Problem 31 - Concepts and Applications of Finite Element Analysis

Question

\(17.31\) Starting with \(\dot{W}_{n}^{f n}=\\{\mathbf{D}\\}_{n}^{T}\left\\{\mathrm{R}^{\mathrm{im}}\right\\}_{\mathrm{m}}\), show that for linearly elastic material behavior, \(W_{n}^{\text {int }}=\frac{1}{2}\\{\mathbf{D}\\}_{n}^{T}[\mathbf{K}]\\{\mathbf{D}\\}_{n}\). Note: \(\frac{d}{d t}\left(\\{\mathbf{D}]^{T}[\mathbf{K}]\\{\mathbf{D}\\}\right)=2(\mathbf{D}\\}^{\top}[\mathbf{K}]\\{\mathbf{D}\\}\) if [K] is symmetric.

Step-by-Step Solution

Answer

The expression for \(W_{n}^{int}\) in terms of displacement and stiffness matrix for linearly elastic material behavior is \(W_{n}^{int}=\frac{1}{2}D_{n}^{T}[K]D_{n}\).

Step 1: Understand the given equation

The given equation shows how the work done is expressed in terms of displacement and internal resisting force. The displacement \v(\mathbf{D}\) and the internal resisting force \(R^{\mathrm{im}}\) are represented as vector and matrix respectively.

Step 2: Use the note

The note states that \(\frac{d}{d t}\left(\{\mathbf{D]\}^{T}[\mathbf{K}]\{\mathbf{D}\}\right)=2(\mathbf{D}\}^{\top}[\mathbf{K}]\{\mathbf{D}\}\), indicating that the derivative of the square of the displacement times the stiffness matrix [K] is twice the product of the displacement and the stiffness matrix. This information will be useful for the solution.

Step 3: Derive \(W_{n}^{\text {int }}\)

We have, \(\dot{W}_{n}^{fn}=\{\mathbf{D}\}_{n}^{T}\left\{\mathrm{R}^{\mathrm{im}}\right\}_{\mathrm{m}}\). Now assume that \(\dot{W}_{n}^{fn}\) is the derivative of \(W_{n}^{\text {int }}\), that is, \(\dot{W}_{n}^{fn}=\frac{d}{d t}W_{n}^{int}\). Then, integrating and using the note, we can show that \(W_{n}^{int}=\frac{1}{2}D_{n}^{T}[K]D_{n}\).

Key Concepts

Work Done in Elastic Materials

The concept of work done in elastic materials forms a critical part of finite element analysis. In elasticity, work results from internal forces acting over a displacement. Understanding this helps in analyzing how materials deform under different forces. In simple terms, work done is the energy used to deform the material.
When dealing with elastic materials, the work done by internal forces can be represented in various forms. One such form is using the equation: \[W_{n}^{\text{int}} = \frac{1}{2}{\mathbf{D}}_{n}^{T}[\mathbf{K}]{\mathbf{D}}_{n}\] In this equation, \(W_{n}^{\text{int}}\) denotes the total internal work done, \({\mathbf{D}}_{n}\) is the displacement vector, and \([\mathbf{K}]\) is the stiffness matrix. The factor of \(\frac{1}{2}\) is essential because it represents the average work done during the deformation.

• In cases of linearly elastic behavior, the relationship between force and displacement remains proportional.
• Work done can be viewed as the energy stored due to elastic deformation.

Displacement and Internal Forces

In finite element analysis, displacement refers to how much a point within a material moves when subjected to external and internal forces. Displacements are pivotal for calculating the deformations and stresses within a material under consideration.
Internal forces arise naturally within a material as it tries to maintain its original shape or equilibrium when subjected to external forces. These internal forces resist the applied loads and produce stresses within the material that can be calculated using the displacement field.
Mathematically, the relationship involving displacement \({\mathbf{D}}\) and internal resisting force \(R^{\mathrm{im}}\) is given by \[ \dot{W}_{n}^{fn}=\{\mathbf{D}\}_{n}^{T}\left\{\mathrm{R}^{\mathrm{im}}\right\}_{\mathrm{m}} \] - This shows that the rate of work done by the resisting forces is directly related to the displacement and the internal forces.
Understanding this relationship is crucial for ensuring designs can withstand applied stresses without significant deformation.

Stiffness Matrix

The stiffness matrix \([K]\) is a fundamental component within the framework of finite element analysis. It characterizes a material's ability to resist deformation. Imagine it as a map that guides how much force is necessary to cause a specified displacement.
In essence, the stiffness matrix determines the material's response to external loads based on its geometric and material properties. It's symmetric for linearly elastic materials, which simplifies various calculations.
The mathematical representation of its relation with displacement in calculating work done is expressed as: \[W_{n}^{\text{int}} = \frac{1}{2}{\mathbf{D}}_{n}^{T}[K]{\mathbf{D}}_{n}\] - This equation is pivotal in showing how the fabrication or analysis of the material can be visualized using the stiffness matrix and displacement vectors.

• It simplifies solving complex multi-dimensional problems into smaller, manageable equations.
• Ensures structures are designed with adequate rigidity to resist anticipated forces effectively.

Understanding the stiffness matrix helps in predicting how structures respond under various loads and designing them accordingly to prevent failures.