8.4 Flamant’s Problem
273
Fig. 8.7 Stresses due to a
vertical line load in
rectangular co-ordinates
But from Fig. 8.7,
r =
x 2 + z 2
cos θ =
z
x 2 + z 2
, sin θ =
x
√
x 2 + z 2
Therefore,
σ z =
2q
π
x 2 + z 2
z
3
√
x 2 + z 2
3
σ ¯
z =
2qz
3
π
x 2 + z 2
2
Similarly,
σ x = σ r sin
2
θ + σ θ cos
2
θ + 2τ r θ sin θ
=
2q
πr
cos θ sin
2
θ + 0 + 0
=
2q
π
√
x 2 + z 2
z
√
x 2 + z 2
x
2
x 2 + z 2
273
Fig. 8.7 Stresses due to a
vertical line load in
rectangular co-ordinates
But from Fig. 8.7,
r =
x 2 + z 2
cos θ =
z
x 2 + z 2
, sin θ =
x
√
x 2 + z 2
Therefore,
σ z =
2q
π
x 2 + z 2
z
3
√
x 2 + z 2
3
σ ¯
z =
2qz
3
π
x 2 + z 2
2
Similarly,
σ x = σ r sin
2
θ + σ θ cos
2
θ + 2τ r θ sin θ
=
2q
πr
cos θ sin
2
θ + 0 + 0
=
2q
π
√
x 2 + z 2
z
√
x 2 + z 2
x
2
x 2 + z 2