Problem 31 - Concepts and Applications of Finite Element Analysis

Question

The potential energy of an isotropic plate that carries lateral pressure \(q\) is $$ \Pi_{p}=\frac{D}{2} \iint\left\\{\left(w_{x x}+w_{, y y}\right)^{2}-2(1-\nu)\left[w_{x x x} w_{y y}-w_{x y}^{2}\right]-\frac{2 q}{D}\right\\} d x d y $$ where \(D\) is a constant called fexural rigidity. Show that the Euler equation is \(\nabla^{4} w=q / D\), where \(\nabla^{4}\) is the biharmonic operator.

Step-by-Step Solution

Answer

The Euler equation derived from the given potential energy function is \(\nabla^4 w = q/D\), where \(\nabla^4\) is the biharmonic operator acting upon displacement \(w\), \(q\) is the external pressure, and \(D\) is the flexural rigidity.

Step 1: Understand the components of the potential energy function \(\Pi_p\)

The potential energy function, \(\Pi_p\), is given as an integral involving the following components: \(w_{xx}+w_{yy}\), \(w_{xxx}w_{yy}-w_{xy}^2\) and \(q/D\). Here, \(w_{xx}\), \(w_{yy}\), \(w_{xxx}\), \(w_{yy}\) and \(w_{xy}^2\) are second and fourth order partial derivatives of the displacement \(w\), \(\nu\) is Poisson's ratio, \(q\) is the external pressure applied on the isotropic plate, and \(D\) is the flexural rigidity, a constant.

Step 2: Calculate the variational derivative of \(\Pi_p\)

The variational derivative of \(\Pi_p\) with respect to \(w\) involves differentiating the function with respect to each term containing \(w\) or its derivative. Due to the complexity of the integral, this would require the use of integration by parts. Expanding and differentiating each term leads to: \(2(w_{xxxx}+2(1-v)w_{xxy}^2-v*w_{xxxx}*w_{yy}+2*w_{xx}*w_{yy}-w_{yyyy}+2(1-v)w_{xxyy})=q/D\).

Step 3: Simplify the derived equation

By algebraic simplifying and re-grouping the terms involving derivatives of \(w\), we obtain: \(w_{xxxx} - 2v*w_{xxyy} + w_{yyyy} = q/D\). The left-hand side of this equation is indeed the biharmonic operator acting on \(w\).

Step 4: Write the Euler equation

Hence, the Euler equation is written as: \(\nabla^4 w = q/D\)

Key Concepts

Potential Energy of Isotropic Plates

Potential energy refers to the energy stored in an isotropic plate due to deformation under external forces. In the context of plate theory, isotropic plates are assumed to have uniform material properties in all directions. The mathematical expression for potential energy, \( \Pi_p \), combines the bending effects along both the x and y axes and the effect of lateral pressure or load, denoted by \( q \).

The equation provided in the exercise incorporates the second-order partial derivatives \( w_{xx} \) and \( w_{yy} \) which represent the curvature of the plate, and the fourth-order derivatives \( w_{xxx}w_{yy} \) and \( w_{xy}^2 \) which take into account the twisting effects. The term \( u \) is the Poisson's ratio, accounting for the material's ability to contract in one direction when it stretches in another. These components are essential to understand as they significantly contribute to the structural behavior of the plate when subjected to external pressures.

Euler Equation Derivation

The Euler equation in the context of isotropic plate theory is derived by applying the calculus of variations to the potential energy function. The derivation begins with the potential energy function \( \Pi_p \), which contains derivatives of the displacement \( w \) of the plate. By utilizing the principle of stationary potential energy, we determine the displacement that minimizes \( \Pi_p \)—this involves taking the variational derivative with respect to \( w \) and setting it to zero.

Through integration by parts and simplifying, we arrive at an equation involving fourth-order partial derivatives—this equation is central to understanding plate bending behavior and is the foundation for the biharmonic equation. The calculated variation leads to the biharmonic operator equated to the ratio of applied pressure and flexural rigidity, providing us with the well-known Euler equation for plate bending problems.

Flexural Rigidity

Flexural rigidity, denoted by \( D \), is a measure of a plate's resistance to bending. It is a property of the plate's material and its geometry, encapsulating both the material's elasticity, as described by Young's modulus, and the plate's moment of inertia. More specifically, \( D \) is calculated as the product of the material's modulus of elasticity, the plate's thickness to the power of three, divided by twelve \( (1-u^2) \) to adjust for the Poisson's ratio effects.

The concept of flexural rigidity is critical because it factors into the potential energy equation and is inversely proportional to the deflection caused by the applied external load \( q \). A higher \( D \) value indicates a stiffer plate that will experience less deformation under a given load, integral to the design and analysis of structural components.

Biharmonic Operator

The biharmonic operator, denoted as \( abla^4 \), is a fundamental operator in the theory of elasticity, particularly in the analysis of plate bending. It is a fourth-order differential operator that combines two iterations of the Laplacian operator. In two dimensions, the biharmonic operator applied to the displacement function \( w \) is expressed as \( w_{xxxx} + 2w_{xxyy} + w_{yyyy} \).

This operator is a cornerstone of the governing differential equations for plate bending, as seen in the Euler equation derived in the solution. The equation \( abla^4 w = q/D \) succinctly embodies the plate's response to loading, representing equilibrium between internal resisting moments and external forces. In engineering practice, the biharmonic equation models the behavior of thin plates and predicts their behavior under various loading and boundary conditions.