In 1940, the Tacoma Narrows Bridge collapsed spectacularly. While the root cause was aeroelastic flutter (a complex dynamic instability), the ultimate mechanism of failure was structural members exceeding their elastic limits. As the bridge twisted violently, the steel suspension cables and girders experienced immense tensile and shear stress. Once the stress surpassed the ultimate tensile strength (UTS) of the steel, the cables snapped. This highlights why civil engineers must design structures not just for static equilibrium, but with a large factor of safety to ensure materials stay well within their linear elastic region under extreme dynamic loads.

A construction crane must maintain static equilibrium while lifting multi-ton loads. Engineers must carefully calculate the center of gravity of the entire system (crane body + load + counterweight). The primary requirement for stability is that the line of action of the total weight must fall within the base of support (the crane's tracks or outriggers). If the load creates a torque that shifts the center of gravity outside this base, the net torque is no longer zero, and the crane will tip over. Movable counterweights are used to dynamically adjust the center of gravity and maintain $\Sigma \tau = 0$.

A 30 kg child sits 2.0 m from the pivot point of a seesaw. Where must a 40 kg child sit on the other side to perfectly balance the seesaw? (Assume the seesaw board is massless).

A uniform wooden beam is 6.0 m long and weighs 500 N. It is supported by two pillars: one at the left end ($x=0$) and one 1.0 m from the right end ($x=5.0 \text{ m}$). A 200 N person stands on the beam at $x=2.0 \text{ m}$. Find the normal forces exerted by both pillars.

A 10 kg uniform ladder of length $L$ rests against a smooth (frictionless) vertical wall at an angle of $60^\circ$ to the horizontal floor. The floor has friction. What is the minimum coefficient of static friction ($\mu_s$) required to keep the ladder from slipping? ($g = 9.8 \text{ m/s}^2$)

A steel wire with a radius of $1.5 \text{ mm}$ supports a heavy chandelier weighing $800 \text{ N}$. Calculate the tensile stress in the wire.

A $10 \text{ m}$ long elevator cable is made of steel (Young's modulus $Y = 200 \times 10^9 \text{ Pa}$) and has a cross-sectional area of $4.0 \times 10^{-4} \text{ m}^2$. How much does the cable stretch when it supports a load of $12,000 \text{ N}$?

A steel bolt with a diameter of $10 \text{ mm}$ connects two metal plates. The plates are pulled in opposite directions with a force of $5000 \text{ N}$, placing the bolt in single shear. The shear modulus for the steel is $S = 80 \times 10^9 \text{ Pa}$. What is the shear strain on the bolt?

A solid brass sphere ($B = 60 \times 10^9 \text{ Pa}$) is dropped to the bottom of the Marianas Trench, where the water pressure is $1.1 \times 10^8 \text{ Pa}$ greater than at the surface. By what fraction does the volume of the sphere decrease?