# Chapter2 - Chapter 2 Two-Dimensional Problems 1.The Overall...

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Chapter 2 1 Chapter 2 Two-Dimensional Problems 1.The Overall Solution Procedure Objective : To focus on the overall solution procedure. Consider the cantilever beam shown in Fig.1.1. The beam is loaded with distributed load T on the top surface. 1 2 1 26 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 3 4 5 7 8 9 10 11 12 1.5 m 1 m J = 012 3 (a) Finite element mesh

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Chapter 2 2 a b 2 3 4 1 1 2 3 4 5 6 7 8 (b) Typical element b ξ η a Figure 1.1 Rectangular beam and a typical rectangular element 1.1 Element Matrices The theoretical basis for the derivation of the element matrices has been presented in Section 3.2 of Chapter 1 and is illustrated in Fig.1.2. Note that in this figure, the total virtual work in a typical element is presented in the format: Total virtual work, W = Internal work - External work. The present application to simple rectangular elements further serves to clarify this theoretical basis. δ ε T E ε + dV − ∫ δ u T dS S et T V e N T f δ q T N T δ q T T B T E B q δ q T δ q T 1 × n k n × n q n × 1 Q n × 1 ) ( W e = W e = W V = δ r T 1 × N K N × N r N × 1 R N × 1 ( ) = 0 Typical element: System: dV V e δ u f T δε T E ε dV V e o T σ o dV V e Virtual Work in Typical Element Strains Initial strains Initial stresses Body forces Surface tractions B T E δ q T ε o B T δ q T σ o ( ) Internal virtual work External work Figure 1.2. Virtual work in a typical element and its contribution to the system
Chapter 2 3 A typical element is shown in Fig.1.1(b). The displacement of a point in the element may be specified by its horizontal and vertical components: u = u ( ξ , η ) v ( ξ , η ) Assume linear variation for u and v : u ( ξ , η ) = a 1 + a 2 ξ+ a 3 η + a 4 ξη v ( ξ , η ) = a 5 + a 6 a 7 η+ a 8 ξη (1.1) In terms of the nodal displacements: u ( ξ , η ) = i = 1 4 N i ( ξ , η ) u i v ( ξ , η ) = i = 1 4 N i ( ξ , η ) v i (1.2) where the shape functions and their derivatives are N 1 = 1 4 ab ( a −ξ )( b −η ) 1 4 ab ( b ) 1 4 ab ( a ) N 2 = 1 4 ab ( a b ) 1 4 ab ( b ) 1 4 ab ( a ) N 3 = 1 4 ab ( a b ) 1 4 ab ( b ) 1 4 ab ( a ) N 4 = 1 4 ab ( a b ) 1 4 ab ( b ) 1 4 ab ( a ) N i N i , ξ = N i ∂ξ N i , η = N i ∂η (1.3) Let q = { u 1 v 1 u 2 v 2 u 3 v 3 u 4 v 4 } u ( ξ , η ) v ( ξ , η ) = N 1 . N 2 . N 3 . N 4 . . N 1 . N 2 . N 3 . N 4 q = Nq (1.4) Strains

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Chapter 2 4 ε ξ ε η γ ξη = u ∂ξ v ∂η u ∂η + v ∂ξ = Bq (1.5) where B = N 1, ξ . N 2, ξ . N 3, ξ . N 4, ξ . . N η . N η . N η . N η N η N ξ N η N ξ . N η N ξ N η N ξ (1.6) Stress-strain relation (Eq.1.3.5): σ = E ( ε − ε o ) = E ( Bq − ε o ) (1.7) where the elasticity matrix for plane stress condition is E = E 1 −ν 2 1.
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Chapter2 - Chapter 2 Two-Dimensional Problems 1.The Overall...

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