Deformation of Axially Loaded
Members - Single Member
P
P
Assumptions:
• uniform cross section (A is a constant)
• homogeneous material (E is a constant)
• loads are applied at ends only


L
   L
  E   


L
E

E
P
A

PL
EA
Axially Loaded Members - Multiple
Members
C
A
B
P2
P1
L2
L1
Rob consists of several portions of
various cross sections and possibly
different materials.
Assumptions:
• uniform cross section and homogeneous
material within each portion
• loads are applied at ends of each portion
n
In general:    Fi Li
i 1 Ei Ai
Fi is the internal axial force inside the ith portion.
Axially Loaded Members - Continuously
Varied Members
L
x
P
Axially Loaded Members - Applications:
Design
Design Considerations:
  o
- Stress
- deformation
  o
Factor of Safety n: n 
Allowable stress:
Allowable load:
Actual strength
1
Re quired strength
 allow 
Actual strength
n
Pallow   allow A
Axially Loaded Members - Applications:
Design
Example 1: A circular bar made of steel is subject to the loading system
indicated in the figure. The maximum elastic deformation has to be limited to
10 mm. And the factor of safety, based on slip failure, is to be 1.8. Determine
the minimum diameter of the bar required to support the loads.
E  200GPa;  Y  600MPa
Axially Loaded Members - Applications:
Statically Indeterminate Problems
Statically indeterminate structures: reactions acting on the structures can’t be
found by solving static equations.
Example 2: Find reactions of the bar.
Axially Loaded Members - Applications:
Statically Indeterminate Problems
Example 3: A horizontal rigid bar AB is pinned at end A and supported by two
wires (CD and EF) at points D and F. A vertical load P acts at end B of the bar.
The length of the bar is 3b and the lengths of wires CD and EF are L1 and L2
respectively. Also wire CD has a diameter of d1 and a modulus of elasticity of E1.
Wire EF has a diameter of d2 and a modulus of E2. Find the allowable load P if the
allowable stresses in wires CD and EF are 1 and 2.
Axially Loaded Members - Temperature
Effects
L
L
Axially Loaded Members - Temperature
Effects
Total strain:
   P  T
L
  0   P  T   P  T
 P  E P
 P   ET
P
P
P   ETA
Axially Loaded Members - Temperature
Effects
Example 4: In the structure (a Thermostat) shown below, find the horizontal
movement of A if the temperature decrease about 45 oC. The thermal expansion
coefficients for steel and aluminum are 11.9 x 10-6 /oC and 22.5 x 10-6 /oC
respectively. Assume the thermal expansion coefficient for bar CE is zero and
it is very soft.
Axially Loaded Members - Temperature
Effects
Example 5: A bar is fixed between two walls. Find the stress inside the bar
when the temperature is decreased about  T .
Al
Steel
Aluminum bar: EAl Al AAl LAl
ESteel Steel ASteel LSteel
Steel bar:
AAl = ASteel
Axially Loaded Members- Stresses in an
Inclined Plane
p
q
Stresses at section pq:
h
P
q
P
q 
F
F
Aq
q 
V
Aq
V
P
F
F
x
 0  F cos q  V sin q  P
y
 0  F sin q  V cos q  0
F  P cos q
A  h W
h
Aq 
W
cos q
Aq 
A
cos q
F P cos2 q
q  =
Aq
A
q 
V
P cos q sin q

Aq
A
V  P sin q
Axially Loaded Members - Maximum Normal
Stress and Maximum Shear Stress
q
q

2
P
P
P
P
P
P

4
q 0
P
A
P

2A
Maximum normal stress:
 q max 
Maximum shear stress:
 q max
Axially Loaded Members- Stresses in an
Inclined plane
Example 6: A circular brass of diameter d is composed of two segments brazed
together on a plane pq making an angle  = 30o with the axis of the bar. The
allowable stresses in the brass are 13,000 psi in tension and 7000 psi in shear.
On the brazed joint, the allowable stresses are 6000 psi in tension and 3000 psi
in shear. If the bar must resist a tensile force P = 6500 lb, what is the minimum
required diameter dmin of the bar? (1 psi = 1 lb/in2)