Page 113 - Shigley's Mechanical Engineering Design

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88 Mechanical Engineering Design
When a material is placed in tension, there exists not only an axial strain, but also
negative strain (contraction) perpendicular to the axial strain. Assuming a linear,
homogeneous, isotropic material, this lateral strain is proportional to the axial strain. If
the axial direction is x, then the lateral strains are y = z =−ν x . The constant of pro-
portionality v is called Poisson’s ratio, which is about 0.3 for most structural metals.
See Table A–5 for values of v for common materials.
If the axial stress is in the x direction, then from Eq. (3–17)
σ x σ x
x = y = z =−ν (3–18)
E E
For a stress element undergoing σ x , σ y , and σ z simultaneously, the normal strains
are given by

1
x = σ x − ν(σ y + σ z )
E
1
y = σ y − ν(σ x + σ z ) (3–19)
E
1
z = σ z − ν(σ x + σ y )
E
Shear strain γ is the change in a right angle of a stress element when subjected to
pure shear stress, and Hooke’s law for shear is given by

τ = Gγ (3–20)
where the constant G is the shear modulus of elasticity or modulus of rigidity.
It can be shown for a linear, isotropic, homogeneous material, the three elastic con-
stants are related to each other by
E = 2G(1 + ν) (3–21)

3–9 Uniformly Distributed Stresses
The assumption of a uniform distribution of stress is frequently made in design. The
result is then often called pure tension, pure compression, or pure shear, depending
upon how the external load is applied to the body under study. The word simple is some-
times used instead of pure to indicate that there are no other complicating effects.
The tension rod is typical. Here a tension load F is applied through pins at the ends of
the bar. The assumption of uniform stress means that if we cut the bar at a section
remote from the ends and remove one piece, we can replace its effect by applying a uni-
formly distributed force of magnitude σA to the cut end. So the stress σ is said to be
uniformly distributed. It is calculated from the equation
F
σ = (3–22)
A
This assumption of uniform stress distribution requires that:

• The bar be straight and of a homogeneous material
• The line of action of the force contains the centroid of the section
• The section be taken remote from the ends and from any discontinuity or abrupt
change in cross section