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.

Two vertical forces are applied to the beam of the cross section shown. Determine the maximum tensile and compressive stresses in portion BC of the beam.

(a) Find the maximum shearing stress caused by a 4.6-kN \cdot m torque T in the 76-mm-diameter solid aluminum shaft shown.

(b) Evaluate part a, assuming that the solid shaft has been replaced by a hollow shaft of the same outer diameter and of 24-mm inner diameter.

Two cylindrical rods, one of steel and the other of brass, are joined at C and restrained by rigid supports at A and E. For the loading shown and knowing that $E_s=200\ \mathrm{GPa}$ and $E_b=105\ \mathrm{GPa}$, determine (a) the reactions at A and E, (b) the deflection of point C.

The nonprismatic cantilever circular bar shown has an internal cylindrical hole of diameter d/2 from 0 to x, so the net area of the cross section for segment 1 is (3/4)A. Load P is applied at x, and load P/2 is applied at x = L. Assume that E is constant.

(a) Find reaction force R$_1$.

(b) Find internal axial forces N$_i$ in segments 1 and 2.

(c) Find x required to obtain axial displacement at joint 3 of $\delta_3$ = PL/EA.

(d) In part (c), what is the displacement at joint $2, \delta_2$ ?

(e) If P acts at x = 2L/3 and P/2 at joint 3 is replaced by $\beta$P, find $\beta$ so that $\delta_3$ = PL/EA.

(f) Draw the axial force (AFD: N(x), 0 $\leq$ x $\leq$ L) and axial displacement diagrams (ADD: $\delta$(x), 0 $\leq$ x $\leq$ L) using results from parts (b) through (d).

A 1.5-m-long tubular steel shaft of 38-mm outer diameter

$$d_1$$

is to be made of a steel for which

$$τ_{all} = 65 MPa$$

and G = 77.2 GPa. Knowing that the angle of twist must not exceed 4° when the shaft is subjected to a torque of 600 N·m, determine the largest inner diameter

$$d_2$$

that can be specified in the design.

The solid shaft shown is made of a steel that is assumed to be elastoplastic with $\tau_Y=145 \mathrm{MPa}$ and $G=77.2 \mathrm{GPa}$. The torque is increased in magnitude until the shaft has been twisted through $6^{\circ}$; the torque is then removed. Determine $(a)$ the magnitude and location of the maximum residual shearing stress, $(b)$ the permanent angle of twist.

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.

The $6 \times 12-\text{in}$. timber beam has been strengthened by bolting to it the steel reinforcement shown. The modulus of elasticity for wood is $1.8 \times 10^6 \mathrm{psi}$ and for steel $29 \times 10^6 \mathrm{psi}$. Knowing that the beam is bent about a horizontal axis by a couple of moment 450 kip. in., determine the maximum stress in (a) the wood, (b) the steel.

The $6 \times 12-\text{in}$. timber beam has been strengthened by bolting to it the steel reinforcement shown. The modulus of elasticity for wood is $1.8 \times 10^6 \mathrm{psi}$ and for steel $29 \times 10^6 \mathrm{psi}$. Knowing that the beam is bent about a horizontal axis by a couple of moment $450\ \text{kip. in.}$, determine the maximum stress in (a) the wood, (b) the steel.

For the beam and loading shown, (a) draw the shear and bending-moment diagrams, (b) determine the equations of the shear and bending-moment curves.

For the beam and loading shown, $(a)$ draw the shear and bending-moment diagrams, (b) determine the equations of the shear and bending-moment curves.

For the beam and loading shown, draw the shear and bending-moment diagrams using method discussed earlier.

Design a solid steel shaft to transmit 100 hp at a speed of 1200 rpm, if the maximum shearing stress is not to exceed 7500 psi.