Problem $1-$ A box beam is constructed from four wood boards that are glued together. The beam cross section shown in Figure $1$ is subjected to a moment $M=40kN-m$ about the neutral axis $($N$.$A$.).$

$1\mathrm{.}1)$ Calculate the normal stress at point $A($top of the cross section$),B($inner face of bottom board$),$ and $C($bottom of cross section$),$ presenting the results in MPa. For each stress result, clearly indicate if it is tensile or compressive by marking it as $($T$)$ or $($C$),$ respectively. $[30$ points$]$

$1\mathrm{.}2)$ Is the cross section capable of resisting the moment $M$ if the wood is a Douglas fir having an allowable compressive strength of $20$ MPa $,$ and an allowable tensile strength of $42$ MPa $?[15$ points$]$

Problem $2-$ The timber beam rectangular cross section in Figure $2$ is subjected to a shear force $V=10$ kip.

$2\mathrm{.}1)$ Calculate the maximum shear stress occurring in the cross section, and clearly indicate all points where it occurs. $[25$ points$]$

$2\mathrm{.}2)$ Assuming that the wood has an allowable shear strength of $250$ psi $,$ calculate the maximum shear force that the cross section can resist. $[10$ points$]$

Problem $3-$ The cantilevered beam shown in Figure $3($free end in A$,$ fixed end in B$)$ is subjected to a concentrated force $P$ applied at the free end $($A$).$

$3\mathrm{.}1)$ Determine the equation of the elastic curve, $v(x),$ knowing that $M(x)=-Px.[15$ points$]$

$3\mathrm{.}2)$ Calculate the maximum vertical displacement assuming that $P=6$kip, and that the beam has a total length $L=15ft,$ a cross section with a moment of inertia $I=204i{n}^4,$ and is made of steel. $[5$ points$]$

Figure $1$

Figure $2$

Figure $3$

Page $1$ of $5$