Solved: Determine the resultant internal normal and shear

The shaft is supported by a smooth thrust bearing at B and a journal bearing at C. Determine the resultant internal loadings acting on the cross section at E.

Determine the resultant internal normal and shear force in the member at (a) section a–a and (b) section b–b, each of which passes through the centroid A.The 500-lb load is applied along the centroidal axis of the member.

Determine the resultant internal loadings acting on section b–b through the centroid C on the beam.

The shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B. Determine the resultant internal loadings acting on the cross section at C.

Determine the resultant internal loadings acting on the cross section at point B.

Determine the resultant internal loadings on the cross section at point D.

Determine the resultant internal loadings at cross sections at points E and F on the assembly.

The beam supports the distributed load shown. Determine the resultant internal loadings acting on the cross section at point C. Assume the reactions at the supports A and B are vertical.

The beam supports the distributed load shown. 1 Determine the resultant internal loadings acting on the cross section at point D. Assume the reactions at the supports A and B are vertical.

The boom DF of the jib crane and the column DE have a uniform weight of 50 lb/ft. If the supported load is 300 lb, determine the resultant internal loadings in the crane on cross sections at points A, B, and C.

Determine the resultant internal loadings acting on the cross sections at points D and E of the frame.

Determine the resultant internal loadings acting on the cross sections at points F and G of the frame.

The blade of the hacksaw is subjected to a pretension force of F = 100 N. Determine the resultant internal loadings acting on section a–a that passes through point D.

The blade of the hacksaw is subjected to a pretension force of F = 100 N. Determine the resultant internal loadings acting on section b–b that passes through point D.

The beam supports the triangular distributed load shown. Determine the resultant internal loadings on the cross section at point C. Assume the reactions at the supports A and B are vertical.

The beam supports the distributed load shown. Determine the resultant internal loadings on the cross section at points D and E. Assume the reactions at the supports A and B are vertical.

The shaft is supported at its ends by two bearings A and B and is subjected to the forces applied to the pulleys fixed to the shaft. Determine the resultant internal loadings acting on the cross section at point D.The 400-N forces act in the -z direction and the 200-N and 80-N forces act in the +y direction. The journal bearings at A and B exert only y and z components of force on the shaft.

The shaft is supported at its ends by two bearingsA and B and is subjected to the forces applied to the pulleys fixed to the shaft. Determine the resultant internal loadings acting on the cross section at point C. The 400-N forces act in the –z direction and the 200-N and 80-N forces act in the +y direction. The journal bearings at A and B exert only y 1 and z components of force on the shaft.

The hand crank that is used in a press has the dimensions shown. Determine the resultant internal loadings acting on the cross section at point A if a vertical force of 50 lb is applied to the handle as shown. Assume the crank is fixed to the shaft at B.

Determine the resultant internal loadings acting on the cross section at point C in the beam. The load D has a mass of 300 kg and is being hoisted by the motor M with constant velocity.

Determine the resultant internal loadings acting on 1 the cross section at point E. The load D has a mass of 300 kg and is being hoisted by the motor M with constant velocity.

The metal stud punch is subjected to a force of 120 N on the handle. Determine the magnitude of the reactive force at the pin A and in the short link BC. Also, determine the resultant internal loadings acting on the cross section at point D.

Determine the resultant internal loadings acting on the cross section at point E of the handle arm, and on the cross section of the short link BC.

Determine the resultant internal loadings acting on the cross section at point C. The cooling unit has a total weight of 52 kip and a center of gravity at G.

Determine the resultant internal loadings acting on the cross section at points B and C of the curved member.

Determine the resultant internal loadings acting on the cross section of the frame at points F and G. The contact at E is smooth.

The pipe has a mass of 12 kg/m. If it is fixed to the wall at A, determine the resultant internal loadings acting on the cross section at B.

The brace and drill bit is used to drill a hole at O. If the drill bit jams when the brace is subjected to the forces shown, determine the resultant internal loadings acting on the cross section of the drill bit at A.

The curved rod AD of radius r has a weight per length of w. If it lies in the horizontal plane, determine the resultant internal loadings acting on the cross section at point B. Hint: The distance from the centroid C of segment AB to point O is CO = 0.9745r.

A differential element taken from a curved bar is shown in the figure. Show that \(d N / d \theta=V\), \(d V / d \theta=-N\), \(d M / d \theta=-T, \text { and } d T / d \theta=M\).

The supporting wheel on a scaffold is held in place on the leg using a 4-mm-diameter pin. If the wheel is 1 subjected to a normal force of 3 kN, determine the average shear stress in the pin. Assume the pin only supports the vertical 3-kN load.

Determine the largest intensity w of the uniform loading that can be applied to the frame without causing either the average normal stress or the average shear stress at section b–b to exceed \(\sigma=15 \mathrm{MPa} \text { and } \tau=16 \mathrm{MPa}\), respectively. Member CB has a square cross section of 30 mm on each side.

The bar has a cross-sectional area A and is subjected to the axial load P. Determine the average normal and average shear stresses acting over the shaded section, which is oriented at \(\theta\) from the horizontal. Plot the variation of these stresses as a function of \(\theta\left(0 \leq \theta \leq 90^{\circ}\right)\).

The small block has a thickness of 0.5 in. If the stress distribution at the support developed by the load varies as shown, determine the force F applied to the block, and the distance d to where it is applied.

If the material fails when the average normal stress reaches 120 psi, determine the largest centrally applied vertical load P the block can support.

If the block is subjected to a centrally applied force of P = 6 kip, determine the average normal stress in the material. Show the stress acting on a differential volume element of the material.

The plate has a width of 0.5 m. If the stress distribution at the support varies as shown, determine the force P applied to the plate and the distance d to where it is applied.

The board is subjected to a tensile force of 200 lb. Determine the average normal and average shear stress in the wood fibers, which are oriented along plane a–a at \(20^{\circ}\) with the axis of the board.

The boom has a uniform weight of 600 lb and is hoisted into position using the cable BC. If the cable has a diameter of 0.5 in., plot the average normal stress in the cable as a function of the boom position u for \(\theta \text { for } 0^{\circ} \leq \theta \leq 90^{\circ}\)

Determine the average normal stress in each of the 20-mm-diameter bars of the truss. Set P = 40 kN.

If the average normal stress in each of the 20-mm-diameter bars is not allowed to exceed 150 MPa, determine the maximum force P that can be applied to joint C.

Determine the maximum average shear stress in pin 1 A of the truss. A horizontal force of P = 40 kN is applied to joint C. Each pin has a diameter of 25 mm and is subjected to double shear.

If P = 5 kN, determine the average shear stress in the pins at A, B, and C. All pins are in double shear, and each has a diameter of 18 mm.

Determine the maximum magnitude P of the loads the beam can support if the average shear stress in each pin is not to exceed 80 MPa. All pins are in double shear, and each has a diameter of 18 mm.

The column is made of concrete having a density of \(2.30 \mathrm{Mg} / \mathrm{m}^{3}\). At its top B it is subjected to an axial compressive force of 15 kN. Determine the average normal stress in the column as a function of the distance z measured from its base.

The beam is supported by two rods AB and CD that have cross-sectional areas of \(12 \mathrm{~mm}^{2} \text { and } 8 \mathrm{~mm}^{2}\), respectively. If d = 1 m, determine the average normal stress in each rod.

The beam is supported by two rods AB and CD that have cross-sectional areas of \(12 \mathrm{~mm}^{2} \text { and } 8 \mathrm{~mm}^{2}\), respectively. Determine the position d of the 6-kN load so that the average normal stress in each rod is the same.

If P = 15 kN, determine the average shear stress in the pins at A, B, and C. All pins are in double shear, and each has a diameter of 18 mm.

The railcar docklight is supported by the \(\frac{1}{8}\)-in.-diameter pin at A. If the lamp weighs 4 lb, and the extension arm AB has a weight of 0.5 lb/ft, determine the average shear stress in the pin needed to support the lamp. Hint: The shear force in the pin is caused by the couple moment required for equilibrium at A.

The plastic block is subjected to an axial compressive force of 600 N.Assuming that the caps at the top and bottom distribute the load uniformly throughout the block, determine the average normal and average shear stress acting along section a–a.

The two steel members are joined together using a \(30^{\circ}\) scarf weld. Determine the average normal and average shear stress resisted in the plane of the weld.

The bar has a cross-sectional area of \(400\left(10^{-6}\right) \mathrm{m}^{2}\). If it is subjected to a triangular axial distributed loading along its length which is 0 at x=0 and 9kN > m at x = 1.5m,and to two concentrated loads as shown, determine the average normal stress in bar as a function of x for \(0 \leq x<0.6 \mathrm{~m}\).

The bar has a cross-sectional area of \(400\left(10^{-6}\right) \mathrm{m}^{2}\). If it is subjected to a uniform axial distributed loading along its length of 9 kN>m, and to two concentrated loads as shown, determine the average normal stress in the bar as a function of x for \(0.6 \mathrm{~m}<x \leq 1.5 \mathrm{~m}\).

The two members used in the construction of an aircraft fuselage are joined together using a \(30^{\circ}\) fish-mouth weld. Determine the average normal and average shear stress on the plane of each weld. Assume each inclined plane supports a horizontal force of 400 lb.

The 2-Mg concrete pipe has a center of mass at point G. If it is suspended from cables AB and AC, determine the average normal stress in the cables. The diameters of AB and AC are 12 mm and 10 mm, respectively.

The 2-Mg concrete pipe has a center of mass at point G. If it is suspended from cables AB and AC, determine the diameter of cable AB so that the average normal stress in this cable is the same as in the 10-mm-diameter cable AC.

The pier is made of material having a specific weight g. If it has a square cross section, determine its width w as a function of z so that the average normal stress in the pier remains constant. The pier supports a constant load P at its top where its width is \(w_{1}\).

Rods AB and BC have diameters of 4 mm and 6 mm, respectively. If the 3 kN force is applied to the ring at B, determine the angle u so that the average normal stress in each rod is equivalent. What is this stress?

The uniform bar, having a cross-sectional area of A and mass per unit length of m, is pinned at its center. If it is rotating in the horizontal plane at a constant angular rate of \(\omega\), determine the average normal stress in the bar as a function of x.

The bar has a cross-sectional area of \(400\left(10^{-6}\right) \mathrm{m}^{2}\). If it is subjected to a uniform axial distributed loading along its length and to two concentrated loads, determine the average normal stress in the bar as a function of x for \(0<x \leq 0.5 \mathrm{~m}\)

The bar has a cross-sectional area of \(400\left(10^{-6}\right) \mathrm{m}^{2}\). If it is subjected to a uniform axial distributed loading along its length and to two concentrated loads, determine the average normal stress in the bar as a function of x for \(0.5 \mathrm{~m}<x \leq 1.25 \mathrm{~m}\).

The prismatic bar has a cross-sectional area A. If it is subjected to a distributed axial loading that increases linearly from w = 0 at x = 0 to w = \(w_{0}\) at x = a, and then decreases linearly to w = 0 at x = 2a, determine the average normal stress in the bar as a function of x for\(0 \leq x<a\).

The prismatic bar has a cross-sectional area A. If it is subjected to a distributed axial loading that increases linearly from w = 0 at x = 0 to w = \(w_{0}\) at x = a, and then decreases linearly to w = 0 at x = 2a, determine the average normal stress in the bar as a function of x for \(a<x \leq 2 a\).

The bars of the truss each have a cross-sectional area of \(1.25 \mathrm{in}^{2}\). Determine the average normal stress in members AB, BD, and CE due to the loading P = 6 kip. State whether the stress is tensile or compressive.

The bars of the truss each have a cross-sectional area of \(1.25 \mathrm{in}^{2}\). If the maximum average normal stress in any bar is not to exceed 20 ksi, determine the maximum magnitude P of the loads that can be applied to the truss.

Determine the largest load P that can be applied to the frame without causing either the average normal stress or the average shear stress at section a–a to exceed \(\sigma=150 \mathrm{MPa} \text { and } \tau=60 \mathrm{MPa}\), respectively. Member CB has a square cross section of 25 mm on each side.

Determine the greatest constant angular velocity \(\omega\) of the flywheel so that the average normal stress in its rim does not exceed \(\sigma=15 \mathrm{MPa}\). Assume the rim is a thin ring having a thickness of 3 mm, width of 20 mm, and a mass of 30 kg>m. Rotation occurs in the horizontal plane. Neglect the effect of the spokes in the analysis. Hint: Consider a 1 free-body diagram of a semicircular segment of the ring. The center of mass for this segment is located at \(\hat{r}=2 r / \pi\) from the center.

The radius of the pedestal is defined by \(r=\left(0.5 e^{-0.08 y^{2}}\right) \mathrm{m}\), where y is in meters. If the material has a density of \(2.5 \mathrm{Mg} / \mathrm{m}^{3}\), determine the average normal stress at the support.

If A and B are both made of wood and are \(\frac{3}{8}\) in. thick, determine to the nearest \(\frac{1}{4}\) in. the smallest dimension h of the vertical segment so that it does not fail in shear. The allowable shear stress for the segment is \(\tau_{\text {allow }}=300 \mathrm{psi}\)

The lever is attached to the shaft A using a key that has a width d and length of 25 mm. If the shaft is fixed and a vertical force of 200 N is applied perpendicular to the handle, determine the dimension d if the allowable shear stress for the key is \(\tau_{\text {allow }}=35 \mathrm{MPa}\)

The connection is made using a bolt and nut and two washers. If the allowable bearing stress of the washers on the boards is \(\left(\sigma_{b}\right)_{\text {allow }}=2 \mathrm{ksi}\), and the allowable tensile stress within the bolt shank S is \(\left(\sigma_{t}\right)_{\text {allow }}=18 \mathrm{ksi}\), determine the maximum allowable tension in the bolt shank. The bolt shank has a diameter of 0.31 in., and the washers have an outer diameter of 0.75 in. and inner diameter (hole) of 0.50 in.

The tension member is fastened together using two bolts, one on each side of the member as shown. Each bolt has a diameter of 0.3 in. Determine the maximum load P that can be applied to the member if the allowable shear stress for the bolts is \(\tau_{\text {allow }}=12 \mathrm{ksi}\) and the allowable average normal stress is \(\sigma_{\text {allow }}=20 \mathrm{ksi}\)

The steel swivel bushing in the elevator control of an airplane is held in place using a nut and washer as shown in Fig. (a). Failure of the washer A can cause the push rod to separate as shown in Fig. (b). If the maximum average shear stress is \(\tau_{\max }=21 \mathrm{ksi}\), determine the force F that must be applied to the bushing. The washer is 1\16 in. thick.

The spring mechanism is used as a shock absorber for a load applied to the drawbar AB. Determine the force in each spring when the 50-kN force is applied. Each spring is originally unstretched and the drawbar slides along the smooth guide posts CG and EF. The ends of all springs are attached to their respective members. Also, what is the required diameter of the shank of bolts CG and EF if the allowable stress for the bolts is \(\sigma_{\text {allow }}=150 \mathrm{MPa} ?\)

The spring mechanism is used as a shock absorber for a load applied to the drawbar AB. Determine the force in each spring when the 50-kN force is applied. Each spring is originally unstretched and the drawbar slides along the smooth guide posts CG and EF. The ends of all springs are attached to their respective members. Also, what is the required diameter of the shank of bolts CG and EF if the allowable stress for the bolts is \(\sigma_{allow} = 150\ MPa\)?

Determine the maximum load P that can be 1 applied to the beam if the bearing plates A? and B? have square cross sections of 2 in. x 2 in. and 4 in. x 4 in., respectively, and the allowable bearing stress for the material is\(\left(\sigma_{b}\right)_{\text {allow }}=400 \mathrm{psi}\).

Determine the required diameter of the pins at A and B to the nearest \(\frac{1}{16}\) in. if the allowable shear stress of the material is \(\tau_{\text {allow }}=6 \mathrm{ksi}\). Pin A is subjected to double shear, whereas pin B is subjected to single shear.

If the allowable tensile stress for wires AB and AC is \(\sigma_{\text {allow }}=200 \mathrm{MPa}\), determine the required diameter of each wire if the applied load is P = 6 kN.

If the allowable tensile stress for wires AB and AC is \(\sigma_{\text {allow }}=180 \mathrm{MPa}\), and wire AB has a diameter of 5 mm 1 and AC has a diameter of 6 mm, determine the greatest force P that can be applied to the chain.

The cotter is used to hold the two rods together. Determine the smallest thickness t of the cotter and the smallest diameter d of the rods. All parts are made of steel for which the failure normal stress is \(\sigma_{\text {fail }}=500 \mathrm{MPa}\) and the failure shear stress is \(\tau_{\text {fail }}=375 \mathrm{MPa}\). Use a factor of safety of \(\left.(\text { F.S. })_{t}=2.50 \text { in tension and (F.S. }\right)_{s}=1.75 \text { in shear. }\)

Determine the required diameter of the pins at A and B if the allowable shear stress for the material is\(\tau_{\text {allow }}=100 \mathrm{MPa} .\) Both pins are subjected to double shear.

The steel pipe is supported on the circular base plate and concrete pedestal. If the thickness of the pipe is t = 5 mm and the base plate has a radius of 150 mm, determine the factors of safety against failure of the steel and concrete. The applied force is 500 kN, and the normal failure stresses for steel and concrete are \(\left(\sigma_{\text {fail }}\right)_{\mathrm{st}}=350 \mathrm{MPa}\) and \(\left(\sigma_{\text {fail }}\right)_{c o n}=25 \mathrm{MPa}\), respectively.

The boom is supported by the winch cable that has a diameter of 0.25 in. and an allowable normal stress of \(\sigma_{\text {allow }}=24 \mathrm{ksi}\). Determine the greatest weight of the crate that can be supported without causing the cable to fail if \(\phi=30^{\circ}\). Neglect the size of the winch.

The boom is supported by the winch cable that has an allowable normal stress of \(\sigma_{\text {allow }}=24 \mathrm{ksi}\). If it supports the 5000 lb crate when \(\phi=20^{\circ}\), determine the smallest diameter of the cable to the nearest \(\frac{1}{16}\) in.

The assembly consists of three disks A, B, and C that are used to support the load of 140 kN. Determine the smallest diameter \(d_{1}\) of the top disk, the largest diameter \(d_{2}\) of the opening, and the largest diameter \(d_{3}\) of the hole in the bottom disk. The allowable bearing stress for the material is \(\left(\sigma_{b}\right)_{\text {allow }}=350 \mathrm{MPa}\) and allowable shear stress is \(\tau_{\text {allow }}=125 \mathrm{MPa}\).

The two aluminum rods support the vertical force of P = 20 kN. Determine their required diameters if the allowable tensile stress for the aluminum is \(\left(\sigma_{b}\right)_{\text {allow }}=150 \mathrm{MPa}\)

The two aluminum rods AB and AC have diameters of 10 mm and 8 mm, respectively. Determine the largest vertical force P that can be supported. The allowable tensile stress for the aluminum is \(\left(\sigma_{b}\right)_{\text {allow }}=150 \mathrm{MPa}\).

Determine the required minimum thickness t of member AB and edge distance b of the frame if P = 9 kip and the factor of safety against failure is 2. The wood has a normal failure stress of \ and a shear failure stress of \(??\tau_{\text {fail }}=1.5 \mathrm{ksi}\)

Determine the maximum allowable load P that can be safely supported by the frame if t = 1.25 in. and b = 3.5 in. The wood has a normal failure stress of \(\sigma_{\text {fail }}=6 \mathrm{ksi}\), and a shear failure stress of \(\tau_{\text {fail }}=1.5 \mathrm{ksi}\). Use a factor of safety against failure of 2.

The compound wooden beam is connected together by a bolt at B. Assuming that the connections at A, B, C, and D exert only vertical forces on the beam, determine the required diameter of the bolt at B and the required outer diameter of its washers if the allowable tensile stress for the bolt is \(\left(\sigma_{t}\right)_{\text {allow }}=150 \mathrm{MPa}\) and the allowable bearing stress for the wood is \(\left(\sigma_{b}\right)_{\text {allow }}=28 \mathrm{MPa}\). Assume that the hole in the washers has the same diameter as the bolt.

The hanger is supported using the rectangular pin. Determine the magnitude of the allowable suspended load P if the allowable bearing stress is \(\left(\sigma_{b}\right)_{\text {allow }}=220 \mathrm{MPa}\), the allowable tensile stress is \(\left(\sigma_{t}\right)_{\text {allow }}=150 \mathrm{MPa}\), and the allowable shear stress is \(\text { tallow }=130 \mathrm{MPa}\). Take t = 6 mm, a = 5mm and b = 25mm.

The hanger is supported using the rectangular pin. Determine the required thickness t of the hanger, and dimensions a and b if the suspended load is P = 60 kN. The allowable tensile stress is \(\left(\sigma_{t}\right)_{\text {allow }}=150 \mathrm{MPa}\), the allowable bearing stress is \(\left(\sigma_{b}\right)_{\text {allow }}=290 \mathrm{MPa}\), and the allowable shear stress is \(\tau_{\text {allow }}=125 \mathrm{MPa}\).

The rods AB and CD are made of steel. Determine their smallest diameter so that they can support the dead loads shown. The beam is assumed to be pin connected at A and C. Use the LRFD method, where the resistance factor for steel in tension is \(\phi=0.9\), and the dead load factor is \(\gamma_{D}=1.4\). The failure stress is \(\sigma_{\text {fail }}=345 \mathrm{MPa}\).

The aluminum bracket A is used to support the centrally applied load of 8 kip. If it has a thickness of 0.5 in., determine the smallest height h in order to prevent a shear failure. The failure shear stress is \(\tau_{\text {fail }}=23 \mathrm{ksi}\). Use a factor of safety for shear of F.S. = 2.5.

If the allowable tensile stress for the bar is \(\left(\sigma_{t}\right)_{\text {allow }}=21 \mathrm{ksi}\), and the allowable shear stress for the pin is \(\tau_{\text {allow }}=12 \mathrm{ksi}\), determine the diameter of the pin so that the load P will be a maximum. What is this load? Assume the hole in the bar has the same diameter d as the pin. Take \(t=\frac{1}{4} \mathrm{in}\). and w = 2 in.

The bar is connected to the support using a pin having a diameter of d = 1 in. If the allowable tensile stress for the bar is \(\left(\sigma_{t}\right)_{\text {allow }}=20 \mathrm{ksi}\), and the allowable bearing stress between the pin and the bar is \(\left(\sigma_{b}\right)_{\text {allow }}=30 \mathrm{ksi}\), determine the dimensions w and t so that the gross area of the cross section is \(w t=2 \mathrm{in}^{2}\) and the load P is a maximum. What is this maximum load? Assume the hole in the bar has the same diameter as the pin.