Russell C. Hibbeler

Chapter 8: Combined Loading

Review of the stress analysis that has been developed in the previous chapters regarding,

- axial load

- torsion

- bending

- shear

In this chapter : Solution of problems where several of these internal loads occur simultaneously on a member’s cross section.

State Of Stress Under Axial Loading

State of Stress Under Pure Torsion

State of Stress Under Pure Bending

State Of Stress Under Transverse Load

State Of Stress Under Combined Loadings

An axial force of 900 N and a torque of 2.50 N.m are applied to the shaft as shown below. If the shaft has a diameter of 40 mm, determine the state of stress at a point P on its surface.

Example 1

kPa9.19804.0

32

02.05.2

4

J

Tr

kPa2.716

04.04

900

2

A

P

The beam shown below is subjected to the distributed loading of w = 120 kN/m. Determine the state of stress at point P which lies at the top of the web. Neglect the size of the fillets and stress concentrations at this point. INA = 67.4 x 10-6 m4.

Example 2

mkN6.30kN84 MV

on)(compressi MPa4.45

104.67

1.0106.306

3

I

My

MPa2.35

01.0104.67

015.0175.01075.010846

3

It

VQ

The solid rod shown below has a diameter of 15 mm. If it is subjected to the loading shown, determine the state of stress at point A.

Example 3

Example 4The joint is subjected to a force of 250 N as shown. Determine the state of stress at points A and B, and sketch the results on differential elements located at these points. The members has a rectangular cross-sectional area of width 0.5 cm and thickness 0.75 cm

Example 5

Determine the maximum and minimum normal stress in the bracket at section a when the load is applied at (a) x = 0, (b) x = 50 mm

X = 0

X = 50

Example 6

The bar has a diameter of 40 mm. If it is subjected to a force of 800N as shown, determine the stress components that act at point A and B and show the results on a volume element located at these point.

Example 7

The beam supports the loading shown. Determine the state of stress at points E and F at section a – a.

Example 8

The tine ABC of the fork lift is subjected to a uniform distributed loading as shown. If it is pin connected at C and Roller supported at B, determine the state of stress at points D and E. The tine is 120 mm wide and 20 mm thick.

© 2008 Pearson Education South Asia Pte Ltd

Chapter 8: Combined LoadingsMechanics of Material 7th Edition

Thin-Walled Pressure Vessels

Thin wall refers to a vessel having an inner-radius to-wall-thickness ratio of 10 or more.

For cylindrical vessels under normal loading, there are normal stresses in the circumferential or hoop direction and in the longitudinal or axial direction.

10/ tr

direction allongitudinin stress normal 2

direction hoopin stress normal

2

1

t

prt

pr

© 2008 Pearson Education South Asia Pte Ltd

Chapter 8: Combined LoadingsMechanics of Material 7th Edition

Thin-Walled Pressure Vessels

The following is a summary of loadings that can be applied onto a member:a) Normal Force

b) Shear Force

c) Bending Moment

d) Torsional Moment

e) Thin-Walled Pressure Vessels

f) Superposition

© 2008 Pearson Education South Asia Pte Ltd

Chapter 8: Combined LoadingsMechanics of Material 7th Edition

Example 1

The tank has an inner radius of 600 mm and a thickness of 12 mm. It is filled to the top with water having a specific weight of γw = 10 kNm3. If it is made of steel having a specific weight of γst = 78 kNm3, determine the state of stress at point A. The tank is open at the top.

kN 56.311000

600

1000

61278

22

ststst VW

The pressure on the tank at level A is

kPa 10110 zp w

The weight of the tank is

For circumferential and longitudinal stress,

(Ans) kPa 9.7756.3

(Ans) kPa 50010

2

10006002

10006122

100012

1000600

1

st

st

A

W

t

pr