By Yu-Qiu Long, Song Cen, Zhi-Fei Long
Complicated Finite point process in Structural Engineering systematically introduces the study paintings at the Finite aspect strategy (FEM), which used to be accomplished by way of Prof. Yu-qiu lengthy and his learn staff long ago 25 years. Seven unique theoretical achievements - for example, the Generalized Conforming point procedure, to call one - and their functions within the fields of structural engineering and computational mechanics are mentioned intimately. The booklet additionally exhibits the recent options for heading off 5 problems that exist in conventional FEM (shear-locking challenge of thick plate parts; sensitivity challenge to mesh distortion; non-convergence challenge of non-conforming parts; accuracy loss challenge of pressure ideas by way of displacement-based parts; pressure singular aspect challenge) by using foregoing achievements.
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2-51), we have 33c(b ) 33p( b ) ³ (b) (b) ª º wM ns · § § ww · w( b ) » ds ¦ w(b ) ('M ns )( b ) M (b ) ¨ ¸ ¨ Qn « ¸ Cab ws ¹ © wn ¹ © J1 J 2 ¬« ¼» Substitution of this equation into (2-38) yields 33 33p( a ) 33p(b ) H pp G1pp G2pp (2-55) where Hpp, G1pp and G2pp are the additional potential energy terms on the interface Cab and the nodes J1 and J2: H pp (b) ° (b ) ª§ ww ·( a ) § ww ·(b ) º § ½ wM ns · (a) (b ) ° ( ) M Q w w « » ® ¾ ds n n ¨ ¸ ¨ ¸ ¨ ³ Cab ws ¸¹ © wn ¹ »¼ © «¬© wn ¹ ¯° ¿° (2-56a) G1pp ¦ [('M ns )( a ) ( w( a ) w) ('M ns )( b ) ( w(b ) w)] (2-57) J1 G2pp ¦ [('M ns )(b ) ( w( a ) w(b ) ) Rw( a ) ] (2-58a) J2 Equations (2-55), (2-56a), (2-57) and (2-58a) are the functional expressions of the sub-region three-field generalized potential energy principle.
The positive deflection w is downward; the positive normal rotation \n rotates from n to z; the positive tangent rotation \s rotates from s to z; the normal bending moment Mn is positive when the bottom of the plate is under tension; the twisting moment Mns 41 Advanced Finite Element Method in Structural Engineering is positive when it produces positive shear stress Wns along the positive direction of s at the bottom of the plate; and the positive transverse shear force Qn is also downward. The key points of the sub-region generalized mixed variational principle can be listed as follows.
2-74) is equivalent to all field equations, boundary conditions, interface conditions, corner point and node conditions of the thin plate system with multi-regions. The procedure for deriving the node conditions of the node J from the stationary condition (2-77) is given in Appendix B. If all the sub-regions are potential energy regions, the functional of the sub-region potential (or generalized potential) energy principle can be obtained from Eq. (2-74): 3 ¦ : 3 p ¦ H pp ¦ G1pp ¦ G2pp Cpp p J1 (2-78) J2 where G1pp and G2pp can be obtained from Eqs.