Problem $4.$

You need to design a timber beam for jacking up a house. The beam will rest on two hydraulic

jacks, and will support eight floor joists, as shown. The joists are at ${18}^{\text{'}\text{'}}$ centers, and each one is

estimated to carry a load of $500$ lbs $.$ Timber beams are available in nominal $2^{\text{'}\text{'}}$ increments $($i$.$e$.$

$42,44,46,$ etc.$),$ and you have determined that the timber can safely be stressed to $1000$ psi

in $($elastic$)$ bending under the given $($service$)$ loads.

a$)$ Where would you place the jacks to minimize the largest bending moment in the beam?

b$)$ What is the smallest beam that you can use without violating the stress limits$?$ Note that

you will provide supports to ensure that the beam does not topple out of the page, so the

beam can have an out$-$of$-$plane width is less than the height.

c$)$ How much does the beam weigh? $($It will have to be carried under the house by hand$)$

Remember that real dimensions of timber members are always $($for these purposes$)\frac{1}{2^{\text{'}\text{'}}}$ less

than the nominal dimensions, so a $68$ is actually ${5\mathrm{.}5}^{\text{'}\text{'}}{7\mathrm{.}5}^{\text{'}\text{'}}$ in cross$-$section. The timber

weighs $35l\frac{b}{f}{t}^3.$