The Plane Stress Problem

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1 14 The Plane Stress Problem IFEM Ch 14 Slide 1

2 Plate in Plane Stress Thickness dimension or transverse dimension z Top surface Inplane dimensions: in, plane IFEM Ch 14 Slide 2

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

4 Plane Stress Phsical Assumptions Plate is flat and has a smmetr plane (the midplane) All loads and support conditions are midplane smmetric 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 zz z z Unessential but used in this course: Plate fabricated of homogeneous material through thickness IFEM Ch 14 Slide 4

5 Notation for Stresses, Strains, Forces, Displacements Thin plate in plane stress z In-plane internal forces h d d p p p d d + sign conventions for internal forces, stresses and strains d d h In-plane stresses σ d d σ = σ σ In-plane bod forces h b d d b In-plane strains d d h e e e = e In-plane displacements h u d d u IFEM Ch 14 Slide 5

6 Inplane Forces are Obtained b Stress Integration Through Thickness Inplane stresses Introduction to FEM z h σ σ = σ σ h p p p Inplane internal forces (also called membrane forces) IFEM Ch 14 Slide 6

7 Plane Stress Boundar Conditions ; ; Γu ; ; ^ + u = 0 Boundar displacements u ^ are prescribed on Γu (figure depicts fiit condition) Γ t t^ Boundar tractions ^t or boundar forces q^ are prescribed on Γ t σ n σ σ nt nn ^t n t n (unit eterior normal) ^t ^t t Stress BC details (decomposition of forces q ^ would be similar) IFEM Ch 14 Slide 7

8 The Plane Stress Problem Introduction to FEM Given: geometr material properties wall fabrication (thickness onl for homogeneous plates) applied bod forces boundar conditions: prescribed boundar forces or tractions prescribed displacements Find: inplane displacements inplane strains inplane stresses and/or internal forces IFEM Ch 14 Slide 8

9 Matri Notation for Internal Fields [ ] u (, ) u(, ) = u (, ) e (, ) e(, ) = e (, ) 2e (, ) σ (, ) σ(, ) = σ (, ) σ (, ) displacements strains (factor of 2 in e simplifies "energ dot products") stresses IFEM Ch 14 Slide 9

10 Governing Plane Stress Elasticit Equations in Matri Form e e 2e σ σ σ = = / 0 0 / / / E 11 E 12 E 13 E 12 E 22 E 23 E 13 E 23 E 33 [ ] σ / 0 / σ 0 / / σ [ u u ] e e 2e + [ b b ] = Introduction to FEM [ ] 0 0 or e = Du σ = Ee D T σ + b = 0 IFEM Ch 14 Slide 10

11 Strong-Form Tonti Diagram of Plane Stress Governing Equations Introduction to FEM Prescribed displacements u^ Displacement BCs u = u^ on Γ u Displacements u Bod forces b Γ Ω Kinematic e = D u in Ω T D σ + b = 0 in Ω Equilibrium Strains e σ = E e or e = Cσ in Ω Constitutive Stresses σ Force BCs T t ^ σ n = t or T ^ p n = q on Γ Prescribed tractions t or forces q IFEM Ch 14 Slide 11

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

13 Total Potential Energ of Plate in Plane Stress = U W U = 1 2 W = h σ T e d = 1 2 h e T Ee d h u T b d + h u T ˆt dɣ Ɣ t bod forces boundar tractions IFEM Ch 14 Slide 13

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

15 Plane Stress Element Geometries and Node Configurations n = 3 n = 4 n = 6 n = IFEM Ch 14 Slide 15

16 Total Potential Energ of Plane Stress Element Ω e Γ e U e = 1 2 W e = e e e γ = U e W e h σ T e d e = 1 2 h u T b d e + Ɣ e e h e T Eed e h u T t dɣ e IFEM Ch 14 Slide 16

17 Constructing a Displacement Assumed Element Node displacement vector n nodes, n=4 in figure u e = [ u 1 u 1 u 2... u n u n ] T Displacement interpolation over element [ ] [ e u (,) N 1 0 N e Nn e 0 u(, ) = = u (,) 0 N e 1 0 N e N e n ] u e = N u e N is called the shape function matri It has order 2 2n IFEM Ch 14 Slide 17

18 Element Construction (cont'd) Differentiate the displacement interpolation wrt, to get the strain-displacement relation e(, ) = N e 1 0 N e 1 0 N e 1 N e 1 N e 2 0 N e N e 2 N e 2 N e n N e n 0 N e n N e n u e = B u e B is called the strain-displacement matri It has order 3 2n IFEM Ch 14 Slide 18

19 Element Construction (cont'd) Introduction to FEM Element total potential energ e = 1 2 u e T K e u e u e T f e Element stiffness matri K e = e h B T EBd e Consistent node force vector f e = h N T b d e + e Ɣ e h N T ˆt dɣ e due to: bod force due to: surface tractions IFEM Ch 14 Slide 19

20 Requirements on Finite Element Shape Functions Interpolation Condition N takes on value 1 at node i, 0 at all other nodes i Continuit (intra- and inter-element) and Completeness Conditions are covered later in the course (Chs ) IFEM Ch 14 Slide 20

The Plane Stress Problem

. 4 The Plane Stress Problem 4 Chapter 4: THE PLANE STRESS PROBLEM 4 TABLE OF CONTENTS Page 4.. INTRODUCTION 4 3 4... Plate in Plane Stress............... 4 3 4... Mathematical Model.............. 4 4