Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

514
STRESS TRANSFORMATIONS
mecmovies 12.15 presents an
animated derivation of the Mohr’s
circle stress transformation
equations.
The German civil engineer Otto Christian Mohr (1835–1918) developed a useful
graphical interpretation of the stress transformation equation. This method is known as
Mohr’s circle. Although it will be used for plane stress transformations here, the Mohr’s
circle method is also valid for other transformations that are similar mathematically, such
as area moments of inertia, mass moments of inertia, strain transformations, and threedimensional
stress transformations.
Derivation of the circle Equation
Mohr’s circle for plane stress is constructed with the normal stress σ plotted along the
horizontal axis and the shear stress τ plotted along the vertical axis. The circle is constructed
such that each point it represents is a combination of σ and τ that acts on one
specific plane through a point in a stressed body. The general plane stress transformation
equations, expressed with double-angle trigonometric functions, were presented as Equations
(12.5) and (12.6), respectively, in Section 12.7:
σ
n
σx + σ y σx − σ y
= + cos2θ + τxy
sin2θ
2 2
τ
nt
σx
=−
− σ y
sin2θ + τxy
cos2θ
2
Equations (12.5) and (12.6) can be rewritten with terms involving 2θ on the right-hand side:
σ
n
σx + σ y σx − σ y
− = cos2θ + τxysin2θ
2 2
σx
− σ y
τ nt =− sin2θ + τxy
cos2θ
2
Both equations can be squared, then added together, and simplified to give
⎛
σ
⎝
⎜
n
σx
−
2
2
+ σ y⎞
σ − σ
2 ⎛ x y⎞
2
τ
τ
⎠
⎟ + nt =
⎝
⎜
⎠
⎟ + xy
(12.20)
2 2
This is the equation of a circle in terms of the variables σ n and τ nt . The center of the circle
is located on the σ axis (i.e., τ = 0), at
C
σx
=
+ σ
2
y
(12.21)
The radius of the circle is given by the square root of the right-hand side of Equation (12.20):
mecmovies 12.16 presents a
step-by-step guide to constructing
Mohr’s circle for plane stress.
R =
Equation (12.20) can be written in terms of C and R as
2
⎛ σx
− σ y⎞
2
τ xy
⎝
⎜
2 ⎠
⎟ + (12.22)
2 2 2
( σ − C)
+ τ = R
(12.23)
n
which is the standard algebraic equation for a circle with radius R and center C.
nt