ultimate strength of stiffened symmetrical welded steel beam-to-column flange connections

C~mpr~rrr d S~rucrwcr Vol. 16 No. 5, pp. 749-760. 1987 O&67949/87 53.C0+O.W

Phcd UI Gru.c Brium. 0 1987 Prrp~on lournab Ltd.

ULTIMATE STRENGTH OF STIFFENED SYMMETRICAL WELDED STEEL BEAM-TO-COLUMN

FLANGE CONNECTIONSt

H. A. EL-GHAZALY$ and A. N. SHERBOURNE~ SDepartment of Civil Engineering, Kuwait University, Kuwait

@epartment of Civil Engineering, University of Waterloo, Ontario, Canada N2L 3G1

(Received 23 July 1986)

Ahabaet-Several EtUdieE have confirmed the importance, in beam-cohmn connections, of reinforcing the

column web with horizontal stilfeners opposite the beam flanges welded to the column web and flanges. Present design codes, besides being conservative, do not adequately reflect the role of the stiffeners as lateral support to the column web aginst buckling. Non-linear finite element formulations, employing an incremental version of Stowell’s deformation theory and a scaled version of the inverse power method, allowed accurate predictions of the bifurcation load of general concentrically stiffened stal plates in the elasto-plastic range. In this paper, the analytical procedure developed has been used to provide qualitative and quantitative design information on the strength of this type of connection by conducting a parameteric study, the parameters considered being the effect of stiffener size on column web buckling capacity and the effect of column axial load on the stiffening requirements of a connection. It has been demonstrated that proper reinforcing of the column web can, in practice, diminish the harmful effects of high column axial loads on connections.

NOTATION

beam flange area column flange area column web area total cross-sectional area of the pair of stiffeners width of the column flange width of one stiffener force. in the beam compression llange critical beam flange compression force causing column web buckling ultimate force from the beam compression flange beam compression flange force causing column web buckling at P = 0.0 and I, = 0.0 beam flange compression force causing first yield in column web beam depth between flange centroids column overall depth modulus of elasticity tangent modulus of a uniaxial stress-strain relationship clear height of the column web length of stiffener distance from the top of the flange to the toe of the adjacent tillet distance from web centre to end of fillet beam plastic moment column axial load aaial load causing yield in column thickness of beam flange thickness of column flange stitIener thickness force in the beam tension flange global displacements along X, Y and Z axes, respective1 y beam direct shear web thickness rate of change of (w) with respect to X and Y axes, respectively Cartesian global rectangular axes

~AccepasdforpmaentationlnpartattheRriAcStnn%ual Steel Conference, Auckland, New Zealand, August 1986.

Greek a, V

0,

$,)mu

CnhlCO Poisson’s ratio critical stress corresponding to buckling material yield stress maximum value for residual stresses stress components in the X and Y directions, respectively

1. INTRODUCTTON

Limit states design [1] addresses the problem of structural behaviour at ultimate loading conditions which necessitates a thorough understanding of the capacity of the various structural elements under these conditions. Because structural steel possesses sufficient ductility, the ultimate behaviour of steel components in compression is, generally, governed by elastic-plastic buckling rather than the equilibrium limit load [2]. Present design codes provide an abundance of information concerning the elastic and plastic design of beams, columns, beam-columns and other structural elements that may be involved in a steel structure. An important structural component that has not been given enough attention is beam-to- column moment connections. Dit5culties associated with theoretical analysis, re5ected by the lack of an accepted generalized theory for plate plastic bifurcation and complications of experimental testing are among the major reasons for inadequately defining the problem. Needless to say, any design procedure becomes invalid if connections fail to transmit the forces which the members are designed to resist.

Several studies on beam-to-column 5ange OOMCC- tions[3-4 have confinned the importance of reinforcing the column web with horizontal stifieners

750 H. A. EL-GHAZALY and A. N. SHER~URNE

opposite the beam flanges and welded to the column flanges and web. These stiffeners effectively increase the load-carrying capacity of the column web and generally improve the overall rotational behaviour of the connection. A recent investigation by the authors 161 resulted in design curves controlling the slender- ness of stiffener plates in moment connections. This allowed appropriate selection of stiffener thickness to prevent premature buckling until the framed beams in the connection attain the full plastic moment, Mp, and undergo sufficient rotation for a mechanism to form. Reference [7j suggested the following two equations to describe the ultimate capacity of the compression zone in the column web in a symmetrical beam-to-column flange connection

C,,,, = uO.w .(I,” + Sk) (unstilfened) (1)

If a horizontal stiffener is placed opposite the beam compression flange, eqn (1) is then modified to read

C,, = uO.w .(r,” + Sk) + Q.A, (stiffened). (2)

It may be noted that the role of the stiffener, accord- ing to (2), is limited to participation in resisting the beam flange force by bearing. This may be true if the stiffener is not welded to the column web; however, if the stiffener is attached to the web, usually by welding, it also tends to stabilize the web plate against buckling.

Experimental [8] and recent analytical results [9, lo] showed that the prediction of eqn (1) was conservative in all cases and ranged between 50 and 70% of the experimentaily observed ultimate values [8]. The effect of stiffener eccentricity was discussed in [7j and a conclusion was drawn that stiffeners with eccentricity larger than 2 in. with respect to the beam flange may be considered, in practice, as ineffective in inmasing the loadcarrying capacity of the unstiffened column web.

An extensive analytical and experimental research programme was launched at Lehigh University in 1972 on beam-to-column moment connections. The results of this research are available in several publications [l l-l 31. Tests were conducted on both welded and bolted connections and on connections where mixed techniques are used. Because of the large increase in maximum strength of the bolted connections compared to welded connections, it was recommended that column stiffening requirements might require modification. As for the design of the stiffeners, it was suggested that column web compression stiffeners may be designed to carry a beam flange force qua1 to (Mp/db+) minus the force taken by the column web as given in (1). The tension stiffeners should be designed to take the beam flange yield force given by (4r.u.J minus the force taken by the column web (1). This implies that the connection is relying on the strain-hardening of the tension stiffeners to carry the plastic limit load. It was also

recommended that fillet welds may be used in lieu of groove-welds in connecting horizontal stiffeners to column flanges.

The report [14] by Sherboume ef 01. proposed the following equation for detining the ultimate capacity of the stiffened column web opposite the beam compression flange

c ,,,, = a,{~ (r:, + 6k) + 0.5A,}. (3)

The effect of column axial load on the stiffened column web capacity has been reflected through a parameter or,, where

ac- LOO-o.,(5)-0.5(&I. (4)

Hence eqn (3) takes the following form in the presence of axial load, P

Cd, = [a,{~: + 6k) + O.SA,)].a,. (5)

It may be further added that the role of the stiffener is still limited to direct bearing. Its contribution in stabilizing the column web has been discarded.

Another study by Sherboume and Murthy [4] included testing the compression zone in full-scale stiffened and unstiffened symmetrical moment connections using a simulated testing technique origin- ally described in [7j. The finite element analytical solution, which was based on a flow theory, always gave conservative results varying from 6 to 36% of the measured load. For unstiffened column webs, the following limits were suggested

Cm={%} for @isO

1 c_=

CYE

I

(7) 50(/i&) - 2000 b,

for (h&) > 50. J No design recommendations were given for the

stiffened web but the experimental results obtained from testing stiffened specimens were used to verify the analytical solution.

2. PROBLEM STATEMENT

Present design codes [l] for stiffened beam-to- column flange moment connections are based on limited semi-empirical evidence because of the great expense of a totally experimental investigation and the lack of an accepted general theory for plate plastic buckling. An integrated tinite element analytical procedure has been recently developed in [6,9,1 l] for

Strength of welded beam-to-column flange connectlons 751

the evaluation of the elastic-plastic buckling loads and modes in general concentrically stiffened metal plates. The procedure employs a deformation theory by Stowell [ 151, after casting it in an incremental form to delineate the stress distribution, which embraced sequential loading-unloading-reloading situations by conducting a multi-stage analysis [la] and using the modified Newton-Raphson technique [17] for loading and reloading and the initial stress method [18, 191 for unloading. The bifurcation load and mode due to the non-uniqueness of the solution has been predicted by using a scaled inverse iterative version of the power method [20,21].

Three examples are shown to demonstrate the applicability of the procedure to the designated problem of plastic stability of stiffened connections. A parametric study is then illustrated where the emphasis has been directed towards assessing the present specifications, eqn (2), and eqn (5) (141 which reflects the influence of axial load on the column web buckling capacity.

3. MODELLING

Figure 1 shows a free body diagram of the connection zone and the forces acting on it. The column, in the connection zone of a symmetric connection, is generally subject to vertical axial loads (P) plus horizontal forces (C, 7’) from the framed-in beams plus a distributed force (P) due to direct beam shear. The general planar and bending boundary conditions are shown in Fig. 2. The stability of the column web has been studied considering the web as a thin plate of constant thickness (w) reinforced by beam elements representing the siffeners. The column flanges have been regarded as membrane elements of thickness (bf). Fillets may also be approximated as elements of enlarged thickness [lo]. This type of modelling reduces the stability of the column web to a two-dimensional bifurcation problem in the plane of the web. It also reflects the function of the column

P

4. =0 k u=o_I

I=

w=o - w,.- -0

Wtt- -0

w=o - WI1 =0

-C y, v t

*Z,W L / / x4

-1

w=o - w,s- -0

%- -0

Fig. 2. General loading and boundary conditions for the parametric study of the stiffened connection.

flanges as bearing plate diffusing the beam flange forces (C, Z’) into the column web. The flexural stiffness of the column flanges is adequately repre- sented as evidenced by successful comparison with test data [9, 11,221.

Fig. 1. Loads on a symmetrical connection.

152 H. A. EL-GHAZALY and A. N. SHERB~URNE

tack welded to the

Fig. 3. Experimental set-up for the simulated compression zone of a stiffened connection [4].

4. EXPERIMENTAL AND NUMERICAL EVALUATION

The experiments were concerned with testing column sections of finite length placed horizontally and loaded by two equal and opposite compressive vertical forces acting on the column flanges. Figure 3 represents a schematic of the tests. Notice that two steel bars were attached to the outside of the column flanges to simulate the beam flanges through which the compression force is acting. There was no axial load in the column and the column ends were free as shown. Three column sections were tested including (W8 x 17, W14 x 22 and W12 x 14). The bars simu- lating beam flanges were equal in length to column flange width and had a square cross-section equal to the thickness as the column flanges and the combined width of stiffeners and column web is equal to the width of the column flange. Table 1 summarizes the real dimensions of the tested specimens and the stiffeners used as given in the experimental report [4]. In all cases, the stiffeners were fillet welded along

the three sides to the column web and flanges. The specific type and grade of steel used in these experiments was not given in the experimental report; however, the average of the coupon tests given in [4] was used to obtain the yield stress (at,) which was estimated as 49.71 ksi. The Rambergasgood parameters [23] identifying the tangent modulus, E,, in the strain-hardening range, obtained from [4], indicated that E, is equal to 2400 ksi. Figure 4 represents the uniaxial stress-strain relationship for this type of steel used in the analytical solution. Residual stresses were assumed in the column web having the distribution pattern shown in Fig. 5 with (~3, equal to f 13 ksi [24] in the column web and an average value of zero residual stresses in the column flanges.

Figures 6-8 show the experimentally recorded load-deflection behaviour of the three stiffened sections [4] and the prediction of the analytical solution using the finite element method in conjunction with a deformation theory. In all three cases the prediction of the analytical solution lies between 4.0

Table 1. Dimensions of the tested stiffened specimens [4]

Section b; G &

W8x17 5.292 0.308 8.125 WI2 x 14 4.00 0.225 11.958 w14 x 22 5.0586 0.317 13.838

Dimensions shown in inches (1 in. = 25.4 mm).

w - 0.228 0.223 0.257

k k, 2b+w 1,

0.808 0.489 5.292 0.308 0.725 0.549 4.00 0.225 0.942 0.629 5.0586 0.317

Strength of welded beam-to-column flange connections

u (ksi)

753

Fig. 4. Stress-strain curves for the tests in [4] (I ksi = 6.894 MPa).

and 5.6% of the experimentally observed buckling load. Figures 9-l 1 show the gross finite element idealization for the three columns and the analytically predicted buckling mode for each. The figures also illustrate the extent of plasticity within the column web and tinges and along the stiffener at buckling. The buckling modes revealed some important facts about the behaviour and testing techniques of stiffened connections. The buckling modes of sections (W8 x 17) and (WI4 x 22) do not involve rotation of the stiffener but show bending about its strong axis as a result of overall lateral buckling of the assembly. Large relative lateral deflections were noticed at the plate ends where no supports were provided. Therefore, the results of these two experiments do not reflect, but rather underestimate, the buckling capacities of stiffened connections and thus tend to show the bifurcation limit of stiffened plates free along two sides. For these two experiments to give information about the buckling capacity of similar full-scale connections it may be necessary to either increase, significantly, the specimen length to mini- mize the weakening effects of the free ends of the column or support the ends against possible lateral

Fig. 5. Assumed residual stress pattern.

0 IO 20 30 40 50 60

Lohrol deflection (IO? (h.) I

Fig. 6. Load-deflection curve for stiffened specimen

W8 x 17 (1 kip = 4.448 kN, I in. = 25.4 mm).

160.

I40 -

C,(Onolylicol~~114.43 kips

B a 60 - v

60 -

40 -

20 -

I I I I I I I

0 IO 20 30 40 50 60 70 I LoMrol deflection (IO? (in.)

I

Fig. 7. Load-deflection curve for stiRned specimen WI2 x 14 (I kip = 4.448 kN, 1 in. = 25.4 mm).

754 H. A. EL-GHAWLY and A. N. SHERBOIJRNE

0 4 8 12 16 20 24 20

Loterol deflection (lOI (in 1

Fig. 8. Load-deflection curve for stiffened specimen W14 x 22 (1 kip = 4.448 kN, 1 in. = 25.4 mm).

buckling. It should be mentioned that the column length used in these experiments was always twice its depth.

The test on the (Wl2 x 14) section was successful and represents, to a great extent, the buckling capacity of a similar full-size stiffened connection. The buckling mode for this column, shown in Figure 10, indicates twisting of the web-stiffener assembly about the line juncture rather than overall lateral displace- ment as observed in the other two tests. The reason that this type of buckling was observed, in this case, seems to be due to the relatively large aspect ratio of each stiffener (I./b, = 6.1) which caused bifurcation to be governed by stiffener twist rather than overall lateral buckling. It may also be mentioned, in this regard, that eqn (2) underestimates the buckling

capacity by more than 20% of the experimentally observed value, while eqn (3) grossly underestimates the web buckling capacity by more than 33% of the experimental value.

The experimental report [4] did not give details about the buckling mode observed; however, the general characteristics of the experimental load- deflection curves shown in Figures 6-8 suggest similar behaviour for (W8 x 17) and (Wl4 x 22) and different behaviour for (Wl2 x 14). It is interesting to note the corresponding agreement in the buckling mode, predicted analytically, for (W8 x 17) and (wl4 x 22) and the different buckling mode for (W12 x 14). The planar and bending boundary conditions used in the analysis are shown in Fig. 12. Notice that although the problem has double symmetry about the centroidal axes, one-half of the column in the longitudinal direction, rather than a quadrant, was used in the bifurcation analysis to give freedom for the first general possible out-of-plane equilibrium configuration to appear.

5. PARAhiElXIC STUDY

Buckling of a properly proportioned stiffened connection occurs, essentially, as a result of buckling of the web opposite the beam compression flange or buckling by flexure of the stiffener plates. The problem of stiffener buckling, assuming the stiffener to be pinned to the column web, has been covered in more detail elsewhere [6] and design charts are obtained. To be able to use these charts effectively, however, the appropriate load-distribution curve [9] showing the proportion of the beam flange force taken by the stiffener and by the web must be known.

In this parametric study, buckling of the column web is investigated. The following two effects are investigated. (1) Effect of changing stiffener thickness on the buckling capacity of the connection. (2) Effect

@ Plastic zones in column web C and flanges buckling Imminent

- Plostlc ports In the stiffener buckling imminent

1 -m-

Column C.L

Buckling mode along coknnn CL

Fig. 9. Analysis of stiffened specimen WS x 17 (1 kip = 4.448 kN).

Strength of welded urn-to~olumn flange connations

@ Ptostic 2w# in column web and flaqe buckiii imminent

- Pfostk ports In t&e ttlffener buckling Imminent

F e

Bucklkq mode OlOnq column C L

Fig. IO. Analysis of stiffened specimen Wl2 x 14 (I kip =4.448 kN).

of column axial load on the stiffening equipments of a connection. A colon of section (?VlZ x 14) is selected for this parametric study. StZeners are provided opposite the compression and tension beam flanges and are welded afong three sides to the column section.

The material stress-strain model is shown in Fig. 13 for a typical (G40.214tW) steel. The cross- sectional dimensions are obtained from [25]. The general loading and boundary conditions are shown in Fig. 2 where it may be noted that the beam shear is neglected based on the results shown in [9]. Also, no web residual stresses are assumed, because, for such a deep section (WI2 x 14), the residual stress pattern shown in Fig. 5 is uniikely to exist, Moreover, the results obtained in [9] support the insensitivity of the web plastic buckling capacity to the residual stresses in the web. Figure 14 shows

@ Plastic zonee in column web ad flangee buckling lmmlnent

- Ploetic part8 in the stlffenrr buckling imminent

the buckling modes along the column eentre line for various tjt; ratios [(stifIener ~ckne~~~(~Iu~ flange thickness)] at zero column axial load. In Fig. 14 the buckling modes are also plotted when the column is subject to a uniform compressive stress of (O.?& prior to welding the stiffeners and applying the beam loads. The range of t&f selected in this study is 0, 0.25, 0.5 and 1; consequently rJw takes on the values 0, l/3, 2/3 and 4/3 which covey a reasonable range of practical cases.

Figure 16 shows a nondimensional plot of (C&C,) versus (r&) for the column axial load (P) equal to zero and (Q.‘?Py). Inv~ti~ting the curve for (P) equal to zero, a gradual non-linear increase in the buckling capacity of the ~nn~tion is observed at increasing stiffener thickness. In fact, the connection capacity is almost &~&led by adding a stiffener equal to the eolumn tIange thickness. The buckling modes

H. A. EL-GHAZALY and A. N. SHERBOURM:

Y

t

Ic

Fig. 12. Boundary conditions used for the analysis of the tests in [4].

u (ksi)

shown in Fig. 14 reflect the significant difference in the behaviour of an unstiffened and a stiffened connection at buckling. A deep buckle is formed opposite the compression flange in an unstiffened connection while the presence of a stiffener enforces the formation of a nodal line (zero lateral deflection) along the stiffener and the buckling mode has two major buckles on either side of the compression stiffener. Severe twisting occurs to the compression stiffener while relatively insignificant twist occurs to the tension stiffener. Figure 17 shows the stress distribution along the column centre-line in the X and Y directions at incipient buckling when I,= t;. The results conform with engineering intuition, where compression and tension zones form in the X direction

000152 0027 00608 c

Fig. 13. Stress-strain relationship (G40.21-44W) (1 k.s.i. = 6.894 MPa).

Fig. 14. Effet of stitTener size on the buckling cohunn capacity for WI; x 14 at zero column axiai load (1 kip = 4.448 kN).

Strength of welded beam-to-column ffange connections 151

P=O7 Py

!3mB

+t:

=+C

-.-

=Z+T

Fig. 15. Effect of stiffener size on the buckling column capacity for W12 x 14 at P = 0.7Py (1 kip = 4.448 kN).

opposite the beam compression and tension flanges, the formation of a tension zone and compression respectively. No interaction between the two zones zone opposite the beam compression and tension can be-observed [IO] since the beam depth is large flanges, respectively. As may be seen, there is (# = d,). In the Y direction, Poisson’s effects cause relatively low compressive stress in the vicinity of

_+_ Eqn (S), P.O.0 -.- Eqn (51, PxO.7 Py --- Eqn (2) -..- Analyrlc, Ps0.7 Py - Analysis, P=O.O

c~~crltlcol beam flanga farm at P=O.O ard whm t,-0.0

Fig. 16. Effect of stiffener size on column W12 x 14 buckling capacity.

158 H. A. EL-GHAWLY and A. N. SHFXEKIURNE

Distribution of Q, along column CL , buckling lmmment

Dlstributlon of vr along column C L , buckling Imminent

@ Plostlc zones In column web buckling tmminent

I - Plastic zones 8n column

flongcs ond stiffcoers buckling imminent

Fig. 17. Stress distribution for W12 x 14 column at P = 0.0 (1 ksi = 6.894 MPa).

the tension zone which justifies the insignificant distortion of the tension stiffener that was observed in the resulting buckling mode. The yielded zones at buckling show similar distributions opposite the tension and compression beam flanges with insignificant yielding in the column flanges.

If the same column is subject to an axial load of 0.7Py, the resulting buckling modes due to increasing beam forces for the stiffened connection differ significantly from the zero axial load case as may be seen by comparing Figs 14 and 15. When large column axial loads are present, they create a zone of high longitudinal compressive stresses (by) opposite the beam tension flange, Fig. 18, which, in turn, causes significant distortion at buckling in the column tension zone.

From Fig. 16 it may be observed that the presence of high axial load causes a significant drop of 26% in the strength of the unstl@‘ened connection as compared to the same unstiffened connection at zero axial load. The reason is that the slender column web (h/w =I 60.6) is on the verge of buckling locally under the<ffect of axial loads only. When beam forces are applied, a zone of high longitudinal compression (a,) is formed opposite the beam tension flange which causes interaction between the longitudinal buckling mode in the axial direction and the cross-buckling

mode opposite the beam compression flange, resulting in earlier buckling [lo, 111. A similar comparison between the buckling capacity of stiffened connections in the case of t, equal to r; for P = 0.0 and P = 0.7Py shows only about 5% reduction in the buckling capacity. The reason may be attributed to the fact that buckling in the two cases is governed by longitudinal wave formation with less significant interaction between the longitudinal and the cross- buckling modes. Notice from Fig. 18 that a zone of severe plastification exists opposite the beam tension flange because of Von Mises’ interpretation of the stress field in this zone. Notice also that extensive plastification has occurred to the column flange opposite the beam tension flange.

It can be concluded that adding stiffeners opposite the beam flanges will in general increase the connection capacity. In the presence of high column axial loads adding stiffeners may, in fact, increase the connection capacity to reach, almost, the capacity of a similar connection under no axial load. Moreover, in cases of low axial loads there is, apparently, no need to weld the tension stiffener to the column web since very little distortion takes place in this zone, while under high axial loads providing such con- tinuity between the column web and tension stiffener is essential.

Strength of welded beam-to-column flange connections 159

Distribution of u, along column CL, buckling imminent

Distribution of or olong column C L , buckling imminent

@ Plastic zones in column web buckling imminent

- Plastic zones in column f longa and stiffeners buckling imminent

Fig. 18. Stress distribution for WI2 x 14 column at P = 0.7Py (1 ksi = 6.894 MPa).

6. CONCLUSIONS

(1) The role of the stiffener in a symmetrical beam-to-column flange is twofold. It takes part of the beam tlange force by direct bearing and also acts as a lateral support to the column web against buckling. Present design formula, eqn (2), seems to consider only the first function of the stiffener.

(2) It has been demonstrated that the relationship between the stiffener thickness (I,) and column web buckling capacity (C,) is non-linear in the range of light stiffening r, s 0.5 rr The present design expression (2) prescribes a linear relationship for these two variables which may be justified in the case of heavy stiffening (f, > 0.523.

(3) The modification to eqn (2) given in eqn (3) which reduced the effectiveness of the stiffener appears to be inadequate and eqn (2) still reflects the effectiveness of the stiffener more correctly as may be seen from the curves of Fig. 16.

(4) In cases of columns under high axial loads, adding horizontal stiffeners that are welded to column web and flanges is very useful as indicated in Fig. 16. The presence of such stiffeners diminishes the harmful effects of high column axial load. Severe distortion, however, may occur opposite the beam

tension flange because of the severity of conditions and plasticity in this zone.

the stress

(5) The authors believe that eqn (2) conservatively represents the plastic buckling capacity of the column web in the stiffened connection zone with a safety factor anywhere between 1.38 and I. 1. More sections should be tested before a more refined expression is suggested.

Acknowledgement-The authors are grateful to NSERC of Canada for assistance received under grant A 1582 which made possible the work presented in this paper.

REFERENCES

1. CISC, ifundbook of Steel Construcfion, 3rd edn (1980). 2. J. A. Yara, T. V. Galambos and M. K. Ravindra, The

bending resistance of steel beams. J. Sfrucr. Div. AXE, no. ST9 (1978).

3. A. N. Sherboume, D. N. S. Murthy, J. C. Jofriet and P. S. Varma, Rehaviour and design of moment connections. Submitted to CSICC, Toronto (March 1975).

4. A. N. Sherboume and D. N. S. Murthy, Plastic design of beam-column moment connections. Can. Sttuct. Engng Cot&, Montreal (1976).

5. A. N. Sherboume and D. N. S. Marthy. Computer simulation of column webs in structural steel connections. Comput. Sfrucf. 8, 479-490 (1978).

760 H. A. EL-GHAWLY and A. N. S-URNE

6. H. A. El-Ghazily, R. N. Dubey and A. N. Sherboume, IS. Elasto-plastic buckling of stiffener plates in beam-to- column flange connections. Cornput. Struct. l&201-213 16. (1984).

7. J. D. Graham, A. N. Sherboume, R. N. Khabbaz and C. D. Jensen, Welded Interior Beam-to-Column , Connections. AISC. New York (1959). 17.

8. W. F. Chen and I. J. Gppenheim, Web buckling strength of beam-to-column connections. Lehigh 18. University, Fritz Engineering Laboratory, Report no. 333.10 (1970).

9. H. A. El-Ghamly, Elastic-plastic bifurcation analysis of column webs in moment connections of steel structures, 19. Ph.D. thesis, Civil Engineering Department, University of Waterloo (1984).

IO.

11.

12.

13.

14.

H. A. El-Gh&ly and A. N. Sherboume, Plastic behaviour of unstiffened symmetrical beam.to-column flange 20. connections. Can. 1. Ciuil Etnrnn 12. 821-837 (1985). H. A. El-Ghazaly and A. N. Sh&ourne, Deformation 21. theory for elastic-plastic buckling analysis of plates under non-proportional planar loading. Compur. Sfruct. 22, 131-149 (1986). W. F. Chen, J. S. Huang and L. S. Beedit, Recent 22. results on connection research at Lehigh. IABSE ASCII AIA AIP IFHP UIA, Regional Conference on Tall Buildings, Bangkok (1974). 23. J. Par&t and W. F. Chen, Tests of welded steel beam- to-column moment connections. J, Stnrcl. Dia. AXE, 102, No. STI, 189-202 (1976). 24. A. N. Sherboume, G. M. McNeice and S. K. Bose, Analysis and design of column webs in steel beam-to- 25. column connections. Submitted to CISC, Toronto (1970).

E. Z. Stowell, A unified theory of plastic bucklmg of columns and DhtCS. NASA Tech. Note 1556 11948). D. E. Grierson, A. Franchi, 0. Dedonato ‘and’ L. Corradi, Mathematical programming and nonlinear finite element analysis. Comput. Merho& Appl. Mech. Engng 17/18, 497-518 (1979). 0. C. Zienkiewia, The Finite Element Method VI Engineer@ Science. McGraw-Hill, New York (1971). 0. C. Zienkiewin, S. Valliappan and 1. P. King, Elasto-plastic solutions of engineering problems ‘Initial Stress’, finite element approach. fni. 1. Numer. Met&x& Engng 1, 75-100 (1969). G. C. Nayak and 0. C. Zienkiewicz, Elasto-plastic stress analysis. A generalization for various constitutive relations including strain softening. Int. J. Numer. Methodr Engng 5, 113-135 (1972). E. Isaacson and H. B. Keller, Analysir of Numerical Metho&. Wiley, New York (1966). H. El-Ghazaly and A. N. Sherboume, The power method in fhtite element analysis of plastic bifurcation plate problems. Solid Mechanics Division, University of Waterloo, Gntario, Canada, Paper no. 176 (1983). A. N. Shcrbourne, H. A. El-Ghamly and M. Crisinel, Physical simulation of steel beam-to-column connections. To be published. W. Ran&erg-and W. R. Osgood, Description of stress- strain curves bv three oarameters. NASA Tech. Note 902 (1943). - - L. S. Beedle and L. Tall, Basic column strength. J. Srrucr. Diu. ASCE Sfi, NO. ST7, 139-173 (1960). AISC, Manual of Steel Construction, 7th edn (1970).

ultimate strength of stiffened symmetrical welded steel beam-to-column flange connections

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