Chapter 6. Plane Stress / Plane Strain Problems Element types: Line elements (spring, truss, beam, frame) – chapters 2-5 2-D solid elements – chapters 6-10 3-D solid elements – chapter 11 Plate / shell elements – chapter 12 1 2-D Elements Triangular elements – plane stress/plane strain: CST – “constant strain triangle” – chap. 6 LST – “linear strain triangle” – chap. 8 Axisymmetric elements – chap. 10 Isoparametric elements – chap. 11 4-node quadrilateral element (linear interpolation) 8-node quadrilateral element (quadratic interpolation) 2 Plane stress x 0 y 0 xy 0 z xz yz 0 3 Plane Strain z xz yz 0 x 0 y 0 xy 0 z 0 xz yz 0 4 2-D Stress States Matrix form: x x y xy 5 Principal Stresses 1, 2 x y 2 x y 2 2 1 xy p tan 1 2 x y 2 2 xy 6 Displacements and Strains Displacement field u ( x, y ) u v ( x , y ) Strains x u v u v ,y , xy x y y x x y xy 7 Stress-Strain Relations D Recall: E – Young’s modulus - Poisson’s Ratio G – Shear modulus 8 Stress-Strain Relations (cont.) Plane stress x E y 2 1 xy 1 0 x 1 0 y 1 0 0 xy 2 Plane strain x 1 0 x E 1 0 y y 1 1 2 0 0 1 2 xy xy 2 Note, in both cases D33 E G 21 9 Derivation of “Constant Strain Triangle” (CST) Element Equations Step 1 – Select element type ui v i u j d v j u m vm Note – x-y are global coordinates (will not need to transform from local to global 10 Displacement Interpolation Assume “bi-linear” interpolation – guarantees that edges remain straight => inter-element compatibility u ( x, y ) a1 a2 x a3 y v( x, y ) a4 a5 x a6 y 11 Displacement Interpolation (cont.) As before, rewrite displacement interpolation in terms of nodal displacements (see text for details) u ( x , y ) N i ( x , y ) u i N j ( x , y ) u j N m ( x, y ) u m v( x, y ) N i ( x, y ) vi N j ( x, y ) v j N m ( x, y ) vm where 1 i i x i y 2A 1 Nj j jx j y 2A 1 m m x m y Nm 2A Ni 12 Displacement Interpolation (cont.) and i x j ym y j ym i y j ym i xm x j 1 xi 1 A 1 xj 2 1 xm yi yj ym 13 Displacement Interpolation (cont.) u N i v 0 0 Nj 0 Nm Ni 0 Nj 0 ui v i 0 u j N d N m v j u m vm 14 Displacement Interpolation (cont.) 1 i i x i y 2A 1 Nj j jx j y 2A 1 m m x m y Nm 2A Ni Graphically: 15 Step 3 – Strain-Displacement and StressStrain Relations x u v u v ,y , xy x y y x x y xy From which it can be shown i 1 0 2A i 0 j 0 m i i 0 j j 0 j m ui v 0 i u j m B d vj m u m vm 16 Strain-Displacement Relations (cont.) • Note – the strain within each element is constant (does not vary with x & y) • Hence, the 3-node triangle is called a “Constant Strain Triangle” (CST) element 17 Stress-Strain Relations D DBd 3x1 3x3 3x6 6x1 18 Step 4 – Derive Element Equations p U p S B which will be used to derive k tA B DB T 6x6 6x3 3x3 3x6 19 Derive Element Equations (cont.) Strain energy: 20 Derive Element Equations (cont.) Potential energy of applied loads: 21 Derive Element Equations (cont.) Potential energy: 22 Derive Element Equations (cont.) Substitute to yield 23 Derive Element Equations (cont.) Apply principle of minimum potential energy To obtain 24 Derive Element Equations (cont.) Element stiffness matrix 25 Steps 5-7 5. Assemble global equations 6. Solve for nodal displacements 7. Compute element stresses (constant within each element) 26 Example – CST element stiffness matrix 27 CST Element Stiffness Matrix k tA B DB T where [B] – depends on nodal coordinates [D] – depends on E, See text for details 28 Body and Surface Forces Replace distributed body forces and surface tractions with work equivalent concentrated forces. { fb } { fs } 29 Work Equivalent Concentrated Forces – Body Forces For a uniformly distributed body forces Xb and Yb: f ix X b f iy Yb f jx X b At f b f jy Yb 3 f mx X b f my Yb 30 Work Equivalent Concentrated Forces – Surface Forces For a uniform surface loading, p, acting on a vertical edge of length, L, between nodes 1 and 3: f six pLt / 2 f 0 siy f sjx 0 At fb f sjy 0 3 f smx pLt / 2 f smy 0 31 Example 6.2 32 Example 6.2 - Solution d 3 x 609 .6 d 3 y 4.2 6 10 in d 4 x 663 .7 d 4 y 104 .1 Element 1 Element 2 x 1005 301 y 2.4 xy x 995 1 . 2 y 2.4 xy 33 In-class Abaqus Demonstrations • • Example 6.2 Finite width plate with circular hole (ref. “Abaqus Plane Stress Tutorial”) 34 Chapter 7 - Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis Discussion of Example 6.2: 35 Example 6.2 - discussion 36