Problem $2$

Consider a beam with a pinned $($left$)$ and simple roller $($right$)$ supports at each end, respectively.

The beam is subjected to three concentrated forces $($P$,$P$,$ and $3$P$)$ as shown in the figure below.

Question $141$ pts

Find the location of the neutral axis as measured from the bottom of the cross$-$section. $($answer in mm$)$

$$

Flag question: Question $15$

Question $152$ pts

Find the moment of inertia about the neutral axis. $($answer in mm$4)$

$$

Flag question: Question $16$

Question $162$ pts

Determine how much P is given the information provided. $($answer in N$)$

$$

Flag question: Spacer

Draw the shear force and bending moment diagrams using the graphical method with your submitted work. This should be included in the submission at the end of this assignment.$$

$$

Flag question: Question $17$

Question $172$ pts

Find the maximum value of the shear diagram. $($answer in N$)$

$$

Flag question: Question $18$

Question $182$ pts

Find the minimum value of the shear diagram. $($answer in N$)$

$$

Flag question: Question $19$

Question $192$ pts

Find the maximum value of the moment diagram. $($answer in N$*$m$)$

$$

Flag question: Question $20$

Question $202$ pts

Find the minimum value of the moment diagram. $($answer in N$*$m$)$

$$

Flag question: Question $21$

Question $211\mathrm{.}5$ pts

Find the shape of the moment diagram for region $1.$

Group of answer choices

Zero

Constant

Linear, Positive Slope

Linear, Negative Slope

Quadratic, Positive Slope, Concave Up

Quadratic, Positive Slope, Concave Down

Quadratic, Negative Slope, Concave Up

Quadratic, Negative Slope, Concave Down

Cubic

None of the Above

$$

Flag question: Question $22$

Question $221\mathrm{.}5$ pts

Find the shape of the moment diagram for region $2.$

Group of answer choices

Zero

Constant

Linear, Positive Slope

Linear, Negative Slope

Quadratic, Positive Slope, Concave Up

Quadratic, Positive Slope, Concave Down

Quadratic, Negative Slope, Concave Up

Quadratic, Negative Slope, Concave Down

Cubic

None of the Above

$$

Flag question: Question $23$

Question $231\mathrm{.}5$ pts

Find the shape of the moment diagram for region $3.$

Group of answer choices

Zero

Constant

Linear, Positive Slope

Linear, Negative Slope

Quadratic, Positive Slope, Concave Up

Quadratic, Positive Slope, Concave Down

Quadratic, Negative Slope, Concave Up

Quadratic, Negative Slope, Concave Down

Cubic

None of the Above

$$

Flag question: Question $24$

Question $241\mathrm{.}5$ pts

Find the shape of the moment diagram for region $4.$

Group of answer choices

Zero

Constant

Linear, Positive Slope

Linear, Negative Slope

Quadratic, Positive Slope, Concave Up

Quadratic, Positive Slope, Concave Down

Quadratic, Negative Slope, Concave Up

Quadratic, Negative Slope, Concave Down

Cubic

None of the Above

$$

Flag question: Question $25$

Question $252$ pts

Calculate the maximum tensile stress in the beam. $($answer in MPa$)$

$$

Flag question: Question $26$

Question $262$ pts

Find the location where the maximum tensile stress in the beam occurs. $($answer in mm$)$

$$

Flag question: Question $27$

Question $272$ pts

Calculate the maximum compressive stress in the beam. $($answer in MPa$)$

$$

Flag question: Question $28$

Question $282$ pts

Find the location where the maximum compressive stress in the beam occurs. $($answer in mm$)$

$$

Flag question: Question $29$

Question $291\mathrm{.}5$ pts

Calculate the maximum tensile stress in the beam if the cross$-$section is flipped $180.($answer in MPa$)$

$$

Flag question: Question $30$

Question $301\mathrm{.}5$ pts

Calculate the maximum compressive stress in the beam if the cross$-$section is flipped $180.($answer in MPa$)$

The beam cross section with detailed dimensions is shown in the figure, right.

$($a$)$ Find the location of the neutral axis on the beam cross section and the moment of inertia

about the neutral axis.

$($b$)$ If the strain is epsilon$\_($x$)$ at point A$,$ determine how much P is given that the applied forces are P$,$

P$,$ and $3$P$.$ The point A $($see the red dot, right$)$ is located at a distance s from the top edge

of the cross section and the section containing the point A is shown in the figure, i$.$e$.,$ at a

distance of $3$d from the roller support.

$($c$)$ Draw the shear and bending moment diagrams for the beam using the value of P you found

in part b$.$ Use the graphical method to get full credit. Label your diagrams with appropriate

values and justify them showing your calculations.

$($d$)$ Find the maximum tensile and compressive stresses $($sigma$\_($max$))$ in the beam and indicate their

locations.

$($e$)$ If the cross$-$section is flipped $180^($@$)($from the inversed $"$ T $"$ to $"$ T $"$ shape$),$ how much would

the maximum compressive and maximum tensile stresses be$?$