Castellated Beams

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Proc. Instn. Ciu. Engrs, Part 1,1991, W, June, 521-536

PAPER 9513 STRUCTURAL AND BUILDING BOARD

Castellated beams

P. R. KNOWLES, MA, MPhil, MICE, FIHT*

The process of castellation was patented in 1939. Applied mainly to rolled sections for use as beams, it has been for many years a significant feature in steel construction. The Paper describes the steps leading up to the invention and the early attempts to devise methods of calculating the load carrying capacity and deflexion of castellated beams. Both elastic and plastic methods of analysis are examined, the basis and use of interactive design charts is explained, and the requirements of BS 5950 are outlined.

Notation notional web area for shear deflexion calculation

beam flange width area of tee.

depth of castellated section serial depth of original section shear modulus notional second moment of area for deflexion calculation second moment of area through an opening constant related to tee constant related to tee bending moment on tee maximum permitted bending moment plastic moment of tee positive bending moment on tee negative bending moment on tee axial force on tee squash load of tee shear force on tee maximum permitted shear shear failure load of tee shear force on upper tee shear force on lower tee elastic section modulus of beam at opening length of horizontal cut forming castellation castellation dimension castellation dimension depth of web between fillets sum of stressesf, andf, stress in tee from bending moment on beam

Written discussion closes 15 August 1991 ;for further details see p. ii. *Principal, Peter Knowles and Associates

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f, stress in tee from shear force on beam h distance between centroids of tees p spacing of openings p y material design strength t web thickness 9 angle of sloping side of castellation to horizontal

Castellated beams Innovations in civil and structural engineering are not common; the invention of the castellated beam was the result of a rare flash of inspiration which occurred to a designer faced with an apparently insoluble problem.

2. Fifty years ago, on 4 January 1939, British Patent number 498281 was granted to Geoffrey Murray Boyd, at that time living at 11 Burwood Avenue, Hayes, Kent, for a specification related to improvements in built-up structural members ‘of the kind comprising two parts with pairs of projections extending towards one another and welded along a line of sinuous or toothed nature’. The rather involved phraseology of the specification refers, in fact, to what is now known as a castellated steel section, although at the time of the patent application it was called the Boyd beam.’

3. The basis of the beam’s method of construction, described by a writer in The Shipbuilder’ as ‘both simple and ingenious’, had occurred to Boyd in 1935 when he was working in Buenos Aires, Argentina, as a structural engineer for the British Structural Steel Company, the South American subsidiary of Dorman Long. He was faced with the problem of designing a beam for a monorail hoist. The maximum beam flange width was, of necessity, restricted by the width of the hoist opening, but the choice of rolled beams in stock was such that the only ones available within the restricted flange width were insuficiently stiff for the required span. Boyd was musing on the possibility of strengthening a beam by welding another below it-a rather crude solution-when he thought of cutting and welding the beam web in such a way as to increase its depth and, consequently, its stiffness. An experiment with a cardboard model quickly showed the feasibility of the idea, and thus the Boyd beam was invented. Much development work was, understandably, still required.

4. The patent specification is concerned not only with a simple castellation of rolled beams but also with techniques of tapering, of forming Z-shaped sections from channels and of forming cruciform and star-shaped sections for use as columns. Applications are claimed for ships, aircraft and vehicles as well as for buildings. In the specification, the use of flame cutting and welding is stressed. Various geometries of castellation opening are discussed, based on the angle of the sloping sides and the length of the horizontal portion. It was, however, the Boyd beam type C which went into general use and whose geometry has become the standard for castellated beams in the United Kingdom (Fig. 1). The depth of cut c is half the serial depth D , of the original section. Therefore, the depth of the castellated beam D , = D + O.5Ds z 1.5DS, where D is the actual depth of the original section.

5. For various reasons, one of which was the 1939-1945 World War and another Boyd’s position as a Civil Servant, he was unable to exploit his invention commercially. Boyd therefore assigned his patent rights to the Appleby Froding- ham branch of the United Steel Companies Limited, which marketed steelwork 522


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CASTELLATED

1 B E A M S

t P , b L a , b L a 1 1 1 l t /Web

D [ [ : C @

+

(b)

Fig. 1 . (a ) Universal beam cut along web; (b) two halves welded together to form castellated beam

fabricated in accordance with the Boyd principles as Appleby Frodingham Castel- lated Construction. Regrettably, the inventor’s name was superseded by a more cumbersome but nevertheless descriptive word meaning ‘castle-like or battle- mented’, a clear reference to the toothed appearance of the flame-cut section before welding. The extended patent expired many years ago, allowing castellated sections to be produced freely by any steelwork fabricator. The fact remains, however, that, with the exception of the Lirska beam in which the depth of the original section is further increased by a plate welded between the castellated teeth (Fig. 2), no significant improvement has been made to Boyd’s original concept.

L l Fig. 2. Extended (Litska) castellated beam

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Design considerations 6. Designers have long laboured under the difficulty of not having a generally

accepted design method for castellated beams. The inventor did produce safe load charts (Fig. 3), which were based on what was then considered the prudent

BOYD BEAMS Shear: Max. reaction = 1/1.52 of max. safe reaction with uniformly distributed loads

Bending: Max. extreme fibre stress 8 t /In2 Deflexion: Not exceeding L1325. € = 12000 t /m2

on original beam at 5 t /in2 on web

Lateral support assumed adequate Type 'c' openings

Span: ft 10 X 4% BB 15 X 4% (6 25 Ibs Means. Boyd Beam 15 X 4% 25 Ibs

made from BSB 10 X 4'12 (ir 25 Ibs

Copyright by G. M. Boyd.

Fig. 3. Safe load chart for Boyd beam

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CASTELLATED B E A M S

Fig. 4. Panel deformation in shear

assumption that the beam properties to be used in calculation should be those minimum values obtained for the portions above and below the opening in the web; in effect, two tee sections. Subsequent testing has shown that this apparently safe procedure is not correct; stresses and, in particular, deflexions can be seriously underestimated. The reason lies in the flexibility of the region at each opening, which, deforming in shear (Fig. 4), produces stresses and deflexions which augment those produced by simple beam action.

Elastic stress distribution 7. Elastic analysis of a beam with openings in the web has been carried out by

a number of methods which include finite difference3 and finite element4 techniques. However, a simple analogy in which the castellated beam is considered to be a Vierendeel girder with points of contraflexure at the mid-line of the openings and at mid-height of the web posts (Fig. 5), and with the shear force at the centre-line of the opening divided equally between top and bottom tees, is an attractive substitute structure which is statically determinate. The stress at the tee-to-post junction can then be calculated.

8. Experimental verification of the validity of the Vierendeel analysis has been reported from many sources. A detailed summary is given in Kerdal and Nether- cot,’ but it must be pointed out that many of these tests have the weakness that they have been carried out on sections whose castellation profiles are significantly different from those of current UK standard type. Early tests in the UK were on British Standard beams, which became obsolete many years ago, and most of the tests were on beams of small size. It may be concluded that further tests on castellated universal beams of intermediate or larger size could usefully be carried out.

9. At each hole, bending and shear are transmitted by the top and bottom tees. On the assumptions of Vierendeel action, the longitudinal stress at the junction of tee and post (A-A in Fig. 6 ) consists of two components: the component due to bending f, ; the component due to shear f,. The total longitudinal stress f =f, +f,. If the tee properties are area A , and elastic section modulus Z,, then for

applied bending moment M and shear force V

f, = M / A , h = K , M

f v = %,/2Z, = K, V 525


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KNOWLES

where

K , = 1/A,h

K , = a /2Z ,

In calculations, the smaller of the two values of Z , is used. The section constants K , and K , have been tabulated.'j

10. On examination, K , will be seen to be approximately equal to the recipro- cal of the castellated beam elastic modulus Z at an opening

I = 2[A, x(h/2)'] = A , h2/2

(neglecting the small value of I for each tee about its own centroidal axis).

I (C)

Fig. 5 . Vierendeel analogy: (a) castellated beam; (b) equivalent Vierendeel girder; (c) location of points of contraflexure

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CASTELLATED BEAMS

( 4 (b) (C)

Fig. 6. Stresses at junction of tee and post (A-A): ( a ) direct stress due to bending f, ; (b) bending stress due to shear fv (c) combined stressf

Therefore, the Vierendeel analysis gives a flange stress larger than that found by using the minimum elastic section modulus.

11. If maximum elastic stress be the design criterion then it will be necessary, in design, to check the sum of stresses at each cross-section A-A (Fig. 6) along the length of the castellated beam, unless the point of maximum stress is immediately apparent. Properties K , and K , are not known until a beam has been selected; direct design is not possible, and a trial and error process is needed. Some assist- ance can be gained from published material: load tables have been calculated for uniformly distributed and central point loads on a range of spans for all the universal beams.6 For certain load cases, use can be made of the results in figure 6 of reference 6.

12. For other load cases, interaction curves are available which are based on the linear interaction between shear force and bending. In the absence of bending moment, the shear force which can be carried by the beam, in terms of the bending stress generated by the shear force, is limited by the material strength py. The maximum shear force V is then given by

v0 = P&

MO = P y K l

Similarly, in the absence of shear force, the maximum bending moment

13. For any general combination of bending moment M and shear force V, the resulting sum of bending stresses must not exceed the material design strength py

Interaction curves (Fig. 7) have been plotted for all castellated universal beams, with X and y axes (l /K2) and (l /K,) respectively. A cornbination(M/p,) and (VIP,) which lies on, or nearer to the origin than, the interaction line for a particular beam is a combination which produces a maximum stress not greater than the allowable stress for that beam material. It will, of course, be necessary to examine combinations of bending moment and shear to determine the worst case. The interaction curves can also be used to check the stress in an existing beam.

14. The shear stress caused by the shear force in the tee webs will also interact with the stress in combined bending, and therefore a check should be made of the

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Section

(458 X 127 X 48) (ex 305 X 127 X 48 UB)

(381 X 146 X 43) (ex 254 X 146 X 43 UB)

(458 X 102 X 33) (ex 305 X 102 X 33 UB)

(381 X 102 X 28) (ex 254 X 102 X 28 UB)

0 2 4 6 8 l iK, Viallowable stress mm2 X 10'

Fig. 7. Interaction chart for elastic design

resulting equivalent stress if design is on an elastic basis. It will generally be found, however, that this combination is not critical.

Deflexion 15. The calculation of deflexion was, as has already been explained, based

initially on considering a castellated beam as a conventional beam having an unperforated web with a second moment of area equal to the minimum value for the castellated beam. This simplification, however, leads to an underestimate of the real deflexion by a significant amount. An early improvement, which resulted from a series of tests carried out by United Steel,7 was to apply a correction factor to the deflexion calculated using the minimum second moment of area, based on tests on small beams loaded at the quarter points. The calculated elastic deflexion was multiplied by a factor

c = 1 + (2OImi,/L2)

where Imin is the second moment of area through an opening (in4) and L is the span (in). Converting to metric

C = 1 + (0~031011,iJZ?)

Later tests produced a revised formula, incorporating the width B and depth D , of the beam. In its metricated version, this becomes

C = 0.94 + 0.03l5(BDc/L)

(all dimensions in mm). 528


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CASTELLATED BEAMS

16. The obvious drawback is that these formulae are derived from a very small number of tests on relatively small beams of a now obsolete pattern, none of which exceeded 375 mm in depth. Nevertheless, as suggested in the 1958 United Steel report,* data for a range of sizes and spans ‘could readily be obtained from beams in production in the workshops. These could be set up on improvised supports on the shop floor, loaded by dead weights and the deflection measured by dial gauges’. The conclusion from the tests was that shear deflexion will be greatest on heavily loaded short-span beams, where the increase in deflexion may be as much as 40%. This is, in fact, less serious than the effect on longer spans, for which the increase, although perhaps only 10%, will be added to an absolute value which is already large.

17. The increased deflexion of a castellated beam is the consequence of its low resistance to shearing deformation (Fig. 4). Calculation of the deflexion caused by shear can be made for each panel of the beam, assuming Vierendeel type action, and the results summed to provide the total shear deflexion. Analysis of the shear deflexion of an individual panel of a castellated beam is contained in reference 6, Appendix 2. The total shear deflexion of the beam can then be found by summa- tion of each panel deflexion; a tedious task for hand calculation if the beam has more than a small number of openings. However, a good approximation can be made by considering an analogous beam with a continuous solid web and thus constant section values Ifict and Afis,. The first of these is used the normal way to find the bending deflexion

y , = K w L ~ J E I ~ ~ ~ ~

where K is the relevant elastic factor for the type of loading, and Ificl may conve- niently be taken as the minimum second moment of area lmin. The shear deflexion is calculated using the relationship

where A is the total shear deflexion between panels 1 and n ; n is the panel number; v is the panel shear in panel i ; G is the shear modulus; and p is the spacing of openings. Now

is the bending moment at panel n. Therefore, the total shear deflexion at a panel in a simply supported castellated beam, with panel 1 adjacent to a support, is equal to the bending moment at that panel divided by CAfic1. The physical property Afict is analogous to the web area of a conventional beam; its derivation is set out in reference 6. The value of Afic1 has been tabulated for each castellated section.6

Webs 18. The capacity of the web will be limited by failure in

(a) shear yield in the web (b) shear yield at the weld (c) buckling (d) bearing.

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lnflll

/

T k’45”

Fig. 8. Infill plate at a point of concentrated load or reaction (Note: It is normal practice to$ll hole completely)

Shear yield

considering average stresses in web and weld 19. Some qualitative appreciation of the effects of shear force can be gained by

(a) at a web post gross web area = tD, = 1.5D, t average shear stress = V/1.5DSt = 0.67V/Ds t

gross web area = t X 0.5D, average shear stress = V/O,5D, t = 2.0V/D,t

area of weld = 2at = 0.250, t shear force = Vp/h = V(1.08DJ.40,) = 0.77V average shear stress = 0.77V/0.25DSt = 3.0V/Ds t

(b) at a hole

(c) at a weld

Comparing these values, it appears that failure in shear yield will occur at a weld. In fact, the effect of welding is to increase the strength of the material in the weld area so that the joint is less likely to be a critical factor.’

Buckling 20. Because the web of a castellated beam consists of a series of relatively

isolated posts separated by the openings, there is a general possibility of a post buckling in a lateral torsion mode. This mode has been investigated by Aglan and Redwood” but they observe that, for beams of standard British castellation profile, yield of the web at the weld should occur before a lateral torsional buckle forms. At points where there is concentrated load or reaction, the web may fail by conventional interactive buckling, a condition which may be checked using the same assumptions as those for solid web beams, namely

(a) that the web acts as a column of slenderness 2.5d/t for which the compressive strength can be obtained from the appropriate column curve

(b) that the compressive capacity of the web post is the compressive strength multiplied by the minimum cross-sectional area of the post at the weld.

Tests have shown that these are safe assumptions.” 530


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C A S T E L L A T E D B E A M S

21. It is important to appreciate that shear force is assumed to be transmitted across the opening in a castellated beam by equal division between top and bottom tees. At the end of a beam, therefore, half of the shear force passing through the lower tee does not exert a compressive buckling load on the end post, and so the end post can resist an external reaction equal to twice its compressive capacity. If the web capacity is inadequate for the applied load or reaction infill, plates may be welded into the appropriate openings in order to increase the loaded area at the beam centre-line, a conventional 45 degree spread of load being assumed (Fig. 8).

Bearing 22. The dispersion of concentrated loads or reactions is assumed to be com-

plete at the nearer root radius, and so the web bearing strength of a castellated beam is identical to that of its parent section.

Plastic design 23. Plastic design of castellated beams may be interpreted in two ways

(a) with the castellated beam as part of a continuous structure (b) with the beam simply supported, collapsing by the formation of either a

parallelogram mechanism or by plastic extension and contraction of the bottom and top chords respectively.

24. The first condition cannot be safely recommended to designers because of the imponderable nature of the beam’s reaction to rotation, which may be accom- panied by premature web-post buckling in a torsional mode. However, plastic design of a simple beam, showing collapse at only one hinge position, is possible.” Nevertheless, it will still be necessary to ensure that web and flanges remain stable up to the formation of the plastic hinge. The two basic modes of plastic collapse of a castellated beam are

(a) plastic extension and compression of the lower and upper tees respectively

(b) parallelogram action of plastic hinge at the corners of an opening in a in a region of high bending moment

region of high shear force (Fig. 9).

The location of both the opening which is critical (at which the collapse mechanism forms) and the corresponding load factor is complicated by the need to consider the response of an unsymmetrical tee section to three stress resultants: axial force N caused by the applied external bending moment; bending moment M ; shear force V caused by the applied external shear force.13 For each of these stress resultants acting alone, a tee has limiting values

and, adopting the Von Mises yield criterion

V, = (web area of tee) X p,,/ J 3

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Yield in tension /

Fig. 9. Plastic collapse: (a) in region of high bending moment; (b) in region of high shear force

25. For a tee under combined factored stress resultants M , N and V , failure surfaces can be derived which give the set of ratios NIN,, M / M p and V J V , at which failure occurs. A two-dimensional interaction diagram illustrating this is shown in Fig. 10. The significant aspect of the diagram is that for fixed values of the ratios N I N , and VlV, , the value of the associated ratio M / M p is dependent on the sign of M . The line A-A in Fig. 10 illustrates this point: for a given magnitude and sign of N and V , the positive value of MIM, is not equal to the corresponding value of the negative ratio of M I M , . The interaction relationship shows the com- putational difficulty of locating the critical point in a beam, where the load factor is smallest. If it is assumed that shear stress is not a significant interactive component, the surface can be reduced to the single curve for V/V, = 0.

26. With this simplified relationship, consideration can be given to a typical bay in which collapse has occurred at the root of the four tees surrounding a hole. Noting that for a given compressive axial force, the moment capacity of a tee is greater for positive (compression in the tee flange) than negative bending implies that the point of inflexion is not central (Fig. 11).

M + = V,X

M - = V , y

M + + M - = V,(X + y ) = V,(a)

and similarly for the lower pair of tees

M + + M - = V,(a) 532


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BEAMS

-1 .2 ' ' I I 1 I -0.8 -0.4 0 0.4 I 0.8

A

Fig. 10. Interaction diagram for bending, shear and axial force on tee

The total shear across the opening

V = V , + r / ;

Therefore

2 ( M + + M - ) = V a

In the limiting case of zero bending moment acting at the opening under consideration, thereby producing a parallelogram failure, there is no axial force N in the

Fig. 11. Forces in tee at collapse 533


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tees, the point of contraflexure is on the opening centre-line, and

M + = M _ = M p

The maximum shear force for this condition, V,,, , can be found from

M + + M _ = 2Mp = V,,,a/2

The interaction curve for the individual tees of a castellated beam can be trans- formed into a curve for the combined tees, forming the beam by a transformation based on

S M + + M - -- SF,,, 2Mp

-

M N - BM,,, N p

where M is the applied bending moment; S is the applied shear force; BM,,, and SF,,, are the maximum values of bending moment or shear force acting alone which the beam can resist.

27. Interaction charts have been prepared for the range of castellated beams6 Their use permits a trial section to be readily checked to determine its adequacy to carry a given factored loading. For concentrated loads, a few trials are necessary to find the critical combination of shear force and bending moment, although the locations which require checking are usually obvious. The common case of uniformly distributed loading can be simplified by comparing a typical beam strength curve with the general uniformly distributed loading curve as shown in Fig. 12. From this it can be seen that a section adequate in shear is inadequate in bending

Applied uniformly distributed load MIml,, = [l - (S~S,,,)I2

Strength ratio MIBM,,, or moment ratio MIM,,, 1 .o

Fig. 12. Beam strength compared with applied unijormly distributed load 534


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Table I . Castellated beams with compact tee stalks

Grade 43 I Grade 50

Original

305 X 127 X 48 254 X 146 X 43 203 X 133 X 30 203 X 133 X 25

Castellated

381 X 146 X 43 305 X 133 X 30 305 X 133 X 25

Castellated

305 X 133 X 30

by a small amount (about 5%). Therefore, a trial section can be selected from the table of ultimate bending and shear force capacities that have a bending moment capacity a little larger than the maximum applied bending moment on the beam.

Design to BS 5950 28. The application of BS 595014 to castellated beams requires a consideration

of width to thickness ratios so that the section can be correctly classified for local buckling (clause 4.15.5.3). Compression flange outstands are unchanged by castellation, and therefore fall into the same class as the parent section. The only castellated beams with semi-compact flanges are in grade 50 steel: 381 X 146 X 31 (ex 254 X 146 X 31 UB) and 305 X 133 X 25 (ex 203 X 133 X 25 UB). The remainder are plastic or compact.

29. However, the expansion of universal beams has an important effect on the web depth to thickness ratio in two respects.

(a) Between openings, the web slenderness d/t is increased by a factor of

(b) At an opening, the beam consists of two tee sections, the webs (stalks) of

All web posts are at least semi-compact, but many have d/t > 63(275/fy)''2 (BS 5950, clause 3.6.2) and therefore will have to be checked for shear buckling. The classification of the tee chord downstand is not clear: should it be treated as a tee stalk or as part of a web? On the former assumption, calculation of the slenderness ( A / t ) ratio of the stalk shows that only a few sections are compact (see Table 1); the remainder are semi-compact.

30. From these considerations it is clear that true plastic design of castellated beams is possible only for a very small number of sections ; the remainder must be designed on an elastic basis, as semi-compact sections, using the net section properties with due allowance for secondary Vierendeel effects of shear at the openings and for the local effects of point loads, if any, at any point in the beam (clause 4.15.3.2).

about 1.5.

which are unsupported at their lower edges.

Conclusion 31. Fifty years after their invention, castellated beams continue to meet a need

for an efficient element to provide for moderately loaded longer spans, as they have useful apertures for the passage of services and an attractive appearance. The design of the beams for simply supported structures by elastic methods is well- documented. Some restriction on their use in continuous structures is inevitable

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given the problems of slenderness which lead to restricted rotation capacity. Experimental work to determine the limits of their use is to be encouraged.

References 1 . PATENT SPECIFICATION. Improvements in built-up structural members. HMSO, 1939, Jan.

2. A new method of girder construction. Shipbldr Mar. Eng. Bldr, 1949, Oct., 682-683. 3. MANDEL J. A. et al. Stress distribution in castellated beams. J . Struct. Diu. Am. Soc. Ciu.

4. SRIMANI S. L. and DAS. Finite element analysis of castellated beams. Computers Structs,

5. KERDAL D. and NETHERCOT D. A. Failure modes for castellated beams. J . Constr. Steel Res. 1984,4259-315.

6. KNOWLES P. R. Design of castellated beams. CONSTRADO, Croydsn, 1985 (for use with BS 5950 and BS 449).

7. UNITED STEEL COMPANIES. Properties and strengths of castella beams. Defection charac- teristics. United Steel Companies, Rotherham, 1960, Aug., Report D.TS.6/262/2.

8. UNITED STEEL COMPANIES. Properties and strengths of castella beams. Further tests. Research and Development Department, Swinden Laboratories, 1958, July, Report D.GE. 71/262/1.

9. HOSAIN M. U. and SPIERS W. G. Failure of castelated [sic] beams due to rupture of welded joints. Acier, Stahl, Steel, 1971,36 ( l ) , Jan., 34-40.

10. AGLAN A. A. and REDWOOD R. G. Web buckling in castellated beams. Proc. Instn Ciu. Engrs, Part 2,1974,57, June, 307-320.

1 1 . OKULW T. and NETHERCOT D. A. Web post strength in castellated beams. Proc. Instn Ciu. Engrs, Part 2, 1985,79, Sept., 533-557.

12. HALLEUX P. Limit analysis of castellated beams. Acier, Stahl, Steel, 1967, 32, Mar.,

13. SHERBOURNE A. N. and O ~ ~ T R O M J. van. Plastic analysis of castellated beams, 1. Inter- action ofmoment, shear and axial force. Computers Structs, 1972,2,79-109.

14. BRITISH STANDARDS INSTITUTION. Structural use of steelwork in building. Part 1 : Code of practice for design in simple and continuous construction: hot rolled sections. BSI, London, 1985, BS 5950.

4, Patent Specification 498,281.

Engrs, 1971,97(7), 1947-1967.

1978,9, Aug., 169-174.

133-144.

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