MIT2_094S11_lec4

MIT2_094S11_lec4 - 2.094 — Finite Element Analysis of...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2.094 — Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 4 - Finite element formulation for solids and structures Prof. K.J. Bathe MIT OpenCourseWare We considered a general 3D body, Reading: Ch. 4 The exact solution of the mathematical model must satisfy the conditions: • Equilibrium within t V and on t Sf , • Compatibility • Stress-strain law(s) I. Differential formulation II. Variational formulation (Principle of virtual displacements) (or weak formulation) We developed the governing F.E. equations for a sheet or bar We obtained t F = tR (4.1) where t F is a function of displacements/stresses/material law; and t R is a function of time. Assume for now linear analysis: Equilibrium within 0 V and on 0 Sf , linear stress-strain law and small displacements yields t F = K · tU (4.2) We want to establish, K U (t) = R(t) (4.3) 14 MIT 2.094 4. Finite element formulation for solids and structures Consider ˆ UT = � U1 V1 W1 U2 · · · WN � (N nodes) (4.4) ˆ where U T is a distinct nodal point displacement vector. Note: for the moment “remove Su ” We also say � ˆ U T = U1 U2 U3 · · · Un � (n = 3N ) (4.5) We now assume ⎤(m) u = ⎣ v ⎦ w ⎡ ˆ u(m) = H (m) U , u(m) (4.6a) ˆ where H (m) is 3 x n and U is n x 1. ˆ �(m) = B (m) U (4.6b) where B (m) is 6 x n, and T �(m) = e.g. γxy � �xx �yy ∂v ∂ u = + ∂x ∂y �zz γxy γyz γzx � We also assume u(m) � (m) = ˆ H (m) U (4.6c) = (m) ˆ U (4.6d) B 15 MIT 2.094 4. Finite element formulation for solids and structures Principle of Virtual Work: � � T T ˆ � τ dV = U f B dV V (4.7) V (4.7) can be rewritten as �� m T �(m) τ (m) dV (m) = �� V (m) m ˆ U (m) T fB (m) dV (m) (4.8) V (m) Substitute (4.6a) to (4.6d). � � � � T (m) T (m) (m) ˆ U B τ dV = V (m) m ˆ U T � �� H (m) T f B (m) (4.9) � dV (m) V (m) m ˆ τ (m) = C (m) �(m) = C (m) B (m) U Finally, � � T �� ˆ U � B (m) T (4.10) � C (m) B (m) dV (m) ˆ U= V (m) m � � T �� ˆ U � m H (m) T (4.11) � fB (m) dV (m) V (m) with T T ˆ �(m) = U B (m) T (4.12) ˆ KU = RB (4.13) where K is n x n, and RB is n x 1. Direct stiffness method: � K= K (m) (4.14) m RB = � (m) RB (4.15) m K (m) (m) RB � T B (m) C (m) B (m) dV (m) = �V V (4.16) (m) (m) T H (m) f B = (m) dV (m) (4.17) 16 MIT 2.094 4. Finite element formulation for solids and structures Example 4.5 textbook E = Young’s Modulus Mathematical model Plane sections remain plane: F.E. model ⎡ ⎤ U1 U = ⎣ U2 ⎦ U3 (4.18) Element 1 ⎤ U1 0 ⎣ U2 ⎦ �U 3 ⎡ u(1) (x) = � � 1− x 100 x �� 100 H (1) � (4.19) 17 MIT 2.094 4. Finite element formulation for solids and structures ⎤ U1 0 ⎣ U2 ⎦ �U 3 (4.20) x 80 (4.21) ⎡ �(1) (x) = xx � − 11 00 � 1 ��100 B (1) � Element 2 u(2) (x) = � 0 � �(2) (x) xx = � � 0 x 1 − 80 �� H (2) − 810 810 �� B (2) � U � � U (4.22) � Then, ⎡ ⎤ 1 −1 0 0 13E ⎣ E⎣ −1 1 0 ⎦ + 0 K= 240 100 0 00 0 ⎡ ⎤ 0 0 1 −1 ⎦ −1 1 (4.23) where, � � A� < η =0 1 � � AE E (1) ≡ L 100 �� 13 E E · 13 = 3 80 3 · 80 � �� � A∗ � � A∗ < A� (4.24) (4.25) (4.26) η =80 < 4.333 < 9 18 MIT OpenCourseWare http://ocw.mit.edu 2.094 Finite Element Analysis of Solids and Fluids II Spring 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. ...
View Full Document

This note was uploaded on 12/29/2011 for the course ENGINEERIN 2.094 taught by Professor Prof.klaus-jürgenbathe during the Spring '11 term at MIT.

Ask a homework question - tutors are online