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Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM 14 The Plane Stress Problem IFEM Ch 14 Slide 1 Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Plate in Plane Stress or

  1. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM 14 The Plane Stress Problem IFEM Ch 14 – Slide 1

  2. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Plate in Plane Stress or transverse dimension Thickness dimension z y x Inplane dimensions: in x,y plane IFEM Ch 14 – Slide 2

  3. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Plane Stress Physical Assumptions Plate is flat and has a symmetry plane (the midplane) All loads and support conditions are midplane symmetric Thickness dimension is much smaller than inplane dimensions Inplane displacements, strains and stresses uniform through thickness Transverse stresses σ , σ and σ negligible yz xz zz Unessential but used in this course: Plate fabricated of homogeneous material through thickness IFEM Ch 14 – Slide 3

  4. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Notation for stresses, strains, forces, displacements In-plane internal forces y p yy h p yy y p p xx p xy xy x p xx x z In-plane stresses h σ y yy σ = σ σ y xy yx x xx x In-plane displacements In-plane strains h h y y u e y yy e = e u e x xx yx xy x x IFEM Ch 14 – Slide 4

  5. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Mathematical Idealization as a Two Dimensional Problem y Midplane Γ Plate x Ω IFEM Ch 14 – Slide 5

  6. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Inplane Forces are Obtained by Stress Integration Through Thickness Inplane stresses σ yy h y σ xy = σ x z σ xx y x y x h p yy y p xx p xy x Inplane internal forces (also called membrane forces) IFEM Ch 14 – Slide 6

  7. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Plane Stress Boundary Conditions n (unit t exterior normal) ��� �� � ^ σ t n ^ t nt t ��� �� �� � � � � σ ^ t n ^ Γ t + Γ t ��� � � � � � u σ nn Stress BC details ��� � �� �� � � � � � (decomposition of forces ^ q would be similar) ^ u = 0 ��� � � � ^ Boundary tractions t or ^ ^ boundary forces q Boundary displacements u are prescribed on Γ are prescribed on Γ t u IFEM Ch 14 – Slide 7

  8. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM The Plane Stress Problem Given: geometry material properties wall fabrication (thickness only for homogeneous plates) applied body forces boundary conditions: prescribed boundary forces or tractions prescribed displacements Find: inplane displacements inplane strains inplane stresses and/or internal forces IFEM Ch 14 – Slide 8

  9. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Matrix Notation for Internal Fields � u x ( x , y ) � u ( x , y ) = displacements u y ( x , y )   e xx ( x , y ) e ( x , y ) = e yy ( x , y ) strains   2 e xy ( x , y )   σ xx ( x , y ) σ ( x , y ) = σ yy ( x , y )   stresses σ xy ( x , y ) IFEM Ch 14 – Slide 9

  10. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Governing Plane Stress Elasticity Equations in Matrix Form     � u x e xx ∂/∂ x 0 �  = e yy 0 ∂/∂ y    u y 2 e xy ∂/∂ y ∂/∂ x       σ xx E 11 E 12 E 13 e xx  = σ yy E 12 E 22 E 23 e yy      σ xy E 13 E 23 E 33 2 e xy � ∂/∂ x � b x � 0 �   σ xx � � 0 ∂/∂ y  + = σ yy  0 ∂/∂ y ∂/∂ x b y 0 σ xy or D T σ + b = 0 e = Du σ = Ee IFEM Ch 14 – Slide 10

  11. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Strong Form Tonti Diagram of Plane Stress Governing Equations Displacement Prescribed BCs Body forces Displacements displacements ^ b u = u u Γ ^ u on Γ Ω u σ D + b = 0 e = D u Kinematic Equilibrium in Ω in Ω σ = E e in Ω Prescribed Force BCs Strains Stresses tractions t σ e σ ^ Constitutive T or forces q n = t T or ^ p n = q on Γ t IFEM Ch 14 – Slide 11

  12. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM TPE-Based Weak Form Diagram of Plane Stress Governing Equations Displacement Prescribed BCs Body forces Displacements displacements ^ b u = u Γ u ^ u on Γ Ω u δΠ = 0 e = D u Equilibrium Kinematic in Ω in Ω (weak) σ = E e Force BCs in Ω (weak) Prescribed Strains Stresses tractions t σ δΠ = 0 e Constitutive or forces q on Γ t IFEM Ch 14 – Slide 12

  13. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Total Potential Energy of Plate in Plane Stress � = U − W � � h σ T e h e T Ee d � U = 1 1 d � = 2 2 � � � � h u T ˆ h u T b d � + W = t d Ŵ � Ŵ t IFEM Ch 14 – Slide 13

  14. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Discretization into Plane Stress Finite Elements (a) (b) (c) Ω (e) Γ (e) Γ Ω IFEM Ch 14 – Slide 14

  15. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Plane Stress Element Geometries and Node Configurations 3 4 3 3 8 3 4 9 5 2 12 10 7 6 2 11 6 1 1 2 1 1 5 2 4 n = 3 n = 4 n = 6 n = 12 IFEM Ch 14 – Slide 15

  16. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Total Potential Energy of Plane Stress Element (e) Ω Γ (e) � ( e ) = U ( e ) − W ( e ) � � U ( e ) = 1 h σ T e = 1 h e T Ee d � ( e ) 2 2 � ( e ) � ( e ) � � W ( e ) = h u T b d � ( e ) + Ŵ ( e ) h u T t d Ŵ ( e ) � ( e ) IFEM Ch 14 – Slide 16

  17. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Constructing a Displacement Assumed Element n nodes, n= 4 in figure Node displacement vector: u ( e ) = [ u x 1 u yn ] T u y 1 u x 2 . . . u xn Displacement interpolation � u x ( x , y ) � N ( e ) N ( e ) N ( e ) � � 0 0 0 . . . u ( e ) 1 2 n u ( x , y ) = = N ( e ) N ( e ) N ( e ) u y ( x , y ) 0 0 0 . . . 1 2 n = N u ( e ) N is called the shape function matrix IFEM Ch 14 – Slide 17

  18. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Element Construction (cont'd) Differentiate the displacement interpolation wrt x,y to get the strain-displacement relation ∂ N ( e ) ∂ N ( e ) ∂ N ( e )   1 2 n 0 0 . . . 0 ∂ x ∂ x ∂ x   ∂ N ( e ) ∂ N ( e ) ∂ N ( e )   u ( e ) = Bu ( e ) e ( x , y ) = n 1 2  0 0 . . . 0  ∂ y ∂ y ∂ y     ∂ N ( e ) ∂ N ( e ) ∂ N ( e ) ∂ N ( e ) ∂ N ( e ) ∂ N ( e )   1 1 2 2 n n . . . ∂ y ∂ x ∂ y ∂ x ∂ y ∂ x B is called the strain-displacement matrix IFEM Ch 14 – Slide 18

  19. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Element Construction (cont'd) Element total potential energy 2 u ( e ) T K ( e ) u ( e ) − u ( e ) T f ( e ) � ( e ) = 1 Element stiffness matrix � K ( e ) = � ( e ) h B T EB d � ( e ) Consistent node force vector � � f ( e ) = � ( e ) h N T b d � ( e ) + Ŵ ( e ) h N T ˆ t d Ŵ ( e ) body force surface force IFEM Ch 14 – Slide 19

  20. Department of Engineering Mechanics PhD. TRUONG Tich Thien Introduction to FEM Requirements on Finite Element Shape Functions Interpolation Conditions: N takes on value 1 at node i , 0 at all other nodes i Continuity (intra- and inter-element) and Completeness Conditions are covered later in the course (Chs. 18-19) IFEM Ch 14 – Slide 20

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