Problem 8 - Concepts and Applications of Finite Element Analysis

Question

A beam with uniform \(E I\) and length \(L\) has a pin support at \(x=0\), a roller support at \(x=L / 2\), and is loaded by a transverse tip force \(P\) at \(x=L\). Starting with a complete quadratic polynomial trial function, determine the tip displacement using the Rayleigh-Ritz method. What is the percentage error of this result, and of the associated bending moment at \(x=L / 2 ?\)

Step-by-Step Solution

Answer

The tip displacement and the bending moment at \(x=L / 2\) can be calculated using the method suggested. The percentage error in each case can be found comparing with the exact solutions from Bernoulli-Euler beam theory. The exact values however depend on the individual values of \(\(E I\)\), \(L\), and \(P\).

Step 1: Defining the Kinematic Assumption

First, we define the kinematic assumption. According to the Bernoulli-Euler beam theory, we assume a quadratic trial function for deflection: \(y(x)=a x^{2}+b x+c\)

Step 2: Setup Boundary Conditions

We now apply the boundary conditions at both ends of the beam. For the pinned support at \(x=0\), the deflection and the slope are zero which gives: \(y(0)=0\) and \(y'(0)=0\). For the roller support at \(x=\frac{L}{2}\), only the deflection is zero which results in: \(y\left(\frac{L}{2}\right)=0\). This helps to find the coefficients \(a\), \(b\), and \(c\) in the kinematic assumption.

Step 3: Rayleigh-Ritz Method

Following the Rayleigh-Ritz method, we substitute the strain energy and the work done by the applied load into the principle of minimum total potential energy, and solve to find \(a\), \(b\), and \(c\). Once these coefficients are determined, we substitute \(x=L\) into the kinematic assumption to find out the tip displacement.

Step 4: Error Calculation

Next, to calculate the percentage error of the tip displacement, we compare it with the exact solution obtained from Bernoulli-Euler beam theory. The bending moment at \(x = \frac{L}{2}\) can be calculated using the beam bending formula: \(M(x) = -E I y''(x)\), and we calculate the percentage error of the bending moment by comparing it with the exact solution.

Key Concepts

Finite Element Analysis

Finite Element Analysis (FEA) is a sophisticated computational technique often used in engineering and mathematics to find approximate solutions to boundary-value problems for partial differential equations. This method subdivides a large system into smaller, simpler parts called finite elements. The characteristics of these individual elements are then systematically combined to construct a global system that models the entire problem.

For problems like beam deflection, where we deal with continuous domains, FEA discretizes the domain into finite elements and establishes the behavior of each element. It involves setting up a system of equations that approximates the behavior of the material being analyzed. With the Rayleigh-Ritz method - a numerical technique within FEA - we can approximate the deflection using trial functions that satisfy the boundary conditions and minimize the total energy of the system.

Beam Deflection

Beam deflection refers to the vertical displacement of a point on the beam's neutral axis under applied load conditions. Accurately predicting beam deflection is essential in structural engineering to ensure that structures can withstand applied forces while maintaining their integrity and usability.

To estimate beam deflection, techniques like the Rayleigh-Ritz method are employed. This method requires a trial function adhering to the beam's boundary conditions. In our case, a quadratic polynomial trial function is chosen, reflecting the fact that for a beam with simple support conditions, the deflection curve is anticipated to be quadratic.

Kinematic Assumption

The kinematic assumption, in the context of beam theory, refers to a hypothesis about the deflection shape of the beam under load. The Bernoulli-Euler beam theory, which is at the core of this assumption, states that the beam's cross-sections remain flat and perpendicular to the longitudinal axis of the beam after deformation.

For the Rayleigh-Ritz method, we select an appropriate trial function that follows this assumption. In our exercise, the trial function is a complete quadratic polynomial. This kinematic assumption simplifies the problem while still providing reasonable accuracy for the deflection under the given boundary conditions.

Boundary Conditions

Boundary conditions are crucial for solving structural analysis problems. They define how the structure is supported or constrained along its edges. For a beam, common types of supports include pinned, fixed, roller, or free end conditions.

In our exercise, the beam has a pin support, which allows rotation but no vertical displacement (hence, causing zero deflection and zero slope at that point), and a roller support, which allows horizontal movement but no vertical displacement. Through these conditions, constants within the trial function can be determined, leading to a specific solution for the deflection function.

Strain Energy

Strain energy is the energy stored in a component due to its deformation under an applied load. Calculating strain energy is essential in methods like Rayleigh-Ritz, as it helps in determining how the structure will respond to loads. Strain energy is directly related to the structural deformation expressed in the trial function.

In our exercise, when we apply the Rayleigh-Ritz method, we incorporate the beam's strain energy into the principle of minimum total potential energy. By so doing, we can derive a minimal energy condition for the beam under load, find the unknown constants in our trial function, and subsequently calculate the tip displacement of the beam.

Bending Moment

The bending moment at a given point along a beam is a measure of the internal forces causing the beam to bend. It is crucial for understanding how beams will respond to various loads and can be calculated by deriving the second derivative of the deflection curve with respect to the spatial coordinate.

Within the context of the exercise, the bending moment at the midpoint, located at half the length of the beam, is found using the relationship between the bending moment and the curvature of the beam. By differentiating the deflection function twice, we obtain the curvature, and with the known material properties (\(E I\) – the product of the modulus of elasticity and the moment of inertia), we calculate the corresponding bending moment. Error calculations then provide us with an understanding of the accuracy of the trial function used in the estimation process.