P$2$ Ans: c$)-1\mathrm{.}5555in,+0\mathrm{.}23044$rad,$+0\mathrm{.}34559in$

Stepped cantilever beam with a square cross section has a free

end at $A,$ is fixed at $D$ and loaded at A with the bending moment

$M_0$ and at $C$ with point load $P.$ Given: $L=12\mathrm{.}0in,M_0=400\mathrm{.}0lb-$

in$,P=100\mathrm{.}0lb,E=16\mathrm{.}0$Msi,$v=0\mathrm{.}3,I=0\mathrm{.}001736i{n}^4.$

a$)$ with Principle of Stationary Potential energy derive

symbolically $($by hand$)$ expression for vertical deflection and

rotation at point A and vertical deflection at point $C.$

b$)$ use Matlab $($Symbolic Math Toolbox$)$ to verify your results from step $($a$)($show the code and output$),$

c$)$ use the values given in the problem and calculate values of vertical deflection and rotation at point A and vertical

deflection at point $C$ with the expressions derived in step $($a$).$

P$3$ Ans: c$)+0\mathrm{.}081002$ rad

Statically indeterminate beam with a square cross section is fixed

at $C,$ supported by roller at $B$ and loaded at its free end A with

the bending moment $M_0.$ Given: $L=36\mathrm{.}0in,M_0=100\mathrm{.}0lb-$in$,E=$

$16\mathrm{.}0$Msi,$v=0\mathrm{.}3,I=0\mathrm{.}001736i{n}^4.$

a$)$ with Principle of Stationary Potential energy to derive

symbolically $($by hand$)$ expression for rotation at point $A.$

b$)$ use Matlab $($Symbolic Math Toolbox$)$ to verify your result from step $($a$)($show the code and output$),$

c$)$ use the values given in the problem and calculate value of rotation at point A with the expression derived in step $($a$).$

P$4$ Ans: c$)6\mathrm{.}75lb,-117\mathrm{.}0lb-in,-0\mathrm{.}0778in$

Statically indeterminate beam with square cross section is fixec

at $D,$ supported by roller at A and loaded at $B$ with point load $P$

Point $C$ is able to move vertically, but not rotate. Given: $L=$

$48\mathrm{.}0in,P=15\mathrm{.}0lb,E=10\mathrm{.}0$Msi,$v=0\mathrm{.}3,I=0\mathrm{.}0108i{n}^4.$

a$)$ with Principle of Stationary Potential energy to derive

symbolically $($by hand$)$ expression for vertical reaction at $A$

and moment reaction at $C.$ Next, obtain expression for

vertical displacement at $B,$

c$)$ use the values given in the problem and calculate the reactions at A and $C$ and the displacement at $B$ with the expressions derived in step $($a$).$