Evaluation of The Stiffness Matrix of An Indeterminate Truss Using Minimization Techniques. Samir H. Helou*

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1 83 An-Najah J. Res., Vol. 4, No. 10, ( 1996) Samir Helou Evaluation of The Stiffness Matrix of An Indeterminate Truss Using Minimization Techniques Samir H. Helou* ABSTRACT For an existing reinforced concrete or steel structure the evaluation of the stiffness matrix may be hampered by certain physical limitations such as material deterioration resulting from prolonged use an adverse or in a corrossive environment. The following is a method that allows the determination of the member stiffness of an indeterminate truss through a minimization technique of a properly constructed Error Function. Thus exact sectional and material properties do not have to be known a priori. 00-4w.L. lite 63L.2.11 Ii5164.4,)! j1,...71 lay. 3.-4,3, ,010.c IiiY1 (3. pilaks,1 /IX 4,11 t.t...qp L.C.71.7 I c.6 :7; yl-iil cjr.c.532.!rs411 oast, ia 4.-1: 5 Assistant Professor, Civil Engineering, An-Najah National University, Nablus.

2 84 An-Najah J. Rei., Vol.4, No. 10, ( 1996) Sainir Helou Introduction Analysis of an existing old structure is usually difficult to perform due to the fact that material properties change over lime. This limitation is frequently encountered in industrial buildings housing a corrossive environment. Particularly relevant are reinforeced concrete buildings in which moisture cause steel rusting and eventual spalling of concrete, specially if this is coupled with lack of effective maintenance. Therefore standard structural analysis methods become inapplicable if accurate results are sought, because such methods hinge upon the availability of the member properties and the geometry of the structure. The following method overcomes such a difficulty through the application of known forces at the nodes and the subsequent measurement of the associated displacements. Ealier presentation of the method(1) were limited to determinate trusses. The follolaing is an extension of the same principles, albeit in more general terms, in order to make the method equally applicable to indeterminate systems. It should be noted, however, that this solution in its present form is conceptual in nature and requires further development in order to make it suitable for the industrial community. Problem Statement and Solution : In structural mechanics the force, displacement equilibrium equation is written in the following form In which { F. } is the force vector applied at the nodes, [K] is a global stiffness matrix, (F) [K](X) (1) {X} is the associated displacement vector at the nodes.

3 85 An-Najah J. Res., Vol. 4, No. 10, ( 1996 ) Samir Helou true For the exact solution of equation 1, the following statement is {F}-11C1 {X}=0 (2) And when equation 1 is not exact an error vector E may be introduced as follows: {E}={F}-[K}{X} (3) A typical element in the error of equation 3 is of the form E. ;.F1 EKi1Xi i=i where n is the number of degrees of freedom. The problem is now reduced to minimizing to zero the error vector of equation 4. For this to be achieved an error function has to be constructed. This is done by squaring both sides of equation 4, i.e by forming the inner product of the right hand side of equation 4 with itself and carrying out the summation over the number of loading conditions. The necessity of using more than one load vector will be made clear later on in the text. The error function takes the form in which m is the number of loading cases, (4) m n EF.LE[Fi3K ik x, (5) j=i i=1 k=1 The solution proceeds by taking the first derivative of the error function, EF, with respect to each unknown element of the stiffness matrix and setting it equal to zero, i.e. aef 5E1 0 (6)

4 86 An-Najah J. Res., Vot. 4, No. 10, ( 1996) Samir Belau This operation will result in a set of linear simultaneous equations equal in number to the elements of the structure m I[P] -1 ((F)' [K](X) 1 ) = 0 (7) in which [J] is a Jacobean matrix defined as follows a EF, a k a EF. a EF, k, a EF a k na k m (8) Equation 7 yields rn ypilt(f)=z[nt[k](xp J= 1 LPITP1(K) j=1 (9) where (k) is the vector of element stiffnesses in local coordinates. Furthermore with Un T[.T] invertible the solution for (k) is written formally as (k) (Dliff [J][k]) -1 1 PftF) ) (10) j=i

5 87 An-Najah J. Res., Vol.4, No. 10, ( 1996) Samir Relou From the solution it remaines to be shown that {K) {K } { X }. This will be shown in the course of the illustrative example. Illustrative Example : Figure 1-a Load case No. 1 4K

6 88 An-Najah J. Res., Vol.4, No. 10, ( 1996) Sarnir ReIon Figure 1-b Lead Case No. 2 K 114 For the indeterminate truss shown in Figure 1 -a and b. All elements have an area of 4 in and modulus of elasticity =30000 ksi The global reduced stiffness k 1+.64k6 k4+.36k6 (lc} = ki 0 kl+.64k k5 k2+.36k5 SYM.48k6 0 0 k3+.64k6

7 89 An-Najah J. Res., Vol.4, No. 10, ( 1996) Sa mir Belau To assure the existence of a solution two loading cases are used. The following are the loading cases together with the associated displacements used in the present numerical experiments {F 1 = 0 {X' and (F 2 ) = {X 2 0 } The error vector is written as EL2 E3 4 ce5 - F3 F4 5 ki+.64k k6 k4+.36k6 k/ 0 kl+.64k k5. 6 4k6. 4 8k6 0.. k2+.36k5 0 SRI 2. k3+.64k6 i345 1

8 90 An-Najh J. Res., Vol. 4, No. 10, ( 1996) Sam ir HeJou The Jacobean matrix is - x1 x x1.48x2.64x x2 0.48x1+.36x2+.48x5 xl + x x3+.48x4 0 0 x x3+.36x x5 0 Upon performing the operation described in equation 10 the unknown elements stiffnesses are readily obtained. They are the same as would be obtained by evaluating EA/L for each element, kip / in Concluding Remarks : From the previous presentation and example it is apparent that the proposed method requires a complete set of data i. e. A displacement reading must be available at every degree of freedom of the structure. This is a shortcoming that perhaps can be avoided through further research.

9 9I Ao-Najah.1. Res., Vol. 4, No. 10, ( 1996) Samir HeIon References Helou, A. FL, Retrieval of System Properties of Existing Structures, International Association for Bridge and Structural Engineering, Colloquium 1993, Copenhagen, Denmark. Matzen, V. C. And NcNiven, H. D., Investigation of the Inelastic Characteristics of a single story Steel Structure Using System Identification and Shaking Table Experiments, Report No. EERC Earthquake Engineering Research Center, University of California, Berkeley, August Shield, P. C., Elementary Linear Algebra, Worth Publisher INC. Touqan, A. R., Najah9l, An Engineering Analysis program for Framed Structures, November White, R. N., Gergely, P. Sexsmith, R. C., Structural Engineering, Combined Edition, John Wiley and Sons.

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

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