Building Design and Construction Handbook - Merritt - Ventech!

5.22 SECTION FIVE
5.2.5 Poisson’s Ratio
Within the elastic limit, when a material is subjected to axial loads, it deforms not
only longitudinally but also laterally. Under tension, the cross section of a member
decreases, and under compression, it increases. The ratio of the unit lateral strain
to the unit longitudinal strain is called Poisson’s ratio.
For many materials, this ratio can be taken equal to 0.25. For structural steel, it
is usually assumed to be 0.3.
Assume, for example, that a steel hanger with an area of 2 in 2 carries a 40-kip
(40,000-lb) load. The unit stress is 40,000/2, or 20,000 psi. The unit tensile strain,
taking the modulus of elasticity of the steel as 30,000,000 psi, is 20,000/
30,000,000, or 0.00067 in/in. With Poisson’s ratio as 0.3, the unit lateral strain is
�0.3 � 0.00067, or a shortening of 0.00020 in/in.
5.2.6 Thermal Stresses
When the temperature of a body changes, its dimensions also change. Forces are
required to prevent such dimensional changes, and stresses are set up in the body
by these forces.
If � is the coefficient of expansion of the material and T the change in temperature,
the unit strain in a bar restrained by external forces from expanding or contracting
is
According to Hooke’s law, the stress ƒ in the bar is
where E � modulus of elasticity.
5.2.7 Strain Energy
� � �T (5.26)
ƒ � E�T (5.27)
When a bar is stressed, energy is stored in it. If a bar supporting a load P undergoes
a deformation e the energy stored in it is
1 U � ⁄2Pe (5.28)
This equation assumes the load was applied gradually and the bar is not stressed
beyond the proportional limit. It represents the area under the load-deformation
curve up to the load P. Applying Eqs. (5.20) and (5.21) to Eq. (5.28) gives another
useful equation for energy:
where ƒ � unit stress
E � modulus of elasticity of the material
A � cross-sectional area
L � length of the bar
2 ƒ
U � AL (5.29)
2E