CEE 456 Structural Analysis HW 6 Assigned 05/9/2015 Due 5/19/2015

Problem 1

Analyze the truss below using MatLab. For this example you may use code based on the

TrussSolver scripts on the course webpage. Note you may need to turn the truss element display

off. All members are steel with E = 29,000 ksi. All diagonals have a cross-sectional area of 2 in2,

all vertical elements have a cross-sectional area of 5 in2, and all horizontal elements have a cross-

sectional area of 4 in2. All supports are pins. Analyze and report results for the following

loadings:

a) Earthquake in the x-direction: -20 kips in the positive x-direction at nodes 1, 2, 3, and 4.

b) Earthquake in the y-direction: -20 kips in the positive y-direction at nodes 1, 2, 3, and 4.

c) Torsional earthquake: -15 kips in the x-direction at node 1, -15 kips in the y-direction at

node 4, 15 kips in the x-direction at node 3, and 15 kips in the y-direction at node 2.

d) The combined effect of earthquake in the x and earthquake in the y (i.e., a and b).

Comment on the relationship between cases a), b), and d). What does this say about

superposition of loads for elastic systems?

Problem 2

Beginning with the fundamental differential equation for beams: EIv = q(x), determine an expression

for the displaced shape of the beam, v(x), with boundary conditions v(0) = (0) = (L) = 0, v(L) = vL.

and a linearly varying distributed load q(x) = -w0x. Sketch the displacement, rotation, moment, and shear

diagrams.

15 ft

30 ft 30 ft

1

2

3

4

5

6

7

8

x y

z

CEE 456 Structural Analysis HW 6 Assigned 05/9/2015 Due 5/19/2015

Problem 3

Develop the symbolic stiffness matrix for the free degrees of freedom [Kff] for each beam (recall

you must put a node under the point load). This should be in terms of the stiffness coefficients,

kfv, kmv, etc. It may help if you assign element letters to the beam elements.

Problem 4

For the statically determinate structure shown in the top figure for Problem 3, assume P = 10

kips, M = 250 kip-ft, E = 10,000 ksi, I = 125 in.4, and L = 8 ft. Determine the displacements and

rotations under the point loads and the rotations at the pins, and solve the reactions at the pins

using the stiffness method (i.e. {Ps}=[Ksf]{uf}+[Kss]{us}).

CEE 456 Structural Analysis HW 6 Assigned 05/9/2015 Due 5/19/2015

Problem 5

Consider the beam below.

a) Label and identify the free degrees of freedom and write the symbolic displacement

matrix {uf}.

b) Develop the symbolic stiffness matrix for the free degrees of freedom [Kff]. This

should be in terms of the stiffness coefficients, kfv, kmv, etc. It may help if you assign

element letters to the beam elements.

c) Write the symbolic load matrix for the free degrees of freedom {Pf}.

d) M = 400 kip-ft, E = 29,000, L = 36 ft, and the beam cross-section shown, determine

the deflection and rotations at the pins using the stiffness method. Note you will have

to review parallel axis theorem and the calculation of the moment of inertia (second

moment of area) from statics.

e) Using the results and known displacements at the supports, determine the

displacement along the length of the beam and plot the deformed shape. You will

need to use the shape functions N1(x), N2(x), N3(x) and N4(x) etc.

f) Using the differential relationship between the displacement along the beam and the

rotation, shear, and moment, plot the rotation, moment and shear diagrams.

g) What is the maximum strain in the beam, and where does it occur?

h) What is the maximum stress in the beam, and where does it occur?

i) At the point of maximum stress, what is the stress at the top of the web (i.e.

immediately below the top flange)?

j) If the yield strength is 36 ksi, is the beam yielding?

Problem 6

Repeat Problem 5, assuming that the internal pin support is removed.