Exercises on finite element method

S. N. Khaderi

August 18, 2015

1 Direct stiffness method

Questions marked with as asterisk are coding assignments that should be submitted

before 15 September .

1. PMPE Write the potential for a spring–force system. Plot the potential energyas a function of x for various values of F .

2. Assembly Populate the global stiffness matrix for the below two nodal num-

bering. What do you observe? Try also playing with the local node numbering toconvince yourself that no matter how you locally number the nodes of an element,the global stiffness matrix remains the same for each of these meshes.

1 2 3

1 2 3 4

1 2 3

1 3 4 2

3. 1D bar elements Consider a bar of length L with a tapering cross-section

A(x) = 1 + x/L. The left end is fixed and to right end a force of unit magnitudeis applied. Find the exact solution for this problem. Solve this problem using

one, two and three finite elements, assuming that the area is constant within eachelement, and compare with analytical solution.

4. 1D bar elements* Write a finite element program to solve the above problem.Verify the output of program by comparing with analytical solution.

5. Truss elements* Consider a truss structure shown below. Assume that the

stiffness of all the members is equal to unity. Write a finite element program tosolve for the axial forces in the members labelled A,B, C. Verify the output of

program by comparing with analytical solution.

1

1

A

BC

6. Beam elements* Modify the above finite element program, using beam ele-

ments, to solve for the deflection of a simply supported beam, loaded by a pointforce at the mid-span. Verify the obtained solution by solving the same problem

using symmetry boundary condition.

7. Frame elements* Consider a portal frame structures shown below. Write afinite element program to compute the displacements and rotations at the point

of application of load. Verify your results by solving the same problem usingsymmetry/antisymmetry boundary conditions at the point of application of theload/moment. Does the solution improve if more than one element is used to

discretize each member of the frame.

L

L L

2

L

L L

2 Calculus of variations

8. Some basics. Why is that in calculus of variation, we do not vary a functional,∫

F (x, y, y′)dx, with respect to the independent variable, x? Why do we consider

y and y′ as independent variables?

9. Laws of reflection. Consider reflection of light from a mirror. The laws ofreflection can be stated as ‘a light ray incident upon a reflective surface will be

reflected at an angle equal to the incident angle’. Derive this law from Fermat’sprinciple.

10. Optical Path Near Event Horizon of a Black Hole. Determine the

optical path in an atmosphere in which the velocity of light increases in proportionto the height, v(y) = y/b, with b > 0 some parameter describing the light speed.

Therefore, v = 0 at y = 0, which simulates the conditions at the surface of a blackhole, called its event horizon, where the gravitational force is so strong that the

velocity of light goes to zero, thus even trapping light. (Source: Weber and Arfken)

11. Electrostatics. The electrostatic potential energy can be written, in two-dimensions, as

Π =ǫ02

∫(

∂φ

∂x

)2

+

(

∂φ

∂y

)2

dxdy,

where φ is the electrostatic potential, ǫ0 is the dielectric permittivity in vacuum.

Derive the conditions for which the electrostatic potential energy is stationary.

12. Boundary conditions. Consider a one-dimensional bar. The extreme endsof the bar are fixed. A concentrated force F is applied at the middle. Derive,

using variational method, the conditions that have to be satisfied at the point ofapplication of force.

3

13. Catenoid Among all the curves joining two points (x1, y1) and (x2, y2), findthe one which generates the surface of minimum area when rotated about the x

axis.

4