Read e-book online Computational mechanics in structural engineering : recent PDF

By Franklin Y Cheng; Yuanxian Gu

ISBN-10: 0080430082

ISBN-13: 9780080430089

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Through complicated finite element analysis, see [3,9], we can obtain the following results: the ~ _k'~_~'9... ~ I I I I. l~,~%~kNI I IkN~'~ ~ I I I I i I I I i I I tJ I ! ~-N~I~X'-'~ll I I ',,~~k"~ ' ~ ~ k ' ~ III III II! ill l l l l l l l ~ ~ lllllll~l~-~C~ I I I I 1 ~ ] ~ 1 I ! 6697 175017 1143604 0 Fig. ,~,m L2(a) where C and A are constants independent of (26) h, e and ~ , and (28) FE COMPUTATION OF u Mh(x) IrE Computation o f Uho° (X) According to Eqn. 12, the next step of computing u Mh(x) is to solve the homogenized FE vitual work equation ( 19 )to obtain ,0h°(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.

The first step is to introduce the Hamiltonian system theory into the fundamental equations, and the longitudinal coordinate x is treated analogous to the time coordinates (Zhong and Yang, 1991); thus (') represents 0( ) / & . X. A. TABLE 1 ANALOG Y RELATIONSHIP BETWEEN PLANE ELASTICITY AND PLATE BENDING Plane elasticity Airy stress function Plate bending Deflection w ( x , y ) tp Displacement vector u,v Strain Bending moment fimction vector qbx,~y Bending moment e x , ~,y ,'~ xy ; Ou Ov Ou Ov My OdOx M x = M y , M x ,2 Mxy ; O' y ,2 M xy = O, x + : Ox' er x 02(1) 02(p 02(p ~y 2 ' er Y O~ 2 ' T"xy ~X~ 02W 02W Ky "-" OY2 ,K x : ~~X, 2 I~2xY -- Stress-strain relation, ex = (erx - Very) / E ,ey = (Cry - Verx) / E Ox O2W OxOy Bending moment-curvature relation M y = O(K:y + VKx), Mx = D ( K x + V K y ) , 2 M xy = 2 ( 1 - v ) DK xy 7xy = Xxy2(1 + v) / E The principle of minimum potential energy Determined displacement boundary s u, u=u-, Oy Deflection-curvature relation Stress function-stress relation The principle of minimum" complementary energy Given force boundary s m , m v=v Cs=¢s, Determined force boundary s~, Given displacement boundary s w, Ky COSO~ -1- l(xy sincz = 0 cr x cosc~ + Xxy sina = 0 K xy cos or + ~:x sin cz = 0 x xy cosa + ~y sin a = 0 Pro-H-R variational principle H-R variational principle Null moment functions H-R variational principle Pro-H-R variational principle Rigid body translation of Eqn.

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Computational mechanics in structural engineering : recent developments by Franklin Y Cheng; Yuanxian Gu

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