As many as three axial loads each of magnitude $P=10 \mathrm{~kips}$ can be applied to the end of a $\mathrm{W}8$ $\times 21$ rolled-steel shape. Detertnine the stress at point $A(a)$ for the loading shown, $(b)$ if loads are applied at points $1$ and $2$ only.

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.

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 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.

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 $500 \mathrm{~mm}$.

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

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.

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

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 composite bar indicated, determine the radius of curvature caused by the couple of moment $35 \mathrm{~N} \cdot \mathrm{m}$.