Finite element method for structural dynamic and stability analyses Assignments Prof C S Manohar Department of Civil Engineering IISc, Bangalore 560 012 India

Problem 1. Formulate the equation of motion for the systems shown below. Do not assume small displacements.

Problem 2 A beam with circular cross section fixed at one end and on roller at the other is hinged in between as shown in figure below. Deduce the equation of motion for the beam and obtain the appropriate boundary conditions.

Problem 3 Derive the equation of motion and appropriate BCs for the beam that rotates about the y-axis as shown in figure below.

Problem 4

Problem 7. For the structure shown below, formulate the governing equation of motion. The discretization scheme shown in figure on the next slide needs to be adopted. Obtain the natural frequencies and mode shapes. The beam is traversed by a concentrated load P=Mg with a velocity 120 kmph. Using the free vibration results obtained in the preceding step, and by using a one mode approximation, estimate the maximum midspan displacement.

Problem 8. Consider the beam structure as in problem 7 Problem 8. Consider the beam structure as in problem 7. Formulate the governing PDE for the system. By employing a three term series solution in terms of trigonometric trial functions, and by using Galerkin’s approximation deduce the discretized equation of motion. Compare the prediction from this model with the results from the FE model in problem 7. . Problem 9. Consider the beam structure as in problem 7. The spring element at x=b, is suddenly removed at t=t*. Analyse the ensuing oscillations. It may be assumed that the structure is acted upon by gravity.

Problem 10 Obtain the damped eigensolutions (natural frequencies and mode shapes) for the system shown below. Demonstrate the orthogonality properties satisfied by these functions. Derive the matrix of impulse response functions and frequency response functions. Plot the receptance, mobility, and accelerance functions (Bode’s and Nyquist’s diagrams). m 1.5m

Problem 11

Problem 12

Problem 13 For the system considered in problem 10, obtain the elements of the first row of the matrix of impulse response functions by numerical integration of the governing equation using: Euler’s forward method, Euler’s backward method, Central difference method, Newmark beta method, and HHT-alpha method. In each case discuss how you have chosen the algorithmic parameters. Using the exact solutions obtained in problem 10, analyse the accuracy of the solutions obtained.

Problem 14

Problem 15

All edges simply supported Problem 17. Analyse the plate structure shown for its natural frequencies and compare the results obtained with the exact solutions provided. All edges simply supported

Problem 18. The beam-column AC is loaded as shown below by two static loads P and Q. The load Q Is suddenly removed. Analyse the ensuing oscillations and its stability.

Problem 19. Investigate the influence of load P on the first two natural frequencies of the Euler-Bernoulli beam shown below.

Determine critical value of P. Problem 20 Determine critical value of P. Plot the axial thrust diagram at the critical loading condition Plot the corresponding buckling mode shape

Problem 21. Analyse the beam column problem shwon below using FEM and compare FE solutions with the exact solutions

Problem 22 A cantilever beam is supported through a cable and carries an axial load P and a transverse load Q as shown below. Determine the critical value of P. Determine the beam response when the axial load is half of the critical load.  

Problem 24. Refer to the discussion on behaviour of circular arch under uniform pressure (Lecture 32). Analyse the structure using FEM and compare the solutions obtained with analytical solutions.

Problem 25. Refer to Lecture 32 and the governing equations for a stack subjected to bi-axial Earthquake ground motions and effect of self-weight. Starting from the governing PDE, develop a FE model for the structure.

Vehicles and the beam interact Problem 24. Extend the FE analysis presented in Lecture 34 for the vehicle-structure Interactions for the problem shown below. Vehicles and the beam interact

Problem 25. Consider the discussion on behaviour of pipe conveying fluid presented in Lecture 34. Formulate the problem using FEM. Investigate the role played by damping of the beam structure.

Problem 26. For the structure considered in problem 4, determine the sensitivity of the first two natural frequencies and mode shapes with respect to EI, m and the coupling spring.