Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM 13 Variational Formulation of Plane Beam Element IFEM Ch 13 – Slide 1
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Beams Resist Primarily Transverse Loads IFEM Ch 13 – Slide 2
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Transverse Loads are Transported to Supports by Flexural Action Compressive stress Neutral surface Tensile stress IFEM Ch 13 – Slide 3
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Beam Configuration Spatial (General Beams) Plane (This Chapter) Beam Models Bernoulli-Euler Timoshenko (advanced topic not covered in class) IFEM Ch 13 – Slide 4
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Plane Beam Terminology y , v q ( x ) y , v Beam cross section x, u z Neutral axis Symmetry plane L Neutral surface IFEM Ch 13 – Slide 5
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Common Support Conditions ������ Simply Supported ������ ������ Cantilever IFEM Ch 13 – Slide 6
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Basic Relations for Bernoulli-Euler Model of Plane Beam � − v ′ � � − θ � u ( x , � − ∂v( x ) � y y y � ) � y ∂ x = = = y v( x , ) v( x ) v( x ) v( x ) ∂ x = − ∂ 2 v ∂ x 2 = − d 2 v e = ∂ u y y dx 2 = − κ y σ = Ee = − E d 2 v y dx 2 = − E κ y M = E I κ Plus equilibrium equation M'' = q (not used specifically in FEM) IFEM Ch 13 – Slide 7
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Tonti Diagram for Bernoulli-Euler Model of Plane Beam (Strong Form) Displacement Prescribed Distributed Transverse BCs end transverse load displacements displacements q(x) v(x) κ = v'' Kinematic M''=q Equilibrium Bending M = EI κ Force BCs Prescribed Curvature moment end loads κ (x) Constitutive M(x) IFEM Ch 13 – Slide 8
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Total Potential Energy of Beam Member � = U − W � L � L � 2 � ∂ 2 v � U = 1 σ xx e xx dV = 1 M κ dx = 1 d x E I 2 2 2 ∂ x 2 V 0 0 � L E I κ 2 dx = 1 2 0 � L W = q v dx . 0 IFEM Ch 13 – Slide 9
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Degrees of Freedom of Beam Element θ j v j θ i v i j i v i θ i u ( e ) = v j θ j IFEM Ch 13 – Slide 10
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Bernoulli-Euler Kinematics of Plane Beam Element y ,v P ′ ( x + u , y + v) θ j E = E ( e ) , I = I ( e ) v j θ i v i x, u j i x ( e ) P ( x , y ) ℓ = L IFEM Ch 13 – Slide 11
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Plane Beam Element Shape Functions ξ = 1 ξ = − 1 v ( e ) N ( e ) v i (ξ) = 1 4 ( 1 − ξ) 2 ( 2 + ξ) = 1 i θ ( e ) = 1 i N ( e ) θ i (ξ) = 1 8 ℓ( 1 − ξ) 2 ( 1 + ξ) v ( e ) = 1 N ( e ) v j (ξ) = 1 4 ( 1 + ξ) 2 ( 2 − ξ) j N ( e ) θ j (ξ) = − 1 8 ℓ( 1 + ξ) 2 ( 1 − ξ) θ ( e ) = 1 j IFEM Ch 13 – Slide 12
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Shape Functions in Terms of Natural Coordinate ξ v ( e ) i θ ( e ) i v ( e ) = [ N ( e ) N ( e ) N ( e ) = Nu ( e ) N ( e ) θ j ] v i θ i v j v ( e ) j θ ( e ) j ξ = 2 x ℓ − 1 N ( e ) N ( e ) v i = 1 4 ( 1 − ξ) 2 ( 2 + ξ), = 1 8 ℓ( 1 − ξ) 2 ( 1 + ξ), θ i N ( e ) N ( e ) v j = 1 4 ( 1 + ξ) 2 ( 2 − ξ), = − 1 8 ℓ( 1 + ξ) 2 ( 1 − ξ). θ j IFEM Ch 13 – Slide 13
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Element Stiffness and Consistent Node Forces B = 1 6 ξ − 6 ξ � � 3 ξ − 1 3 ξ + 1 ℓ ℓ ℓ 2 u ( e ) T K ( e ) u ( e ) − u ( e ) T f ( e ) � ( e ) = 1 � ℓ � 1 E I B T B dx = E I B T B 1 K ( e ) = 2 ℓ d ξ 0 − 1 � ℓ � 1 N T q dx = N T q 1 f ( e ) = 2 ℓ d ξ 0 − 1 IFEM Ch 13 – Slide 14
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Analytical Computation of Prismatic Beam Element Stiffness 36 ξ 2 − 36 ξ 2 6 ξ( 3 ξ − 1 )ℓ 6 ξ( 3 ξ + 1 )ℓ � 1 ( 9 ξ 2 − 1 )ℓ 2 ( 3 ξ − 1 ) 2 ℓ 2 − 6 ξ( 3 ξ − 1 )ℓ K ( e ) = E I d ξ 2 ℓ 3 36 ξ 2 − 6 ξ( 3 ξ + 1 )ℓ − 1 ( 3 ξ + 1 ) 2 ℓ 2 symm 12 6 ℓ − 12 6 ℓ 4 ℓ 2 2 ℓ 2 = E I − 6 ℓ ℓ 3 12 − 6 ℓ 4 ℓ 2 symm IFEM Ch 13 – Slide 15
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Mathematica Script for Symbolic Computation of Prismatic Plane Beam Element Stiffness ClearAll[EI,l, Ξ ]; B={{6* Ξ ,(3* Ξ -1)*l,-6* Ξ ,(3* Ξ +1)*l}}/l^2; Ke=(EI*l/2)*Integrate[Transpose[B].B,{ Ξ ,-1,1}]; Ke=Simplify[Ke]; Print["Ke for prismatic beam:"]; Print[Ke//MatrixForm]; Ke for prismatic beam: EI EI � EI EI � � � � � l l � �������� ������� �������� l ���� �������� ������� �������� l ���� � � � � � � � � � � EI EI � EI EI � � � � � � �������� l ���� �������� ����� �������� l ���� �������� ���� � � l l � � � � � � � � � EI � EI EI � EI � � � � l l � �������� ������� �������� l ���� �������� ������� �������� l ���� � � � � � � � � � EI EI EI � EI � � �������� l ���� �������� ���� �������� l ���� �������� ����� l l Corroborates the result from hand integration. IFEM Ch 13 – Slide 16
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Analytical Computation of Consistent Node Force Vector for Uniform Load q 1 4 ( 1 − ξ) 2 ( 2 + ξ) � 1 � 1 8 ℓ( 1 − ξ) 2 ( 1 + ξ) 1 f ( e ) = 1 N d ξ = 1 2 q ℓ 2 q ℓ d ξ 1 4 ( 1 + ξ) 2 ( 2 − ξ) − 1 − 1 − 1 8 ℓ( 1 + ξ) 2 ( 1 − ξ) 1 2 1 12 ℓ = q ℓ "fixed end moments" 1 2 − 1 12 ℓ IFEM Ch 13 – Slide 17
Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Mathematica Script for Computation of Consistent Node Force Vector for Uniform q ClearAll[q,l, Ξ ] Ne={{2*(1- Ξ )^2*(2+ Ξ ), (1- Ξ )^2*(1+ Ξ )*l, 2*(1+ Ξ )^2*(2- Ξ ),-(1+ Ξ )^2*(1- Ξ )*l}}/8; fe=(q*l/2)*Integrate[Ne,{ Ξ ,-1,1}]; fe=Simplify[fe]; Print["fe^T for uniform load q:"]; Print[fe//MatrixForm]; fe^T for uniform load q: l q l q � l q l q � ������� ����������� ������� � ����������� Force vector printed as row vector to save space. IFEM Ch 13 – Slide 18
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