2.21 Equilibrium Equations in Polar Co-ordinates (Two-Dimensional …
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2.21 Equilibrium Equations in Polar Co-ordinates
(Two-Dimensional State of Stress)
While discussing the problems with circular boundaries, it is more convenient to use
the cylindrical co-ordinates such as r, θ and z. In the case of plane stress or plane strain
problems, we have τ r z = τ θ z = 0 and the other stress components as functions of
r and θ only. Hence, the cylindrical co-ordinates reduce to polar co-ordinates in this
case. In general, polar co-ordinates are used advantageously where a degree of axial
symmetry exists. Examples include a cylinder, a disc, a curved beam and a large thin
plate containing a circular hole.
The polar co-ordinate system (r, θ ) and the cartesian system (x, y) are related by
the following expressions:
x = r cos θ, r
2
= x
2
+ y
2
y = r sin θ, θ = tan
−1
y
x
(2.44)
Consider the state of stress on an infinitesimal element a b c d of unit thickness
described by the polar co-ordinates as shown in Fig. 2.16. The body forces denoted
by F r and F θ are directed along r and θ directions, respectively.
Resolving the forces in the r-direction, we have for equilibrium, ΣF r = 0,
− σ r × r dθ +
σ r +
∂σ r
∂r
dr
(r + dr )dθ − σ θ dr sin
dθ
2
+ F r −
σ θ +
∂σ θ
∂θ
dθ
dr sin
dθ
2
− τ r θ dr cos
dθ
2
+
τ r θ +
∂τ r θ
∂θ
dθ
dr cos
dθ
2
= 0
Since dθ is very small,
Fig. 2.16 Stresses acting on
an element