Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros

Problem 2.4-14

A rigid bar

ABCD

is pinned at point

B

and supported by springs at

A

and

D

(see figure). The springs at

A

and

D

have stiffnesses

k

1 = 10 kN/m and

k

2 = 25 kN/m, respectively, and the dimensions

a

,

b

, and

c

are 250 mm, 500 mm, and 200 mm, respectively.

A load

P

acts at point

C

. If the angle of rotation of the bar due to the action of the load

P

is limited to 3 ° , what is the maximum permissible load

P

max?

Problem 2.5-5

A bar

AB

of length

L

is held between rigid supports and heated nonuniformly in such a manner that the temperature increase Δ

T

at distance

x

from end

A

is given by the expression Δ

T

=

T B

(

x^

3/

L^

3), where Δ

T B

is the increase in temperature at end

B

of the bar (see figure).

Derive a formula for the compressive stress

c

in the bar. (Assume that the material has modulus of elasticity

E

and coefficient of thermal expansion .)

2.6: Stresses on inclined sections

• Up to now we have considered normal stresses acting on cross-sections • Provided that the bar is prismatic, the material is homogeneous, the axial force P acts at the centroid of the cross-sectional area and the cross-section is away from any localized stress concentrations

(a) Bar with axial forces P (b) Three-dimensional view of the cut bar showing the normal stresses (c) Two-dimensional view

FIG. 2-30

Prismatic bar in tension showing the stresses acting on cross section mn

Copyright 2005 by Nelson, a division of Thomson Canada Limited

2.6: Stress elements

• Isolating a small element C (fig. 2-30 c) • We have a

stress element

(fog 2-31 a) • The only stresses acting are the normal stresses σ x • Because it is more convenient, we usually draw a 2-D view of the stress element (fig. 2-31 b)

FIG. 2-31

Stress element at point C of the axially loaded bar shown in Fig. 2-30c: (a) three-dimensional view of the element, and (b) two-dimensional view of the element

Copyright 2005 by Nelson, a division of Thomson Canada Limited

2.6: Stresses on inclined sections

• In order to obtain a more complete picture, we need to investigate the stresses acting on

inclined sections (plane pq

fig. 2-32 a) • Uniform distribution of stresses as shown in figs 2-32 b and c

(a) Bar with axial forces P (b) Three-dimensional view of the cut bar showing the stresses (c) Two-dimensional view

FIG. 2-32

Prismatic bar in tension showing the stresses acting on an inclined section pq

Copyright 2005 by Nelson, a division of Thomson Canada Limited

2.6: Stresses on inclined sections

•We need to specify the orientation of the

section pq.

Define an angle

θ

between the x-axis and the normal

n

to the section • We need to find the stresses acting on the

section pq

• Load P, which is the stress resultant

,

can be resolved with respect to N and V • N is associated with normal stresses and V is associated with shear stresses

FIG. 2-33

Prismatic bar in tension showing the stresses acting on an inclined section pq

Copyright 2005 by Nelson, a division of Thomson Canada Limited

2.6: Stresses on inclined sections

Establish standard notation and sign convention • Normal stresses σ θ are positive in tension and shear stresses τ θ when they tend to produce counterclockwise rotation of the material where σ x = P/A, in which σ x is the normal stress on a cross-section

FIG. 2-34

Sign convention for stresses acting on an inclined section (Normal stresses are positive when in tension and shear stresses are positive when they tend to produce counterclockwise rotation)

Copyright 2005 by Nelson, a division of Thomson Canada Limited

2.6: Stresses on inclined sections

FIG. 2-35

stress

 

Graph of normal stress verses angle

  

and shear of the inclined section (see Fig. 2-34 and Eqs. 2-29a and b)

Copyright 2005 by Nelson, a division of Thomson Canada Limited

2.6: Stresses on inclined sections

• Element A: The only stresses are the maximum normal stresses (θ = 0) • Element B: This is a special case where all four faces have the same magnitude normal and shear stress

(σ x /2)

FIG. 2-36

stresses acting on stress elements oriented at and



= 45 tension Normal and shear

 

for a bar in = 0

 Copyright 2005 by Nelson, a division of Thomson Canada Limited

2.6: Stresses on inclined sections

The shear stress may cause failure if the material is much weaker in shear than in tension

FIG. 2-37

Shear failure along a 45



plane of a wood block loaded in compression

Copyright 2005 by Nelson, a division of Thomson Canada Limited

FIG. 2-38

tension Slip bands (or Lüders’ bands) in a polished steel specimen loaded in

Copyright 2005 by Nelson, a division of Thomson Canada Limited