Question

Which of the following statement about Airy's stress function \(\phi(x,y)\) is correct?

• A. It is a vector function used to represent displacement components
• B. It automatically satisfies the equilibrium equations in two-dimensional elasticity
• C. It only applies to three-dimensional elasticity problems
• D. It does not need to satisfy any differential equation

Hint:

Airy's stress function is a \[ \boxed{\text{scalar function}} \] used in plane elasticity. It automatically satisfies the equilibrium equations and must satisfy the biharmonic equation \[ \boxed{\nabla^4\phi=0.} \]

Answer 1

The Correct Option is B

Step 1: Recall Airy's stress function. Airy's stress function, \[ \phi(x,y), \] is a scalar function used in two-dimensional elasticity. The stress components are obtained as \[ \sigma_x=\frac{\partial^2\phi}{\partial y^2}, \] \[ \sigma_y=\frac{\partial^2\phi}{\partial x^2}, \] \[ \tau_{xy}=-\frac{\partial^2\phi}{\partial x\partial y}. \]

Step 2: Identify its main property. When stresses are expressed using Airy's stress function, \[ \boxed{ \text{the equilibrium equations are automatically satisfied.} } \] Therefore, \[ \boxed{\text{It automatically satisfies the equilibrium equations in two-dimensional elasticity.}} \] Thus, \[ \boxed{(B)} \] is the correct answer.