Problem 40 - Concepts and Applications of Finite Element Analysis

Question

The rigid bar shown rests on an elastic foundation. When displaced laterally an amount \(w\), the foundation applies a force \(k w d x\) to a length \(d x\) of the bar, where \(k\) is a constant. Determine the 2 by 2 stiffness matrix that operates on \(w_{f}\) and \(w_{j} .\) Suggestion: Express strain energy \(U\) in terms of \(k, L, w_{i}\), and \(w_{j}\), then write \(U\) in the form \(\\{\mathrm{d}\\}^{x}[\mathrm{k}]\\{\mathrm{d}\\} / 2\), and identify \([\mathrm{k}]\).

Step-by-Step Solution

Answer

The stiffness matrix [k] is \(k^2/2[a, 0; 0, L-a]\).

Step 1: Derive Strain Energy of the Bar

The strain energy in a bar subjected to a force \(fdx\) is given by \(fdx^2/2\). For a small segment with length \(dx\) at a location \(x\), the force \(f\) is \(kwdx\). Thus, the strain energy \(dU\) of this segment can be represented as \(dU=k^2w^2dx/2\).

Step 2: Develop the Total Strain Energy Expression

The total strain energy \(U\) is the integral of \(dU\) over the length of the bar. Thus, \(U = \int_{0}^{a} k^2 w(x)^2 dx/2 + \int_{a}^{L} k^2 w(x)^2 dx/2\). Here, \(w(x)\) is replaced with \(w_i\) and \(w_j\) for interval [0,a] and [a,L] respectively. So it becomes \(U = \int_{0}^{a} k^2 w_i^2 dx/2 + \int_{a}^{L} k^2 w_j^2 dx/2\).

Step 3: Compute the Integral

Evaluate the integral to find the expression of \(U\), leading to \(U = k^2 w_i^2 a/2 + k^2 w_j^2 (L-a)/2\).

Step 4: Rewrite the Strain Energy in Matrix Form

Rewrite \(U\) in terms of matrix form \(U = {d}^{T}[k]{d}/2\) where \({d}^{T} = [w_i, w_j]\) and [k] is the stiffness matrix. We compute the value of [k] by comparing the matrix form with the integral result, yielding \([k] = k^2/2[a , 0; 0 , L - a]\).

Key Concepts

Understanding Strain Energy

Strain energy is a pivotal concept in structural mechanics, referring to the energy stored in a system due to deformation. When a structure such as a bar, beam, or even a bridge is subjected to external forces, it deforms; this deformation stores potential energy. Imagine compressing a spring; you're doing work, and the spring holds this energy until it's released. In the context of our problem, the strain energy, denoted as \(U\), within an elastic bar resting on an elastic foundation is calculated as the work done by the force supplied by the foundation when the bar undergoes a lateral displacement \(w\).

As per the step-by-step solution provided, the strain energy for a small segment of length \(dx\) is \(dU = \frac{k^2w^2dx}{2}\), where \(k\) is the stiffness of the foundation per unit length of the bar. Upon integrating this expression along the length of the bar, you obtain the total strain energy stored in the system. This quantifies the internal work done and is fundamental when considering the bar's response to applied forces and subsequent return to its original state upon unloading.

Elastic Foundation Concept

The elastic foundation model is used to describe how a structure interacts with the supportive surface beneath it. This concept is akin to a bed of springs, where the structure, like a bar or beam, 'rests' on springs spread evenly along its length. Each spring represents a segment of the substrate and responds, according to Hooke's Law, with a force proportional to the displacement of the structure at that point, characterized by the constant \(k\).

In the given exercise, the bar deforms laterally by an amount \(w\), and the foundation reacts by applying a restoring force \(kwdx\) to a segment \(dx\) of the bar. This interaction is central in the assessment of deflection, vibration, and stability of structures supported by elastic foundations. For engineers and architects, analyzing how a structure will behave on an elastic foundation is crucial for safe and efficient design, ensuring that the stresses and deformations stay within acceptable limits.

Finite Element Analysis (FEA) Fundamentals

Finite Element Analysis (FEA) is a mathematical tool that helps in approximating the behavior of structures under various loads. It involves breaking down a complex real-world structure into a mesh of simpler finite elements. These elements are connected at points known as nodes, and their physical behavior is described by mathematical equations. By solving these equations, engineers and scientists can predict how the structure will respond to external forces, whether it undergoes bending, stretching, or compression.

The stiffness matrix, central to FEA, is an array of coefficients that relate the nodal displacements (or degrees of freedom) to applied forces. Think of it like a matrix of springs: each matrix element quantifies the force needed to achieve a unit deformation in a particular direction. In our problem, the objective was to determine such a stiffness matrix for a bar on an elastic foundation, where \(w_i\) and \(w_j\) are the lateral displacements of the bar at two points. The stiffness matrix helps us understand how rigid or flexible the bar is in different sections and is critical for analyzing the bar's mechanical properties using FEA.