Elementos Finitos

Review of Elasticity Equations
Linear, homogeneous, isotropic material behavior.

3D Isotropic Stress/Strain Law
Three-dimensional Hooke’s Law: stress/strain relationships for an isotropic material

y

σy

σz σx

As you recall, an isotropic body can have normal stresses acting on each surface: σx, σy, σz When the only normal stress is σx this causes a strain along the x- axisaccording to Hooke’s Law

σx z σz σy

x

εx =

σx
E

3D Isotropic Stress/Strain Law
Note, that a tensile stress in the x direction, produces a negative strains in the y and z directions This is called the Poisson effect.

σx σx

These negative strains are computed via: where:
εy = εz = −
E

νσ x

E is Young’s Modulus ν is Poisson’s ratio

3D Isotropic Stress/Strain Law
Sincethe material is isotropic, application of normal stresses in the x, y, and z directions generates, a total normal strain in the x direction:

σy
σx σx

σz

+

σy

+ σz

εx =

σx
E

−ν

σy
E

−ν

σz
E

3D Isotropic Stress/Strain Law
The total normal strains in the y and z directions can be determined in a similar manner:
εx = σx
E −ν

σy
E +

−ν

σz
E

ε y =−ν ε z = −ν

σx
E

σy
E

−ν

σz
E

σx
E

−ν

σy
E

+ν

σz
E

3D Isotropic Stress/Strain Law
Rearranging the above equations and yields 3 equations relating normal stresses and strains :

E [ε x (1 − v) + νε y + νε z ] (1 + ν )(1 − 2ν ) E σy = [νε x + (1 − ν )ε y + νε z ] (1 + ν )(1 − 2ν ) E σz = [νε x + νε y + (1 − ν )ε z ] (1 + ν )(1 − 2ν )

σx =

These equationscan also be written in matrix notation: {σ}=[D]{ε}

Shear Stress/Strain Relationships
Hooke’s law also applies for shear stress and strain: τ=Gγ where G is the shear modulus, τ is a shear stress, and γ is a shear strain. For 3-D this results in a further 3 equations.

y

σy τyx

τ xy = Gγ xy = 2Gε xy
σz σx

τ yz = Gγ yz = 2Gε yz τ zx = Gγ zx = 2Gε zx

σx z σz σy

τxy

xStress-Strain Relationships
0 0 0 ⎤ ⎡1−ν ν ν ⎢ ν 1−ν ν ⎥ ε 0 0 0 ⎥⎧ x ⎫ ⎧σx ⎫ ⎢ ⎪σ ⎪ ⎢ ν ν 1−ν 0 0 0 ⎥⎪εy ⎪ ⎪ y⎪ ⎢ ⎥⎪ ⎪ ⎪σ ⎪ ⎢ 0 0 0 1−2ν 0 ⎥⎪εz ⎪ E ⎪ z⎪ 0 ⎪ ⎪ = ⎨τ ⎬ ⎢ ⎥⎨γ ⎬ 2 ⎪ xy ⎪ (1+ν)(1−2ν) ⎢ ⎥⎪ xy ⎪ 1−2ν ⎪τ ⎪ ⎢ 0 0 0 0 0 ⎥⎪γyz ⎪ ⎪ yz ⎪ 2 ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎩τzx ⎭ 1−2ν⎥⎩γzx ⎭ 0 0 ⎢ 0 0 0 ⎥ ⎢ 2 ⎥ ⎣ ⎦

3D Stress-Strain Matrix
ν 0 0 0 ⎤ ⎡1 −ν ν ⎢ ν 1 −ν ν 0 0 0 ⎥ ⎢ ⎥ ⎢ ν 0 0 0 ⎥ ν 1 −ν ⎢ ⎥ 1 − 2ν⎢ 0 E 0 0 0 0 ⎥ [ D] = ⎥ 2 1 +ν)(1 − 2ν) ⎢ ( ⎢ ⎥ 1 − 2ν ⎢ 0 0 0 0 0 ⎥ 2 ⎢ ⎥ ⎢ 1 − 2ν ⎥ 0 0 0 0 ⎢ 0 ⎥ ⎢ 2 ⎥ ⎣ ⎦ E Note : G = 2(1 +ν)

Strain-Displacement
∂u εx = ∂x ∂v εy = ∂y ∂u ∂v γ xy = + ∂y ∂ x ∂u ∂w γ xz = + ∂z ∂ x

∂w ∂w ∂v εz = γ yz = + ∂z ∂y ∂z (u,v,w) are the x, y and z components of displacement

Stress Equilibrium Equations
∂σ x ∂τ xy ∂τ xz + + + Xb = 0 ∂x ∂y ∂z ∂τ xy ∂σ y ∂τyz + + + Yb = 0 ∂x ∂y ∂z ∂τ xz ∂τ yz ∂σ z + + + Zb = 0 ∂x ∂y ∂z
σy τyz τzy σz τzx τxz τxy τxy σx

Two-dimensional Elements Plane Stress/Strain Stiffness Equations

Two-dimensional Elements
1. 2. 3. 4. 5.

Thin 2D elements . Two coordinates to define position. Elements connected at common nodes and/or along common edges. Nodal compatibility enforced to obtain equilibrium equations Twobasic types
1. 2. Plane stress Plane Strain

Introduction to 2-D Elastic Stress Analysis
Two-dimensional stress analysis allows the engineer to determine detailed information concerning deformation, stress and strain, within a complex shaped two-dimensional elastic body. Assumptions Deformations and strains are very small Material behaves elastically – stress and strain are related by Hooke’sLaw. Hooke’s Law is a matrix equation relating 3 normal stresses and one shear stress to 3 normal strains and one shear strain

{σ} = [D]{ } ε

or {ε} = [D]−1{σ}

Introduction to 2-D Elastic Stress Analysis
2-D Stress analysis allows the engineer to model complex 2-D elastic bodies by discretizing the geometry with a mesh of finite elements.

Modeled as

Introduction to 2-D Stress...