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

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

3. Please submit the 4th homework after the class

4. 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

5. 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:

6. 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?

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

8. 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

9. 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

10. Let us go back to finite element!

11. 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

12. 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

13. 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

14. As a result,

15. Written in matrix form
Thus we can write

16. 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

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

18. The strain field
We can write

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

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

21. The virtual displacement
The virtual strain
The principle of virtual work

22. The virtual strain energy

23. The virtual work

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

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

26. 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

27. 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?

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

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

30. 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

31. 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.

32. See Lecture 5 OHP Slides