Knowing that the couple shown acts in a vertical plane, determine the stress at $(a)$ point $A,(b)$ point $B$.

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Using an allowable stress of 155 MPa, determine the largest bending moment $\mathbf{M}$ that can be applied to the wide-flange beam shown. Neglect the effect of fillets.

Ajay S.

Numerade Educator

Solve Prob. $4.3,$ assuming that the wide-flange beam is bent about the $y$ axis by a couple of moment $M_{y^{\prime}}$

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Using an allowable stress of $16 \mathrm{ksi}$, determine the largest couple that can be applied to each pipe.

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Knowing that the couple shown acts in a vertical plane, determine the stress at $(a)$ point $A,(b)$ point $B$

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Two $\mathrm{W} 4 \times 13$ rolled sections are welded together as shown. Knowing that for the steel alloy used $\sigma_{U}=58$ ksi and using a factor of safety of $3.0,$ determine the largest couple that can be applied when the assembly is bent about the $z$ axis.

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Two $\mathrm{W} 4 \times 13$ rolled sections are welded together as shown. Knowing that for the steel alloy used $\sigma_{U}=58 \mathrm{ksi}$ and using a factor of safety of $3.0,$ determine the largest couple that can be applied when the assembly is bent about the $z$ axis.

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

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Knowing that a beam of the cross section shown is bent about a horizontal axis and that the bending moment is $6 \mathrm{kN} \cdot \mathrm{m},$ determine the total force acting on the shaded portion of the web.

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Knowing that a beam of the cross section shown is bent about a horizontal axis and that the bending moment is $4 \mathrm{kN} \cdot \mathrm{m}$, determine the total force acting on the shaded portion of the beam.

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Solve Prob. 4.13 , assuming that the beam is bent about a vertical axis by a couple of moment $4 \mathrm{kN} \cdot \mathrm{m}$

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Knowing that for the extruded beam shown the allowable stress is 12 ksi in tension and 16 ksi in compression, determine the largest couple $\mathbf{M}$ that can be applied.

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The beam shown is made of a nylon for which the allowable stress is $24 \mathrm{MPa}$ in tension and $30 \mathrm{MPa}$ in compression. Determine the largest couple $\mathbf{M}$ that can be applied to the beam.

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Knowing that for the beam shown the allowable stress is 12 ksi in tension and 16 ksi in compression, determine the largest couple $\mathbf{M}$ that can be applied.

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Knowing that for the extruded beam shown the allowable stress is $120 \mathrm{MPa}$ in tension and $150 \mathrm{MPa}$ in compression, determine the largest couple $\mathbf{M}$ that can be applied.

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Straight rods of $6-\mathrm{mm}$ diameter and $30-\mathrm{m}$ length are stored by coiling the rods inside a drum of 1.25 -m inside diameter. Assuming that the yield strength is not exceeded, determine ( $a$ ) the maximum stress in a coiled rod, $(b)$ the corresponding bending moment in the rod. Use $E=200 \mathrm{GPa}$.

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A $900-\mathrm{mm}$ strip of steel is bent into a full circle by two couples applied as shown. Determine

(a) the maximum thickness $t$ of the strip if the allowable stress of the steel is $420 \mathrm{MPa}$

(b) the corresponding moment $M$ of the couples. Use $E=200 \mathrm{GPa}$

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Straight rods of 0.30 -in. diameter and 200 -ft length are sometimes used to clear underground conduits of obstructions or to thread wires through a new conduit. The rods are made of highstrength steel and, for storage and transportation, are wrapped on spools of 5 -ft diameter. Assuming that the yield strength is not exceeded, determine $(a)$ the maximum stress in a rod, when the rod, which is initially straight, is wrapped on a spool, (b) the corresponding bending moment in the rod. Use $E=29 \times 10^{6}$ psi.

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A $60-\mathrm{N} \cdot \mathrm{m}$ couple is applied to the steel bar shown. $(a)$ Assuming that the couple is applied about the $z$ axis as shown, determine the maximum stress and the radius of curvature of the bar. $(b)$ Solve part $a$, assuming that the couple is applied about the $y$ axis. Use $E=200 \mathrm{GPa}$.

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(a) Using an allowable stress of $120 \mathrm{MPa}$, determine the largest couple $\mathbf{M}$ that can be applied to a beam of the cross section shown. (b) Solve part $a$, assuming that the cross section of the beam is an 80 -mm square.

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A thick-walled pipe is bent about a horizontal axis by a couple $\mathbf{M}$. The pipe may be designed with or without four fins. $(a)$ Using an allowable stress of $20 \mathrm{ksi}$, determine the largest couple that may be applied if the pipe is designed with four fins as shown. $(b)$ Solve part $a,$ assuming that the pipe is designed with no fins.

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A couple $\mathbf{M}$ will be applied to a beam of rectangular cross section that is to be sawed from a log of circular cross section. Determine the ratio $d / b$ for which $(a)$ the maximum stress $\sigma_{m}$ will be as small as possible,

(b) the radius of curvature of the beam will be maximum.

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A portion of a square bar is removed by milling, so that its cross section is as shown. The bar is then bent about its horizontal axis by a couple M. Considering the case where $h=0.9 h_{0},$ express the maximum stress in the bar in the form $\sigma_{m}=k \sigma_{0}$ where $\sigma_{0}$ is the maximum stress that would have occurred if the original square bar had been bent by the same couple $\mathbf{M},$ and determine the value of $k$

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In Prob. $4.28,$ determine $(a)$ the value of $h$ for which the maximum stress $\sigma_{m}$ is as small as possible, $(b)$ the corresponding value of $k$.

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For the bar and loading of Concept Application $4.1,$ determine $(a)$ the radius of curvature $\rho,(b)$ the radius of curvature $\rho^{\prime}$ of a transverse cross section, $(c)$ the angle between the sides of the bar that were originally vertical. Use $E=29 \times 10^{6}$ psi and $\nu=0.29$.

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The couple $\mathrm{M}$ is applied to a beam of the cross section shown in a plane forming an angle $\beta$ with the vertical. Determine the stress at $(a)$ point $A,(b)$ point $B$ $(c)$ point $D.$

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A $\mathrm{W} 200 \times 31.3$ rolled-steel beam is subjected to a couple $\mathrm{M}$ of moment $45 \mathrm{kN} \cdot \mathrm{m} .$ Knowing that $E=200 \mathrm{GPa}$ and $\nu=0.29,$ deter$\operatorname{mine}(a)$ the radius of curvature $\rho,(b)$ the radius of curvature $\rho^{\prime}$ of a transverse cross section.

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It was assumed in Sec. $4.1 \mathrm{B}$ that the normal stresses $\sigma_{y}$ in a member in pure bending are negligible. For an initially straight elastic member of rectangular cross section, $(a)$ derive an approximate expression for $\sigma_{y}$ as a function of $y,(b)$ show that $\left(\sigma_{y}\right)_{\max }=$ $-(c / 2 \rho)\left(\sigma_{x}\right)_{\max }$ and, thus, that $\sigma_{y}$ can be neglected in all practical situations. (Hint: Consider the free-body diagram of the portion of beam located below the surface of ordinate $y$ and assume that the distribution of the stress $\sigma_{x}$ is still linear.)

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A bar having the cross section shown has been formed by securely bonding brass and aluminum stock. Using the data given below, determine the largest permissible bending moment when the composite bar is bent about a horizontal axis.

$$\begin{array}{lcl} & \text { Aluminum } & \text { Brass } \\\hline \text { Modulus of elasticity } & 70 \mathrm{GPa} & 105 \mathrm{GPa} \\\text { Allowable stress } & 100 \mathrm{MPa} & 160 \mathrm{MPa} \\\hline\end{array}$$

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$$\begin{array}{lcl} & \text { Aluminum } & \text { Brass } \\\hline \text { Modulus of elasticity } & 70 \mathrm{GPa} & 105 \mathrm{GPa} \\\text { Allowable stress } & 100 \mathrm{MPa} & 160 \mathrm{MPa} \\\hline\end{array}$$

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For the composite bar indicated, determine the largest permissible bending moment when the bar is bent about a vertial axis.

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Wooden beams and steel plates are securely bolted together to form the composite member shown. Using the data given below, determine the largest permissible bending moment when the member is bent about a horizontal axis.

$$\begin{array}{lll} & \text { Wood } & \text { Steel } \\\hline \text { Modulus of elasticity } & 2 \times 10^{6} \mathrm{psi} & 29 \times 10^{6} \mathrm{psi} \\\text { Allowable stress } & 2000 \mathrm{psi} & 22 \mathrm{ksi} \\\hline\end{array}$$

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$$\begin{array}{lll} & \text { Wood } & \text { Steel } \\\hline \text { Modulus of elasticity } & 2 \times 10^{6} \mathrm{psi} & 29 \times 10^{6} \mathrm{psi} \\\text { Allowable stress } & 2000 \mathrm{psi} & 22 \mathrm{ksi} \\\hline\end{array}$$

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A copper strip $\left(E_{c}=105 \mathrm{GPa}\right)$ and an aluminum strip $\left(E_{a}\right.$ $=75 \mathrm{GPa}$ ) are bonded together to form the composite beam shown. Knowing that the beam is bent about a horizontal axis by a couple of moment $M=35 \mathrm{N} \cdot \mathrm{m}$, determine the maximum stress in $(a)$ the aluminum strip, $(b)$ the copper strip.

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

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

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For the composite beam indicated, determine the radius of curvature caused by the couple of moment 450 kip $\cdot$ in.

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A concrete slab is reinforced by $\frac{5}{8}$ -in.-diameter steel rods placed on 5.5 -in. centers as shown. The modulus of elasticity is $3 \times 10^{6} \mathrm{psi}$ for the concrete and $29 \times 10^{6}$ psi for the steel. Using an allowable stress of 1400 psi for the concrete and 20 ksi for the steel, determine the largest bending moment in a portion of slab 1 ft wide.

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Solve Prob. $4.47,$ assuming that the spacing of the $\frac{5}{8}$ -in.-diameter steel rods is increased to 7.5 in.

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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 GPa for the concrete and 200 GPa for the steel determine $(a)$ the stress in the steel, $(b)$ the maximum stress in the concrete.

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Solve Prob. $4.49,$ assuming that the 300 -mm width is increased to $350 \mathrm{mm}$.

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Knowing that the bending moment in the reinforced concrete beam is +100 kip $\cdot \mathrm{ft}$ and that the modulus of elasticity is $3.625 \times 10^{6} \mathrm{psi}$ for the concrete and $29 \times 10^{6}$ psi for the steel, determine $(a)$ the stress in the steel, (b) the maximum stress in the concrete.

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A concrete beam is reinforced by three steel rods placed as shown. The modulus of elasticity is $3 \times 10^{6}$ psi for the concrete and $29 \times 10^{6}$ psi for the steel. Using an allowable stress of 1350 psi for the concrete and 20 ksi for the steel, determine the largest allowable positive bending moment in the beam.

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The design of a reinforced concrete beam is said to be balanced if the maximum stresses in the steel and concrete are equal, respectively, to the allowable stresses $\sigma_{s}$ and $\sigma_{c}$. Show that to achieve a balanced design the distance $x$ from the top of the beam to the neutral axis must be

\[x=\frac{d}{1+\frac{\sigma_{s} E_{c}}{\sigma_{c} E_{s}}}\]

where $E_{c}$ and $E_{s}$ are the moduli of elasticity of concrete and steel respectively, and $d$ is the distance from the top of the beam to the reinforcing steel.

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For the concrete beam shown, the modulus of elasticity is $25 \mathrm{GPa}$ for the concrete and 200 GPa for the steel. Knowing that $b=200 \mathrm{mm}$ and $d=450 \mathrm{mm},$ and using an allowable stress of 12.5 MPa for the concrete and 140 MPa for the steel, determine $(a)$ the required area $A_{s}$ of the steel reinforcement if the beam is to be balanced, $(b)$ the largest allowable bending moment. (See Prob. 4.53 for definition of a balanced beam.)

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Five metal strips, each $0.5 \times 1.5$ -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, $15 \times 10^{6}$ psi for the brass, and $10 \times 10^{6}$ psi for the aluminum. Knowing that the beam is bent about a horizontal axis by a couple of moment 12 kip $\cdot$ in. determine $(a)$ the maximum stress in each of the three metals, (b) the radius of curvature of the composite beam.

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The composite beam shown is formed by bonding together a brass rod and an aluminum rod of semicircular cross sections. The modulus of elasticity is $15 \times 10^{6}$ psi for the brass and $10 \times 10^{6}$ psi for the aluminum. Knowing that the composite beam is bent about a horizontal axis by couples of moment 8 kip -in., determine the maximum stress $(a)$ in the brass, $(b)$ in the aluminum.

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A steel pipe and an aluminum pipe are securely bonded together to form the composite beam shown. The modulus of elasticity is 200 GPa for the steel and 70 GPa for the aluminum. Knowing that the composite beam is bent by a couple of moment $500 \mathrm{N} \cdot \mathrm{m}$ determine the maximum stress $(a)$ in the aluminum, $(b)$ in the steel.

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The rectangular beam shown is made of a plastic for which the value of the modulus of elasticity in tension is one-half of its value in compression. For a bending moment $M=600 \mathrm{N} \cdot \mathrm{m}$ determine the maximum ( $a$ ) tensile stress, (b) compressive stress.

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A rectangular beam is made of material for which the modulus of elasticity is $E_{t}$ in tension and $E_{c}$ in compression. Show that the curvature of the beam in pure bending is

\[\frac{1}{\rho}=\frac{M}{E_{r} I}\]

where

\[E_{r}=\frac{4 E_{t} E_{c}}{(\sqrt{E_{t}}+\sqrt{E_{c}})^{2}}\]

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Knowing that $M=250 \mathrm{N} \cdot \mathrm{m}$, determine the maximum stress in the beam shown when the radius $r$ of the fillets is $(a) 4 \mathrm{mm},(b) 8 \mathrm{mm}$.

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Knowing that the allowable stress for the beam shown is $90 \mathrm{MPa}$ determine the allowable bending moment $M$ when the radius $r$ of the fillets is $(a) 8 \mathrm{mm}$ (b) $12 \mathrm{mm}$

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Semicircular grooves of radius $r$ must be milled as shown in the sides of a steel member. Using an allowable stress of $8 \mathrm{ksi}$, determine the largest bending moment that can be applied to the member when $(a) r=\frac{3}{8}$ in, $(b) r=\frac{3}{4}$ in.

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Semicircular grooves of radius $r$ must be milled as shown in the sides of a steel member. Knowing that $M=4$ kip $\cdot$ in., determine the maximum stress in the member when the radius $r$ of the semicircular grooves is $(a) r=\frac{3}{8}$ in, $(b) r=\frac{3}{4}$ in.

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A couple of moment $M=2 \mathrm{kN} \cdot \mathrm{m}$ is to be applied to the end of a steel bar. Determine the maximum stress in the bar $(a)$ if the bar is designed with grooves having semicircular portions of radius $r=10 \mathrm{mm},$ as shown in Fig. $a,(b)$ if the bar is redesigned by removing the material to the left and right of the dashed lines as shown in Fig. $b$.

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The allowable stress used in the design of a steel bar is $80 \mathrm{MPa}$ Determine the largest couple $\mathbf{M}$ that can be applied to the bar $(a)$ if the bar is designed with grooves having semicircular portions of radius $r=15 \mathrm{mm},$ as shown in Fig. $a,(b)$ if the bar is redesigned by removing the material to the left and right of the dashed lines as shown in Fig. $b$

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The prismatic bar shown is made of a steel that is assumed to be elastoplastic with $\sigma_{Y}=300 \mathrm{MPa}$ and is subjected to a couple $\mathbf{M}$ parallel to the $x$ axis. Determine the moment $M$ of the couple for which (a) yield first occurs, (b) the elastic core of the bar is $4 \mathrm{mm}$ thick.

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Solve Prob. 4.67 , assuming that the couple $\mathbf{M}$ is parallel to the $z$ axis.

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A solid square rod of side 0.6 in. is made of a steel that is assumed to be elastoplastic with $E=29 \times 10^{6} \mathrm{psi}$ and $\sigma_{Y}=48 \mathrm{ksi} .$ Knowing that a couple $\mathbf{M}$ is applied and maintained about an axis parallel to a side of the cross section, determine the moment $M$ of the couple for which the radius of curvature is $6 \mathrm{ft}$.

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For the solid square rod of Prob. $4.69,$ determine the moment $M$ for which the radius of curvature is $3 \mathrm{ft}$

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The prismatic rod shown is made of a steel that is assumed to be elastoplastic with $E=200 \mathrm{GPa}$ and $\sigma_{Y}=280$ MPa. Knowing that couples $\mathbf{M}$ and $\mathbf{M}^{\prime}$ of moment $525 \mathrm{N} \cdot \mathrm{m}$ are applied and maintained about axes parallel to the $y$ axis, determine $(a)$ the thickness of the elastic core, $(b)$ the radius of curvature of the bar.

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Solve Prob. 4.71 , assuming that the couples $\mathbf{M}$ and $\mathbf{M}^{\prime}$ are applied and maintained about axes parallel to the $x$ axis.

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A beam of the cross section shown is made of a steel that is assumed to be elastoplastic with $E=200 \mathrm{GPa}$ and $\sigma_{Y}=240 \mathrm{MPa}$ For bending about the $z$ axis, determine the bending moment at which $(a)$ yield first occurs, (b) the plastic zones at the top and bottom of the bar are $30 \mathrm{mm}$ thick.

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$4.76 \quad$ A beam of the cross section shown is made of a steel that is assumed to be elastoplastic with $E=29 \times 10^{6} \mathrm{psi}$ and $\sigma_{Y}=42$ ksi. For bending about the $z$ axis, determine the bending moment at which $(a)$ yield first occurs, $(b)$ the plastic zones at the top and bottom of the bar are 3 in. thick.

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For the beam indicated, determine $(a)$ the plastic moment $M_{p},(b)$ the shape factor of the cross section.

Beam of Prob. 4.73.

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For the beam indicated, determine $(a)$ the plastic moment $M_{p},(b)$ the shape factor of the cross section.

Beam of Prob. 4.74.

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For the beam indicated, determine $(a)$ the plastic moment $M_{p},(b)$ the shape factor of the cross section.

Beam of Prob. 4.75.

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For the beam indicated, determine $(a)$ the plastic moment $M_{p},(b)$ the shape factor of the cross section.

Beam of Prob. 4.76.

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Determine the plastic moment $M_{p}$ of a steel beam of the cross section shown, assuming the steel to be elastoplastic with a yield strength of $240 \mathrm{MPa}$.

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Determine the plastic moment $M_{p}$ of a steel beam of the cross section shown, assuming the steel to be elastoplastic with a yield strength of 42 ksi.

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Determine the plastic moment $M_{p}$ of the cross section shown when the beam is bent about a horizontal axis. Assume the material to be elastoplastic with a yield strength of 175 MPa.

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Determine the plastic moment $M_{p}$ of a steel beam of the cross section shown, assuming the steel to be elastoplastic with a yield strength of 36 ksi.

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For the beam indicated, a couple of moment equal to the full plastic moment $M_{p}$ is applied and then removed. Using a yield strength of 240 MPa, determine the residual stress at $y=45 \mathrm{mm}$.

Beam of Prob. 4.73

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For the beam indicated, a couple of moment equal to the full plastic moment $M_{p}$ is applied and then removed. Using a yield strength of 240 MPa, determine the residual stress at $y=45 \mathrm{mm}$.

Beam of Prob. 4.74

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A bending couple is applied to the bar indicated, causing plastic zones 3 in. thick to develop at the top and bottom of the bar. After the couple has been removed, determine ( $a$ ) the residual stress at $y=4.5$ in., $(b)$ the points where the residual stress is zero, $(c)$ the radius of curvature corresponding to the permanent deformation of the bar

Beam of Prob. 4.75

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A bending couple is applied to the bar indicated, causing plastic zones 3 in. thick to develop at the top and bottom of the bar. After the couple has been removed, determine ( $a$ ) the residual stress at $y=4.5$ in., $(b)$ the points where the residual stress is zero, $(c)$ the radius of curvature corresponding to the permanent deformation of the bar

Beam of Prob. 4.76

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A bending couple is applied to the beam of Prob. $4.73,$ causing plastic zones $30 \mathrm{mm}$ thick to develop at the top and bottom of the beam. After the couple has been removed, determine $(a)$ the residual stress at $y=45 \mathrm{mm},(b)$ the points where the residual stress is zero, $(c)$ the radius of curvature corresponding to the permanent deformation of the beam.

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A beam of the cross section shown is made of a steel that is assumed to be elastoplastic with $E=29 \times 10^{6} \mathrm{psi}$ and $\sigma_{Y}=42 \mathrm{ksi}$ A bending couple is applied to the beam about the $z$ axis, causing plastic zones 2 in. thick to develop at the top and bottom of the beam. After the couple has been removed, determine ( $a$ ) the residual stress at $y=2$ in., $(b)$ the points where the residual stress is zero, $(c)$ the radius of curvature corresponding to the permanent deformation of the beam.

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A rectangular bar that is straight and unstressed is bent into an arc of circle of radius $\rho$ by two couples of moment $M$. After the couples are removed, it is observed that the radius of curvature of the bar is $\rho_{R}$. Denoting by $\rho_{Y}$ the radius of curvature of the bar at the onset of yield, show that the radii of curvature satisfy the following relation:

\[\frac{1}{\rho_{R}}=\frac{1}{\rho}\left\{1-\frac{3}{2} \frac{\rho}{\rho_{Y}}\left[1-\frac{1}{3}\left(\frac{\rho}{\rho_{Y}}\right)^{2}\right]\right\}\]

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A solid bar of rectangular cross section is made of a material that is assumed to be elastoplastic. Denoting by $M_{Y}$ and $\rho_{Y},$ respectively, the bending moment and radius of curvature at the onset of yield, determine $(a)$ the radius of curvature when a couple of moment $M=1.25 M_{Y}$ is applied to the bar, $(b)$ the radius of curvature after the couple is removed. Check the results obtained by using the relation derived in Prob. 4.93.

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The prismatic bar $A B$ is made of a steel that is assumed to be elastoplastic and for which $E=200$ GPa. Knowing that the radius of curvature of the bar is $2.4 \mathrm{m}$ when a couple of moment $M=350 \mathrm{N} \cdot \mathrm{m}$ is applied as shown, determine $(a)$ the yield strength of the steel, $(b)$ the thickness of the elastic core of the bar.

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The prismatic bar $A B$ is made of an aluminum alloy for which the tensile stress-strain diagram is as shown. Assuming that the $\sigma-\epsilon$ diagram is the same in compression as in tension, determine $(a)$ the radius of curvature of the bar when the maximum stress is $250 \mathrm{MPa},(b)$ the corresponding value of the bending moment. (Hint: For part $b$, plot $\sigma$ versus $y$ and use an approximate method of integration.

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The prismatic bar $A B$ is made of a bronze alloy for which the tensile stress-strain diagram is as shown. Assuming that the $\sigma-\epsilon$ diagram is the same in compression as in tension, determine $(a)$ the maximum stress in the bar when the radius of curvature of the bar is 100 in., $(b)$ the corresponding value of the bending moment. (See hint given in Prob. $4.96 .)$

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A prismatic bar of rectangular cross section is made of an alloy for which the stress-strain diagram can be represented by the relation $\epsilon=k \sigma^{n}$ for $\sigma>0$ and $\epsilon=-\left|k \sigma^{n}\right|$ for $\sigma<0 .$ If a couple M is applied to the bar, show that the maximum stress is

\[\sigma_{m}=\frac{1+2 n}{3 n} \frac{M c}{I}\]

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Knowing that the magnitude of the horizontal force $\mathbf{P}$ is $8 \mathrm{kN}$ determine the stress at $(a)$ point $A,(b)$ point $B$

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A short wooden post supports a 6 -kip axial load as shown. Determine the stress at point $A$ when $(a) b=0$ $(b) b=1.5$ in. $(c) b=3$ in

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Two forces $\mathbf{P}$ can be applied separately or at the same time to a plate that is welded to a solid circular bar of radius $r$. Determine the largest compressive stress in the circular bar, $(a)$ when both forces are applied, $(b)$ when only one of the forces is applied.

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A short $120 \times 180-\mathrm{mm}$ column supports the three axial loads shown. Knowing that section $A B D$ is sufficiently far from the loads to remain plane, determine the stress at $(a)$ corner $A,(b)$ corner $B$

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As many as three axial loads, each of magnitude $P=50 \mathrm{kN},$ can be applied to the end of a $\mathrm{W} 200 \times 31.1$ rolled-steel shape. Determine the stress at point $A,(a)$ for the loading shown, $(b)$ if loads are applied at points 1 and 2 only.

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Two $10-\mathrm{kN}$ forces are applied to a $20 \times 60$ -mm rectangular bar as shown. Determine the stress at point $A$ when $(a) b=0$ $(b) b=15 \mathrm{mm}$ $(c) b=25 \mathrm{mm}$.

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Portions of a $\frac{1}{2} \times \frac{1}{2}$ -in. square bar have been bent to form the two machine components shown. Knowing that the allowable stress is 15 ksi, determine the maximum load that can be applied to each component.

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Knowing that the allowable stress in section $A B D$ is $80 \mathrm{MPa}$ determine the largest force $\mathbf{P}$ that can be applied to the bracket shown.

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A milling operation was used to remove a portion of a solid bar of square cross section. Knowing that $a=30 \mathrm{mm}, d=20 \mathrm{mm}$ and $\sigma_{\text {all }}=60 \mathrm{MPa}$, determine the magnitude $P$ of the largest forces that can be safely applied at the centers of the ends of the bar.

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A milling operation was used to remove a portion of a solid bar of square cross section. Forces of magnitude $P=18 \mathrm{kN}$ are applied at the centers of the ends of the bar. Knowing that $a=30 \mathrm{mm}$ and $\sigma_{\text {all }}=135 \mathrm{MPa}$, determine the smallest allowable depth $d$ of the milled portion of the bar.

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The two forces shown are applied to a rigid plate supported by a steel pipe of 8 -in. outer diameter and 7 -in. inner diameter. Determine the value of $\mathbf{P}$ for which the maximum compressive stress in the pipe is 15 ksi.

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An offset $h$ must be introduced into a solid circular rod of diameter $d .$ Knowing that the maximum stress after the offset is introduced must not exceed 5 times the stress in the rod when it is straight, determine the largest offset that can be used.

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An offset $h$ must be introduced into a metal tube of 0.75 -in. outer diameter and 0.08 -in. wall thickness. Knowing that the maximum stress after the offset is introduced must not exceed 4 times the stress in the tube when it is straight, determine the largest offset that can be used.

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A short column is made by nailing four $1 \times 4$ -in. planks to a $4 \times 4$ -in. timber. Using an allowable stress of 600 psi, determine the largest compressive load $\mathbf{P}$ that can be applied at the center of the top section of the timber column as shown if $(a)$ the column is as described, $(b)$ plank 1 is removed $(c)$ planks 1 and 2 are removed $(d)$ planks $1,2,$ and 3 are removed, $(e)$ all planks are removed.

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A vertical rod is attached at point $A$ to the cast iron hanger shown. Knowing that the allowable stresses in the hanger are $\sigma_{\text {all }}=+5$ ksi and $\sigma_{\text {all }}=-12$ ksi, determine the largest downward force and the largest upward force that can be exerted by the rod.

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Solve Prob. 4.113 , assuming that the vertical rod is attached at point $B$ instead of point $A$

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Knowing that the clamp shown has been tightened until $P=400 \mathrm{N},$ determine $(a)$ the stress at point $A,(b)$ the stress at point $B,(c)$ the location of the neutral axis of section $a-a$.

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The shape shown was formed by bending a thin steel plate. Assuming that the thickness $t$ is small compared to the length $a$ of each side of the shape, determine the stress $(a)$ at $A,(b)$ at $B,(c)$ at $C$.

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Three steel plates, each of $25 \times 150$ -mm cross section, are welded together to form a short H-shaped column. Later, for architectural reasons, a $25-\mathrm{mm}$ strip is removed from each side of one of the flanges. Knowing that the load remains centric with respect to the original cross section, and that the allowable stress is $100 \mathrm{MPa},$ determine the largest force $\mathbf{P}(a)$ that could be applied to the original column,

(b) that can be applied to the modified column.

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A vertical force $\mathbf{P}$ of magnitude 20 kips is applied at point $C$ located on the axis of symmetry of the cross section of a short column. Knowing that $y=5$ in., determine $(a)$ the stress at point $A,(b)$ the stress at point $B,(c)$ the location of the neutral axis.

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A vertical force $\mathbf{P}$ is applied at point $C$ located on the axis of symmetry of the cross section of a short column. Determine the range of values of $y$ for which tensile stresses do not occur in the column.

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The four bars shown have the same cross-sectional area. For the given loadings, show that $(a)$ the maximum compressive stresses are in the ratio $4: 5: 7: 9,(b)$ the maximum tensile stresses are in the ratio $2: 3: 5: 3 .$ (Note: the cross section of the triangular bar is an equilateral triangle.)

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An eccentric force $\mathbf{P}$ is applied as shown to a steel bar of $25 \times 90-\mathrm{mm}$ cross section. The strains at $A$ and $B$ have been measured and found to be

\[\epsilon_{A}=+350 \mu \quad \epsilon_{B}=-70 \mu\]

Knowing that $E=200 \mathrm{GPa}$, determine $(a)$ the distance $d,(b)$ the magnitude of the force $\mathbf{P}$

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Solve Prob. 4.121 , assuming that the measured strains are

\[\epsilon_{A}=+600 \mu \quad \epsilon_{B}=+420 \mu\]

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The C-shaped steel bar is used as a dynamometer to determine the magnitude $P$ of the forces shown. Knowing that the cross section of the bar is a square of side $40 \mathrm{mm}$ and that the strain on the inner edge was measured and found to be $450 \mu$, determine the magnitude $P$ of the forces. Use $E=200 \mathrm{GPa}$.

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A short length of a rolled-steel column supports a rigid plate on which two loads $\mathbf{P}$ and $\mathbf{Q}$ are applied as shown. The strains at two points $A$ and $B$ on the centerline of the outer faces of the flanges have been measured and found to be

\[\epsilon_{A}=-400 \times 10^{-6} \text {in. } / \text { in. } \quad \epsilon_{B}=-300 \times 10^{-6} \text {in. } / \text { in. }\]

Knowing that $E=29 \times 10^{6} \mathrm{psi}$, determine the magnitude of each load.

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A single vertical force $\mathbf{P}$ is applied to a short steel post as shown. Gages located at $A, B,$ and $C$ indicate the following strains:

\[\epsilon_{A}=-500 \mu \quad \epsilon_{B}=-1000 \mu \quad \epsilon_{C}=-200 \mu\]

Knowing that $E=29 \times 10^{6} \mathrm{psi}$, determine $(a)$ the magnitude of $\mathbf{P},(b)$ the line of action of $\mathbf{P},$

$(c)$ the corresponding strain at the hidden edge of the post, where $x=-2.5$ in. and $z=-1.5$ in.

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The eccentric axial force $\mathbf{P}$ acts at point $D,$ which must be located $25 \mathrm{mm}$ below the top surface of the steel bar shown. For $P=60 \mathrm{kN}$ (a) determine the depth $d$ of the bar for which the tensile stress at point $A$ is maximum, $(b)$ the corresponding stress at $A$

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The couple $\mathbf{M}$ acts in a vertical plane and is applied to a beam oriented as shown. Determine $(a)$ the angle that the neutral axis forms with the horizontal, $(b)$ the maximum tensile stress in the beam.

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The couple $\mathbf{M}$ acts in a vertical plane and is applied to a beam oriented as shown. Determine the stress at point $A.$

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The tube shown has a uniform wall thickness of $12 \mathrm{mm}$. For the loading given, determine $(a)$ the stress at points $A$ and $B,(b)$ the point where the neutral axis intersects line $A B D$.

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A horizontal load $\mathbf{P}$ of magnitude $100 \mathrm{kN}$ is applied to the beam shown. Determine the largest distance $a$ for which the maximum tensile stress in the beam does not exceed 75 MPa.

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Knowing that $P=90$ kips, determine the largest distance $a$ for which the maximum compressive stress does not exceed 18 ksi.

Ajay S.

Numerade Educator

Knowing that $a=1.25$ in., determine the largest value of $\mathbf{P}$ that can be applied without exceeding either of the following allowable stresses:

\[\sigma_{\text {ten }}=10 \mathrm{ksi} \quad \sigma_{\text {comp }}=18 \mathrm{ksi}\]

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A rigid circular plate of 125 -mm radius is attached to a solid $150 \times 200-\mathrm{mm}$ rectangular post, with the center of the plate directly above the center of the post. If a $4-\mathrm{kN}$ force $\mathbf{P}$ is applied at $E$ with $\theta=30^{\circ},$ determine $(a)$ the stress at point $A,(b)$ the stress at point $B,(c)$ the point where the neutral axis intersects line $A B D$.

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In Prob. 4.148 , determine $(a)$ the value of $\theta$ for which the stress at $D$ reaches its largest value, $(b)$ the corresponding values of the stress at $A, B, C,$ and $D$.

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A beam having the cross section shown is subjected to a couple $\mathbf{M}_{0}$ that acts in a vertical plane. Determine the largest permissible value of the moment $M_{0}$ of the couple if the maximum stress in the beam is not to exceed 12 ksi. Given: $I_{y}=I_{z}=11.3$ in $^{4}, A=$ 4.75 in $^{2}, k_{\min }=0.983$ in. (Hint: By reason of symmetry, the principal axes form an angle of $45^{\circ}$ with the coordinate axes. Use the relations $\left.I_{\min }=A k_{\min }^{2} \text { and } I_{\min }+I_{\max }=I_{y}+I_{z} .\right)$

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Solve Prob. 4.150 , assuming that the couple $\mathbf{M}_{0}$ acts in a horizontal plane.

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The Z section shown is subjected to a couple $\mathbf{M}_{0}$ acting in a vertical plane. Determine the largest permissible value of the moment $M_{0}$ of the couple if the maximum stress is not to exceed 80 MPa. Given: $I_{\max }=2.28 \times 10^{-6} \mathrm{m}^{4}, I_{\min }=0.23 \times 10^{-6} \mathrm{m}^{4},$ principal

axes $25.7^{\circ}$ ? and $64.3^{\circ} \mathcal{L}$.

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Solve Prob. 4.152 assuming that the couple $\mathbf{M}_{0}$ acts in a horizontal plane.

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An extruded aluminum member having the cross section shown is subjected to a couple acting in a vertical plane. Determine the largest permissible value of the moment $M_{0}$ of the couple if the maximum stress is not to exceed 12 ksi. Given: $I_{\max }=0.957$ in $^{4}$ $I_{\min }=0.427$ in $^{4},$ principal axes $29.4^{\circ} \angle$ and $60.6^{\circ}$

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A beam having the cross section shown is subjected to a couple $\mathbf{M}_{0}$ acting in a vertical plane. Determine the largest permissible value of the moment $M_{0}$ of the couple if the maximum stress is not to exceed 100 MPa. Given: $I_{y}=I_{z}=b^{4} / 36$ and $I_{y z}=b^{4} / 72$.

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Show that, if a solid rectangular beam is bent by a couple applied in a plane containing one diagonal of a rectangular cross section, the neutral axis will lie along the other diagonal.

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(a) Show that the stress at corner $A$ of the prismatic member shown in Fig. $a$ will be zero if the vertical force $\mathbf{P}$ is applied at a point located on the line

\[\frac{x}{b / 6}+\frac{z}{h / 6}=1\]

(b) Further show that, if no tensile stress is to occur in the member, the force $\mathbf{P}$ must be applied at a point located within the area bounded by the line found in part $a$ and three similar lines corresponding to the condition of zero stress at $B, C,$ and $D,$ respectively. This area, shown in Fig. $b$, is known as the kern of the cross section.

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A beam of unsymmetric cross section is subjected to a couple $\mathbf{M}_{0}$ acting in the horizontal plane $x z$. Show that the stress at point $A$ of coordinates $y$ and $z$ is

\[\sigma_{A}=\frac{z I_{z}-y I_{y z}}{I_{y} I_{z}-I_{y z}^{2}} M_{y}\]

where $I_{y}, I_{z},$ and $I_{y z}$ denote the moments and product of inertia of the cross section with respect to the coordinate axes, and $M_{y}$ the moment of the couple.

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A beam of unsymmetric cross section is subjected to a couple $\mathbf{M}_{0}$ acting in the vertical plane $x y .$ Show that the stress at point $A$ of coordinates $y$ and $z$ is

\[\sigma_{A}=-\frac{y I_{y}-z I_{y z}}{I_{y} I_{z}-I_{y z}^{2}} M_{z}\]

where $I_{y}, I_{z},$ and $I_{y z}$ denote the moments and product of inertia of the cross section with respect to the coordinate axes, and $M_{z}$ the moment of the couple.

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(a) Show that, if a vertical force $\mathbf{P}$ is applied at point $A$ of the section shown, the equation of the neutral axis $B D$ is

\[\left(\frac{x_{A}}{r_{z}^{2}}\right) x+\left(\frac{z_{A}}{r_{x}^{2}}\right) z=-1\]

where $r_{z}$ and $r_{x}$ denote the radius of gyration of the cross section with respect to the $z$ axis and the $x$ axis, respectively. ( $b$ ) Further show that, if a vertical force $\mathbf{Q}$ is applied at any point located on line $B D,$ the stress at point $A$ will be zero.

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For the curved bar shown, determine the stress at point $A$ when $(a) h=50 \mathrm{mm},(b) h=60 \mathrm{mm}$.

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For the curved bar shown, determine the stress at points $A$ and $B$ when $h=55 \mathrm{mm}$.

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For the machine component and loading shown, determine the stress at point $A$ when $(a) h=2$ in., $(b) h=2.6$ in.

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For the machine component and loading shown, determine the stress at points $A$ and $B$ when $h=2.5$ in.

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The curved bar shown has a cross section of $40 \times 60 \mathrm{mm}$ and an inner radius $r_{1}=15 \mathrm{mm} .$ For the loading shown, determine the largest tensile and compressive stresses.

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For the curved bar and loading shown, determine the percent error introduced in the computation of the maximum stress by assuming that the bar is straight. Consider the case when $(a) r_{1}$ $=20 \mathrm{mm},(b) r_{1}=200 \mathrm{mm},(c) r_{1}=2 \mathrm{m}$.

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Steel links having the cross section shown are available with different central angles $\beta .$ Knowing that the allowable stress is $12 \mathrm{ksi}$ determine the largest force $\mathbf{P}$ that can be applied to a link for which $\beta=90^{\circ}$.

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The curved bar shown has a cross section of $30 \times 30 \mathrm{mm}$. Knowing that the allowable compressive stress is 175 MPa, determine the largest allowable distance $a$.

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For the split ring shown, determine the stress at $(a)$ point $A,(b)$ point $B$.

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Three plates are welded together to form the curved beam shown. For $M=8$ kip $\cdot$ in., determine the stress at $(a)$ point $A,(b)$ point $B$ $(c)$ the centroid of the cross section.

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Three plates are welded together to form the curved beam shown. For the given loading, determine the distance $e$ between the neutral axis and the centroid of the cross section.

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Knowing that the maximum allowable stress is $45 \mathrm{MPa}$ determine the magnitude of the largest moment $M$ that can be applied to the components shown.

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The split ring shown has an inner radius $r_{1}=0.8$ in. and a circular cross section of diameter $d=0.6$ in. Knowing that each of the $120-$ lb forces is applied at the centroid of the cross section, determine the stress $(a)$ at point $A,(b)$ at point $B$.

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Solve Prob. $4.175,$ assuming that the ring has an inner radius $r_{1}=0.6$ in. and a cross-sectional diameter $d=0.8$ in.

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The bar shown has a circular cross section of 14 -mm diameter. Knowing that $a=32 \mathrm{mm}$, determine the stress at $(a)$ point $A$, $(b)$ point $B$.

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The bar shown has a circular cross section of 14 -mm diameter. Knowing that the allowable stress is 38 MPa, determine the largest permissible distance $a$ from the line of action of the $220-\mathrm{N}$ forces to the plane containing the center of curvature of the bar.

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The curved bar shown has a circular cross section of $32-\mathrm{mm}$ diameter. Determine the largest couple $\mathbf{M}$ that can be applied to the bar about a horizontal axis if the maximum stress is not to exceed 60 MPa.

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Knowing that $P=10 \mathrm{kN}$, determine the stress at $(a)$ point $A$, $(b)$ point $B$.

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Knowing that $M=5$ kip $\cdot$ in., determine the stress at $(a)$ point $A,(b)$ point $B$.

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Knowing that $M=5$ kip $\cdot$ in., determine the stress at $(a)$ point $A,(b)$ point $B$.

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Knowing that the machine component shown has a trapezoidal cross section with $a=3.5$ in. and $b=2.5$ in., determine the stress at $(a)$ point $A,(b)$ point $B$.

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Knowing that the machine component shown has a trapezoidal cross section with $a=2.5$ in. and $b=3.5$ in., determine the stress at $(a)$ point $A,(b)$ point $B$.

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For the curved beam and loading shown, determine the stress at $(a)$ point $A$ $(b)$ point $B$.

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For the crane hook shown, determine the largest tensile stress in section $a-a$.

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Using Eq. $(4.66),$ derive the expression for $R$ given in Fig. 4.61 for $*4.187$ A circular cross section.

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Using Eq. $(4.66),$ derive the expression for $R$ given in Fig. 4.61 for A trapezoidal cross section.

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Using Eq. $(4.66),$ derive the expression for $R$ given in Fig. 4.61 for A triangular cross section.

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Show that if the cross section of a curved beam consists of two or more rectangles, the radius $R$ of the neutral surface can be expressed as

\[R=\frac{A}{\ln \left[\left(\frac{r_{2}}{r_{1}}\right)^{b_{1}}\left(\frac{r_{3}}{r_{2}}\right)^{b_{2}}\left(\frac{r_{4}}{r_{3}}\right)^{b_{3}}\right]}\]

where $A$ is the total area of the cross section.

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For a curved bar of rectangular cross section subjected to a bending couple $\mathbf{M}$, show that the radial stress at the neutral surface is

\[\sigma_{r}=\frac{M}{A e}\left(1-\frac{r_{1}}{R}-\ln \frac{R}{r_{1}}\right)\]

and compute the value of $\sigma_{r}$ for the curved bar of Concept Applications 4.10 and $4.11 .$ (Hint: consider the free-body diagram of the portion of the beam located above the neutral surface.)

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A steel band saw blade that was originally straight passes over 8-in.- -diameter pulleys when mounted on a band saw. Determine the maximum stress in the blade, knowing that it is 0.018 in. thick and 0.625 in. wide. Use $E=29 \times 10^{6}$ psi.

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A couple of magnitude $M$ is applied to a square bar of side $a$. For each of the orientations shown, determine the maximum stress and the curvature of the bar.

Prashant B.

Numerade Educator

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In order to increase corrosion resistance, a 2 -mm-thick cladding of aluminum has been added to a steel bar as shown. The modulus of elasticity is 200 GPa for steel and 70 GPa for aluminum. For a bending moment of $300 \mathrm{N} \cdot \mathrm{m}$, determine $(a)$ the maximum stress in the steel, $(b)$ the maximum stress in the aluminum, the radius of curvature of the bar.

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The vertical portion of the press shown consists of a rectangular tube of wall thickness $t=10 \mathrm{mm} .$ Knowing that the press has been tightened on wooden planks being glued together until $P=$ $20 \mathrm{kN},$ determine the stress at $(a)$ point $A,(b)$ point $B$.

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The four forces shown are applied to a rigid plate supported by a solid steel post of radius $a .$ Knowing that $P=24$ kips and $a=1.6$ in., determine the maximum stress in the post when $(a)$ the force at $D$ is removed (b) the forces at $C$ and $D$ are removed.

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The curved portion of the bar shown has an inner radius of $20 \mathrm{mm} .$ Knowing that the allowable stress in the bar is $150 \mathrm{MPa}$ determine the largest permissible distance $a$ from the line of action of the 3 -kN force to the vertical plane containing the center of curvature of the bar.

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Determine the maximum stress in each of the two machine elements shown.

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Three $120 \times 10$ -mm steel plates have been welded together to form the beam shown. Assuming that the steel is elastoplastic with $E=200 \mathrm{GPa}$ and $\sigma_{Y}=300 \mathrm{MPa}$, determine $(a)$ the bending moment for which the plastic zones at the top and bottom of the beam are $40 \mathrm{mm}$ thick, $(b)$ the corresponding radius of curvature of the beam.

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A short length of a $\mathrm{W} 8 \times 31$ rolled-steel shape supports a rigid plate on which two loads $\mathbf{P}$ and $\mathbf{Q}$ are applied as shown. The strains at two points $A$ and $B$ on the centerline of the outer faces of the flanges have been measured and found to be

\[\epsilon_{A}=-550 \times 10^{-6} \text {in./in. } \quad \epsilon_{B}=-680 \times 10^{-6} \text {in. } / \text { in }\]

Knowing that $E=29 \times 10^{6} \mathrm{psi}$, determine the magnitude of each load.

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Two thin strips of the same material and same cross section are bent by couples of the same magnitude and glued together. After the two surfaces of contact have been securely bonded, the couples are removed. Denoting by $\sigma_{1}$ the maximum stress and by $\rho_{1}$ the radius of curvature of each strip while the couples were applied, determine $(a)$ the final stresses at points $A, B, C,$ and $D$ $(b)$ the final radius of curvature.

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