Five metal strips, each of $0.5 \times 1.5-\text{in.}$ cross section, are bonded together to form the composite beam shown. The modulus of elasticity is $30 \times 10^6$ psi for the steel, $15 \times 10^6 \mathrm{psi}$ for the brass, and $10 \times 10^6\ \mathrm{psi}$ for the aluminum. Knowing that the beam is bent about a horizontal axis by a couple of moment $12\ \text{kip.in}$ determine (a) the maximum stress in each of the three metals, (b) the radius of curvature of the composite beam.

The shaft is supported by smooth journal bearings at A and B that only exert vertical reactions on the shaft. Determine its smallest diameter $d$ if the allowable bending stress is $\sigma_{\text {allow }}=180 \mathrm{MPa}$.

Which of the following statements about parallel circuits is false? a. The voltage is the same across all branches in a parallel circuit. b. The equivalent resistance, $R_\text{EQ}$, of a parallel circuit is always smaller than the smallest branch resistance. c. In a parallel circuit the total current, $I_\text{T}$, in the main line equals the sum of the individual branch currents. d. The equivalent resistance, $R_\text{EQ}$, of a parallel circuit decreases when one or more parallel branches are removed from the circuit.

Knowing that the bending moment in the reinforced concrete beam is $+150 \mathrm{~kip} \cdot \mathrm{ft}$ and that the modulus of elasticity is $3.75 \times$ $10^{6}$ $\mathrm{psi}$ for the concrete and $30 \times 10^{6}$ $\mathrm{psi}$ for the steel, determine (a) the stress in the steel, (b) the maximum stress in the concrete.

A concrete beam is reinforced by three steel rods placed as shown. The modulus of elasticity is $3 \times 10^6 \mathrm{psi}$ for the concrete and $30 \times$ $10^6 \mathrm{psi}$ for the steel. Using an allowable stress of $1350\ \text{psi}$ for the concrete and $20\ \text{ksi}$ for the steel, determine the largest permissible positive bending moment in the beam.

Knowing that the bending moment in the reinforced concrete beam is $+150 \mathrm{~kip} \cdot \mathrm{ft}$ and that the modulus of elasticity is $3.75 \times$ $10^{6}$ $\mathrm{psi}$ for the concrete and $30 \times 10^{6}$ $\mathrm{psi}$ for the steel, determine (a) the stress in the steel, (b) the maximum stress in the concrete.

Solve a concrete slab is reinforced by $16 -\mathrm{mm}$-diameter steel rods placed on $180-\mathrm{mm}$ centers as shown. The modulus of elasticity is $20 \mathrm{~GPa}$ for the concrete and $200 \mathrm{~GPa}$ for the steel. Using an allowable stress of $9 \mathrm{~MPa}$ for the concrete and $120 \mathrm{~MPa}$ for the steel, determine the largest allowable positive bending moment in a portion of the slab $1 \mathrm{~m}$ wide. assuming that the spacing of the $16-\mathrm{mm}$-diameter rods is increased to $225 \mathrm{~mm}$ on centers.

A cylindrical spring consists of a rubber annulus bonded to a rigid ring and shaft. If the ring is held fixed and a torque $T$ is applied to the rigid shaft, determine the angle of twist of the shaft. The shear modulus of the rubber is $G$. Hint: As shown in the figure, the deformation of the element at radius $r$ can be determined from $r d \theta=d r \gamma$. Use this expression along with $\tau=T /\left(2 \pi r^2 h\right)$ from Prob. 5-26, to obtain the result.

Solve earlier problem by assuming that the wide-flange beam is bent about the y axis by a couple of moment $M_y$.

A concrete slab is reinforced by $16 -\mathrm{mm}$-diameter steel rods placed on $180-\mathrm{mm}$ centers as shown. The modulus of elasticity is $20 \mathrm{~GPa}$ for the concrete and $200 \mathrm{~GPa}$ for the steel. Using an allowable stress of $9 \mathrm{~MPa}$ for the concrete and $120 \mathrm{~MPa}$ for the steel, determine the largest allowable positive bending moment in a portion of the slab $1 \mathrm{~m}$ wide.

For the composite bar indicated, determine the radius of curvature caused by the couple of moment $450\ \text{kip - in}$.

Solve the reinforced concrete beam shown is subjected to a positive bending moment of $175 \mathrm{~kN} \cdot \mathrm{m}$. Knowing that the modulus of elasticity is $25 \mathrm{~GPa}$ for the concrete and $200 \mathrm{~GPa}$ for the steel, determine (a) the stress in the steel, (b) the maximum stress in the concrete. assuming that the $450-\mathrm{mm}$ depth of the beam is increased to $500 \mathrm{~mm}$.

For the composite bar indicated, determine the radius of curvature caused by the couple of moment $450\ \text{kip - in}$.

The reinforced concrete beam shown is subjected to a positive bending moment of $175 \mathrm{~kN} \cdot \mathrm{m}$. Knowing that the modulus of elasticity is $25 \mathrm{~GPa}$ for the concrete and $200 \mathrm{~GPa}$ for the steel, determine (a) the stress in the steel, (b) the maximum stress in the concrete.