7/31/2019 SE130B-HW1

1/4

1

SE 130B Homework 1

Problems 1+2 due 7/10/2012, Problem 3 due 7/13/2012

Problem 1

For the systems shown below, formulate the differential equations and

obtain the displacement field, u(x), the axial force, N(x), and the

reactions. Also, draw the variation of functions u(x) and N(x).

(a) (b)

NOTE for case b: When in a member we have a change of

temperature, T, compared to the temperature that the member has

during its fabrication (e.g., if the temperature of the member was

equal to 20o C when the member was constructed and the current

temperature of the member is 30o C, then = 30 20 = 10 C) then

the temperature-induced strains in the member are given by: =

aT. The only differences compared to the case of a truss member

without any temperature change (see prof. Contes notes, p. 1-1 and 1-

2) are in the compatibility equation and in the constitutive law:

Compatibility equation: M T

du

dx

= = +, where M is the mechanical

strain, that is, the strain which causes stress.

Constitutive equation: M Tdu

N A E A E dx

= =

x

wx(x)

x

T = constant

E,A =constant

E,A, aT =constant

7/31/2019 SE130B-HW1

2/4

2

Problem 2

For the truss member shown in the Figure below:

a. Determine the transformation matrices [ROT], [RBM], and [].

b. Determine matrices k ' , [ ]k ' and [ ]k

c.

{ }

1

2

3

4

0

0.1 in

0.05 in

0.05 in

= =

If the nodal displacement vector is known as shown above, calculate{ }F' and { }F' ( = the axial force of the member).

X

E = 10000 ksiA = 1 in2

YNode 1

X1 = 10 in

Y1 = 5 in

Node 2

X1 = 16 in

Y1 = 13 in

1

2

3

4

7/31/2019 SE130B-HW1

3/4

3

Problem 3

a. Develop a Matlab routine, called truss.m.

The routine must be able to make all the computations pertaining

to a truss member. More specifically, the routine must take as input

arguments the 2-dimensional vectors X and Y, containing the nodal

abscissas and ordinates of a member in the global coordinate system,

the 4-dimensional vector U, containing the nodal displacements in the

global coordinate system, and the values E and A. Additionally, another

parameter, itask, must be given as input in the member. This integer

will determine what kind of task the element routine must conduct. If

itask = 1, the routine must calculate the element stiffness matrix in

the GLOBAL coordinate system. If itask = 2, the routine must calculate

the element end forces in the GLOBAL coordinate system, {F}, as well

as the element basic deformation, du and basic force, N (that is, the

routine must determine the axial deformation and axial force). In

summary, a call command for the routine in Matlab must read:

[k,F,du,N]=truss(X,Y,U,E,A,itask)

If itask = 1, then the routine returns k, while the vector F and the

scalars du and N will have zero values.

If itask =2, then [k] will have zero values, while the routine will return

vector F and scalars du and N.

7/31/2019 SE130B-HW1

4/4

4

b. Use the routine to repeat (and verify) the calculations of [k] and

{ }F' in Problem 2.

c. Also, use the routine to calculate the stiffness matrices [k] for the

following two members:

d. Finally, determine the axial deformation and axial force of the above

two members, corresponding to the following displacement vector in

the global coordinate system:

{ }

1

2

3

4

0

0.1 in

0.05 in

0.05 in

= =

X

E = 29000 ksiA = 1 in2

YNode 1

X1 = 16 in

Y1

= 3 in

Node 2

X2

= 16 in

Y2 = 13 in E = 29000 ksiA = 1 in2

Node 1

X1

= 7 in

Y1 = 3 in

Node 2

X2 = 17 in

Y2 = 3 in