Problem $2(35$ points, Core Course Outcomes $6$ & $11)$

A cantilevered beam of length $L=16$ is made of a hybrid material that has a spatially varying product of the area moment of

Figure $1$: Sketch of cantilevered beam with varying load $p(x)$ and varying $EI(x).$

inertia and Young's modulus, $EI(x,),$ where $$ is a parameter describing the hybrid material composition with $25.$ The beam is subjected to a varying load $p(x),$ see Figure $1($positive values of $p$ are pointing in the positive $y$ direction$).$ Recall that for the beam shown, the shear force $F(x_i)$ at location $x_i$ is

$F(x_i)={\displaystyle \underset{x_i}{\overset{L}{}}}p(x)dx$

$1$

the bending moment $M(x_i)$ is

$M(x_i)={\displaystyle \underset{x_i}{\overset{L}{}}}F(x)dx$

the beam's slope $(x_i)$ as a result of the bending moment is

$(x_i)={\displaystyle \underset{0}{\overset{x_i}{}}}\frac{M(x)}{EI(x,)}dx$

and the beam's deflection $y(x_i)$ is

$y(x_i)={\displaystyle \underset{0}{\overset{x_i}{}}}(x)dx$

The load $p(x)$ can be determined by calling the content hidden function loadbeam $($ x $)$ available in the Canvas assignment. The input argument can be either a scalar value of a single location $x$ along the beam, or a column vector containing multiple $x$ locations. The output is $p(x)$ at the given input location$($s$)$ as either a scalar $($for a scalar input $x)$ or a column vector.

$EI(x,)$ can be determined by calling the content hidden function EIbeam $(x,)$ available in the Canvas assignment. Here alpha is the scalar value of $$ and the input argument x can be either a scalar value of a single location $x$ along the beam, or a column vector containing multiple $x$ locations. The output is $EI(x,)$ at the given input location$($s$)$ as either a scalar $($for a scalar input $x)$ or a column vector.

The content hidden Matlab function files loadbeam. p and EIbeam. p are available on Canvas in the exam assignment.

For $=3$ and $N=32$ equally sized intervals along the beam, determine the beam's deflection at it's end, $y(L)$ using only integration methods covered in this class. Graph in separate plots the load $p(x),$ the beam's shear force $F(x),$ bending moment $M(x),$ slope $(x),$ and deflection $y(x).$

Note that all quantities are dimensionless and Eq$.(3)$ does require component$-$wise division $./$ when calculating the integrand using vectors. Use column vectors throughout. State which integration method you have used.

On Canvas, you will find a link to a MatlabGrader non$-$credit test for the script you code. The test let's you check whether your script will calculate the deflection at the beam's end correctly using Newton$-$Cotes integration techniques covered in class. It is entirely optional and will not earn any credit. Please see the description of the MatlabGrader problem for further details.