Last course Bar structure Equations from the theory of elasticity Strain energy Principle of virtual work

Today Bar element Principle of virtual work Formulation of bar element using the principle of virtual work

Please submit the 4th homework after the class

δw is the virtual displacement X, u Z, w w δw δw is the virtual displacement The principle of the virtual work δU: the virtual strain energy of the internal stresses δW: the virtual work of external forces on the body

The virtual strain energy for a bar x, u b=b(x) L X=0 X=L P The virtual strain energy for a bar δε : virtual strain, which is the strain associated with the virtual displacement, i.e. The external virtual work:

The virtual strain energy for a bar x, u b=b(x) L X=0 X=L P The virtual strain energy for a bar δε : virtual strain, which is the strain associated with the virtual displacement, i.e. Notice that Why is it not equal to the definition of actual strain energy?

The external virtual work: x, u b=b(x) L X=0 X=L P The external virtual work: The principle of virtual work:

Problem statements of the bar Strong or classical form Given geometrical and material properties L, E, A, α, and external actions P, b(x), T, and support displacement u0, find u(x) such that x, u b=b(x) L X=0 X=L P

Weak or variational form x, u b=b(x) L X=0 X=L P Weak or variational form Given geometrical and material properties L, E, A, α, and external actions P, b(x), T, and support displacement u0, find compatible u(x) such that for all δu(x) satisfying homogeneous boundary condition, Finite element methods are based upon the weak statement of the problem. The basis of the FEM

Let us go back to finite element!

Formulation of bar element Consider the two-node bar element as follows b(x) : body force along the bar element in the unit of force per unit length (e.g. kN/m) u=u(x) : displacement field within the element, i.e. displacement as a function of point x within the element

Displacements at nodes 1 and 2: Forces at nodes 1 and 2: The first and fundamental step in the finite element formulation is to assume the displacement field within the element in terms of its nodal displacements

Here it is assumed that the displacement field is a linear function Let us now express the assumed function in terms of nodal displacements d1 and d2

As a result,

Written in matrix form Thus we can write

Where N is also called the matrix of interpolation functions, because it interpolates the displacement field u=u(x) from the nodal displacements Matrix of shape functions Vector of nodal displacements

The graphs of the shape functions The important property of the shape functions is the partition of unity, i.e.

The strain field We can write

Strain-displacement matrix where The stress field Strain-displacement matrix

Application of the principle of virtual work Imagine that the bar undergoes a linear virtual nodal displacement δd1 and δd2 The virtual nodal displacement

The virtual displacement The virtual strain The principle of virtual work

The virtual strain energy

The virtual work

The equivalent nodal force vector Thus, using the principle of virtual work we obtain The stiffness matrix The equivalent nodal force vector

It can be concluded that Let us rename the vector of applied nodal forces The equivalent nodal force vector due to body force

The stiffness matrix of the bar element with cross-sectional area A (may vary along the bar) If the bar is prismatic and made up from homogeneous material, i.e. E and A are constant, then we can evaluate the stiffness matrix as follows

What if the bar is not prismatic, e. g What if the bar is not prismatic, e.g. if the cross-sectional area varies linearly along the bar? Can you evaluate the stiffness matrix?

The equivalent nodal force Now suppose that the body force varies linearly along the element as shown here

The equivalent nodal force L/6 (2b1+b2) L/6 (b1+2b2)

The effect of temperature change Suppose that now there is no mechanical loads and the bar undergoes temperature change T0C. Assume that the coefficient of thermal expansion of the bar is constant α /0C. Using the principle of virtual work and assuming the displacement field is linear as before, show that the equivalent nodal force vector due to the temperature change is

Example of the DSM including ‘element load’ Given a steel bar of the diameter 16 mm. Axial distributed force 5 kN/m acts along the bar and a concentrated force 10 kN acts at the right end. If the bar is discretized as in the figure, determine the internal stress along the bar. Compare the finite element solutions to the exact solutions.

See Lecture 5 OHP Slides