollers. Because the outer frame is relatively stiff and because the clearance between the two frames is only 2 mm, the deviation of the test loads from the vertical does not exceed about 1/600.

Restraint conditions Every point on a beam has six degrees of freedom consist-

ing of three translational and three rotational displacement modes about the longitudinal, vertical, and lateral axes. When attempting to simulate flange restraint conditions commonly found in practice, there is the difficult problem of preventing a particular set of displacements while maintaining the freedom of others. The presence of additional and undesirable restraints can alter the behaviour of an idealized specimen.

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ALBER T ET AL. 769

,Outer frame I

Inner frame I

I ) ) Loading I I rods I

2440

FIG. 5. Loading frame and rods.

(a) Top flange restraint Depending on the construction details, the three distinct

top flange restraint conditions of no restraint (free), lateral restraint, and lateral and torsional restraint shown in Fig. 7 may exist. In Fig. 7a, vertical displacements are constrained to be consistent with the loading frame, while lateral and longitudinal displacements are permitted by the longitudinal and lateral rollers respectively. The hemispherical rocker allows rotation about the longitudinal and lateral axes at a load point. The thrust bearing, shown in Figs. 7b and 7c, allows rotation about the vertical axis, but was considered unnecessary when a hemispherical rocker was used, as shown in Fig. 7a, because the very limited contact area causes a negligible moment resistance about a vertical axis. When the vertical loads are large, however, the hemispherical rocker does not suffice to carry the loads and a pair of semicylindrical rockers or knife edges oriented at right angles to each other and a thrust bearing are required to allow an unsupported flange to rotate about all three orthogonal axes.

By omitting the longitudinal rollers as shown in Fig. 7b, lateral restraint is provided simply through friction. The lateral rollers still allow longitudinal movement, while rotation about the three axes is ensured by the thrust bearing and the two semicylindrical rockers. With the removal of the longitudinal rocker, Fig. 7c shows that the top flange is prevented from rotating about its longitudinal axis as torsional or twisting restraint is developed. As the bottom flange moves sideways, the web is bent in a distortional buckling mode.

Due consideration must be given to the displacement capacity of the loading apparatus. The presence of long unsupported spans or the ductile behaviour of lightly restrained beams can give rise to large lateral displacements. In particular, the large rotations about a longitudinal axis can be difficult to accommodate by the longitudinal rockers. Assuming a static friction coefficient of p = 0.3, static equilibrium limits the rotational capacity of a rocker to a value of 0 = t a n p 1 p = 17'.

(b) Restraint a t the columns At column locations, the test specimen could be prevented

from twisting about its longitudinal axis by a "fork" support, as shown in Fig. 8a, which simulates the case of a joist with its top chord welded to the top flange of the supporting beam and its bottom chord extension connected either

FIG. 6. Loading frame.

to the bottom flange or the column. The fork support consisted of a pair of T-sections mounted on both sides of the web, providing lateral support to the web near the top and bottom flanges through four short cylindrical stubs. It was considered that longitudinal translation and rotation about a lateral axis were not impeded by the stubs as the transverse forces exerted by them, and hence the longitudinal frictional forces are small and were further reduced by lubrication. The rest of the reaction assembly from the bottom up in this figure provides for longitudinal movement (lateral rollers), rotation about a vertical axis (thrust bearing), measurement of the reaction force (load cell), and rotation about a lateral axis (semicylindrical lateral rocker). The three degrees of freedom that are inhibited are the rotation about a longitudinal axis (fork support), lateral movement (fork support), and vertical movement (pedestal).

Although both flanges can deflect sideways at a column location when a joist and its bottom chord extension are not present, a rigid moment connection between the beam and supporting column provides torsional restraint to the bottom flange and the buckled shape will be accompanied by web distortions, as shown in Fig. 8b. This configuration was simulated by removing the fork support and adding longitudinal rollers under the bottom flange to allow sidesway. A pair of load cells was used to measure the reaction and the restraining moment on the bottom flange. The broad width of the two load cells also ensured longitudinal rotational stability. The support next to the cantilever was modelled this way in some tests. With the lateral rollers removed at this location only, a single point of longitudinal fixity is provided along the length of the beam.

In those tests where the cantilever load was the only applied load, the downward reaction force at the opposite end of the beam was provided by a reaction beam bolted to columns on both sides of the pedestal.

Instrumentation The applied forces were measured by a load cell at each

load and reaction point. Statics could then be used to con- firm that the loads were balanced, that the load cells were functioning properly, and that frictional losses were minimal. The bending and warping strains were monitored by four longitudinal strain gauges on each flange at five locations along a test specimen. Lateral bending strains in the web due to distortion were monitored by a vertical strain

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770 CAN. J. ClV. ENG. VOL. 19, 1992

I Loading frame I

Longitudinal rollers

Hemispherical rocker

Lateral rollers

Thrust bearing

Semi-cylindrical lateral rocker

Semi-cylindrical ' longitudinal rocker

Lateral rollers

latekl rocker L FIG. 7. Top flange restraint conditions at load points: ( a ) no

restraint (free); ( b ) lateral restraint; (c) lateral and torsional restraint.

gauge mounted on each side of the web at points of torsional restraint. The buckling displacements, in view of their spatial nature, required the use of cable transducers. To establish the lateral displacements of each flange and the vertical displacement of all four flange tips, six cable transducers were required at a cross section, as shown in Fig. 6. This arrangement was used at the cantilever tip and near each of the third points of the interior span. In those tests where the fork support was omitted at a column, a pair of LVDTs was used to measure the lateral displacements of each flange at the column. All measurements were recorded on a FLUKE data acquisition system.

Material properties The test program included W360 x 39 and W3 10 x 39

beams of grade 300W steel. These relatively light sections were selected so that the testing frames would be neither too heavy nor too large. They also provided reasonable span- to-depth ratios for this type of construction, width-to-depth ratios of the two beams (0.36 and 0.53) in the proportions of beams used in real structures, and one Class 1 and one Class 2 beam. Material properties used in the analyses were obtained from tensile coupons. The mean value of the flange yield stress, Fy, and the overall modulus of elasticity, E , were 288 MPa and 204 500 MPa, respectively, for W360 x 39 sections, and 352 MPa and 209 300 MPa for W3 10 x 39 sections. The width-to-thickness ratios, based on the measured

( b ) ..-..

Test beam Semi-cylindrical

lateral rocker ___I_-.

Thrust bearing Load cells

Longitudinal rollers Suppon pedestal

a FIG. 8. Restraint conditions at columns: ( a ) lateral restraint,

both flanges (fork support); (b ) torsional restraint, bottom flange.

properties, correspond to classes 1 and 2, respectively. The shear modulus, G, was set equal to E/(2(1 + u)), where Poisson's ratio u = 0.3.

Residual stresses determined by sectioning are shown in Fig. 9. The flange of the W360x 39 had tensile residual stresses varying on the average from 5 MPa at the tip to 196 MPa at the flange-web junction, with a compressive residual stress of 198 MPa at mid-depth of the web. The corresponding values for the W3 10 x 39 sections were - 20, 73, and - 130 MPa (tensile stresses are positive). The average residual stress distributions were incorporated in the analyses.

Analytical method Finite element model

The boundary conditions encountered in cantilever- suspended span systems give rise to buckling modes whose prediction requires a refined numerical analysis. A finite element model comprised of four-node plate elements for the web and two-node line elements for the flanges, shown in Fig. 10, was implemented on a microcomputer to predict lateral-torsional buckling on the basis of bifurcation theory. There are three degrees of freedom at each node associated with out-of-plane buckling displacements: the lateral displacement along the z-axis, w; rotation about the x-axis, 0,;

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ALBERT ET AL. 771

l W MPa

0

2W MPa

o Measured residual stress

W 360 x 39 - Average residual sness

FIG. 9. Residual stresses. (Numbers in parentheses indicate average residual stresses, positive in tension.)

Top flange element

/ Degrees of freedom Web element w

,----------------------

d Bottom flange element z FIG. 10. Finite element model.

and rotation about the y-axis, 0,. The sign convention for all rotations is in accordance with the right-hand rule. All nodes are located at the intersection of the middle surfaces of the flanges and web plates. The in-plane stresses due to applied gravity loading are also shown in the figure. The present analytical method computes the tangent modulus buckling load based on the extent of yielding just prior to buckling (Galambos 1968). With a flexible web representa- tion, distortional buckling modes can also be modelled. Local buckling and initial imperfections are not considered. Also, the fillets at the flange-web junctions are neglected.

A typical test specimen, illustrated in Fig. 1 la, is loaded by six concentrated forces. The corresponding finite element mesh is shown in Fig. 1 lb. This mesh contains a single row of 15 web elements over the depth of the web, 30 flange elements, 32 nodes, and a total of 96 degrees of freedom in the entire mesh. The top and bottom flange elements coincide with the upper and lower edges of the web elements. The mesh is refined in the vicinity of the column support nearest the cantilever span where the bending moment is a

FIG. 11. Finite element discretization: (a) typical beam specimen; (b) finite element mesh.

maximum and yielding sometimes occurs. A mesh refine- ment is also required at the cantilever tip to model cross- sectional distortions when torsional restraint is involved.

The buckling strength of a structure discretized as a finite element mesh is obtained from the governing equation:

where [K] is the global structural stiffness matrix, [Kg] is the global geometric stiffness matrix, X is an eigenvalue representing a load intensity, and { r ) is an eigenvector of nodal displacements representing a buckled shape. The global stiffness matrices are assembled from the component element stiffness matrices of the web, flange, and stiffener plates.

Element stiffness (a) Structural stiffness The structural stiffness matrix of an element is generally

expressed as follows:

where [B] is a matrix containing derivatives of shape functions, [Dl is a constitutive matrix containing material properties, Vis the element volume, and the superscript t denotes a transposed matrix. The lateral stiffness of a flange was modelled with a two-node frame element derived from a cubic shape function for the lateral displacement, w (Akay et al. 1977). The [B] matrix serves to interpolate w" = d2w/dw2 in the form: w" = [B]{a} , where { a ) contains nodal displacements for w and 8,. [Dl contains a single term E Z ~ which, when integrated over the cross- sectional area, produces the effective lateral rigidity EI,, in which I,, is the moment of inertia about a vertical axis. The torsional stiffness of a flange is constructed from a linear shape function for the twisting displacement, Ox, and is also expressed in the form of [2]. In this case, [B] interpolates 19: = dOx/dw over the element length and [Dl relates to the torsional rigidity GJ, in which J i s the St. Venant torsional constant. It was assumed in the formulation that the lateral and torsional rigidities varied linearly over the element length. An identical two-node frame element was used to mo>el the stiffeners.

The structural stiffness of the web was modelled by a rectangular plate bending element derived from a two-

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dimensional cubic shape function for the lateral displacements, w (Zienkiewicz 1977). The constitutive matrix is of order 3 and contains plate rigidity coefficients expressed in terms of the web thickness and the material properties, E, G, and v. The resulting element stiffness matrices of the web and flanges involve, as shown in Fig. 10, all three degrees of freedom per node.

The stiffness matrices expressed by [2] were evaluated by numerical integration. Nine integration points across the flange width and two integration points over the length of an element were considered sufficient to model the structural stiffness of the flanges, considering the presence of yielded regions, which will be discussed subsequently. A rectangular arrangement consisting of nine integration points over the depth and three integration points over the length of an element was implemented to represent the web.

In some tests, either intentionally or because of the finite size of the restraint devices, lateral restraints were applied above the top flange and were modelled by modifying the global structural stiffness matrix to simulate a fictitious two- node vertical element, similar to a stiffener, with one node located on the top flange and the other at the elevated restraint point.

(b) Geometric stiffness The geometric stiffness matrix is computed by differen-

tiating the work done by in-plane stresses as component plates shorten during buckling, with respect to the nodal displacements. The stresses considered are shown in Fig. 10 and consist of the longitudinal stresses, a, (including bending and residual stresses), the shear stress, T,,, and the vertical stress, a,, due to vertical gravity loads. Shear and vertical stresses are considered to occur only in the web. For simplicity, the shear stress distribution is assumed to be constant over the depth of the web.

The geometric stiffness of a flange element involves only the longitudinal stresses, a,. The work done by a force, a, dA, through a displacement, dr, is given by

where L and A are the element length and cross-sectional area, respectively. The displacement, dr, can be expressed by expanding the arc length, ds, as follows (Galambos 1968):

where v ' and w' are derivatives with respect to x of the displacements along the y and z axes, respectively. After substituting the relationship v = -0,z into 141, and dif- ferentiating [3] with respect to the nodal displacements, w and 13, (and assuming linear shape functions in both cases), we obtain [5] and 161:

V

where d V = dA dx, [KT] is the geometric stiffness matrix of a flange element relating the nodal forces and displacements in the lateral direction, and [K$] is the geometric stiffness matrix of a flange element relating the nodal moments and rotations about the longitudinal axis. Commas denote differentiation with respect to x. ($1 is a vector of linear shape functions for interpolating the displacements between the two nodes of a flange element:

and the natural coordinate is defined by [ = (2x/L - 1). The geometric stiffness of a web element is given by

Johnson and Will (1974):

where [Kg] is the geometric stiffness matrix relating the nodal forces and displacements in the lateral direction; {a} is a vector of bilinear shape functions for the lateral displacement, w; and commas denote differentiation with respect to the indicated coordinate (x or y). Rotational displacements, 13, and O,, are not included in the formulation. The above expression contains the longitudinal, vertical, and shear stresses in the web normally obtained by a preliminary in-plane finite element analysis. Taking advantage of a statically determinate system, a more direct approach adopted in the present study consisted in determining the longitudinal and shear stresses from the bending moment and shear diagrams using ordinary flexure theory. The a, and T,, stresses thus obtained can be substituted in 181. Because of the complexity of the distribution of vertical stresses, they are treated separately with a simplified approach described below.

The expressions of 151, 161, and [8] were evaluated with the same integration points as for the structural stiffness. In the formulation, it was assumed that the integral of the stress terms a, and z2a, (151 and 161) over the cross-sectional area varied linearly over the length of a flange element and that the bending strains used to construct the longitudinal stress distribution in [8] varied linearly over the length of a web element.

Vertical stresses are induced by vertical loads; therefore, the geometric stiffness follows from the work done by vertical forces applied above or below the shear centre, as their point of application undergoes a vertical displacement during twisting of the cross section ( U = PA,), as shown in Fig. 12. For loads applied above the shear centre but between the flanges, the angle of twisting rotation, Ox, is expressed in terms of the nodal lateral displacements, w, and wb, of the top and bottom flanges, respectively, and the vertical displacement is approximated by the first term of its polynomial expansion:

where h ' is the distance between the middle surfaces of the

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ALBERT ET AL. 773

Tension flange - ,

Bending + Residual - Combined - Stresses Stresses Stresses

(Tensile suesses are positive)

FIG. 13. Extent of yielding. FIG. 12. Vertical loads. A, = vertical displacement of load

point; a = height of load application above the shear centre. Extent o f Y ieldinn

flanges and a is the height of load application with respect to the shear centre. It is assumed that the location of the shear centre remains unaffected by yielding. For loads applied above the top flange, the additional vertical displacement undergone by the load is expressed in terms of the nodal rotation of the top flange, Ox, to reflect the influence of web distortion, and the total vertical displacement is given by

[lo] A,, = - - 2 h' 2 ( W t Y wb)i

Web distortions were ignored in [9], as their effects are less significant for loads applied between the flanges. Similar expressions were derived for loads and reactions applied below the shear centre.

Inelastic behaviour Structural steel members usually contain residual stresses

resulting from uneven cooling after hot-rolling. Under the influence of residual stresses, portions of a beam cross section will become inelastic before the loading intensity actually reaches the yield moment. Although residual stresses generally do not significantly affect the cross sectional strength, premature yielding leads to a loss of stiffness and thus a reduction in buckling resistance. Inelastic material behaviour affects both structural and geometric stiffnesses. When evaluating the constitutive matrix [Dl in 121, the modulus of elasticity is considered to assume a value of zero at integration points within a yielded region. Similarly, when evaluating the geometric stiffness ([S], [6], and [8]), the longitudinal stress, ox, is set equal to the yield stress at integration points within the yielded region. The present elastic-plastic constitutive model neglects strain hardening and the computed solutions are bounded by the theoretical plastic moment, M,. Based on the measured properties and the measured dimensions and neglecting the fillets, the plastic moment had values of 188 kN . m for the W360 x 39 and 208 kN.m for the W310 x 39. The shear modulus is considered to be the same in both the elastic and inelastic regions (Galambos 1968).

A typical longitudinal stress distribution is illustrated in Fig. 13. Regions of tensile stresses are indicated as positive and regions of compressive stresses as negative. Under combined bending and residual stresses, a tension flange begins to yield in the middle while a compression flange begins to yield at the tips. The shaded areas indicate yielding where the longitudinal stresses are limited to Fy, the yield stress.

Determining the extent of yielding in a cross section began by assuming a curvature and a neutral axis location for a given bending moment (Bradford 1986). At each iteration, the longitudinal stress distribution was constructed from the combined bending and residual strains. The force and moment equilibrium were evaluated from the longitudinal stress distribution by numerical integration, with the cross section discretized with an arrangement of integration points similar to that used for computing the stiffness matrices. Iterations were performed by changing the curvature and neutral axis until the stress distribution satisfied both the moment and axial force equilibrium; and once the extent of yielding was established, the stiffness matrices were evaluated.

Solution technique Finding the buckling load involves the solution of the gov-

erning equation [ l ] for the loading intensity, A, and the corresponding buckled shape, { r } , using inverse iteration with eigenvalue shifting (Bathe 1982). Because the stiffness matrices are non-linear functions of the loading intensity even in the elastic range, they must be computed on the basis of an assumed loading intensity and iterations performed until the computed intensity agrees with the assumed value.

The solution technique is summarized below: 1. The beam geometry, restraint, and loading conditions are

defined. 2. A loading intensity is assumed. 3. The extent of yielding in the beam is determined at each

cross section, iterating on the curvature and neutral axis with the bisection method.

4. The structural and geometric stiffnesses of each element are evaluated and added to the global stiffness matrices.

5. The buckling load and buckled shape are computed using inverse iteration.

6. The computed loading intensity is compared with the assumed intensity. If the discrepancy exceeds the desired

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TABLE 1. Beam tests

Test load (kN)

Predicted load (kN)

Test Test no. Predicted Loading and restraint diagramf Section

P (95.95) C - x B W360 x 39

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TABLE 1. (concluded)

Test Predicted Test Test . load load no. Loading and restraint diagramt Section ( k N ) ( k N ) Predicted

P

22 * - fi

P f W310x39 133.3 129.6 1.03

23 W360x 39 45.9 49.8 0.92

P (0,60) 5 @ PI5 (145,205)

24 - - - - - - W 3 6 0 ~ 39 40.7 43.9 0.93 X

9.6 10.4 p (0,60)

25 - x B W310x39 55.9 54.1 1.03

P (0.60) 5 @ PI5 (145,205)

26 - - - - - 8 W310x39 46.7 45.4 1.03

P 5 @ PI5 11.2 10.0

27 - - - - - W310X39 127.5 142.8 0.89 X

P (0,60) 5 @ P/5 (145,205) 26.7 29.9

28 - - - - - - 8 W360x 39 128.8 136.7 0.94

P 5 @ PI5 (145,205) 28.2 29.9

29 x- W360x 39 76.3 72.4 1.05

P (0.60) 5 @ PI5 (145,205) 17.5 16.6

30 - - - - - X

W360x 39 42.2 43.0 0.98

P 5 @ PI5 10.7 10.9

f - - - 3 1 - W310x39 127.8 143.9 0.89 X

p ( o m 27.2 30.6 (- 150)

32 W360x 39 72.3 70.9 1.02

33 (O>; (O,?

W360x 39 73.4 70.0 1.05 - 1219 3657 mm

NOTES: Excluding tests 1 , 6, and 8, the mean test-to-predicted ratio, p , is 0.99 with a standard deviation, a, of 0.062.

*These values are excluded in the calculation of the mean and the standard deviation. Legends: 0, lateral restraint; 0, torsional restraint; e , lateral and torsional restraints; 11, web stiffener;

A , V , reaction; and the values in parentheses are the height of load application and the height of lateral restraint above top flange in millimetres, if applicable.

tolerance, a new intensity is assumed. Return to step 3. The final buckling load is obtained with the bisection method.

Results of the experiments and finite element analyses Beam tests

The maximum experimental loads and the predicted loads are presented in Table 1 for the 33 full-scale tests. The six columns in the table contain in order: the test number, the load and restraint diagram, the section designation, the test load, the theoretical prediction of this load, and the test- to-predicted ratio.

The loading and restraint diagrams show where loads were applied and the conditions of restraint that existed. The test loads correspond to the maximum load, P , applied at the cantilever tip when lateral-torsional buckling occurred.

Buckling was deemed to occur when a load-deflection curve (the lateral deflection of one of the flanges or the rotation of the cross section at the cantilever tip) reached a horizontal asymptote. As the maximum load was approached, deformation control was used in applying the loads. The magnitude of the additional loads applied between supports (interior loads) is given in the second line in both the test and predicted load columns (as in test 6). The test-to- predicted ratios refer to the cantilever tip loads only. The test loads include the dead weight of the loading frames (3 kN each); thus, the ratios of interior loads to the cantilever load may not coincide exactly with the nominal ratios shown in the table (e.g., P/5), which refer to the applied jack loads only. The actual load ratios were used in obtain- ing the theoretical predictions, which were calculated from the measured material properties and the measured dimensions.

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TABLE 2. Sectional dimensions of test beams

Section

W360 x 39

Measured Nominal

Dimension Coefficient of (mm) Nominal Mean variation

Depth, d 353 0.999 0.0048 Width, b 128 0.993 0.0045 Flange thickness, t 10.7 1 .OOO 0.014 Web thickness, w 6.5 1.023 0.025

Measured Nominal

Coefficient of Nominal Mean variation

310 0.996 0.0077 165 0.990 0.001 1

9.7 0.967 0.012 5.8 1.072 0.01 1

The white dots, white squares, and black dots on the load and restraint diagram represent different restraint conditions at load and reaction points as given in the table footnotes. A white dot represents a point where only lateral restraint was provided to a flange; a white square where only torsional restraint existed; and a black dot where both lateral and torsional restraints existed. When no restraint was provided, none of these symbols appear. Reactions are denoted by triangles pointing upward or downward in the direction of application. As fork supports, at column locations, provide lateral restraint to both flanges, they appear as pairs of white dots one above the other.

The numbers in parentheses following the nominal load designation (e.g., P) indicate, in order, the height in millimetres above the top flange where the load was applied and the height where lateral restraint was provided. When the flange is unrestrained, only the height of load application is shown; and where no numbers are given, the loads, reactions, and restraints were applied directly to the flange sur- face. In general, the points of load application and lateral restraint are the same height above the top flange. This occurs in cases of torsional restraint or when the rotational freedom about the longitudinal axis is ensured by a longitudinal knife-edge. In some instances, e.g., test 8 for interior loads, the point of lateral restraint is 60 mm above the point of load application. This occurs when a semicylindrical or hemispherical rocker of 60 mm radius is used to ensure rotational freedom about the longitudinal axis. The point of lateral restraint is located at the top of the rocker but, as the latter rotates with the test beam, the spherical shape of the rocker is such that the line of load application extends downward through the rocker to the centre of its base 60 mm below.

Full-depth stiffeners (tests 5 and 19), indicated by a pair of vertical lines, had a thickness of 10 mm and a width equal to the flange width and were welded symmetrically on both sides of the web.

The 33 tests were performed on a total of 11 beam specimens, comprised of seven W360x 39 sections and four W310 x 39 sections only. By testing a given beam in a sequence of generally increasing restraints, the beam could be used repeatedly because in earlier tests it buckled elastically and returned to its original shape upon unloading. A specimen was replaced after undergoing inelastic buckling that resulted in noticeable residual deformations. All of the

specimens, except for the one used in tests 1 through 4 with a sweep of 11700th of the span, met the specification limit of 1/1000th of the span. The sectional dimensions of the test beams are given in Table 2. The table also contains the measured-to-nominal dimension ratio and the coefficient of variation of the measured dimensions (equal to the standard deviation divided by the mean). Overall, the measured-to- nominal ratios are close to 1.0, indicating that the deviations from the nominal dimensions were minimal.

The load of a suspended span is usually transferred to the cantilever tip through a shear connection with the load applied near the shear centre of the beam. In these tests, the cantilever tip loads were applied at or above the top flange and this was, of course, considered in the analyses. The test situation represents a more severe loading condition because of the destabilizing effect of loads applied above the shear centre. Discussion of results

(a) Beams with lateral restraint at column locations In tests 1 through 5, a single load was applied at the can-

tilever tip of the W360 x 39 and fork supports were provided at both column locations, as shown in Table 1. The residual stress distribution of the W360 x 39 (Fig. 9) with its flanges entirely in tension tends to have a stabilizing effect in the elastic range, as the geometric stiffness of the flanges ([5] and [6]) is increased by the added tension. This effect was noted previously by Kitipornchai and Trahair (1975).

The top flange was laterally restrained at the load point in tests 1 and 2, unrestrained in test 3, and restrained laterally and torsionally in tests 4 and 5. Tests 1 through 3 behaved elastically, but tests 4 and 5 displayed inelastic behaviour with significant residual deformations upon unloading. Test 3, without restraint, had the lowest test load and exhibited the greatest lateral displacements at the bottom flange with little movement of the top flange. In test 2, lateral restraint was provided 40 mm above the top flange where the load was also applied. The test load was slightly increased over that of test 3, though the predicted load was not. Test 1, also laterally restrained and predicted to have a lesser capacity because of the higher point of load application (95 mm), actually carried a slightly greater load than in tests 2 and 3. The greater test load is attributed to the longitudinal knife-edge at the cantilever tip, which did not rotate smoothly and developed a small frictional torsional restraint.

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It is interesting to note how the behaviour of an unrestrained overhanging beam as tested here in test 3 dif- fers from that of an ordinary single-span cantilever with warping prevented at its root. The ordinary cantilever displays a greater displacement at the top flange than at the bottom (Galambos 1988). Because the opposite is observed in an overhanging beam (e.g., test 3), it follows that the latter benefits little from lateral restraint at the top flange of the cantilever tip (the predicted loads for tests 2 and 3 are both about 79 kN). This observation does not apply to interior loads, as will be seen later.

A substantial increase in strength occurred in test 4, where lateral and torsional restraints were provided to the top flange and considerable cross-sectional distortions similar to Fig. 7c were observed at the cantilever tip. In test 5, a stiffener was welded to the web at the cantilever load point where the top flange was laterally and torsionally restrained. The stiffener prevented the cross-sectional distortions previously observed in test 4 and, in conjunction with the lateral and torsional restraints at the top flange, also prevented any lateral or twisting displacement of the bottom flange, resulting in a further improvement in strength. A plan view of the longitudinal buckled shape of test 5 can be seen in Fig. 14. The photograph shows the main span with the fork support close to the cantilever located at the bottom. It can be seen that the bottom flange, painted in white, has undergone a much greater lateral deflection than the top flange, with the maximum rotation occurring near the middle of the main span, at the top of the photograph.

Tests 6 through 8 were conducted with five additional loads, each about one-fifth of the cantilever load, applied between the supports. This load configuration models the limit of the unbalanced condition in which the cantilever- suspended span is loaded at full intensity and the main span is loaded at half intensity. This results in a shorter length of the bottom flange in compression as compared to the previous tests 1 through 5 , where the entire bottom flange was in compression. The addition of lateral restraint at all load points in test 8 significantly increased the predicted buckling strength as compared to tests 6 and 7 which were unrestrained. This increase is in marked contrast to tests 1 through 3 where no interior loads were applied and where lateral restraint of the top flange at the cantilever tip had no appreciable effect. The difference is that the bottom flange showed little movement while the top flange displayed significant lateral displacements at the interior load points in tests 6 and 7, where it is in compression. The added top flange restraint in test 8 greatly altered the buckled shape, leading to an increase in the maximum load. Both tests 6 and 8 on a W360 x 39 gave test-to-predicted ratios much in excess of 1 .O, indicating that unwanted restraint may have been present. In tests 6 and 7, which are duplicates, an array of ball bearings was used at the interior load points to provide lateral, longitudinal, and rotational freedom about the vertical axis. The only difference between test 7 with a test- to-predicted ratio of 1.16, and test 6 with a test-to-predicted ratio of 1.47, was that in the former a small lateral disturb- ing force estimated at 0.6 kN was applied by hand to the specimen after each load step. This suggests that in test 6 there was sufficient friction between the ball bearings and the plates to develop partial restraint and induce a higher energy buckling mode. The ball bearings were not used

FIG. 14. Buckled shape, test 5.

subsequently. In test 8, the inflated buckling strength is attributed to the improper functioning of the longitudinal knife-edge at the cantilever tip, which did not allow the specimen to rotate fully about a longitudinal axis. When using a knife-edge, the twisting rotation is permitted by a prismatic pivot rolling within a longitudinal groove. The similar radii of curvature of the pivot and groove in the particular knife- edge used here may have forced the rolling movement to be coupled with sliding and frictional restraint. Therefore, the specimen again carried more than the anticipated load. These two tests, together with test 1, demonstrate that small unwanted restraints result in higher energy buckling modes. In practice, the corollary is that relatively small and unac- counted for extraneous restraint may be beneficial and prevent buckling at low loads.

Tests 9 through 15 all had loads applied at the cantilever tip and a second load at mid-span of a W 3 6 0 ~ 3 9 , with various ratios of the two loads and various restraint conditions. Although the height of application of the cantilever tip load was increased between tests 9 and 10, which would tend to decrease the failure load, the test and predicted loads actually increased because of the lateral restraint provided in test 10 at the mid-span load point. Tests 10 and 11 are duplicates of each other and comparable failure loads were reached. Test 13 shows a slight increase in strength compared to test 12 because of the added torsional restraint at mid-span. The elevated application point of the cantilever load in tests 11 through 13 showed a destabilizing effect, which tended to increase twisting in the cross section and

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778 CAN. J . CIV. ENG. VOL. 19, 1992

induce noticeable distortion in the web (in a mode similar to that shown in Fig. 12). As the height of load application is reduced from test 11 to test 12, both the predicted and test loads increased. The provision of lateral and torsional restraints in test 14 resulted in a sizable strength increase as compared with test 12, although the reduced height of load application would also contribute. A slight further increase was achieved in test 15 with respect to test 14 by lowering the height of the mid-span load.

Tests 16 through 22 were performed on W310 x 39 sections and are mostly counterparts of previous tests on W360 x 39 sections. The residual stress distribution of the W310x 39, with its wider flanges, results in less tension in the flanges (Fig. 9) and is not as favourable as that of the W360 x 39. The compression flange of the former will yield earlier at the tips, causing a reduction in the effective lateral rigidity, EI,.

Consider tests 16, 17, 18, 19, and 22 as a set with a load applied only at the cantilever tip. The tests, in order of increasing restraint at the point of load application, are 17, no restraint; 16, lateral restraint only; 18, torsional restraint only; 22, lateral and torsional restraints; and 19, lateral and torsional restraints with a web stiffener to prevent web distortion and restrict movement of the bottom (compression) flange. Both the test and predicted loads show an orderly increase in loads through this progression - 77.6, 84.0, 118.9, 133.3, and 152.6 kN for the test loads and 78.0, 83.9, 111.5, 129.6, and 160.7 kN for the predicted or theoretical loads. Test 18 was the only test in which torsional restraint alone was provided without lateral restraint of the top flange and resulted in a greater strength, in this case, than in test 16 which was provided only with lateral restraint. When both lateral and torsional restraints are provided as in test 22, there is a significant improvement in the strength as compared with providing one or the other. This increase is further enhanced when a web stiffener is added because the bottom flange is restrained from moving laterally. This sequence of tests therefore clearly demonstrates experimentally and theoretically the enhancement of strength by providing translational and rotational restraints to the top flange and by minimizing web distortion. The relative magnitude of the increase in strength is, of course, dependent on the cross-sectional properties (including residual stresses) and the overall geometry of the beam and loading.

The addition of lateral restraint at mid-span in test 20, as compared to test 17, would normally have increased the strength but the failure load actually dropped to 52.7 kN from 77.6 kN because of the higher point of application of the cantilever load. Test 21 was loaded and laterally restrained at the cantilever tip and at the five interior points. The failure load is much increased as compared to test 16 because of the reduced length of the unsupported bottom flange in compression and the support of the top compression flange between the supports. The shape of the moment diagram and the conditions of support have a significant effect.

The load of 154.5 kN at the cantilever tip in test 21 cor- responds to 91% of the calculated plastic moment for the W310 x 39 section. Therefore, this test presented an oppor- tunity to assess the analytical method well into the inelastic range. The predicted load of 155.4 kN is in good correspon- dence with the test load, with a test-to-predicted ratio of 0.99.

Test 28 on a W360x 39 is a duplicate of test 8, but with a higher point of load application at the interior points and with the longitudinal knife-edge replaced by a longitudinal semicylindrical rocker at the cantilever top to remove the unintended small restraint that the longitudinal knife-edge may have provided. This proved to be true and the test-to- predicted ratio in test 28 was 0.94, as compared to 1.33 in test 8.

(b) Beams without lateral restraint of the column at the cantilever root

The omission of lateral bracing at a column support was simulated in tests 23 through 27 and 29 through 31. Longi- tudinal rollers were provided at the pedestal assembly nearest the cantilever, as illustrated in Fig. 8b, to allow lateral movement of the bottom flange there. The absence of a bottom chord extension on the bottom flange allows the beam to move laterally with the top of the column, while a rigid connection to the column (denoted by a white square) develops torsional restraint and forces the web into a distortional buckling mode. Note that the column cap plate to beam connection must be capable of developing the distortional moment of the web. Furthermore, in an actual structure the relative distortion of the web would be greater than that depicted in Fig. 8b due to the inclination of the column. The pedestal at the opposite end of the beam was detailed as in Fig. 8a with a fork support. The beneficial effect of the tensile residual stresses in the elastic range under these circum- stances is undermined. There is less flange participation in the overall behaviour because of the web distortion. In test 23, a W360x39 beam was loaded and laterally restrained at the cantilever tip. A test load of 45.9 kN was observed as compared to 78.9 kN in test 2, which was braced at both columns. These results clearly indicate that torsional restraint alone at the column, as may be supplied by a rigid connection between the beam and the column, does not com- pensate for the omission of bracing or alternatively of web stiffeners.

In test 24, the loads and lateral restraint points were applied at the cantilever and between columns. Again, the beam was free to translate at the column near the cantilever root, and the test load at the cantilever tip was only 40.7 kN. This represents less than a third of the capacity achieved in test 28, in which bracing was supplied at the column but was otherwise identical. Tests 25 and 26, performed on W310 x 39 beams, again without bracing at the column near the cantilever, are similar to tests 23 and 24, respectively, on W360 x 39 beams. Tests 25 and 26 reached only 67% and 30'70, respectively, of the capacities achieved in tests 16 and 21 on W310 x 39 beams where bracing was provided at both columns. The reductions when lateral bracing at the column is not provided are generally similar for the two beam cross sections. Beams with interior loads (tests 24 and 26) were more severely affected by the lack of bracing because of the destabilizing effect of the additional loads applied between the supports above the top flange. The relatively low test- to-predicted ratios in tests 23 and 24 are considered to be due to incomplete torsional restraint at the column. The nar- row bottom flange width of the W360 x 39 was insufficient to prevent rotation about the longitudinal axis, and a slight separation between the edge of the flange and the supporting assembly was observed at failure in these two tests.

Providing torsional restraint to the top flange in addition

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to lateral restraint has a marked effect on the load the beam can carry. Essentially, identical tests 27 and 3 1 with lateral and torsional restraints at all load points have a test load of about 128 kN at the cantilever tip, while test 26 with lateral restraint only at the load points but otherwise identical reached a failure load of only 46.7 kN, only 37% of the former. The only difference between tests 24 and 29 is that test 29 had lateral and torsional restraints at the cantilever tip while 24 had lateral restraint only. At all other load points, only lateral restraint was provided. Test 24 had a failure load of 40.7 kN, only 53% of the 76.3 kN of test 29. Torsional restraint mobilizes the distortional strength of the web whenever it is provided and thus enhances the resistance of the beam. The distortional buckling mode at the cantilever tip in test 31 can be seen in Fig. 15. The photograph shows that the top flange has remained horizontal while the bottom flange has rotated and translated laterally, with considerable distortion in the web. When joist shoes are properly welded to the top flange of beams to develop a fixed connection between the joists and the beam, the joists provide torsional restraint to the beam flange and increase the stability of the beam. While the tests simulate complete torsional restraint, the actual degree of fixity is proportional to the flexural stiffness of the joists.

In tests 5 and 19, the provision of stiffeners at the cantilever tip resulted in significant increases in the failure load of 20% (125.8 versus 104.9 kN) and 14% (152.6 versus 133.3 kN) when compared to tests 4 and 22 respectively where stiffeners were not provided. This suggests that the provision of web stiffeners at the column location would be beneficial. The stiffeners anchored by the torsional restraint at the bottom flange would prevent web distortions and inhibit twisting of the entire section at this location, as was observed at the cantilever tip in tests 5 and 19. Some of this restraint would be lost in proportion to the flexibility of the column.

(c) Double cantilever beams Tests 32 and 33 were tested in a double cantilever con-

figuration to examine the stabilizing effect of a lower load point (test 32) and the effect of span length (test 33). A single fork support was provided at the common root of the two cantilevers of unequal span. The negative height of load application in test 32 at the right-hand end, as shown in Table 1, means that the downward load was applied 150 mm below the unrestrained bottom flange. This models the loading of a monorail beam. In test 33, the top flange was laterally restrained at both cantilever tips. The test loads are in good agreement with the predicted loads.

(d) Summary A measure of the accuracy of the analytical method is

afforded by computing the test-to-predicted ratios for each test and the mean value for all the tests. In such a compari- son, experimental errors contribute to deviations from a value of 1.0 and increase the variation. As discussed previously, the results of tests 1, 6, and 8 were considered unreli- able because of unwanted frictional restraints and these tests were repeated in tests 2, 7, and 28, respectively, with more reliable results. Therefore, excluding tests 1, 6, and 8 (indicated by asterisks in Table I), the mean test-to-predicted ratio obtained for 31 tests is 0.99 with a standard deviation of 0.062. The mean value close to 1.0 indicates the analytical model has good predictive capacity for a wide range of

FIG. 15. Web distortion, test 31.

boundary conditions. The standard deviation of 0.062 related both to experimental errors and model simplifications is quite small. It arises from variations in residual stress patterns, yield strengths, moduli of elasticity of the beam from the measured values, experimental errors in calibration of load cells and the like, and unwanted friction in the reac- tive devices as well as the model simplifications. It is interesting to note that Mirza and MacGregor (1982) suggested a coefficient of variation of 0.040 to account solely for errors in measurement when assessing the strength of reinforced and prestressed concrete beams.

Buckled shape In addition to the buckling load, the theoretical solution

gives the buckled shape contained in the eigenvector {r). The governing equation [I] does not give the magnitude of the displacements but only a normalized shape. The predicted buckled shape of test 4 is shown in Fig. 16, where the lateral displacements of the top flange, mid-web, and bottom flange are plotted. There are no lateral displacements at the columns, where fork supports were provided. Due to the lateral and torsional restraints of the top flange at the cantilever tip, the top flange displaces the least laterally and the bottom flange displaces the greatest amount, with maximum displacements occurring between the columns. At the cantilever tip, in particular, the uneven spacing between the lines representing the top flange, mid-web, and bottom flange displacements indicates web distortion similar to that shown in Fig. 7c. In general, the predicted buckled shapes agreed with those observed during the tests.

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,Fork support Fork support.

Top flange - Mid-web - Bottom flange - Initial position

Cantilever span Main soan

FIG. 16. Buckled shape, test 4 (plan view).

Conclusions provision of lateral restraint to the top of the column at the cantilever root significantly increases the buckling strength.

The web Such restraint can be provided economically by the extension stresses, and inelastic behaviour predicts failure loads that of a bottom chord of a joist. If it is impracticable to extend are in good agreement with the experimental results. The the bottom chord to provide this restraint, then web stif- ex~erimental results are very sensitive to unforeseen feners connected to both flanges with a moment connection restraints. Even minimal amounts of friction will force a between the bottom flange of the beam and the column can beam into a higher energy buckling mode. be designed to prevent twisting of the cross section. Stif-

The boundary conditions used in the programme reflect feners may also be desirable in other situations. typical design situations. The applicability of the model con- The welds connecting joists to the supporting beams ditions to design situations depends on such factors as the should ideally be designed and made to provide effective live load pattern, the location and the flexural stiffness of lateral and torsional restraints to the beams. It is expected the joists, the strength of the joist shoe connection, and the such would not be significantly different from welds presence of stiffeners and bottom chord extensions. currently specified and therefore would not increase costs.

Both the experimental and theoretical results of this study show that therestraint conditions dominate the behavioir of steel beams in cantilever-suspended span construction. At the same time the shape of the moment diagram, that is to say, which flange and how much of each is in compression, is significant. Residual stresses play a major role in determining when inelastic action begins. When the flanges have a favourable tensile residual stress pattern, the geometric stiffness of the flanges is increased and the onset of lateral instability is delayed. There is a stabilizing effect in the elastic range. In the inelastic range, residual stresses adversely affect the buckling strength.

When lateral restraint is not supplied to the column at the cantilever root, otherwise identically loaded and restrained single cantilever beams have failure loads reduced to as low as 30% of those when such restraint is provided. Lateral top flange restraint is particularly effective in increasing the buckling strength when it is provided where the flange is in compression. The provision of torsional restraint to the top flange further improves the buckling strength by forcing the beam into a distortional buckling mode. When open-web steel joists are properly welded to supporting beams, they provide both lateral restraint to the top flange and torsional restraint through their flexural action.

When a web stiffener is introduced at a section where one flange is torsionally restrained, the overall stability is enhanced by eliminating web distortions and preventing twisting of the cross section about its longitudinal axis.

The influence of initial imperfections on the buckling strength does not appear to be significant as long as they do not exceed rolling and fabrication tolerances.

The formulation of these conclusions in the form of spe- cific design recommendations is the subject of further work now underway. These will be developed from the analytical model, as verified by the tests, for a number of different boundary and loading conditions, that is, different design situations. In general terms, however, it is noted that the

Acknowledgements The financial assistance provided by the Natural Sciences

and Engineering Research Council and the Canadian Institute of Steel Construction to D.J.L. Kennedy is greatly appreciated. E.Y.L. Chien of the Canadian Institute of Steel Construction was responsible for the conceptual and preliminary design of the load frames.

Akay, H.U., Johnson, C.P., and Will, K.M. 1977. Lateral and local buckling of beams and frames. ASCE Journal of the Struc- tural Division, 103(ST9): 1821-1832.

Bathe, K.-J. 1982. Finite element procedures in engineering analysis. Prentice-Hall, Englewood Cliffs, N.J.

Bradford, M.A. 1986. Inelastic distortional buckling of I-beams. Computers and Structures, 24(6): 923-933.

Elias, Z.M. 1985. Buckling with enforced axis of twist. ASCE Jour- nal of Engineering Mechanics, l l l (12): 1539-1 543.

Galambos, T.V. 1968. Structural members and frames. Prentice- Hall, Englewood Cliffs, N.J.

Galambos, T.V. Editor. 1988. Guide to stability design criteria for metal structures. 4th ed. John Wiley and Sons, New York, N.Y.

Johnson, C.P., and Will, K.M. 1974. Beam buckling by finite element procedure. ASCE Journal of the Structural Division, lOO(ST3): 669-685.

Kitipornchai, S., and Trahair, N.S. 1975. Inelastic buckling of simply supported steel I-beams. ASCE Journal of the StructuraI Division, lOl(ST7): 1333-1347.

Mirza, S.A., and MacGregor, J.G. 1982. Probabilistic study of strength of reinforced concrete members. Canadian Journal of Civil Engineering, 10: 431-448.

Nethercot, D.A. 1973. The effective lengths of cantilevers as governed by lateral buckling. The Structural Engineer, 51(5): 161-168.

Zienkiewicz, O.C. 1977. The finite element method. McGraw-Hill Book Company, London, United Kingdom.

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