Problem 6 - Concepts and Applications of Finite Element Analysis

Question

Let loads \(P_{1}\) and \(P_{2}\) be functions of displacements \(D_{1}\) and \(D_{2}\), that is, \(P_{1}=f_{1}\left(D_{1}, D_{2}\right)\) and \(P_{2}=f_{2}\left(D_{1}, D_{2}\right)\). Let \(D_{A}\) and \(D_{B}\) be exact values of \(D_{1}\) and \(D_{2}\) produced by loads \(P_{A}\) and \(P_{B}\). Let \(D_{\lambda}^{*}\) and \(D_{B}^{*}\) be approximations of \(D_{A}\) and \(D_{B}\). Assume that \(D_{A}=D_{A}^{*}+\Delta D_{A}\) and \(D_{B}=D^{2}+\Delta D_{B}\). Derive the following equations (analogous to Eq. 17.2-9): $$ \left[\begin{array}{ll} \partial P_{1} / \partial D_{1} & \partial P_{1} / \partial D_{2} \\ \partial P_{2} / \partial D_{1} & \partial P_{2} / \partial D_{2} \end{array}\right]_{D_{\lambda}^{\prime}, D_{s}^{\prime}}\left\\{\begin{array}{l} \Delta D_{A} \\ \Delta D_{B} \end{array}\right\\}=\left\\{\begin{array}{l} P_{A}-f_{1}\left(D_{\lambda}^{*}, D_{B}^{*}\right) \\ P_{B}-f_{2}\left(D_{\lambda}^{*}, D_{B}^{\prime}\right) \end{array}\right\\} $$

Step-by-Step Solution

Answer

\(\Delta D_{A}\) and \(\Delta D_{B}\) represent the amounts by which displacement variables \(D_{1}\) and \(D_{2}\) should be adjusted in order that the approximations \(f_{1}\) and \(f_{2}\) of the loads match the exact loads \(P_{A}\) and \(P_{B}\), respectively. The given equations are therefore expressions for these adjustments, obtained by approximating the functions \(f_{1}\) and \(f_{2}\) using first order Taylor series expansion and then by solving the resulting linear system for \(\Delta D_{A}\) and \(\Delta D_{B}\).

Step 1: Apply small perturbations method

The method of small perturbations is a way of approximating the response of a system when its inputs are perturbed slightly. We have two functions \(f_{1}(D_{1}, D_{2})\) and \(f_{2}(D_{1}, D_{2})\), and we are interested in how their outputs (the loads \(P_{1}\) and \(P_{2}\)) change when the inputs \(D_{1}\) and \(D_{2}\) are perturbed by small amounts \(\Delta D_{A}\) and \(\Delta D_{B}\) respectively. This allows us to write the perturbed loads as \(P_{A} = f_{1}(D_{\lambda}^{*} + \Delta D_{A}, D_{B}^{*} + \Delta D_{B})\) and \(P_{B} = f_{2}(D_{\lambda}^{*} + \Delta D_{A}, D_{B}^{*} + \Delta D_{B})\).

Step 2: Approximate using first order Taylor series expansion

A popular way to approximate a function near a point is by using the Taylor series expansion. For small perturbations, we can limit this expansion to first order. This results in the approximations \(P_{A} \approx f_{1}(D_{\lambda}^{*}, D_{B}^{*}) + \Delta D_{A} * (\partial f_{1} / \partial D_{1})|_{D_{\lambda}^{*}, D_{B}^{*}} + \Delta D_{B} * (\partial f_{1} / \partial D_{2})|_{D_{\lambda}^{*}, D_{B}^{*}}\) and \(P_{B} \approx f_{2}(D_{\lambda}^{*}, D_{B}^{*}) + \Delta D_{A} * (\partial f_{2} / \partial D_{1})|_{D_{\lambda}^{*}, D_{B}^{*}} + \Delta D_{B} * (\partial f_{2} / \partial D_{2})|_{D_{\lambda}^{*}, D_{B}^{*}}\).

Step 3: Solve for the perturbations

Now, we have a system of linear equations w.r.t. \(\Delta D_{A}\) and \(\Delta D_{B}\), obtained by equating the expressions derived in Step 2 with \(P_{A}\) and \(P_{B}\). Solving this system gives us the required changes \(\Delta D_{A}\) and \(\Delta D_{B}\), thereby deriving the given set of equations.

Key Concepts

Perturbation Methods

Perturbation methods provide a systematic approach for finding an approximate solution to problems involving small changes, or perturbations, to the control parameters. These methods are commonly used in many fields of science and engineering, especially in situations where an exact solution is difficult or impossible to determine.

Imagine you're holding a smooth ball, representing a system at a stable point, and you nudge it slightly—this nudge is akin to the perturbation. In finite element analysis, perturbation methods allow us to study the effects of slight changes in load or shape on the response of structures or materials. Applying this to our exercise, the displacements \(D_1\) and \(D_2\) are perturbed to create a scenario that is close to the real condition but still manageable to analyze mathematically.

By using perturbation methods, engineers and scientists can predict how a system will respond to small disturbances, which is critical in design and failure analysis. This technique is particularly invaluable when the structure of the problem is known but the exact magnitudes of influencing factors are not.

Taylor Series Expansion

The Taylor series expansion is a powerful mathematical tool that approximates functions using the sum of terms calculated from the values of its derivatives at a single point. Think of it like an artist creating a portrait: starting with a rough sketch (the function at a point) and adding layer upon layer of detail (the derivatives) to capture the full picture.

In the context of the given exercise, Taylor series expansion assists in simplifying the functions \(f_1\) and \(f_2\), which relate the displacements to the loads. When we introduce small changes \(\Delta D_A\) and \(\Delta D_B\), the Taylor series helps us approximate the new loads without recalculating everything from scratch. We can liken this to adjusting a photograph's brightness; a small tweak can predict the outcome without re-processing the entire image.

For small perturbations, a first-order Taylor series expansion is often sufficient. It provides a linear approximation, which simplifies many complex problems in engineering, making them easier to solve and interpret. Especially in finite element analysis, where small changes can significantly affect outcomes, being able to predict behavior under these slight modifications is essential.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. For those just starting out in algebra, think of it like a recipe that involves balancing ingredients perfectly to bake the desired cake—in our case, these ingredients are the variables, and the cake is the solution that satisfies all equations simultaneously.

In our original problem, we derived two linear equations that relate the perturbations \(\Delta D_A\) and \(\Delta D_B\) to the loads \(P_A\) and \(P_B\). Solving this system yields the values of the perturbations that are analogous to finding the right 'mix' of variables. Linear algebra offers various methods to solve these equations, such as substitution, elimination, or matrix operations.

Understanding how to manipulate and solve systems of linear equations is fundamental in engineering, as it often underpins the process of design optimization and stability analysis. In finite element analysis, the ability to solve these systems allows for predicting the behavior of complex structures under various loading conditions.