By Franklin Y Cheng; Yuanxian Gu
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Additional info for Computational mechanics in structural engineering : recent developments
Sensitivity of Cost Function on Structural Probability Failure and Optimum Solution This example shows the influence of cost function on structural probability failure (or safety). , 1994). A ten-story one-bay steel shear-building is optimized with probability failure at Pfo=10-1, 10-3, 105, and 10-7, where Pro is based on normal distribution. The structure is optimized with the earthquake load of equivalent lateral force recommended in UBC. The cost function comprises three components: initial construction cost (Ct), future failure cost (Lf), and system probability of failure (PIT)" They are expressed as C T : Ci+LfPfr (10) in which C I = Cu~LiAi + Cn; Cu = a unit steel volume cost; Cn = nonstructural members cost; Lf = CvC~ + CL; JCv = coefficient to describe the ratio of repair cost to initial cost; CL = business and human losses; and Pz~ = system probability of failure.
Therefore uM(x) is a proper u(x). -, M, G~X-~l::7~'at (12) and u 0h(x) are the FE solutions of the following problems respectively • 1. hk(~)~hk(N:,m(~))d~=E Ic,(v)a,j,~:(~)d~ VveSho(Q) eeSh ; eeS h e (13) S h denotes the FE partition on 1-square Q. t-aijpq ( ~) E pqe ( S hkh (~))~~ 2. From virtual work Eqn. a~a,h(~)Nalhm(~)SO(v)d~ N~,m(~)(ctl,m- It is well-known that the true solution well as the accurate Notl~2m(¢) (a,,a2,m= h values of (16) VI~ sSho (Q) 1,2) of Eqn. al,m 3. ~1m solution of the following virtual work equation Z eeS h VI~ ~ S o ( Q ) = 1,2, (17) 1 = 1...
20 to obtain u0h°(x). 7. ~, in the formulations ( 31 )--( 35 ). 8. Evaluate the approximate displacements M(x) e h~ u Mh(x) in formulation ( 12 ) , the strains in formulation ( 36 ), and the stresses cr~ (x) in formulation ( 37 ). Numerical Results We have coded the computing program of the FE method based on TSA for 2-dimension case, and made some numerical experiments to verify its effectiveness. Here are some numerical results. The structure is a cantilever investigated by us; the macroscopic model is shown in Fig.