PSL-1

Cantilever beam deflection

A cantilever beam experiences a deflection from its original position when uniform load acts upon it. This deflection (y) can be calculated across the distance of the beam (x), given the analytical solution. The beams length, uniform load, moment of inertia, and modulus of elasticity can be found below in Table 1.

Analytical Solution

The analytical approach considers the beams downward deflection (y) by relating to a uniform load(W), its distance along the beam (x), its inertia (I), and its modulus of elasticity (E). The relationship between all these factors can be described by the equation above .

Although this scenario offers a analytical solution this example has helped made me realize how many variables can go into a simple problem.

Numerical Solution

By adding the initial value with the change in x (step size) times the slope (dy/dx) , given by the derivative of the analytical solution in respect to y and x. The derivative provides the slope at the next given step size. By first adding the initial value the first iteration is complete. The following iteration will be that value added to the slope at the next step size.

Pseudocode

First, the parameters such as w are set. Then conditions for x and y are set. The condition for step size can also be added here, but I end up deciding to place it along the numerical solution. This way I can copy and paste the code for the numerical solution with the half step.

Matlab Explanation`

By setting the parameters and initializing conditions (ay) can be calculated using the analytical equation. Then both (ay) and (ax) are plotted, a title and labels are added.

Numerical solution / Step size

I decided to do look at three different step sizes, (.5) , (1) , (5), each done the same way. The step size is set for the x value of the numerical functions from 0 to L. The estimation is then plotted with dotted lines and color coded according to step size. As the step size gets larger the redder the color will get for the graph. A legend is then added at the top left to better characterize the lines.

Results

The graph above provides three different time steps for the numerical solution while also showing the analytical solution. The analytical solution (pictured in blue) depicts an exponential curve for the deflection. On the other hand, the highest step size curve (pictured in red) shows a linear representation for the deflection. As the step size decreases for the numerical solution it begins to curve like the analytical solution.

Error

The numerical solution is written above in Matlab each iteration is one more to the left, similar to a rieman sum estimation from the left, without the rectangles. The numerical solution can also be used to find the analytical solution. This is done by finding the next derivative in the function until it is 0. The error that occurs from the numerical solution comes from the step size and its order O(h).

Discussion

The deflection of steel beams (in meters) is often limited to L/1320, where L is the length of the beam (in meters). Is this threshold under the current load configuration exceeded? If so, how would you modify your beam configuration and/or load to meet the deflection criteria?

(6/1320) = .0045m , First I would try to change any of the parameters, such as its inertia by changing its shape.

I realized mainly by picturing this scenario in my mind that for solving the majority of simple problems much is ignored. Tiny forces that minimally impact the solution are easily ignored considering how much effort can be exhausted to be more accurate. By playing with the time step during the numerical estimations I realized that each time a sig fig was put on the step size the error drastically increased. As we mentioned in class, due to the estimation being derived from a Taylor series, each derivative taken along with step size will provide an order of magnitude of accuracy. When calculating for error compared to the analytical solution.

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