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6 Two-Dimensional Problems in Elasticity …
6.11 The Effect of Circular Holes on Stress Distributions
in Plates
Consider a plate subjected to a uniform tensile stress p as shown in Fig. 6.9. The
plate thickness is small in comparison to its width and length so that we can treat
this problem as a plane stress case. Let a hole of radius “a” is drilled in the middle
of the plate as shown in the figure. This hole will disturb the stress field in the
neighbourhood of the hole. But from St. Venant’s principle, it can be assumed that
any disturbance in the uniform stress field will be localized to an area within a circle
of radius “b”. Beyond this circle, it is expected that the stresses to be effectively the
same as in the plate without the hole.
Now consider the equilibrium of an element ABC at r = b and angle θ with
respect to x-axis.
∴ σ r = p BC
cos θ
AC
= p cos
2
θ
∴ σ r =
p
2
(1 + cos 2θ)
(6.56)
and
τ r θ = −p.BC
sin θ
AC
= −p sin θ cos θ
∴ τ r θ = −
p
2
sin 2θ
(6.57)
These stresses, acting around the outside of the ring having the inner and outer
radii r = a and r = b, give a stress distribution within the ring which may be regarded
as consisting of two parts.
(a) A constant radial stress
p
2
at radius b. This condition corresponds to the ordinary
thick cylinder theory, and stresses σ
r and σ
θ at radius r is given by
σ
r = A +
B
r 2
and σ
θ = A −
B
r 2
Constants A and B are given by boundary conditions,
(i) At r = a, σ r = 0
(ii) At r = b, σ r =
p
2
On substitution and evaluation, we get
6 Two-Dimensional Problems in Elasticity …
6.11 The Effect of Circular Holes on Stress Distributions
in Plates
Consider a plate subjected to a uniform tensile stress p as shown in Fig. 6.9. The
plate thickness is small in comparison to its width and length so that we can treat
this problem as a plane stress case. Let a hole of radius “a” is drilled in the middle
of the plate as shown in the figure. This hole will disturb the stress field in the
neighbourhood of the hole. But from St. Venant’s principle, it can be assumed that
any disturbance in the uniform stress field will be localized to an area within a circle
of radius “b”. Beyond this circle, it is expected that the stresses to be effectively the
same as in the plate without the hole.
Now consider the equilibrium of an element ABC at r = b and angle θ with
respect to x-axis.
∴ σ r = p BC
cos θ
AC
= p cos
2
θ
∴ σ r =
p
2
(1 + cos 2θ)
(6.56)
and
τ r θ = −p.BC
sin θ
AC
= −p sin θ cos θ
∴ τ r θ = −
p
2
sin 2θ
(6.57)
These stresses, acting around the outside of the ring having the inner and outer
radii r = a and r = b, give a stress distribution within the ring which may be regarded
as consisting of two parts.
(a) A constant radial stress
p
2
at radius b. This condition corresponds to the ordinary
thick cylinder theory, and stresses σ
r and σ
θ at radius r is given by
σ
r = A +
B
r 2
and σ
θ = A −
B
r 2
Constants A and B are given by boundary conditions,
(i) At r = a, σ r = 0
(ii) At r = b, σ r =
p
2
On substitution and evaluation, we get