Download CVEEN 6330 Homework Assignment 4 1. Determine the wave

CVEEN 6330
Homework Assignment 4
1.
Determine the wave propagation velocity in feet per second for a compressional wave in
a constrained rod of:
(a)
steel (E = 29 x 106 psi, Poissons Ratio = 0.294, specific gravity = 7.85)
(b)
cast iron (E = 10 x 106 psi, Poissons Ratio = 0.219, specific gravity = 7.02)
(c)
concrete (unconfined compressive strength, fc, equals 4,000 psi, Poissons Ratio
= 0.3, specific gravity = 2.40)
Note: Use the formula that E(psi) = 57000 * (fc)0.5
2.
For the information given above, determine the wave propagation velocity for a torsional
wave in a constrained rod of:
(a)
steel
(b)
cast iron
(c)
concrete
3.
For the materials in problem 2, determine the wavelength of a compressional wave for a
harmonic axial stress that is operating a frequency of 1 Hertz (one cycle per second).
4.
For the materials in problem 2, determine the wavelength of a torsional wave for a
harmonic torsional stress that is operating a frequency of 1 Hertz (one cycle per second).
5.
Using the spreadsheet provided on the course website, you are required to develop a
finite difference solution for the wave equation for a vertically propagating shear wave in
an homogenous, elastic, undamped soil column. Note in doing so, you are only required
to program the cells that pertain to steps (3), (4), (5), (6) and (7). (See columns G, F, H, I
and E, respectively, in the provided spreadsheet). Note also that the required macros
have already been created, so you do not need to recreate them.
Remember that the free surface boundary condition requires the shear strain and shear
stress to be zero at the free surface. It also requires a doubling of the velocity at the free
surface. Thus the velocity at v i,t +t for the last node (i = 11) can be calculated as:
C11+($B$3/($B$2/$B$6/$B$5))*(I12-I11)*2 , where the 2 at the end of this equation
represents the doubling of the velocity at the free surface.
For grading purposes, please provide an acceleration time history for the first five
seconds of elapsed time and a table of the results for t = 5 seconds.
6.
For the earthquake record below, do the following:
a. identify the arrival times of the first arrival times of the p-waves and s-waves (in
seconds) from the Belrus recording below:
b. use the information to calculate the average Poisson’s ratio for the earth’s crust.
c. use the information and calculations in a and b and Figure 5.9 in Lecture 4b to
calculate the Rayleigh wave arrival time.
d. compare this calculation with the actual earthquake record. Does the calculation
agree with the first arrival time of the Rayleigh wave?