Design of steel beams with end copes

J. Construct. Steel Research 25 (1993) 3-22

Design of Steel B e a m s with End Copes J. J. Roger Cheng Department of Civil Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G7

ABSTRACT In steel beams with end copes, the strength and torsional stiffness at the coped section are reduced and a high stress concentration in the web at the cope corner is introduced. Therefore, besides yielding, coped beams can fail in three distinct failure modes: lateral-torsional buckling, local web buckling and fatigue crackinO. Based on the behavior and analytical studies of coped beams, design methods are proposed for these three failure modes. Simple interaction equations, which utilize the individual lateral buckling capacities of the coped region and the uncoped length, are developed for design of the lateral buckling of coped beams. Inelastic lateral bucklin 0 caused by the residual stresses and design of short span coped beams are also considered. As for local bucklin 0 strength at the coped region, a plate buckling model is developed for compression flange coped beams, and a lateral buckling model is adopted for double flange coped beams. Yielding at the coped corner caused by the high local stress concentration is considered in the design. For the fatigue strength of coped beams, the actual stress range, which is the nominal stress range multiplied by the stress concentration factor, could be used along with the category C from the existing S-N curves when one is designing coped beams subjected to fatigue loading.

NOTATION b c

Cb

Flange width Cope length Equivalent moment coefficient depending upon loading conditions

3 J. Construct. Steel Research 0143-974X/93/$06-00 © 1993 Elsevier Science Publishers Ltd, England. Printed in Malta

4

Cw d dc E f

A For Fy G ho Ix ly J k L Lb M

Mcope Mcr Minel Mp

Muncop¢ My R Sx t V

W

Y

~x ~m

K

d.J. Roger Cheng

Warping torsional constant Beam depth Cope depth in each flange Modulus of elasticity for steel Adjustment factor for the plate buckling model Adjustment factor for the lateral buckling model Critical local buckling stress Yield strength of steel Elastic shear modulus Depth of a beam at coped region = d - E d c Moment of inertia of a section about the x - x axis Moment of inertia of a section about the y - y axis Torsional constant of a cross-section Plate buckling coefficient Span length Unbraced length Bending moment Elastic lateral-torsional buckling moment of the coped region Critical lateral-torsional buckling moment; critical local web buckling moment Inelastic lateral-torsional buckling moment of I-beams Plastic moment Elastic lateral-torsional buckling moment of the uncoped region First yield moment Cope radius in mm Elastic section modulus about the x - x axis Flange thickness Shear force Web thickness Distance between centroid of reduced section and extreme fiber of flange Defined in eqn (4) Monosymmetric parameter of lateral-torsional buckling of a tee section Ratio of the smaller moment to the larger moment at the opposite ends of the unbraced length, positive for double curvature, negative for single curvature Poisson's ratio for steel=0.3

Design of steel beams with end copes

5

INTRODUCTION In steel construction beam flanges are frequently coped to provide enough clearance for the supports when the framing beams are at the same elevation either at the top or the bottom flange of the main girders, as shown in Fig. 1. The copes can be at the top, the bottom, or both flanges in combination with the various types of simple connections. Simply supported end conditions are normally assumed in the design of coped beam connections. It is customary in design to check the bending and shear strength (including tearing-out strength) at the reduced section of coped beams. However, beams with copes have a reduced bending and torsional stiffness, so that buckling, both local and lateral, can also be affected. It is also found that, by removing the flanges of the beams, a high stress concentration in the web at the cope corner is introduced because of the geometric discontinuities. High stress concentration could cause localized yield at the coped corner that might cause the beam to fail in inelastic local buckling. Furthermore, this high stress combined with the flame-cut procedure, usually used to make the cope, also can lead to fatigue cracking for coped beams under cyclic loadings. In recent years, a great deal of progress has been made in the design of steel beams with copes. Much of the work is based on the research by the author and other co-authors. 1-6 The purpose of this paper is to present a state-of-the-art summary of the behavior and design of coped steel beams. The discussion will be centered on three distinct failure modes, lateraltorsional buckling, local web buckling, and fatigue cracking, normally encountered in coped beams. These failure modes are shown photographically in Figs 2-4. Because of these three distinct problems, the behavior and design procedure are presented separately for each phenomenon. The connection strength of the coped beams, such as shear strength and t

I

!

I

Bolted Clip A n g l e

S i n g l e Web Plate

Fig. I. Types of coped beam connection.

o j c~r c~

~q

q~

Design of steel beams with end copes

7

Fig. 4. Typical fatigue cracking.

tearing-out strength, which have been well documented in other literature, 7-9 will not be covered here.

LATERAL-TORSIONAL BUCKLING General The basic theoretical formulas for lateral-torsional buckling of steel Ibeams, upon which current design standards are based, assume that the flanges at the ends of the beam are restrained against lateral movement. However, for a laterally unsupported beam with end copes, the lateral end restraint would be reduced because movement at the end of the flange is not positively prevented. The lateral buckling strength of coped beams could be significantly less than that predicted by the current design formulas, especially when the copes are long or the beam span length is short. It is also known that the residual stresses due to hot-rolling or welding processes would reduce the lateral buckling strength of beams by partial yielding. Thus, it is necessay to include this effect in the design of coped beams.

8

J.d. Roger Chen#

Background of proposeddesign model The design model proposed herein for the lateral buckling of coped beams is based on theoretical parametric studies and observation and confirmation from experimental programs. Although the experimental investigation is mainly on one flange (compression flange) coped beams, a theoretical study has been carried out to extend the design model to double flange coped beams. The model recognizes that the lateral buckling strength of a coped beam is governed by the buckling strength of both the uncoped length and the coped region. This can be easily understood from the normalized buckled shapes obtained from the experiment as shown in Fig. 5.6 In the figure, normalized curves of lateral deflection of the compression flange versus span length of the beam are plotted for three different beam span lengths and only the controlled half span is plotted. The beams were loaded and braced at the midspan. A W250 x 33 section (a section 250 mm deep and a mass of 33 kg/m) with Grade 300W steel (specified minimum yield strength 300 MPa) is used for the beams, all three beams have a 250 mm cope length and 25 mm cope depth in both ends. Except for the 5580mm long beam, the maximum deflection was observed at the coped region. The uncoped region of the 1930 mm beam has little curvature at the failure load, which implies very little contribution by the uncoped region. In other words, the coped region, which comprises 13% of the unbraced length of the 1930mm beam, governs the buckling strength of this beam. Very severe crosssectional distortion was observed at the coped region for the 1930 mm and 3130 mm long beams. Based on these observations and further study, an interaction model that considers the buckling strengths of both the coped and uncoped region has been developed for the lateral buckling strength of coped beams, t'3.a'6

°i: °°.o

oh

o'.z

o:s

0:4

o.s

Normalized Span Length Fig. 5. Normalized lateral deflection curves for coped beams. 6

Desion of steel beams with end copes

9

Two interaction formulas are developed, one for beams with copes at both ends of the unbraced length and one for beams with copes at only one end of the unbraced length. It is found that there is little further reduction in the capacity of coped beams due to the partial yielding in the section. Therefore, the lateral buckling strength of coped beams can be determined by the proposed interaction formulas to consider coping effects alone. However, this capacity should not be greater than the inelastic lateral buckling strength of the beams calculated in accordance with the usual design provisions in which coping effects are not considered} ° Design recommendations The following interaction equations are applicable to both compression flange coped beams and double flange coped beams. No resistance factor is included in the formulas. The Mcow in the equations refers to the buckling strength of the coped section (tee or rectangular section, as appropriate). The equations are limited to the case where the cope length is not greater than twice the beam depth and the cope depth is not greater than one-fifth the beam depth. For coping details outside these limitations, modifications to the design equations are necessary to include the tipping effects; these can be found in Ref. 3. Two design formulas applied to the different coping details in the beams are: Copes at both ends of the unbraced len#th

,,, Copes at only one end of the unbraced length

[,

1 12 fu:cope'J-L(LtffC[McopeJ

2

(2)

where Me, is the critical moment of the coped beams. Of course, Me, is to be taken not greater than the inelastic critical moment, Mi,cm, which is determined according to the design of steel beams assuming no copes (e.g. CAN/CSA-S16.1-M891°). Lb is the unbraced length and c is the cope length. Mu,,op, is taken as the elastic critical lateral buckling moment for a doubly symmetrical 1-beam and is given by

, . . . . . c, (:,)

~E 2

(3)

10

J. d. Rooer Cheno

where Cb= 1"75+ 1"05 X+0"3 ~¢2~ 2"5 r = r a t i o of the smaller moment to the larger moment at the opposite ends of the unbraced length, positive for double curvature, negative for single curvature. For loading cases other than linear moment diagrams, Cb can be found in the SSRC Guide. 11 Moor, is the elastic critical lateral buckling moment of the coped region. For compression flange coped beams, the lateral buckling formulas for tee sections with an unbraced length equal to 2c is used:

(4)

where ~,m= ~ 1

X/GJ _

t

b2

For double flange coped beams, the formula for rectangular sections is used:

(5) All the symbols used in these equations are defined in the Notation. Commentary

Squared terms used in eqn (2) are to reduce the coping effects since copes at only one end of the unbraced length was found to have less effect than the copes at both ends. Equations (1) and (2) imply that the Mcooc will govern the design when the unbraced length is short or the cope length is long. Figure 6 illustrates the design method and the effect of the unbraced length. It is plotted on the basis of the procedures outlined above. The section used in the calculation is a W250 x 33 section with Grade 300W steel. The beams have a 100 mm cope length and 25 mm cope depth in the compression flange at both ends. A concentrated load is applied at the mid-span and no bracing is provided except at the supports. Ml,,~ in Fig. 6 is obtained based on the Canadian Standard (CAN/CSA-S16.1-M891°). It

Design of steel beams with end copes

11

2.0

•~

Mp

|.0

................................................................

0.5 DestS~n Envelopt;~~" xxx\\\\\~ 0.0

•

,

2000

•

,

,

,

4000

6000

8000

10000

Unbraced Length. m m Fig. 6. Effect of span length on buckling strength for coped beams,

can be seen that the buckling strength of the coped beam decreases dramatically for span lengths less than about 1200 mm. As illustrated in Fig. 6, there are some instances (such as short beams or long copes) for which the reduction of the lateral buckling capacity can be very significant. Instead of increasing the beam size, it may be practical to reinforce the coped area with stiffeners in order to increase the capacity. The reinforcing details shown in Figs 7(a) and 7(b) can be used for top flange and double flange coped beams, respectively. The size of the stiffeners should be at least the same as those of the flanges. If the cope depth, de, is less than 0.2d, no reduction of bending capacity is required for the reinforced beams. Otherwise, the tipping effect at the coped corner should be considered.

(a)

(b)

Fig. 7. Web reinforcements for lateral buckling of coped beams.

L O C A L WEB B U C K L I N G General When the compression (coped) flange is laterally supported, the capacity of the beam can still be affected by the copes in a local fashion. In practice,

12

J. J.

Roger Cheng

such beams are designed for both shear and moment at the reduced section against material yielding. But, due to the discontinuity at the coped corner, high stress concentrations can occur at this location; their magnitude depends on the coping profile. (Stress concentration will be discussed further in fatigue design.) This high stress concentration can cause localized yield at the region, and conventional shear and bending stress calculations will not give the actual stresses at the coped region. As yielding spreads, failure due to inelastic web buckling occurs since the yielding will reduce the stiffness of the web in the coped region. In the cases of long copes or thin web plate girders, a beam could fail by elastic local web buckling at the coped portion.

Background of proposed design models The web buckling problem is very complex and the number of variables that can affect the buckling strength is quite large. In order to present the research results in a form useful for design, it was decided to start with classical buckling solutions, which would then be altered by factors representing the principal variables such as cope length and cope depth. Because the location of the neutral axis at the coped region is different for compression flange and double flange coped beams, two different buckling models are adopted. The basic plate buckling model, as shown in Fig. 8, is used for the compression flange coped beams, and the lateral-torsional buckling model (Fig. 9) is adopted for double flange coped beams. The plate buckling coefficient k shown in Fig. 8 was obtained from Japanese CRC Handbook t2 and further expanded and checked by numerical analysis. The solutions from these two models are then adjusted by a

40 ¸

PLATE

BUCKLING

COEFFICIENT

F

Free

F

20 F = k'nZE cr iZ(l_.u 2) (~____.)z

1.0

20 c/h o

Fig. 8. Plate buckling model.

5.0

Desi#n of steel beams with end copes

13

A ~

W

ho

M

~

M- Diagram

Fig. 9. Lateral-torsional buckling model.

factor considering the effects of stress concentration, shear stress, cope length, cope depth, and stress variation from the beam end to the end of the cope. It is found that first yielding at the cope corner is generally localized (due to stress concentration) so that deformation is not large enough to curtail the usefulness of the beam. As yielding spreads, however, failure occurs due to inelastic local web buckling. This buckling strength could be conservatively predicted by the conventional bending and shear stress calculations at the reduced section against material yielding.

Design recommendations The recommendations for local web buckling of coped beams described herein are applicable to copes with a length less than twice the beam depth and with a depth less than one half the beam depth (one-fifth in each flange for double copes). An appropriate resistance factor must be applied to these formulas when used in design.

Compressionflange coped beams The bending stress at the reduced coped section must not exceed the critical local buckling stress, F t . given by

Fc,=k 12(1--v 2) ho f

(6)

14

J. J. Roger Cheng

but not more than Fy, where k = the plate buckling coefficient (Fig. 8), and can be given by k= 2.2(ho/c) x~s for c/ho < l.O k= 2.2(hdc) for c/ho>~1.0 f = an adjustment factor, defined as f = 2c/d for c/d < 1.0 f=l+c/d for c/d>11.O In addition to the design buckling equation given by eqn (6), the maximum shear stress at the coped section based on the conventional shear stress calculation (V/how) must not exceed the yield shear strength of the material (1)577Fy).

Double flange coped beams The maximum moment at the reduced coped-region, as shown in Fig. 9, must not be greater than the critical buckling moment at the coped section. The critical moment is determined by the lateral torsional buckling model with an unbraced length equal to c and is given by

(7) but not more than the first yield moment of the coped section, My where fd = an adjustment factor in which moment gradient is included with other factors, taken as

My=FyS, Equation (7) can be rewritten in terms of critical bending stress by substituting Sx = 21,,/ho, Ix = h3ow/12, Iy = how3~12, G = E/2.6, J = how3~3 to yield an equation which is similar in format to eqn (6), the plate buckling model,

F . = 1-95E \ c ]\ho) but not more than Fy.

(8)

Des.ionof steel beams with end copes

15

As for the case of compression flange coped beams, the maximum shear stress (V/how) at the coped section must not exceed the shear yield of the material (0.577Fr).

Commentary Although the two design models used for the local buckling strength of coped beams appear to have very different foundations, the principal variables in the design equations are the same in both cases. The ratio of these two design equations (eqn (6) and eqn (8)) shows the differences between these two models. This ratio is Eqn (6)/eqn (8).~f/fd

for

c/ho >11"0

Eqn (6)/eqn (8),~(ho/c)°65(f/fd)

for

c/ho < 1.0

This indicates that, when c/ho>~1-0, the only difference between the two design models is the adjustment factors, f and fd. The adjustment factor f is a function of cope length, but fd is not much affected by cope length, mainly because of the unsymmetrical section and the higher stress concentration effect at the coped region for the compression flange coped beams. When c/ho < 1-0, the differences between the two models include not only the adjustment factor but also the ratio ho/c. The latter factor reflects the fact that when the cope length decreases, the plate buckling coefficient k (or plate buckling strength), which is a function of ho/c, increases very rapidly (Fig. 8). To give some indication when yielding (or inelastic local buckling) controls the design, the results from eqns (6) and (8) are plotted in Figs 10 and I I, respectively, for different d/w ratios, various c/d ratios up to 2.0, and a dc=0.1d. The two figures show a lot of similarity. For all of the available hot-rolled W-sections in North America (d/w<.60) with yield strength Fy = 250 MPa, the cope length must be greater than the depth of the section for the buckling criterion to govern; otherwise yielding controis. This rule of thumb can be extended to steel strengths up to 350 MPa when the beams are double flange coped. For thin web members, however, web buckling is more likely to control for common coping details. For d/w = 150 and a cope length equal to 0-2d, the buckling stress will be as low as 100MPa for compression flange coped beams. It is interesting to see that, for thin wall members, the critical buckling stresses for double flange coped beams with c/d <<.1.0 are higher than that for compression flange coped beams. This is due to the differences in sectional properties of the

J. J. Roger Cheng

16

::t

¢h

i.

300

~L 0

'

200,

10o ,

-

-

1

~

1

5

0

1

~

Q.J I I , •

0

I I

1 I

' t

0.2

2

I

c/d Fig. 10. Local buckling strength of compression flange coped beams. 5oo

3~"

-2 °

\

\loo

I

b o

i

! !

1 c/d

Fig. 11. Local buckling strength of double coped beams.

coped section and the different adjustment factors for the two types of coped beams. To improve the local behavior of coped beams, the three reinforcing details as shown in Fig. 12 can be used. Type (a) and (b) reinforcing details are recommended for all rolled sections or sections with d/w<.60. The thickness of the doubler plate in Type (a) should be large enough to guard against buckling. The Type (c) detail is recommended for thin web members (d/w> 60); it includes an additional vertical stiffener to prevent out-of-plane buckling of the horizontal stiffeners. No reduction in local web buckling strength is required when such stiffening details are used. However, it is necessary to check the yielding of the reinforced sections.

Design of steel beams with end copes - D o u b l e r Plate

~-- Stiffener

17 ~-

t

Stiffeners

..... ........

Fig. 12. Web reinforcements for local web buckling of coped beams.

FATIGUE STRENGTH

General Discontinuity at the coped region introduces high stress concentrations in the web at the cope corner. In addition, flame cutting is usually employed in fabricating coped beams. This fabrication procedure produces high tensile residual stresses at the cut edge because of the differential temperature during the cooling process. The magnitude of these residual stresses can be as high as the yield strength of the material. Therefore, fatigue cracking can be developed under cyclic loading conditions even when the copes are located in compression zone. Flame cutting also causes metallurgical changes such as the formation of a martensite layer at the cut edge. Hence, initial micro-flaws are believed to exist in the martensite layer which consequently reduce the fatigue life of the coped beams.

Background of proposed design method The design strength of coped beams proposed herein is mainly based on an experimental program in which cope radius and stress range were the two principal test parameters, s All specimens were tested as simply supported beams with one point loading at midspan. Only tension flange copes were considered in the program. Tensile residual stresses due to the flame cutting were eliminated from the specimens by pre-loading the specimens to the maximum load level used during the fatigue test and then unloading. In all specimens cracks were initiated within the cope radius region and propagated at an angle through the web until the test beam reached net section failure (Fig. 4). The degree of stress concentration at

J. J. Roger Cheng

18

the coped corner increased with decreasing cope radius. As expected, the fatigue life of the test specimens decreased with either increasing nominal stress range or decreasing cope radius. It was also found that the tensile residual stresses produced by the flame cutting could be as high as 300 MPa. To consider the high stress gradient and severe local stress concentration produced by cope geometry, the finite element method (FEM) was employed to evaluate this distribution. The maximum stress at the coped region can be expressed as a stress concentration factor (SCF) multiplied by the nominal stress. The SCF is defined as the ratio of the maximum longitudinal stress obtained from the FEM to the nominal stress calculated by the simple bending theory at the cope end. The SCF values obtained from the analysis for single flange coped beams can be plotted against cope radius as shown in Fig. 13. A logarithmic fit to the solution gives the equation Iog(SCF) = 0-937 - 0"285 log(R)

(9)

where R = cope radius in millimeters. The test specimens were then analyzed by linear elastic fracture mechanics (LEFM) theory to predict the fatigue life of the specimens. The predicted fatigue life using LEFM and the tested fatigue life of the test specimens at a final crack size of 30 mm are shown in Fig. 14, using the conventional S-N curve approach. The stress range in the figure is the nominal stress range multiplied by the SCF from eqn (9). The 95%

• u.

oU'J

FEM Results ]

5

3

0

10

20

30

Cope P.~llu. (n.n)

Fig. 13. Stress concentration factors for single flange coped beams.

4

•~ l l u o o

~ailyu

~m r'ul

Q

o o

m.

.

o

c ~r

o_. o__ e~ co

F_ Z

OQ t

Oi

~uuJ

Vu~ , f j ! m ~ u ~ u ~ i~,~j~ j u

uu!~, U

20

J. J. Rooer Cheno

confidence limit regression lines for both experimental results and predicted solutions are plotted along with S - N curves for Categories a, b and c from the CAN/CSA-S16.1-M89 Standard. t° (Categories a, b and c in North American practice correspond approximately to Detail Categories 160, 125 and 90, respectively, in European practice, t3) It can be seen from this figure that all the data points lie on or above the Category b line and that all points lie above both the Category c and the 95% confidence limit for the analytical solution. To investigate the effects of roughness of the burned edges and the martensite layer produced by flame cutting on the fatigue strength of coped beams, three cope fabrication procedures were used in the test program. These were flame cutting, flame cutting followed by grinding the edges smooth, and drilling a hole followed by flame cutting. No improvement was found in the fatigue life for the specimen produced by the hole drilling method. Some improvement was found by smooth grinding the cut edges, but the effect of the grinding procedure cannot be conclusively shown because of the limited test data. Nevertheless, the results showed the detrimental effect of deep cut or notch in the cope edges and such defects should be removed in practice.

Design recommendation Based on the analytical study and the limited number of experimental data, it is recommended that the actual stress range, which is the nominal stress range multiplied by the appropriate SCF given by eqn (9), be used along with Category c from the S - N curves in CAN/CSA-S16.1-M89 t° when designing coped beams subjected to fatigue loading. The flame cut edge should always be ground smooth to avoid local discontinuities. Special attention should be given to avoid deep cuts or notches in the cope edge. The same design rules for the fatigue life as recommended above should be applied for copes fabricated either by the flame cutting method or the hole drilling method.

Commentary The available test data indicate that fatigue Category b provides a reasonable estimate of the fatigue life, as seen in Fig. 14. However, use of Category c might be a better choice because the number of test data is so limited. If a more accurate (and less conservative) estimate of the fatigue life is desired, the 95% confidence limit based on the linear elastic fracture mechanics analysis, as shown in Fig. 14, can be used. The test to predicted (from LEFM) ratios range from 0.37 to 0.75, with a mean value of 0.52.

Design of steel beams with end copes

21

Although test results did not conclusively show the effect of the cope fabrication process, it is advised that flame-cut edges should always be ground smooth in order to avoid local roughness and to remove microflaws existing in the martensite layer. The cope radius should always be made as large as possible in order to minimize the effect of stress concentration. By increasing the cope radius from 12.5 mm to 25 mm, the fatigue life of a coped beam with a nominal stress range of 50 MPa can be increased more than 50%. The SCF developed here (eqn (9)) is based on single flange coped beams with simply supported ends. It is applicable only to tension flange coped beams. For other copes and support conditions, an appropriate analytical method, such as the finite element method, should be used to determine the actual stress range at the cope region. Coped beams subjected to a compressive stress range are also susceptible to fatigue damage because of the high tensile residual stresses produced by flame cutting. Before more research is done on this problem, their fatigue life can be designed according to Category c assuming the tensile residual stresses at the coped region equal to the yield strength of the steel. The recommendations proposed herein are not intended to cover the coped beams subjected to secondary out-of-plane bending stress range. SUMMARY AND C O N C L U S I O N S The design of steel beams with end copes can be subdivided into lateral-torsional buckling, local web buckling, and fatigue strength based on their three distinct failure modes. The lateral-torsional buckling strength of a coped beam can be described by simple interaction equations, which utilize the buckling strengths of the coped region and the uncoped length. However, the buckling strength so calculated should not be greater than the inelastic lateral buckling strength calculated from current design provisions, in which coping effects are not considered. A significant reduction in lateral buckling strength is found in short-span coped beams. A plate buckling model has been used to predict the local web buckling strength for compression flange coped beams and a lateral buckling model is adopted for double flange coped beams. Inelastic local web buckling caused by the yielding at the coped corner, the result of the high stress concentration at this location, can be determined by conventional bending and shear stress calculations against material yielding, using the reduced section.

22

J. J. Roger Cheng

Regarding the fatigue strength of coped beams, it is recommended that the actual stress range, which is the nominal stress range multiplied by the appropriate stress concentration factor, can be used along with Category c from the existing S - N curves in CAN/CSA-S16.1-M89,1° or with Fatigue Detail 90 according to the ECCS rulesJ 3 T h e cope radius should always be made as large as possible to minimize the effect of stress concentration. The flame-cut edges should always be g r o u n d s m o o t h to avoid local discontinuity.

REFERENCES 1. Cheng, J. J. R., Yura, J. A. & Johnson, C. P., Design and behavior of coped beams. Department of Civil Engineering, The University of Texas at Austin, PMFSEL Report No. 84-1, July 1984, 276P. 2. Cheng, J. J. R. & Yura, J. A., Local web buckling of coped beams. Journal of Structural Engineering, ASCE, 112(10) (October 1986) 2314-31. 3. Cheng, J. J. R., Yura, J. A. & Johnson, C. P., Lateral buckling of coped steel beams. Journal of Structural Engineering, ASCE, 114(1)(January 1988) 1-15. 4. Cheng, J. J. R. & Yura, J. A., Lateral buckling tests of coped steel beams. Journal of Structural Engineering, ASCE, i 14(1) (January 1988) 16-30. 5. Yam, M. C. H. & Cheng~ J. J. R., Fatigue strength of coped steel beams. Journal of Structural Engineering, ASCE, i16(9) (September 1990) 2447-63. 6. Cheng, J. J. R. & Snell, W. J., Experimental study of lateral buckling of coped beams. Proceedings, Structural Stability Research Council, Annual Technical Session, Chicago, IL, April, 1991, pp. 181-92. 7. Birkemoe, P. E. & Gilmor, M. I., Behaviour of bearing critical double angle beam connections. Engineering Journal, AICS, 15(4) (1978) 109-15. 8. Ricles, J. M. & Yura, J. A., The behaviour and analysis of double-row bolted shear web connections. Department of Civil Engineering, The University of Texas at Austin, PMFSEL Report No. 80-1, September, 1980. 9. Shelton, B. G. & Yura, J. A., Tests of bolted shear web connections. Department of Civil Engineering, The University of Texas at Austin, PMFSEL, Report No. 81-1, February, 1981. 10. Canadian Standard Association, CAN/CSA S16.1-M89: Limit States Design of Steel Structures. Canadian Standards Association, Rexdale, ON, 1989. 11. Galambos, T. V., Guide to Stability Design Criteria for Metal Structures, 4th edn. Structural Stability Research Council, John Wiley, New York, 1988. 12. Handbook of Structural Stability, Column Research Committee of Japan, Tokyo, 1971." 13. European Convention for Constructional Steelwork, Recommendations for the Fatigue Design of Steel Structures. European Convention for Constructional Steelwork--Technical Committee 6--Fatigue, 1985.