Macro Elements for Variable-section Beams

8/2/2019 Macro Elements for Variable-section Beams

1/7

Computers & Sfructwes Vol. 37, No. 4. pp. 55S559, 1990Printd in Gnat Britain.

0045-7949/90 s3.00 + 0.00Per~Orl Press plc

MACROELEMENTS FOR VARIABLE-SECTION BEAMSF. ARBABI and F. LI

Department of Civil and Environmental Engineering,Michigan Technological University, Houghton, MI 49931, U.S.A.(Received 16 October 1989)

Ahatract-A procedure is described for developing macroelements using distributions, and is applied tovariable-section beams. An element can include any number of step variations in the cross-section withoutincreasing the number of degrees of freedom, which is six for two-dimensional beam elements. Any beamelement with a continuous profile can be approximated to any desired accuracy by one with step variations.A symbolic manipulation system is used to derive the expressions for the terms of the stiffness matrix andload vector.

INTRODUCTION

Variable-section members are often encountered inframes and continuous beams. The variations may becontinuous or have jump discontinuities. Linear andparabolic profiles or haunches are examples of con-tinuous variations. Step changes of the cross-sectionand bar cut-offs of reinforced concrete membersprovide examples of jump discontinuities. An accu-rate representation of such members by finite el-ements would lead to a substantial increase in thenumber of degrees of freedom. Substructuring [l-6] isone way to avoid this problem. However, it requiresadditional programming and computer time forcalculation of the necessary matrices for the sub-structures. Closed-form solutions may be pos-sible for simple cases. However, they do not providea general procedure. In addition, other procedurescan be used, such as determination of the flexibilityby integration. However, such methods do notprovide general procedures for systematic derivationand extension to beam-columns and other elements,as the proposed procedure does.

This paper presents a procedure for the directsubstructuring and development of the stiffnessmatrix and load vector for members with jumpdiscontinuities. Any continuous variations may berepresented by a distribution using step variationssimilar to the approximation in numerical integrationor finite element methods. However, this represen-tation does not increase the number of degrees offreedom when the number of steps is increased forimproved accuracy. Thus, large size elements can beused regardless of the variations of cross-sections andmaterials. The theory of distributions [7-161 providesthe mathematical basis for this development. Theprocedure allows closed-form solutions of memberswith jump discontinuities [17-201. The mathematicaloperations involving jump discontinuities can be car-ried out by symbolic programs [21-261. Using thesymbolic package MUMATH [27] the routines for

handling distributions were written [28] and used fora parametric study. Some examples are presentedhere to demonstrate the efficiency and versatility ofthe method.

DISTRIBUTIONAL REP RES ENTATION OF VARIABLECROSSSECTIONS

Jump discontinuities can be described by Heavisidefunctions. If a functionf(x) has jumps of magnitudesAh,..., Ah a t a , , . . ,k, respectively, it can berepresented by

g =f + i A$ H(x - ai), (1)i-1

where g is piecewise continuous and can be differen-tiated in the distributional sense. Thus, each simplejump discontinuity contributes a term AA 6(x - ai ) toits distributional derivative. Successive differen-tiations result in the Delta function and its deriva-tives. When the cross-section has jumps, the area ofcross-section and moment of inertia can be expressedin the same way. It is more convenient to describe therigidities by distributions as

with

Ji = f (Ai,, - A,),I

~z=~z,(l+$-z,i,) (2b)with

ri=fv,+,-Ii)1CAS 37/4-M 553


2/7

554or flexibilities by distributions as

with&=&(l++,)

with

with Hi = H(x - a,).

F. ARBABI and F. LIcan be found by direct integration, which, aftersubstitution from eqns (2~) and (2d), yields

(24

(24

For a planar beam element with variable cross-sec-tion, including axial load but without consideringbuckling (Fig. la), the differential equations govem-ing the deformation are

(EAu) = pX (W(Elv ) = py ) (3b)

where EA and EZ are continuous functions of dis-tance x.

When EA and EZ are described by eqns (2), (3a)and (3b), each represents a set of differentialequations combined in one expression, with jumpdiscontinuities expressed by step functions.

SHAPE FUNCTIONS

The solution of homogeneous equations(EAu) = 0(EZn) = 0

(4a)

(4b)(a)Y

(blY

Fig. 1. Beam element with step approximation. (a) Actualvariation. (b) Step approximation.

Wand

Integrating eqn (5a) again and applying the boundaryconditions, u(0) = u, , u(f) = u2, leads to

dx)= -+d ~+fH~s,(x-~,) +u, (6)i= I >with

n-lY = l+ 1 Si(l-Ui) .i=l >u(x) can thus be written in terms of the shapefunctions 4, and & as

(7)where

41= 1-$x), 9+(x)withn-lq(x) = x + c Hisi(x - a,).i=l

Substituting eqn (7) into the potential energy of theaxial deformation, we obtain

l-I=; EA(u)dx-s (8)0

From eqn (8) the stiffness matrix can be extracted as

[E]= 4EA+dx=y _1 1 , (9)s

1 -10 [ 1

and the fixed-end force vector as

( 1 0 )

Similarly, integration of eqn (Sb) gives the equationof slope

n-lx2 + c H,r,(x- a:)i- 1n-l+ C2 x + C H, r,(x - UJI + C, (11)i-l


3/7

Macroelements for variable-section beams 555and deflection

0 = -& ; c, x3 + i z-z,r,(x - a#(x + 2a,)I [ ( i-1 )

+;c*( *+y,r,(x -a$ )Ic , x + c , .i - l ( 1 2 )Substituting the boundary conditions, o(O) = D, ,u(l) = u2, u(0) = 0, and u(l) = 02, we obtain

Cl

II I, la, -a3 -a,

c2 a2 la,-a4 -a2c3 =El,a o 1 0

C4 1 0 0

a3

a4 400

Ib0 2

0 2

(13)with

1a= a2a3 - al a4

(n- l

aI = - I + C ri(l - a i)i= I )

a*=; 1*+fr#*-ai)( ,= I )

a3 = -i ( f* + f ri(/ - ai)*i - 1 )a, = i

(1 + i ri(l - a,)*(1 + 2a,) .

i=l )

Thus, the bending displacement at an arbitrarypoint can be written in terms of end displacements as

where

with

N, = 1 +fi(xb2a: -t.h(xha,N2 = x -fi @ha --.Mx)a2a4 = -h(xb2a -_Mxha,N4 = h (x h a +h (x h a

r,(x)=;( +pz,r,(xx,)2 >L(x) = ; ( x3 + i H,r,(x - &)2(X + 2a,)i-1 >

all-a,= -a2 and a3 = -a,1 +a,.

Similar to the case of axial deformation, we can findthe bending stiffness matrix from the bending poten-tial energy as

I[t] =

JEZWTN dx

0and the fixed-end forces as . I

fNIPS dx

0I N,P, dx0I N3py dx0N4py dx

(15)

. (16)

Combining the stiffness matrices for the axial andbending deformations, we obtain a 6 x 6 stiffnessmatrix for a beam macroelement. It should be notedthat this formulation carried out on a symbolicmanipulation system provides the closed-form sol-ution for members with jump discontinuities. Formembers with arbitrarily varying cross-sections, nearclosed-form solutions may be obtained by approxi-mating the profile with one including n steps. Therepresentation can be improved by increasing thenumber of segments in the member, especially inregions of rapid changes. A function written in thesymbolic program MUMATH generates the terms ofthis stiffness matrix. The method presented here isapplicable to any beam component with regular orirregular variations of the cross-section. The devel-oped beam macroelement can describe larger com-ponents without increasing the number of degrees offreedom. Thus, the memory requirement and compu-tation time can be reduced significantly. This tech-nique can be easily extended to three-dimensionalelements. The direct stiffness method of structuralanalysis can be used to solve for the end forces anddisplacements of each macroelement. To describe theforces and displacements within each macroelement,the effects of the in-span loads must be included. Infact, once the end forces and displacements are foundfrom the global analysis, the solution of individualmacroelements is reduced to the solution of simplysupported beams. To make the formulation versatile,we must seek Greens function for beams with jumpdiscontinuities.

GREENSFUNCTIONLet g, and g2 be Greens functions for the axial and

vertical displacements, respectively. They must satisfythe differential equations

and(EAg;)=cS(x - 5)

(EZg;) =6(x - 5)

(1W

(17b)


4/7

5 5 6 F. ANNI and F. LIand all the boundary conditions associated with theproblem. We can derive Greens functions by inte-grating both sides of the above equations. This gives

t?*(d)=&-I (n-lH(x-O tx-t)+ 1 Hi s i ( x -0

I i-1 )n-l

- iT, HisiH(ai- e)(Ui- t) (18)with

B, = -$--[(u 5) + y s,(l- 0)1 1 i - 1

n-l-i~,siH(ai-C)(ui-5)1&M)& Cl-x3(x-mx-5+~n -1

+ C HiriH(X - t)(X - t ) 3i = In-l

n-l- ,c,H,r,y (x-Ui)2(X +2Ui) +fizX

1 (19)

B2=-hl ( n-l(1 + c rJ(l - r)3 - (I - 01*I i=ln-l

- i& riH(Ui- [)(a,- 5)2(3r - 5 -2u,)$1 ,(W>r, I (I - u,)~(Z+ 2u,) .is >

The axial and lateral displacements due to in-spanloads are found from

n(x) = s p,(5)& C) dr (20a)0( 2 0 b )

Concentrated loads and couples can be representedby 6 functions and their derivatives. Partially dis-tributed loads can be expressed by the difference oftwo step functions. The solution can thus be easilyobtained following the integration procedure for 6functions and step functions. For given in-span loads,the integrations can be carried out symbolically byfunctions written in MUMATH.

MACROELEMENT WITH STEP VARIATIONSFrom the above analysis it is clear that we can

formulate macroelements for beams involving jumpdiscontinuities. The exact formulation is obtainedusing the singularity method. We can also use thisprocedure to generate macroelements for any con-tinuously varying profile to any desired accuracywithout increasing the number of degrees of freedom.Any variation can be approximated by a series of npiecewise prismatic segments, i.e. with n jump discon-tinuities. When the number of segments, n, is suffi-ciently large the terms of the element stiffness matrixconverge to their exact values. Some examples of thecommonly used variable section elements are pre-sented here to check the validity and efficiency of thistechnique. The stiffness matrix of variable sectionmembers is determined by a set of symbolic functionswritten for MUMATH [28].

The first example is a beam element with rectangu-lar cross-section and a depth linearly varying from2h, to ho (Fig. 2). The axial deformation is ignored inthis example.

The depth of the beam at x is h = 2h, - h,x/l, andthe moment of inertia is iith*=%.A MUMATH function, II, generates the vector of themoments of inertia at specified intervals. For fivesegments it gives

1, = {6.861,4.911,3.381,2.201, 1.331).Using this vector in the symbolicwe find the stiffness matrix

function SSTIFF,

38.50 25.331 - 38.50 13.171

Kc=? - 25.3318.50 -25.3319.3712 -25.3318.50 -13.17f.9661 .13.171 5.966i2 - 13.171 1.2051*

Increasing the number of segments to 30, we obtain a more accurate stiffness matrix as follows:37.79 25.181 -37.79 12.601

Ke=; ! - 25.1817.79 -25.1819.4512 -25.1817.79 - 12.601.73212 12.601 5.73212 - 12.601 6.87212


5/7

Macroelements for variable-section beams 557The next example is a beam member with a THE FIXED-END FORCESrectangular cross-section and parabolic profile(Fig. 3). Once the stiffness matrices are calculated for the

The depth of the section at x is h = 2h0(Z - macroelements, the global stiffness matrix associated(1 - x)x/Z2), and the moment of inertia is with the degrees of freedom can be established. Thenext step is to generate the fixed-end forces for the

z,=z 2-;(Z-x)x( >3 macroelements. The fixed-end force vector is given by. eqn (16). For any load, the integrals in eqn (16) can

be calculated with a symbolic function written forSubstituting this expression into the symbolic func- MUMATH. For the beam of Fig. 2 under a concen-tion II with five segments, we obtain trated load P at the mid-span, with five segments, wehave

Z, = (4.411, 1.561, Z, 1.561,4.411}. Z, = {6.861,4.911,3.381,2.201, 1.331).The stiffness matrix is

31.62 18.81 - 37.62 18.811

,E18.811 11 2312 - 18.811 l.576i2

K e 1 - 37.62 - 18.811 37.62 -18.811 .18.811 7.57612 - 18.811 ll.2312

I

For 30 segments the stiffness matrix is

I0.65 20.331 -40.65 20.331Kc=7 -40.650.331 -20.3312.0012 - 40.650.331 -20.331.32612 20.331 8.32612 -20.331 12.0012Comparison of the above matrices indicates thatconvergence is rather fast. However, the time increasefor generating the stiffness matrices may be out ofproportion for further increase in the number ofsegments. This is due to the so-called intermediateexpression swell in symbolic manipulation systems.This refers to the situation where the results of infiniteprecision calculations are very long expressions foradditional subdivisions. However, this will not pre-sent a problem for numerical computations. We recallthat, unlike other numerical methods, e.g. finitedifference and finite element, the accuracy can beimproved by merely increasing the number of segments without increasing the number of degrees offreedom.

Fig. 2. Beam with a linearly varying depth.

The concentrated load P can be symbolically rep-resented by

p,=Pd x-;.( >In this case, integration of eqn (16) becomes merelya substitution, leading to

N, f P0N20; PN, 0f P

=

By increasing the number of segments to 30, thefixed-end force vector becomes


6/7

558 F. ARBAIIInd F. LI

Fig. 3. Beam with parabolic profile.

In this process the convergence is fast as the numberof segments increases [28].

Next, we consider the beam of Fig. 2 with adistributed load on the whole span. In this case, thefixed-end force vector with five segments is

=

I % pydx0I Nzp,dx0I

IN,P,~

0

s

I%P, d.x

0

=

and with 30 segments is

For a parabolic beam profile (Fig. 3), the moment ofinertia is represented by

I,=I 2-&x)x( 13 .

segments. Generally, an approximation of 30 seg-ments will result in a very accurate solution. Thefixed-end force vector for 30 segments is

As we expect for tt tis beam, with larger moments ofinertia near the ends, the fixed-end moments arelarger than those of a constant cross-section beamand the values of the moments at the central portionare smaller than the latter. For the beam of Fig. 3under a distributed load, the fixed-end force vector,with 30 segments, is

0.5P0.16383Pl= 0.5P

-0.16383Pl

0.5ql0.102414q120.5ql

-0.102414q12

SOLUTION PROCESS

The solution process is the same as for the finiteelement method, involving the assembly of macroele-ments and the determination of the displacementsand forces. For example, let us consider the three-span beam of Fig. 4 with three identical spans. Thecross-section varies parabolically in each span. Aconcentrated load and a uniform load are applied atthe middle of the left span and over the middle span,respectively.

The moment of inertia is given by

r,=r 2-fO-x)x( >3

The stiffness matrix for each element, with 30ments, was found in the previous example as

40.65 20.331 -40.65 20.33120.331 12.0012 -20.331 8.3261

~ 1 = ~ 2 = ~ 3 - ! ! !e 0 c - I 3 I40.65 - 20.331 40.65 -20.33120.331 8.3261 -20.331 12.0012 :

w3-

Again, we consider two cases of loading; a concen- The structure stiffness matrix can thus be assembledtrated load, P, at the mid-span, and a uniformly asdistributed load. We substitute the function for themoment of inertia into MUMATH symbolic function K = E 23.9977 8.3264II to generate the moments of inertia for n prismatic I 8.3264 13.9977


7/7

Macroelements for variable-section beams 559

Fig. 4. Bridge with parabolic profile.The fixed-end force vector is calculated using theresults of the previous examples. This gives

0.16383Pl - 0.102414q12

The joint rotations are then

=K-R 2 0.007761 P - 0.006535qiEI -0.002693P + 0.006535ql

The deflection in any span can be found by super-position. For example, the equation of deflection forthe middle span can be found by superimposing thedeflections due to the uniformly distributed load andthe above joint rotations.

CONCLUStONSA procedure is described for generating macroele-

ments for variable section beams. The method pro-vides a tool for the analysis of frames and continuousbeams with variable cross-sections and materials. Theanalysis can be performed with any desired accuracywith less time and storage requirement than theconventional finite element method with substructur-ing technique. A symbolic manipulation system isused in deriving the terms of the stiffness matrix andload vector. The theory of distributions is used in theformulation of the element. The procedure can alsobe applied to buckling and vibration problems ofbeams and frames with arbitrary profiles.

REFERENCESJ. S. Przemieniecki, Matrix structural analysis of sub-structures. AZAA Jnl. l(1) (1963).T. Furike, Computerized multiple level substructuringanalysis. Compu~ Srrucr. 2, 1063-1074 (1972).R. H. MacNeal and C. W. McCormick. Comnuterizedsubstructure analysis. Proc. World Co&ess bn FiniteElement Methods in Structural Mechanics, Dorset, UK,1975.A. K. Noor and C. M. Andersen, Computerized sym-bolic manipulation in structural mechanics-progressand potential. Compur. Strucr. 10, 95-118 (1977).

5. A. K. Noor, H. A. Kamel and R. E. Fulton, Substruc-turing techniques--status and projections. Compur.Struct. 8, 621-632 (1977).6. R. D. Cook, Concepts and Applications of Finire ElemenrAnalysis, 2nd Edn. John Wiley, New York (1981).7. P. A: N. Dirac, The physical interpretation of quantummechanics. Proc. R. Sot.. Ser. A 113 (1926).

8. L. Schwarts, Theorie desDisrriburion. kern&n & Cie(1950).9. G. Temule. The theorv of generalized functions. Proc.10.

11.12.13.

14.15.16.17.18.19.

R. SO C.~S~;. A 228, l?S-l& (1955).A. Erdblyi, From delta function to distributions. InModern Mathematics for Engineers, 2nd Series (Editedby E. F. Beckenbach), pp. 5-50 (1961).A. Erdtlyi, Operarional Calculus and Generalized Func-tions. Holt, Rinehart & Winston, New York (1962).I. M. Gelfand and G. E. Shilov, Generalized Functions,Vols 1-5. Academic Press, New York (1964).T. P. G. Liverman, Generalized Functions and DirecrOperational Merhoak. Prentice-Hall, Englewood Cliffs,NJ (1964).J. G. Mikusinski, Operarional Calculus. PergamonPress, Oxford (1959).G. F. Roach, Greens Functions. Cambridge UniversityPress, Cambridge (1970).I. Stackgold, Greens Functions and Boundary ValeProblems. John Wiley, New York (1979).F. Arbabi, Discussion of Macalays method: a general-ization. J. Enrme. Mech. Diu.. ASCE lO@Mll (1980).F. Arbabi, A&&t solution df the equivalent &a-me fdrtwo way slabs. Compur. Srrucr. is(l), 159-164 (1984).F. Arbabi, A singularity method for continuous beamswith variable sections. Proc 1st East Asian Conferenceon Structural Engineering and Construction, Bangkok,1986.20.

21.

22.

23.24.

25.26.27.28.

F. Arbabi, Srrucrural Analysis and Behavior, Chapter 9.McGraw-Hill, New York (1991).M. M. Cecchi and C. Lami, Automatic generation ofstiffness matrices for finite element analysis. Inr. J.Numer. Merh. Enpna 11 (1977).A. R. Komcoff aid-S. J.-Fe&es, Symbolic generationof finite element stiffness matrices. Compur. Srrucr. 10,119-126 (1979).D. R. Stoutemyer, Symbolic computation comes of age.SIAM New s 12 (1979).A. K. Noor and C. M. Andersen, Computerized sym-bolic manipulation in nonlinear finite element analysis.Compur. Struck 13, 37943 (1981).S. Steinberg, Mathematics and symbol manipulation.ACM SIGSAM Bull. 16 (1982).R. J. Fateman, My view of th; future of symbolic andalgebraic computation. SIGSAM Bull. 18 (1984).MUMATH, Microsoft, 10700 Northup Way, Bellevue,WA 98004 (1987).F. Li, The development of macroelements with distri-butions. Ph.D. Dissertation, Michigan TechnologicalUniversity, Houghton, MI (1989).

Macro Elements for Variable-section Beams

Documents

Transcript of Macro Elements for Variable-section Beams