George Mason University

Conor Philip Nelson

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Math 447: Numerical Analysis

Project 1, The Euler-Bernoulli Beam

The Euler-Bernoulli beam equation is a fundamental model for a material bending under stress. In particular, the equation describes the relationship between a static beam's deflection and the applied load placed on the beam. In describing the beam, one relates the vertical displacement of the beam to the Young's modulus of the material, the area moment of inertia, and the applied load in force per unit length. In doing so, the fourth derivative of the vertical displacement is required. As analytic techniques are often impractical to use in solving equations via algorithms, numerical techniques have been applied here. Discretization methods have been applied to approximate the differential equation as a system of linear equations. This system has been represented as a matrix throughout our computations, allowing Matlab to efficiently calculate the vertical displacement of the beam given the applied force.

In exploring this model, the error of approximation has been determined and found proportional to the discretization size used*. Along with this, the model has been used to determine the displacement of the beam when a sinusoidal pile has been added as well as when a diver has been added, thus serving as a model for a diving board. Lastly, the free end of the board was fixed, creating a "clamped-clamped" beam. In this scenario, the model can serve to answer questions regarding the sag in structures such as bridges.

*Assuming large enough discretization, \(h\). We must note that, assuming a numerical method with error on the order of \(h^2\), we would hope the total error to be proportional to this, allowing the error to go to zero as our step size approaches zero. However, each function evaluation will introduce a rounding error on the order of machine epsilon, or machine precision. Thus, the total error \(T\) can be expected to be of the order \(T(h) = Ah^2 +\frac{B\epsilon}{h}\) for some A, B>0 (Computational and Nonlinear Aspects of PDEs- Sander and Wanner).
From this, given A,B, we can determine an optimal step size which will yield minimal error. At minimum though, we can gain an understanding of the fact that error eventually increases drastically near small \(h\).

Project 1: Reality Check # 2: The Diving Board

Step 1: Defining the structure matrix \(A\). Solving, for n=10, the vertical displacements.

Step 2: Plot the results of Step 1 against the correct solution.

Step 3: Solve the system for \(n=10*2^k\) for \(k=1,...,11\), then find for which n is the error the smallest and why. Finally create a table of step size vs error vs cond(a).

Step 4: Prove that a given function \(y(x)\) satisfies the Euler-Bernoulli with Clamped-Free boundary conditions.

Step 5: Repeat Step 3, except now we set \(p=100\frac{kg}{m}\). We will show that \(x=L\) is proportional to \(h^2\).

Step 6: For the optimal \(n\) from Step 5. We will find the deflection of the diving board due to a diver.

Step 7: Fixing the free end of the diving board, we model a clamped-clamped beam under a sinusoidal load and compare to an exact solution.