14 The Plane Stress Problem IFEM Ch 14 Slide 1 Introduction to - PDF document

Introduction to FEM 14 The Plane Stress Problem IFEM Ch 14 – Slide 1

Introduction to FEM Plate in Plane Stress or transverse dimension z Thickness dimension y x Inplane dimensions: in x,y plane IFEM Ch 14 – Slide 2

Introduction to FEM Mathematical Idealization as a Two Dimensional Problem y Midplane Γ x Plate Ω IFEM Ch 14 – Slide 3

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, set to 0 yz zz xz Unessential but used in this course: Plate fabricated of homogeneous material through thickness IFEM Ch 14 – Slide 4

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 Positive sign convention h σ y yy σ = σ σ y yx xy x xx x In-plane displacements In-plane strains h h u y y e y yy e = e u e x xx yx xy x x IFEM Ch 14 – Slide 5

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

Introduction to FEM Plane Stress Boundary Conditions n (unit t exterior normal) �� ��� � ^ σ t n ^ nt t �� �� ��� � � � � t σ ^ t n Γ ^ t Γ + t ��� � � � � � u σ nn Stress BC details ��� �� � � �� � � � � (decomposition of forces ^ ^ u = 0 q would be similar) ��� � � � ^ ^ Boundary tractions t or Boundary displacements u are prescribed on Γ ^ boundary forces q u are prescribed on Γ (figure depicts fixity condition) t IFEM Ch 14 – Slide 7

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

Introduction to FEM Matrix Notation for Internal Fields � u x ( x , y ) � u ( x , y ) = displacements u y ( x , y )   e xx ( x , y ) strains (factor of 2 in e ( x , y ) = e yy ( x , y ) e simplifies "energy xy   dot products") 2 e xy ( x , y ) σ xx ( x , y )   σ yy ( x , y ) stresses σ ( x , y ) =   σ xy ( x , y ) IFEM Ch 14 – Slide 9

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 D T σ + b = 0 or e = Du σ = Ee IFEM Ch 14 – Slide 10

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 Equilibrium Kinematic in Ω in Ω σ = E e in Ω Force BCs Prescribed Strains Stresses tractions t σ e σ T ^ or forces q Constitutive n = t T ^ or p n = q on Γ t IFEM Ch 14 – Slide 11

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

Introduction to FEM Total Potential Energy of Plate in Plane Stress � = U − W � � h σ T e h e T Ee d � 1 U = 1 d � = 2 2 � � � � h u T b d � + h u T ˆ W = t d Ŵ � Ŵ t body forces boundary tractions IFEM Ch 14 – Slide 13

Introduction to FEM Discretization into Plane Stress Finite Elements (a) (b) (c) Ω e Γ Γ e Ω IFEM Ch 14 – Slide 14

Introduction to FEM Plane Stress Element Geometries and Node Configurations 3 4 3 3 8 3 4 9 5 2 10 12 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

Introduction to FEM Total Potential Energy of Plane Stress Element Ω e Γ e � e γ = 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

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 over element � u x ( x , y ) � N e N e N e � � 0 0 . . . 0 1 2 n u e 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 It has order 2 x 2 n IFEM Ch 14 – Slide 17

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 = B u e e ( x , y ) = 2 1 n  0 0 . . . 0  ∂ y ∂ y ∂ y     ∂ N e ∂ N e ∂ N e ∂ N e ∂ N e ∂ N e   2 1 1 2 n n . . . ∂ y ∂ x ∂ y ∂ x ∂ y ∂ x B is called the strain-displacement matrix It has order 3 x 2 n IFEM Ch 14 – Slide 18

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 tractions IFEM Ch 14 – Slide 19

Introduction to FEM Requirements on Finite Element Shape Functions Interpolation Condition 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