Q1. The exact solution to the boundary value
problem
 ,
 ,
for
y(4) is
-234.66
0.0000
16.000
106.66
Q2. Given
 ,
 ,
 ,
the exact value of  is
-72.0
0.00
36.0
72.0
Q3. Given
 ,
 ,
 ,
If one was using shooting method with
Euler’s method with a step size of
h=4,
and an assumed value of  =20,
then the estimated value of y(12) in the first iteration most
nearly is
160.0
496.0
1088
1102
Q4. The transverse deflection, u of a
cable of length, L, fixed at both ends, is given as a solution to
where
T = tension in cable
R = flexural stiffness
q = distributed transverse load
Given are  ,
 ,
 ,
 .
The shooting method is used with Euler’s method assuming a step size of  .
Initial slope guesses at x=0 of
 and
 are
used in order, and then refined for the next iteration using linear
interpolation after the value of u(L) is found. The deflection
in inches at the center of the cable found during the second iteration
is most nearly
0.03583
0.08083
0.08484
0.08863
Q5. The radial displacement, u is a
pressurized hollow thick cylinder (inner radius=5″, outer radius=8″) is
given at different radial locations.
|
Radius |
Radial Displacement |
|
(in) |
(in) |
|
5.0 |
0.0038731 |
|
5.6 |
0.0036165 |
|
6.2 |
0.0034222 |
|
6.8 |
0.0032743 |
|
7.4 |
0.0031618 |
|
8.0 |
0.0030769 |
The maximum normal stress, in psi, on the
cylinder is given by
 
The maximum stress, in psi, with second
order accuracy is
2079.3
2104.5
2130.7
2182.0
Hint:
 ,
and
where
Q6. For a simply supported (at x=0 and
x=L)
beam with a uniform load q, the deflection v(x) is described by
the boundary value ordinary differential equation as
 ,
where
E = Young’s modulus of
elasticity of beam
I = second moment of
cross-sectional area.
This is based on assuming that
 is
small; if  is
not small, then the ordinary differential equation is
|