# Stress and Dynamic Analysis (326MAE)

Stress and Dynamic Analysis (326MAE)

Stress Assignment – For

hand-in on or before Friday 22nd November 2013

Question

1

a) A

hollow shaft is subjected to a bending moment of 10 kNm and a torque of 5 kNm. The internal diameter is 0.75 times the outer

diameter. If the yield stress is 300 MPa

and a safety factor of 1.5 is required, determine the minimum inner and outer

diameters required according to the Maximum Shear Stress theory (Tresca).

(10

marks)

b)

Figure Q1a on the next page shows a jib crane. When a load is lifted by the crane a force is

reacted through the support frame on the left of Figure Q1a.

The support frame basically consists of

two 50 mm x 50 mm square columns made from plates 5 mm thick, joined by a 6 mm thin

web as shown in the X – X cross-section in Figure Q1b. A force of 100 kN is applied to the top of

the support frame at an angle that creates a 3-4-5 triangle with the vertical

and horizontal directions.

There is a point A identified on the

support frame. It is 350 mm from the

top, on the web adjacent to the right hand column.

(i)

Calculate the axial stress at A.

(4 marks)

(ii)

Calculate the bending stress at A.

(4 marks)

(iii) Calculate the shear stress at A.

(4 marks)

(iv)

Show the state of stress on an infinitesimal aligned to the vertical and

horizontal directions.

(3

marks)

.jpg”>

Question

2

The stress tensor for a

three-dimensional system for some x, y z axis system is

.gif”> MPa

a) Find

the normal and shear stress on a plane whose normal makes an angle of 50° with x-axis and 70° with y-axis. Also give the

direction cosines for the shear stress on the plane.

(8

marks)

b) Find

the principal stresses for the stress tensor.

(5 marks)

c) What angles does the y-axis make with the second

principal direction? (3

marks)

d)

Carry out an analysis to determine if this stress tensor fails the von

Mises criterion when the yield stress of the material is 250 MPa? If not, what

is the factor of safety?

(4 marks)

(You

may use MATLAB for this problem if you wish.)

Question

3

A

steel cylinder with closed ends has an internal diameter of 160 mm and external

diameter of 320 mm OD. It is subject to

an internal pressure of 150 MPa.

a) Determine

the radial and tangential stress distributions and plot the results in

EXCEL.

(10

marks)

b) Also determine the maximum shear stress in

the cylinder.

(5 marks)

(You

may use Lame’s equations without proof.)

Question 4

A cantilever beam with length L and flexural rigidity EI has a

rectangular cross-section. The height in

the y-direction is 2c. It has an end

load P as shown in Figure Q4.

The deflection, including shear effects, can be written as:

.gif”>

.gif”>

a)

Determine σx, σy, and τxythroughout

the beam from the displacement field and verify that these results agree with

the basic strength of materials theory for beams.

(This means that you have to work out

normal and shear stresses from.gif”> and .gif”> and show the

answers are the same as from the displacement functions.)

(12 marks)

b) Find

the vertical displacement of the centre line.gif”> and the slope

of the beam along the centreline Ф(x,0).

Compare your answers with the displacement and slope obtained from

Macaulay’s method and discuss any differences.

(8

marks)

.jpg” alt=”stress2.bmp”>

Figure Q4

Question

5

a) Show

that.gif”> is a permissible Airy stress function. (4 marks)

b) Derive expressions for the stresses

from this Airy function. (5 marks)

c) These general stresses may be used to

solve the problem of a taped cantilever beam of unit thickness carrying a

uniformly distributed load q per unit length as shown in Figure Q5. Show that the derived stresses in b) satisfy

all the boundary conditions along the edges θ = 0◦ and θ = α. (6

marks)

d) Hence obtain a value for the constant Cin terms of q and α and therefore that

.gif”> ,when θ = 0◦.

(2 marks)

e) Compare the value obtained from this

formula with the bending stress obtained from simple bending theory.gif”>when α = 30◦. Give the percentage difference between the

stress given by the formula and that from simple bending. Identify clearly which is the bigger.

(3 marks)

q per unit length

.gif”>

Figure Q5