Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 74

φ =

1.72 If G 1, the force formulation for the torsion problem leads to the governing

equation

2 2

∂

y 2 + ∂

ψ ψ

z 2 =−

2

∂ ∂

with the requirement that 0 on the boundary. Show that this problem formulation

involving Poisson's equation can be reformulated in terms of Laplace's equation.

ψ =

ψ = ψ ∗ −

1

y 2 z 2

Hint: Let

2 (

+ )

1
2

Answer: With ψ = ψ ∗ − ( y 2 + z 2 ) , the governing formula becomes

ψ ∗ ψ ∗

∂

2

y 2 + ∂

2

= 0

∂ ∂ z 2

ψ ∗ =

1

y 2 z 2 with

2 (

+ ) on the boundary.