Major Assignment $3$

Due Thursday, December $5,2024$

The purpose of this major assignment is to consolidate your knowledge of the material on integration

methods.

You should submit one Matlab file following this naming format: major$3\_$ID#$.$m

$($e$.$g$.,$ major$3\_123456789.$m$)$

Problem

You are trying to study the forces at several locations in a beam subjected to a uniform load. A free body

diagram showing the relationship between the applied load and the shear and moment forces is provided

below.

For the case of a uniformly distributed load, the shear and moment forces at any point along the span of

the beam are given by:

$()=$

$2(1)$

$()=$

$2()(2)$

In this problem, the length of the beam l $=4$ m$,$ and the applied load w $=10$ kN$/$m$.$

Page $2$ of $2$

Task $1$

Find the area under the moment curve for the first quarter of the beam$$s span $($x $=0$ to $1$ m$).$

$$ To achieve that, prepare a code that utilizes the Multiple$-$Application Trapezoidal Rule as a multi$-$

line function called Trapezoidal to evaluate the integral of the moment. Then, apply it for all step

sizes $1/$n for n $=1,2,3,...,100.$

$$ For all tested step sizes, calculate the true error using the analytical solution as the exact answer.

Then, find out what is the maximum step size needed to reach an error rate $0\mathrm{.}01\%.$

Task $2$

$$ Find the moment at every point along the beam span, which equals the integration of the shear

force, use a step size of $0\mathrm{.}1$ m$.$ Use a multi$-$line function called Romberg, that utilizes the Romberg

Integration method with k values up to $10.$

$$ The function should have built$-$in stopping criteria in terms of the number of iterations $($maxit $=$

$100)$ and the error tolerance $($es $=0\mathrm{.}01\%).$

Task $3$

$$ Plot the moment diagram from Task $2$ along the span of the beam.

$$ Create a summary output to the command window that only contains the following statements

dynamically updated from the code:

The exact moment area for the first quarter of the beam equals $\_\_\_\_$ kN$.$m$2.$ The

Trapezoidal Rule needed a maximum step size of $\_\_\_\_$ to reach a reasonable

estimate of that range.

The maximum bending moment along the span of the beam was $\_\_\_\_$ kN$.$m and

it was applied at x $=\_\_\_\_.$