Plane Problems: Constitutive Equations - UFL · PDF filePlane Problems: Constitutive Equations...
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Plane Problems: Constitutive Equations Constitutive equations for a linearly elastic and isotropic material in plane stress (i.e., z=xz=yz=0):
where the last column has the initial (thermal) strains which are 0 , xy000 === Tyx
Rewriting in a compact form and solving for the stress vector,
Plane Problems: Approximate Strain-Displacement Relations
From the above, by definition
Plane Problems: Strain-Displacement Relations
As the size of the rectangle goes to zero, in the limit,
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.
Stiffness Matrix and strain energy Strain energy density of an elastic material (energy/volume)
Integrating over the element volume, the total strain energy is
( )dEBBdE 21
== dVdVU TTT
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
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.
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)
Constant Strain Triangle (CST)
The sequence 123 in node numbers must go counterclockwisearound the element. Linear displacement field in terms of generalized coordinates i:
Then, the strains are(constant within the element!!)
Constant Strain Triangle (CST): Stiffness MatrixStrain-displacement relation, =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
Linear Strain Triangle (LST)
The element has six nodes and 12 dof. Not available in Genesis!
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.
Plane Problems: Constitutive EquationsPlane Problems: Approximate Strain-Displacement RelationsPlane Problems: Strain-Displacement RelationsPlane Problems: Displacement Field InterpolatedStiffness Matrix and strain energyImportant Note on interpolation (shape) functionsLoads and Boundary ConditionsConstant Strain Triangle (CST)Constant Strain Triangle (CST): Stiffness MatrixLinear Strain Triangle (LST)Linear Strain Triangle (LST)