7.7 prandtl’s Membrane Analogy
239
Fig. 7.6 Stretching of a membrane
When the membrane is subjected to a constant lateral pressure p, it undergoes a
small displacement z where z is a function of x and y.
Consider the equilibrium of an element ABCD of the membrane after deformation.
Let T be the uniform tension per unit length of the membrane. The value of the initial
tension T is large enough to ignore its change when the membrane is blown up by
the small pressure p. On the face AC, the force acting is T.dy. This is inclined at
an angle β to the x-axis. Also, tan β is the slope of the face AB and is equal to
∂z
∂ x
.
Hence, the component of Tdy in z-direction is
−T dy
∂z
∂ x
. The force on face BD is
also Fdy but is inclined at an angle (β + β) to the x-axis. Its slope is, therefore,
∂z
∂ x
+
∂
∂ x
∂z
∂ x
dx
and the component of the force in the z-direction is
T dy
∂z
∂ x
+
∂
∂ x
∂z
∂ x
dx
Similarly, the components of the forces T dx acting on face AB and CD are
−T dx
∂z
∂ y
and T dx
∂z
∂ y
+
∂
∂ y
∂z
∂ y
dy
Therefore, the resultant force in z-direction due to tension T