A.H. Helou Ph.D.~P.E.

Transcription

1 1 EVALUATION OF THE STIFFNESS MATRIX OF AN INDETERMINATE TRUSS USING MINIMIZATION TECHNIQUES A.H. Helu Ph.D.~P.E. :\.!.\STRAC'l' Fr an existing structure the evaluatin f the Sti"ffness matrix may be hampered by certain physical limitatins such as material deteriratin resulting frm prlnged use in a crrsive envirnment. The fllwing is a methd that allws the determinatin f the member stiffness f an indeterminate truss thrugh a Minimizatin tec~nique f an Errr Functin. Thus exact sectinal and material prperties needed fr thrugh structural analysis d nt have t be knwn a priri. * ASSISTANT PROFESSOR AT AN-NAJAH NATIONAL UNIVERSITY 256

2 ACKNOWLEDGMENT The authr wishes t express his gratitude t Mr. Mhammed H. Arafa- fr the invaluabl~ assistance he rendered in the preparatin f this paper. * GRADUATE STUDENT,TEACHING ASSISTANT, AT AN-NAJAH NATIONAL UNIVERSITY 258

3 INTRODUCTION Analysis f existing ld structures is usually difficult t perfrm due t the fact that material prperties change ver time. This limitatin is frequently encuntered in industrial buildings husing a crrsive envirnment. Therefre standard structural analysis methds becme inapplicable if accurate results are sught; this is because presently knwn methds f analysis hinge upn the availability f the member prperties and the gemetry f the structure. The fllwing methd vercmes such a difficulty thrugh the applicatin f knwn frces at the ndes and the subsequent measurement f the assciated displacements. Helu(l) presented the methd but the presentatin was limited t determinate trusses. The fllwing is an extensin f the same principles, albeit in mre general terms, in rder t make the methd equally applicable t indeterminate systems. 259

4 PROBLEM STATEMENT AND SOLUTION In structural mechanics the frce, displacement equilibrium equatin is written in the fllwir.g frm {P} = [K) {X} (1) In which {P} is the frce vectr applied at the ndes, [K) is a glbal stiffness matrix, {X} is the assciated displacement vectr at the ndes. Fr an exact slutin f equatin 1 I the fllwing statement is true (P}-[K) {X} = 0.0 (2) And when equatin 1 is nt exact an errr vectr E may be intrduced as fllws. {E} =(P) - [K) {X} (3) A typical element in the errr vectr f equatin 3 is f the frm where n is the number f degrees f freedm. 260

5 The prblem is nw reduced t minimizing t zer the errr vectr f equatin 4.Fr this t be achieved an errr functin has t be cnstructed. Thii is dne by squaring bth sides f equatin 4,i.e. by frming the inner prduct f the right hand side f equatin 4 with itself and carrying ut the summatin ver the number f lading cnditins. The necessity f using mre than ne lading vectr will be made clear later n in the text. The errr functin takes the frm. (5) In which m is the number f lading cases The slutin prceeds by taking the first derivative f the errr functin, EP, with respect t each unknwn element f the stiffness matrix and setting it equal t zer, i.e. (6) This peratin will result in a set f linear simultaneus equatins equal in numbe~ t the elements f the structure. (7) 261

6 in which (J) is a jacbean matrix defined as fllws [J] - (8) which will yield t ~JD [Jj) T{F}j_~JD [Jj) T[K]{x}J ~j-l ~j-l (9) _~m [Jj) T[J]{k} ~j-l where {k} is the vectr f element stiffnesses in lcal crdinates furthermre with [JJT[J) invertible the slutin fr (k} is written trmally as (10) Frm the slutin it remains t be is shwn that {J}{K} = (K}(X) Thisfwill be. dne in the curse f the illustrative example. 262

7 I 4K ~-----=-----"7I Z~ f------l / Figure l-b LOAD CASE NO. 2 Fr the indeterminate truss shwn in Figure 1 a and b. All elements have an area f 4 in 2 and mdulus f 263

8 elasticity = ksi.the glbal reduced stiffness matrix btained by standard structural ~nalysis methds is kl+.64k6 sm {xl -.48k6 -kl -.64k6 k4+.36k6.48k6 kl+.64k5.48k5 k2+.36ks k3+.64k6 T assure the existence f a slutin tw lading cases are used. The fllwing are the lading cases tgether with the assciated displacements used in the present numerical experiments. 4 ~ {F 1 l 0 {xl} and 4-5 {F2} {x2l

9 The errr vectr is written ~ kl+.64k5 -.48k5 -kl -.64k5 k4+.36k5.48k5 kl+.64ks.48ks SYM k2+.36ks k3+.64k5 The Jacbean Matrix is [J) - Xl-X Xl+X3 0 X4 0 0 X2 0 0 XS 0.64X3+.48X4 48X3+. 36X4. 64Xl-. 48X2-. 64XS -. 48Xl+. 36X2+. 48X Xl+. 48X2+. 64XS Upn perfrming the peratin described in equatin 10 the unknwn elements stiffnesses are the readily btained. They are the same as wuld be btained by evaluating EAjL fr each element. {x} kip/in 265

10 CONCLUDING REMARK Frm the previus presentatin and example it is apparent that the prpsed methd requires a cmplete set f data i.e. A displacement reading must be available at every degree f freedm f the structure. This is a shrtcming that perhaps can be avided thrugh further research in this area. 266

11 REFERENCES 1- Helu,A.H., Retrieval f System Prperties f Existing Structures, Internatinal Assciatin fr Bridge and Structural Engineering, Cllquium 1993, Cpenhagen, Denmark. 2- Matzen,V.C. And McNiven, H.D., Investigatin f the Inelastic Characteristics f a Single Stry Steel Structure Using System Identificatin and Shaking Table Experiments, Reprt N.EERC Earthquake Engineering Research Center, University f Califrnia, Berkeley, August Shield, P.C.,Elementary Linear Algebra,Wrth Publisher INC. 4- Tuqan,A.R.,Najah91, An Engineering Analysis Prgram fr Framed Structures,Nvember White,R.N.,Gergely,P.,Sexsmith,R.C.,Structural Engineering, Cmbined Editin, Jhn Wiley and Sns. 267