OPTIMIZATION OF FRAMES WITH SEMI RIGIDS CONNECTIONS 1995.pdf

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00457 W(%)t-KM27-0 compurers t srrucrures Vol. 60. No. 4, pp. 531539, I996Copyright 0 1996 Elsetier Science LtdPrinted in Great Britain. All rights reserved

0045-7949/96 I 5.00 + 0.00

OPTIMIZATION OF FRAMES WITH SEMI RIGIDCONNECTIONSL. M. C. Sim s

Departamento de Engenharia Civil, Faculdade de Ci&ncias e Tecnologia Universidade de Coimbra,3049 Coimbra Codex, Portugal(Received 29 April 1995)

Abstract-This work describes a computer-based method for the optimum design of steel frameworksaccounting for the behaviour of semi-rigid connections. The procedure explicitly accounls for bothconnections and members by taking connection stiffnesses and member sizes as continuous-valued anddiscrete-valued design variables, respectively. The optimization algorithm minimizes the cost of theconnections and members of the structure subjected to constraints on stresses and displacements underspecified design loads. Two examples are presented to illustrate the features of the optimization method.dopyright a-1996 Elsevier Science Ltd. _

1. INTRODUCTIONTraditional approaches to steel frame design neglectthe actual behaviour of connections. Instead, twoidealizations are used: pinned and fully rigid. Althoughthese models simplify analysis and design procedures,the predicted response of the frame may not berealistic. In practice most connections transmit somemoments and experience some rotations that cancontribute substantially to overall structure displace-ments. The term semi-rigid is commonly used todenote the connection behaviour between these twoextremes. When a beam-column assembly is tested,for a given moment, a corresponding rotation isobtained for the beam plus connection. If the elasticrotation of the beam is subtracted, the applied momentmay be plotted against the characteristic rotation ofthe connection. This is referred as the M/4 curve [l-3].A selection of M/4 curves is represented in Fig. 1.It is apparent that the behaviour is nonlinear, as aninitial stiff phase is followed by a second phase ofmuch reduced stiffness and the shape of the curvedepends on the exact form of connection. Flexibleconnection behaviour affects the internal force distri-bution in the members of a framework. By treatingthe connection as semirigid a more reliable predictionof frame behaviour is obtained. Additional economycan be achieved by making use of the stiffness andstrength of connections that would otherwise betreated as pinned and also by avoiding the stiffeningoften required in rigid connections.In the past three decades, considerable research hasbeen carried out to assess the actual behaviour ofsteel frameworks accounting for the effect of connec-tion flexibility. Although a semi-rigid connectionframework design has been adopted by Eurocode 3and America Institute of Steel Construction specifica-tions, specific guidelines for the design of such frame-

works are not provided or readily available. Currentresearch interest in semi-rigid design in Europe andelsewhere is extensive. Research is ongoing and it ishoped that this will result in sufficient informationbeing available to allow engineers full use of thismethod in the future. However, much of this researchhas been limited to the analysis problem, with littleattention being paid to the important problem ofoptimization [4]. This work presents a systematicmethod for the optimum design of steel frameworksaccounting for the behaviour of semi-rigid connec-tions. The design which has the minimum combinedcost of members and connections is sought, while alsoensuring that stresses and displacements are withinacceptable limits. Members are sized using discretestandard steel sections, while connections are selectedon the basis of their continuous-valued momentrotation stiffnesses. The segmental method [6-71which uses linear programming is adopted for optim-ization purposes. The sensitivity analysis is carriedout by analytic means. Two steel frameworksexamples are presented to illustrate the features ofthis design method.

2. ANALYSIS OF FRAMEWORKS WITHSEMI-RIGID CONNECTIONSSophisticated methods of frame analysis are alreadyavailable. These enable all significant influences such asthe nonlinearities resulting from material behaviour

and the geometry of the structure. The connection isusually represented by fictitious structural elements atthe ends of members. These elements are assigned pre-determined relationship between forces and displace-ments so as to simulate the behaviour of the joint asa whole. The most simple model to analyse frameswith semi-rigid connections is a linear representationof the spring which in many cases will be quiteadequate. The effects of connection flexibility are531


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532 L. M. C. Sim6es

Extended E/P

Fig. 1. M/4 comparisons for various connection types.

I-----------L *Ii

= 11 +3EI-LSjFig. 2. Fix&y factor.

modelled by attaching rotational springs of moduli S,and S, normally measured in kNm rad- to the ends1 and 2 of a member, as shown in Fig. 2.

According to a first-order analysis, the stiffnessmatrix of a planar member with semi-rigid restraintat the ends can be represented by the stiffness matrixK, for the member with rigid connections modified bya correction matrix [5]:

106

5x1053x1052x105

1053 7x104t. 5x104Bz; 3x104i 2x104Ez 104Iz2 70002 50002 30000 2000vi-

103700500300200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Fig. 3. Variation of S, and a for a typical portal frame.

where E is Youngs modulus, and L, A and I are thelength, cross-section area and moment of inertia ofthe member, respectively. The parameters CI, nd t12 neqn (1) are defined as fixity factors at the two ends 1and 2 of the member, and are related to the corre-sponding rotational spring stiffnesses S,, and Sz as,

I 1I= I +3EI/S,L = 1+3EI/S,L (2a,b)

The fixity factor TVefines the stiffness of the connec-tion relative to the attached beam and is perhaps theEA 0 0 EA- --L 0 0L

12EI (a1 + a2 + a, a2)o- 6EI (2u, + c(,Q) 12EI (CL, a2 + rz,a2) 6EI (2a, + c(,c(2)L (4 - El@2) F (4-cc,Qo

L (4-a,ad F (4-a,cr,)0 6EI (2a, + dII (2) 4EZ (3c(,) 0 6EI (2~9 + cl,ozz) 2EI (3cr,~1~)Lz (4-a,@,) --L (4-a,m2) L2 (4-a,a,) L (4-%a,)

3 (1)EA 0 0 EA-- -L 0 0L12EI (c1, + a2 + a,$)o- 6EZ (2a, + a,ct2) 12EI (CL,+ a2 + a, a2) 6EI (2a, + a, az)-- --L (4 - alad L2 (4-a,a,)

oL (4 - u,u2) L (4-u,u,)

0 6EI (201,+ UIu2 2EI (3a, CQ)L2 (4-u,a,) L(4-cc,a,) 06EI (2a, + c$uz) 4EI (3u,)-- L2 (4-cc,X,) I (4-u,a,)


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Optimization of frames with semi-rigid connections 533most powerful and important concept for the analysisof frames with semi rigid joints. It relates quite closelyto how the structure will behave in the context of theconnection, far more so than the absolute value of S,.Figure 3 indicates a typical variation in S, against m,.

As it can be seen, a real pin connection will alwayshave some stiffness and will lead to significantrestraint moments which could be of great benefit toa structure. Also, large reductions in stiffness fromfull fixity are seen to have little effect on the structurein that only a small change in c( will result in a verylarge physical change to S,. The fact that x, smoothsthe variations in 5 confirms that only an approxi-mately accurate M/4 curve is required, as the analysisis directly affected by I and not 5,.In the linear elastic analysis, the connection andmembers are assumed to have linear force-displace-ment relationships and the effect of deformation onthe equilibrium of the frame is disregarded. In itsusual form, the analysis requires the solution of theset of linear equations:

Ku = P. (3)The advantage of this approach is that the overallprocedure is the same as the conventional matrixdisplacement method commonly adopted for rigid-jointed frames. The end-moments for a semi-rigidlyconnected planar member of length L under uniformlydistributed loading p are:

M =pL23~,(2-~~,). 12 (4-a,a,)

M PL 3a2(2-al)2 12 (4-a,a,) .

(44

3 STRUCTURAL OPTIMIZATIONThe optimization problem consists of finding the

minimum combined cost of members and connectionswhile accounting for the semi-rigid behaviour of theconnections. The connection behaviour, as reflectedthrough the fixity factor, has a significant influence onthe response of the structure and, as such, will alsohave considerable effect on the cost of the structure.Herein, the cost of each member i is represented byits weight, while the cost of each connection k is takento be directly related to the fixity factor. Therefore,the total cost of a member i with two end-connectionsk = I, 2 may be expressed as

Z,=wiU,+C,=,,2(V0,+ V kaik+ Viafk), (5)where ai and wi are the member cross-section areaand weight coefficient (w, = pill, i.e. material densityx member length), aik, Vjk and Vi are the fixityfactor and cost coefficients and Vpkis the cost of apinned connection having zero rotational stiffness.

Although there is little or no information in theliterature concerning the value of the cost coefficientsfor the various types of connections used in steelframeworks, representative values for V: and Vi willbe derived in the following section.

Each member i is to be sized using a commercialstandard steel section and, as such, its cross-sectionarea aj is a discrete design variable. On the otherhand, each connection k may be fabricated anywherein the range from being fully-pinned to fully-fixedand, as such, its fixity factor alk is a continuous designvariable.

a,oA,; OCa,k< 1, (60)where eqn (6a) requires the cross-section properties ofmember i to belong to the discrete set A, = {a, , a:, .}prevailing for the standard section shape specified(e.g. IPE shape), while eqn (6b) imposes specifiedlower and upper bounds.

The optimal design of a framework of i = 1,2,., n members having semi-rigid connections subjectto stress and displacement constraints, may begenerally stated as,

Min Z = Xi= ,,n

subject to:af


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534 L. M. C. Sim6esRecognizing that the cost of a member with pin-jointedconnections is function is known explicitly and the derivatives canbe obtained by direct differentiation of eqn (I 1).The first order Taylor series approximation gives:

w ,a,+X :,=,,zV~= 1.2w ,a,, (9)12w a - 0.8wi (a,,iai + a,croi a,,q)

and the fact that VP, Vi, it follows from eqn (9)that the cost of a pin-jointed connection is Vi =0.1 w ,a,. If a linear variation of cost between a pinnedand a fully rigid connection is assumed, Vt l = Vi = 0and V, , = Vfi = 0.2w ,a,.

A more accurate connection cost is obtained byconsidering a nonzero coefficient for the quadraticterm in c(,~, eflecting the increase in cost to provideeither a perfectly fixed or perfectly pinned connection.Average connection cost coefficients are given by:

VP, = O.lw ,a,, V), = Vk = -0.4w ,a,;

+ 1.6w, a,u,f,,+ 2aoic(,,cr, 2aoia,$)= w ,a, + w ,(cp + cf ai + cfm,). (13)

Although the fixity factors may vary between theperfectly pinned and the fully fixed cases, movelimits should be imposed on E, to reduce inaccuraciesassociated with the computation of the explicitapproximations on the stresses and displacements.Therefore the optimation problem expressed explicitlyin terms of the design variables is:

Vf, = Vf, = 0.6w ,a,, (10) Min Z = Xi= ,,,[w ,a, + w ,(cp+ c, a, + cf cc,)], (14a)and the total (material plus connection) cost of amember with semi-rigid joints in eqn (7a) becomesa nonlinear function coupling the design variables a,

subject to

and x,:w ,a, + 0.2w ,a, - 0.8w ,a,a, + I .6w ,aiaf . (11)

3.2. Explicit approximation problem f i S(u,-a,,)


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Optimization of frames with semi-rigid connections 535virtual load procedure. Given some design variable tinuous problem. The continuous optimum designvalues a full analysis of the frame will give numerical forms a lower bound to the discrete optimum and itvalues for all the nodal displacement under all load- is usually assumed that the continuous sizes shoulding cases. The displacement derivatives au/a l/ai and somehow be rounded up or down to discrete sizes.au/al/l, are computed by implicit differentiation of This rounding process turns out also to be a combin-the equilibrium equations: atorial problem. The method described next introducesthe artificial concepts of segmental members and

KO$= +; auQ aKo segmental optimum design and provides a closea, a, KOalir,= -aljl,~o. (15) lower bound to the discrete-continuous optimumproblem.

Since u and K are known from the analysis of the Problem (14) assumes that each member is of knowninitial design, a solution for &/al/a, and au/a l/I, length Li and has unknown, but uniform cross-involves only calculation of the r.h.s. vector of eqn sectional properties (area A modulus W inertia I,).(15) and forward and back substitutions. To compute Consider replacing this assumption by a different one.aK/ax,, only elements of associated with member Assume instead that each member of the frame isi must be considered and all are constants. Assuming composed of a total of D segments, each with a cross-that R, and R, are transformation matrices giving the sectional area equal to one of the discrete sizes S,,,axial and bending components of the normal stress, d = 1,. . D such that all sizes are representedrespectively: among the segments. Let I,, be the unknown length

of the segment of member i which has area S,,, d = 1CT rr,+cr,=R,,u +R,u, (16) . D. This is shown in Fig. 4 for a member which

has three discrete sizes. The areas of all segments80, au,-=R,-.d l/a, aao -R au,,al/a, al/I, a/r (17)

are known, but the segment lengths are unknown.The ordering of segments along a member is im-material. Clearly, to replace a conventional member

The derivatives au/&t, and &r/&x, are computed in a by a segmental one their total lengths must besimilar way. However, it should be remembered that identical. Thus for member i relationship (31) mustthe aP/acr, terms must be considered:

K,%=(?po-i7Kgu0,&, aa, aa,

and sinceIJ = Qu,

the stress gradients are

Z=Q,$.,

5. SEGMENTAL OPTIMUM DESIGN

hold:

(18) Xd= ,,Dl,, = L. ; i = 1, , N. (21)If problem (14) is re-formulated with this segmentalassumption in place of the uniform members thefollowing problem is obtained:

(19)

(20) subject to+ p,(cYl, + cj.rJ,,,+ cZ4a,)l (22a)

An algorithm for the direct solution of problem (14)must include provisions to overcome the stress anddisplacement constraints non-linearly and the discrete-ness requirement stated by eqn (14e). The rigorousdiscrete-continuous optimum design is a NP hardproblem, significantly more difficult than the con-

duo,+ acI (a, - ao,) < u: (22b)

+ i l e 42+ i344 (reaAi AreaS, Area S2 Area S3Fig. 4. Conventional member and segmental member.


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536 L. M. C. SimBes

The design variables of problem (22) are the lengthsIid of all segments of all the members and the fixityfactors. The stress and displacement constraints arelinear in both types of design variables. The discrete-ness requirement, eqn (14e), has been removed andabsorbed into the problem by defining segments whichhave only the discrete sizes of set S. The additionalconstraints in problem (22) which do not appear inproblem (14) are constraints like eqn (21) one foreach member, which ensure length equivalence ofconventional and segmental members, and the non-negativity requirement for all segment lengths whichare clearly necessary. Problem (22) is an LP problemwhich may be solved by any LP algorithm, and willyield what can be termed a segmental optimumdesign.

Several features of the segmental optimum designcan be deduced. Firstly, its weight will be globallyminimum for the explicit approximations on the dis-placements and stresses. This is a property of linearprogramming problems. This indeterminate framewill be solved in an iterative fashion and each of thesequence of LPs will be solved globally, althoughthe sequence of global minima will not necessarilyconverge to a global minimum weight. This is aninherent feature of the iterative process and appliesnot just to segmental optimum design, but to allmethods which solve minimum weight frame designproblems iteratively. Secondly, the weight of thesegmental optimum design, eqn (22), is a lower boundto the weight of the discrete optimum design ofeqn (14).5.1. Achievin g a d iscrete optimum design

The discrete optimum design for a frame must haveonly one segment of discrete size per member. In thesegmental optimum design most members will satisfythis requirement, but there will be a few multi-segment members which do not. Some type of round-ing operation is necessary. An obvious scheme issimply to round up all the multi-segment members,i.e. increase the size of all the smaller size segmentsin each member until they are of the same discretecross-sectional area as the largest discrete size withinthe member. If all multi-segment members are treatedin this way the result will be a feasible discrete designwhich may be the discrete optimum design and willin general be an upper bound to the discrete optimumdesign. Because only a few members of the frame are

concerned in this rounding up operation the percent-age weight increase should be small and the discretedesign thus obtained, though perhaps not optimal,should have a weight only fractionally larger thanthe globally optimum discrete design. This roundingup can be done using the simplex table correspond-ing to the segmental optimum design. The roundup operation is equivalent to setting the segmentlength variables of the smaller size segments within amulti-segment member to zero.

Having achieved a rounded-up discrete design fromthe segmental optimum design this is often as faras the method will go. Occasionally however, furtherrefinement of the discrete design may be possible andit is comparatively easy to check this. The check con-sists of determining whether any complete membersin the discrete design .can be replaced by completemembers of a smaller size without violating anyconstraints. The simplex table for the rounded updiscrete design can be used for this. The slack variablesof the displacement constraints will be in the basic setand must remain there with positive or zero values.The other basic variables are segment lengths, one permember with a value equal to the physical length ofthe member. The objective function coefficients willindicate several candidate segment length variablesin the non-basic set which, if they entered the basis,would reduce the weight of the structure. Eachcandidate can be examined and pivoted into the basisprovided that (a) it pivots a complete segment variableout of the basis, and (b) it does not violate the non-negativity of any other basic variable. These extrachecks and pivots will produce a new discrete designeven closer to the optimum. It should be noted herethat only the optimality of the segmental optimumdesign is guaranteed.

The rounding up and refinement processes arenot rigorous and have no inherent guarantees ofoptimality. However, because the segmental optimumdesign is globally optimal and forms an almostdiscrete design, it is usually a close lower bound to thediscrete optimum design. Because the number of mem-branes involved in the rounding is small, the roundedup segmental optimum design usually forms a closeupper bound to the discrete optimum design. Anyfurther refinements of this upper bound design willtighten further these already close bounds upon thediscrete optimum design.5.2. I ndeterm in ate stru ctur es

Among the several ways in which the optimizationstrategy can be included in an iterative sequence ofanalysis-optimization cycles is to perform only asegmental optimum design in each iterative cycleuntil convergence of the sequence is almost complete.Rounding up and refinement are only added in at thislate stage. This approach raises one small difficultyin that the structural analysis which separates eachoptimization must be carried out on a structure con-taining multi-segment members. For a multi-segment


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Optimization of frames with semi-rigid connections 53138kN/m

36 kN

18 kN

Fig. 5. Example I: one-bay two-storey steel frame.

member with different lengths, material propertiesand areas for each segment, the simplest way ofaccommodating this within the stiffness matrix K isto calculate an equivalent stiffness value for themember rather than to attempt to write Kin terms ofindividual segments. The equivalent stiffness ki of amulti-segment axial force bar will be simply

k, = Z I=,. .L ,_D++ for all l,, 0. (23)d

Using eqn (23) for axial forces and similarly forbending, all elements of K corresponding to a seg-mental design can easily be calculated and the sub-sequent determination of the nodal displacement isstraightforward.

6. NUMERICAL APPLICATIONS6. I Example 1

The one-bay, two-storey steel framework loaded asshown in Fig. 5 has semi-rigid connections of equalfixity factors at the two ends of each of the beammembers. The two column members at each storey

IPE 450

rIPE 550

IPE 450

IPE 450

IPE 450

level are prescribed to be the same size. The frameis to be designed in accordance with the strength-stability (stress) requirements, while, at the sametime, ensuring that the top storey lateral sway atnode 6 does not exceed 18.28mm (i.e. h/400). Thecross-section area a of each member is to be selectedconsistent with the requirement that all column andbeam members are to have IPE sections. Youngsmodulus E = 206 kN mm-* and the material densityp = 7.85 t mm3. As such, the weight coefficient w, ineqn (5) for each column member of length 3.65 m isw, = 28.65 t m-*, while that for each beam memberof length 7.30 m is w, = 57.305 t m-*. A quadraticconnection cost with respect to c( is adopted for eachbeam-to-column joint.A fixity factor c( = 0.9 is used to define the initialrotational stiffness of each of the four semi-rigidconnections for the framework. Upon applying thepreviously described iterative design optimizationprocedure, the optimal design of the frame havingminimum member plus connection cost is found aftertwo design stages. The optimal member IPE-sectionsare depicted in Fig. 6.

IPE 400

IPE 400 IPE 400IPE SO0

Fig. 6. Optimum solution fully rigid joints and optimum solution semi rigid joints.


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538 L. M. C. Sim8es31 kN/m 31 kN/m

~-~66.10m.-~~6.10m qFig. 7. Example 2: two-bay three-storey steel frame.

The optimal fixity factors of the semi-rigidconnections are a, = 0.59 (S, = 5.9 x lo4 kNm rad-)for the lower-storey beam and CQ 0.66 (S, = 3.8 xIO4kNmrad_r) for the upper-storey beam. Alsoshown in Fig. 6 is the optimum solution for the casewhen the connections are assumed to be fully rigid.Interestingly, the design that accounts for semi-rigidconnections weighs 14% less than the design foundwhen only fully-rigid connections are considered.6.2. Example 2

The two-bay, three-storey steel framework loadedas shown in Fig. 7 has semi-rigid connections at theends of the beam members. Connection fixity factorsare specified to be the same for all semi-rigid connec-tions at each storey level. External column membersat each storey level are prescribed to have the sameIPE section, as are beam members at each storeylevel. The frame is to be designed in accordance withthe strength-stability (stress) requirements, while

WE 330

IPE 330

IPE 3S0

IPE 360

IPE 400

IPE 450

IPE 360

IPE 240IPE 400

IPE 330IPE 450

IPE 500

IPE 330

IPE 330

IPE 360

ensuring that the top-storey lateral sway does notexceed 27.38 mm.The material density p = 7.85 t mm3 and, assuch, the weight coefficient for each column memberis wi= 28.65 t m-*, while that for each member iswi = 47.885 t m-. As for Example 1, the connectioncost is also assumed quadratic in a.

A fixity factor a = 0.9 is adopted to define theinitial rotational stiffness of each of the semi-rigidconnections for the framework. Upon applying theiterative design optimization procedure, the optima1design of the frame having a minimum member plusconnection cost is found after four design stages. Theoptimal member IPE-sections are depicted in Fig. 8.The optimal rotational stiffness of the semi-rigidconnections area a, = 0.6 (S, = 2.5 x IO4kN m rad-)for the first storey, a2 = 0.575 (S, = 2.2 x lo4 kN mrad-) for the second storey and c+ = 0.55 (S, = 1.5x IO4kN m rad-) for the third storey. Also shown inFig. 8 is the optimum solution when all connections

IPE 300

IPE 330

IPE 360

IPE 330 IPE 330

IPE 240IPE 360 IPE 360

IPE 330IPE 360 IPE 360

IPE 550

IPE 300

IPE 330

IPE 360

Fig. 8. Optimum solution fully rigid joints and optimum solution semi rigid joints.


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Optimization of frames with semi-rigid connections 539are assumed to be fully rigid. As for Example 1, thefully-rigid connection design weighs 12% more thanthe design found when the influence of semi-rigidconnections is accounted for.

7. CONCLUDIN G REMARKSThe described optimization procedure provides aneffective means to account for the cost of both mem-

bers and connections in the design of steel buildingframeworks. In fact, accounting for the actual semi-rigid behaviour of connections results in designs thatare less costly than when, as is usually done, theconnections are idealized as being fully rigid. This isbecause semi-rigid connections allow for a redistribu-tion of internal member forces (shear, moment, etc.)that results in a more economical use of material toresist the applied loads. This is an important resultbecause it implies that accounting for the actualsemi-rigid behaviour of connections in the design ofsteel frameworks is both realistic and economical.

Acknowledgemen/-The author wishes to thank the financialsupport given by JNICT (Junta National de InvestigaqaoCientifica e Tecnologica), Project STRDA/C/TPR/576/92.

I.

2.

3.4.

5.

6.7.

REFERENCES

D. Anderson, F. Bijlaard, D. A. Nethercot and R.Zandonini, Analysis and design of steel frames withsemi-rigid connections. IABSE SW. 4, 61-78 (1987).D. Anderson and K. Kavianpur, Analysis of steelframes with semi-rigid connections. Strucr. Engng Rev.3, 79-87 (1991).R. Cunningham, Some aspects of semi-rigid connectionsin structural steelwork. Siruci. Engng 68; 85-92 (1990).D. E. Grierson and L. Xu. Outimal design of steelframeworks accounting for semi-rigid &nections.In: Optimization oflarge Structural Sysfems (Edited byG. I. Rozvany). Kluwer, Dordrecht (1992).G. R. Monforton and T. S. Wu, Matrix analysis ofsemi-rigidly connected steel frames. ASCE .struct. Dir.89, 13-42 (1963).L. M. C. Sirnoes, Reliability of portal frames withdiscrete design variables. Struct. Oprim. 5, 76 -83 (1992).A. B. Templeman and D. F. Yates, A segmental methodfor the discrete optimum design of structures. EngngOptim. 6, 145-155 (1983).

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