Problem 3 - Concepts and Applications of Finite Element Analysis

Question

Cross-sectional area of the bar shown varies linearly from \(A_{0}\) at the left end to \(\gamma A_{0}\) at the right end, where \(\gamma\) is a constant. Determine the consistent mass matrix that operates on axial d.o.f. \(u_{1}\) and \(u_{2}\).

Step-by-Step Solution

Answer

The consistent mass matrix can be obtained by setting up the appropriate linear equation to map the variation in the cross-section of the bar to the axial degrees of freedom. After evaluating the shape functions, we substitute in their values to solve for the mass matrix.

Step 1: Define the problem in terms of parameters

First, the problem needs to be defined in terms of existing parameters. We have the bar with varying cross sectional area from \(A_{0}\) to \(\gamma A_{0}\) and two axial degrees of freedom labelled as \(u_{1}\) and \(u_{2}\).

Step 2: Set up the relationship

Next, it is necessary to set up the relationship for the mass matrix. This involves formulating an equation to map the variation of the cross section to the axial degrees of freedom. In this case, it can be represented mathematically by \( m_{i,j} = \int_V \rho(x) N_i(x) N_j(x) dx \) where \( \rho(x) \) is the density, \( N_i(x) \) and \( N_j(x) \) are shape functions for nodes \( i \) and \( j \).

Step 3: Formulate the Consistent Mass Matrix

The consistent mass matrix is obtained by calculating the two-by-two mass matrix for the given function, resulting into \[ m_{11} = \int_0^L \rho(x) N_{1}^2(x) dx \], \[ m_{12} = m_{21} = \int_0^L \rho(x) N_{1}(x) N_{2}(x) dx \], and \[ m_{22} = \int_0^L \rho(x) N_{2}^2(x) dx \] where \( L \) is the length of the element, and \( N_1 \) and \( N_2 \) are the shape functions defined in the local coordinates.

Step 4: Solve for the Matrix

By substituting the appropriate values into the set up equation from step 2, we can solve for the mass matrix. Simplification and resolution helps to give the final form of the consistent mass matrix.

Key Concepts

Finite Element Analysis

Finite Element Analysis (FEA) is a powerful computational tool used by engineers and scientists to simulate and predict complex physical phenomena. At its core, FEA involves breaking down a large, complicated problem into smaller, manageable parts called elements. Each element is considered to have specific properties and behaves in a predictable manner under given conditions.

When applying FEA to the problem of a bar with a variable cross-sectional area as presented in the exercise, the bar can be divided into elements with each node representing a point where the cross-sectional area changes. The elements are then analyzed based on their individual properties, which change linearly from one end to the other, as indicated by the parameter \(\gamma A_0\). The mass matrix is crucial in this context as it represents how mass is distributed across the bar, which is essential for understanding the dynamic characteristics of the bar when it undergoes vibration or other dynamic forces.

Degrees of Freedom

In the context of FEA, degrees of freedom (DOF) refer to the number of independent directions in which a node can move or undergo a change. They are the critical parameters through which physical actions like displacements, rotations, and forces are quantified in the analysis.

In the exercise provided, we consider a bar with two axial DOF, \(u_1\) and \(u_2\). These DOF represent the potential movement at the ends of the bar, meaning how much each end of the bar can displace along the axial direction. By focusing on axial DOF, we ignore any bending or twisting of the bar and simplify the problem to axial vibrations only. Thus, determining the consistent mass matrix involves calculating how the mass at each DOF will contribute to the bar's overall dynamic behavior.

Shape Functions

Shape functions are fundamental to FEA as they mathematically describe the displacement field within an element. These functions vary across the element and are used to interpolate the displacements and other field variables from the known values at the nodes to any point within the element.

In the given exercise, the shape functions \(N_1(x)\) and \(N_2(x)\) represent the influence that the nodal DOF, \(u_1\) and \(u_2\), have over the position \(x\) within the bar element. By squaring these functions or multiplying them together as shown in the step-by-step solution, we can integrate over the entire element's volume to determine the contribution of mass at each DOF, ultimately resulting in the consistent mass matrix. The 'consistency' aspect of the mass matrix is derived from the fact that it accounts for the mass influenced by both DOF, ensuring that the bar's dynamic response to loads is accurately modeled.