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1 Plane Problems: Constitutive Equations Constitutive equations for a linearly elastic and isotropic material in plane stress (i.e., σ z =τ xz =τ yz =0): where the last column has the initial (thermal) strains which are 0 , xy0 0 0 = = = γ α ε ε T y x • Rewriting in a compact form and solving for the stress vector, where

Plane Problems: Constitutive Equations - UFL MAE · Plane Problems: Constitutive Equations ... • Rewriting in a compact form and solving for the stress vector, where. 2 Plane Problems:

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Page 1: Plane Problems: Constitutive Equations - UFL MAE · Plane Problems: Constitutive Equations ... • Rewriting in a compact form and solving for the stress vector, where. 2 Plane Problems:

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Plane Problems: Constitutive Equations■ Constitutive equations for a linearly elastic and isotropic material in plane stress (i.e., σz=τxz=τyz=0):

where the last column has the initial (thermal) strains which are 0 , xy000 === γαεε Tyx ∆

• Rewriting in a compact form and solving for the stress vector,

where

Page 2: Plane Problems: Constitutive Equations - UFL MAE · Plane Problems: Constitutive Equations ... • Rewriting in a compact form and solving for the stress vector, where. 2 Plane Problems:

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Plane Problems: Approximate Strain-Displacement Relations

• From the above, by definition

, ,xv

yu

yv

xu

xyyx ∆∆

∆∆

∆∆

∆∆ +≈≈≈ γεε

Page 3: Plane Problems: Constitutive Equations - UFL MAE · Plane Problems: Constitutive Equations ... • Rewriting in a compact form and solving for the stress vector, where. 2 Plane Problems:

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Plane Problems: Strain-Displacement Relations

■ As the size of the rectangle goes to zero, in the limit,

Page 4: Plane Problems: Constitutive Equations - UFL MAE · Plane Problems: Constitutive Equations ... • Rewriting in a compact form and solving for the stress vector, where. 2 Plane Problems:

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Plane Problems: Displacement Field Interpolated■ Interpolating the displacement field, u(x,y) and v(x,y), in the plane finite element from nodal displacements,

where entries of matrix N are the shape (interpolation) functions Ni.

■ From the previous two equations,

where B is the strain-displacement matrix.

Page 5: Plane Problems: Constitutive Equations - UFL MAE · Plane Problems: Constitutive Equations ... • Rewriting in a compact form and solving for the stress vector, where. 2 Plane Problems:

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Stiffness Matrix and strain energy■ Strain energy density of an elastic material (energy/volume)

εε ET21

• Integrating over the element volume, the total strain energy is

( )dEBBdE 21

21

∫∫ == dVdVU TTT εε

where the term in parantheses is identified to be the element stiffness matrix.

■ The strain energy then becomes

where the term on the right is the total work done on the element.

1 1 r2 2

T TU = =d kd d

Page 6: Plane Problems: Constitutive Equations - UFL MAE · Plane Problems: Constitutive Equations ... • Rewriting in a compact form and solving for the stress vector, where. 2 Plane Problems:

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Important Note on interpolation (shape) functions

■ Observe that, for a given material, stiffness matrix k (and, therefore, the behavior of an element) depends solely on N, the interpolation functions, and ∂. The latter prescribes differentiations which define strains in terms of displacements.

NBEBBk ∂=∫= , dVT

■ The variation of the shape functions in the element compared toactual variations of the true displacements determines element size required for good accuracy. Low-0rder shape functions will require smaller elements than higher-order shape functions.

Page 7: Plane Problems: Constitutive Equations - UFL MAE · Plane Problems: Constitutive Equations ... • Rewriting in a compact form and solving for the stress vector, where. 2 Plane Problems:

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Loads and Boundary Conditions

■ Surface tractions: distributed loads on a boundary of a structure; e.g., pressure.■ Body forces: loads acting on every particle of the structure; e.g., acceleration (gravitaionalor otherwise), magnetic forces.■Concentrated forces and moments.

• Boundary conditions on various segments of the surface:

A to B: free. B to C: normal traction (pressure)

C to D: shear traction. D to A: zero displacements (dofs=0)

Page 8: Plane Problems: Constitutive Equations - UFL MAE · Plane Problems: Constitutive Equations ... • Rewriting in a compact form and solving for the stress vector, where. 2 Plane Problems:

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Constant Strain Triangle (CST)

■ The sequence 123 in node numbers must go counterclockwisearound the element.■ Linear displacement field in terms of generalized coordinates βi:

■ Then, the strains are(constant within the element!!)

Page 9: Plane Problems: Constitutive Equations - UFL MAE · Plane Problems: Constitutive Equations ... • Rewriting in a compact form and solving for the stress vector, where. 2 Plane Problems:

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Constant Strain Triangle (CST): Stiffness Matrix■Strain-displacement relation, ε=Bd, for the CST element

where 2A is twice the area of the triangle and xij=xi- xj , etc.

■ From the general formula

where t: element thickness (constant)

■ NOTE: To represent high strain gradient will require very largenumber of small CST elements

tATEBBk =

Page 10: Plane Problems: Constitutive Equations - UFL MAE · Plane Problems: Constitutive Equations ... • Rewriting in a compact form and solving for the stress vector, where. 2 Plane Problems:

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Linear Strain Triangle (LST)

■ The element has six nodes and 12 dof. Not available in Genesis!

Page 11: Plane Problems: Constitutive Equations - UFL MAE · Plane Problems: Constitutive Equations ... • Rewriting in a compact form and solving for the stress vector, where. 2 Plane Problems:

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Linear Strain Triangle (LST)■ The displacement field in terms of generalized coordinates:

which are quadratic in x and y.

■ The strain field:

which are linear in x and y.