Problem 8 - Concepts and Applications of Finite Element Analysis

Question

Consider a uniform, simply supported beam of length \(L\). For each of the following loadings, use the Galerkin method to determine constant \(a\) in the approximating lateral displacement field \(\bar{w}=a x(L-x)\). Also compute the percentage error of the predicted midspan deflection.(a) Sinusoidal distributed lateral load \(q=q_{0} \sin (\pi x / L)\) over \(0

Step-by-Step Solution

Answer

The constant 'a' is found by applying the Galerkin's method to the differential equation of lateral deflection and comparing this with the actual loading on the beam. The percentage error in the predicted midspan deflection is calculated by comparing the deflection obtained from the trial function and from the exact analytical solution of the beam loading problem.

Step 1: Define the differential equation

The differential equation for the lateral deflection \( w(x) \) of a simply supported beam is: \[EI \frac{d^4 w}{dx^4} = -q(x)\]where:\( EI \) is the flexural rigidity,\( q(x) \) is the load per unit length.

Step 2: Define the trial function

Given that the trial function for lateral deflection is: \(\bar{w}=ax(L−x)\), this function satisfies the boundary conditions since \( \bar{w}(0) = \bar{w}(L) = 0 \). The function form implies that the maximum deflection will be at the mid-span.

Step 3: Apply Galerkin's Method for the first case

In the first part, we are dealing with a Sinusoidal distributed lateral load scenario and thus \( q = q_{0} \sin (\pi x / L) \). Galerkin's method relates the differential equation and the trial function through their integral over the domain. This results in the evaluation of the constant 'a'.

Step 4: Apply Galerkin’s Method for the second case

As for part (b), we are dealing with a Uniformly distributed lateral load scenario, i.e., \( q = q_{0} \). We have to repeat the integral calculations similar to the previous step, which yields a different value for the constant 'a'.

Step 5: Compute percentage error for each case

The midspan deflection calculated from these scenarios can be compared with the exact solution to find the percentage error. The exact solution in these loading conditions can be obtained from beam deflection tables or by solving the governing differential equation. The percentage error will be calculated as \(\frac{|calculated - exact|}{exact}*100\% \).

Key Concepts

Galerkin Method

The Galerkin Method is a weighted residual technique used in the Finite Element Method to approximate solutions to differential equations. It's particularly useful when dealing with complex structures like beams under various loads. The method involves selecting a trial function that approximately satisfies the boundary conditions of the problem. In the context of beam deflection, this trial function is often parameterized by one or more constants. The Galerkin Method helps determine these constants by ensuring that the residual, or the difference between the exact and approximate solutions, is orthogonal to the weight functions over the domain.

This approach transforms a differential equation problem into an integral equation problem, making it easier to handle analytically or numerically. The trial function in this exercise, \(\bar{w}=ax(L-x)\), perfectly fits the simply supported boundary conditions, as it is zero at both ends of the beam. The value of 'a' is found by ensuring the trial function best fits the differential equation for each loading case through integration.

Beam Deflection

Beam Deflection refers to the bending or displacement a beam undergoes when external forces are applied. This concept is crucial in structural engineering, as it determines how much a beam will deviate from its original position under a given load. The deflection is influenced by factors such as the material's properties, the beam's dimensions, and the type and magnitude of the applied load.

In the exercise at hand, the beam is simply supported, meaning it is supported at both ends but free to deflect in the middle. This configuration ensures that the moments at the supports are zero, simplifying the analysis. The governing equation for the deflection of a beam is a fourth-order differential equation: \(\EI \frac{d^4 w}{dx^4} = -q(x)\). Here, \(EI\), or flexural rigidity, denotes the product of the modulus of elasticity and the beam's moment of inertia, which resists bending.

Sinusoidal Load

A Sinusoidal Load is a type of lateral load distribution that varies smoothly and periodically along the beam's length. In this exercise, the load is defined by the function \(\q = q_{0} \sin(\pi x / L)\), representing how the load intensity changes over the span of the beam. This load distribution is common in practical problems where loads might vary cyclically, such as in vibrating structures or wave-induced loads.

Using the Galerkin Method, the trial function \(\bar{w}=ax(L-x)\) is integrated with the sinusoidal load function over the domain to determine the constant 'a'. This ensures that the approximation accounts for the specific nature of the load, allowing for an accurate estimate of the beam’s deflection.

Uniform Load

Uniform Load refers to a constant force per unit length applied over the entire span of the beam. This distribution is one of the simplest and most commonly seen in engineering problems, representing a beam carrying its self-weight or other uniformly distributed loads.In the given exercise, the load is defined as \(\q = q_{0}\). Unlike sinusoidal load, the uniform load does not vary along the length of the beam. To find the beam's deflection under this load, the Galerkin Method is again employed. The uniform nature of the load means the integration will reflect a constant intensity across the beam\By applying the Galerkin Method and performing the necessary integrals, we can easily find the constant 'a' in the trial function. This calculation will allow us to approximate the deflection effectively, providing an insight into how the beam will behave under this straightforward loading condition.