e e For element equilibrium, - PowerPoint PPT Presentation

Discretization     e e For element equilibrium,             e e e e e 0 U W U W 3  4 n     w , w N x y w o o i oi   4 w  1 i  o      u  4 n x           , N x y   x 2 y x i xi  1 i 1  4 n        , N y x y y i yi    1 i w 0 0 N o Concisely,   i     4   n 0 0  N N x i u N u i   i i   0 0  N y  1 i i 34

Finite Element Discretization If we define        t ( , , , , , ,......., , , ) d w w w 1 1 1 2 2 2 o x y o x y oNN xNN yNN and   w w w 0 0 | 0 0 | | 0 0 N N N o o o  1 2  NN          0 0 | 0 0 | | 0 0  N N N N x x x 1 2 NN       0 0 | 0 0 | | 0 0  N y N y N y 1 2 NN Then we can also write,  u Nd 35

Discrete Strains   ε L u ε L u s s b b NN  NN  ; L N u ; L N u s i i b i i i=1  1 i NN  NN ;  B u si i ; B u  1 bi i i    1 i Q B =L N   si s i Q B =L N bi b i ; B d s ; B d b ε=Lu Q NN    ; Q ; B u B =LN Bd i i i i 36 i  1

Finite Element Discretization     N y   i 0 0       w x N o  i  0    N y    N i x x      i 0 0   B B  bi    y  si w N o     i 0   N x      i N N y   x   y i i 0       x y 37

Finite Element Discretization NN NN NN 1     e t e t u K u u f i j i i 2 ij    1 1 1 i j j 1 t e t e d K d - d f 2 where the element stiffness matrix linking node i and j is given by   e t K B DB dA ij i j A 38

Finite Element Discretization and the consistent nodal forces for node I are given by     e t t f N b dA N t dS i i i A S t The complete element matrices may be written as   e t K B DB dA A     e t t f N b dA N t dS A S t 39

Boundary Conditions Clamped Free Symmetric Skew Symmetric     0 0 0 0 w Q Q w o n n o       0 0 0 0 M M n n nt t       0 0 0 0 M M t nt n n Simply Supported 1 2 SS SS   0 0 w w o o    0 0 M nt t   0 0 M M n n 41

Effective Design Of Reissner-Mindlin Plate Bending Element One must consider and study the following points on which considerable body of knowledge is now available.  Locking : shear constraints : Kirchhoff mode criteria  Uniform reduced/selective integrations (Equivalence with mixed methods)  Spurious zero energy modes/rank deficiency 42

P P h L 43

TIP DISPLACEMENT (DEEP CANT. BEAM) N 1 Point Quadrature 2 Points Quadrature 1 0.762 0.416 x 10 -1 2 0.940 0.445 4 0.985 0.762 8 0.996 0.927 16 0.999 0.981 44

TIP DISPLACEMENT (THIN CANT. BEAM) N 1 Point Quadrature 2 Points Quadrature 1 0.750 0.200 x 10 -4 2 0.938 0.800 x 10 -4 4 0.984 0.320 x 10 -3 8 0.996 0.128 x 10 -3 16 0.999 0.512 x 10 -3 45

2-Node Timoshenko Beam Element        0 0 0 0 1 1 2 1 1 2 l l       0 1 0 1 1 2 3 1 2 6 l l l l       EI K b K s Gh      0 0 0 0 1 1 2 1 1 2 l l         0 1 0 1  1 2 6 1 2 3  l l l l   Discrete Bending Discrete Shear 2 l h   1 K K O l h s b Energy Energy   t    w w d b s U U 1 1 2 2 e e 1 1   t t U K d K d d d   2  s b … Contradiction e b s U U O l h 2 2 e e l  b U U s  h   U 0 e e Physics e 46

REMEDIAL MEASURES Use of Discrete Kirchhoff Procedure Element matrix equation is stabilized by typing the two independent dofs. at discrete point according to Kirchhoff mode criterion. Method is effective but implementation is complicated NOT WIDELY USED. Use Of Selective Reduced Integration Total strain energy is split into bending and to shear energies. A one order lower quadrature formula than the normal one is used for evaluating constrained shear energy component. 47

Lagrange Serendipity Heterosis w Serendipity Serendipity Lagrange   Serendipity Lagrange Lagrange , 1 2 X b X b 3 3 3 3 Integration X 2 2 X S X S 2 2 2 2 Spurious zero energy - 1* 0 1 es mod * Not communicable in mesh of two or more elements. 48 Hughes and Cohen(1978), Computers and Structures, 9, 445-450.