The bar has a cross-sectional area A and is subjected to

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 point A. The 500-lb load is applied along the centroidal axis of the member.

The beam AB is fixed to the wall and has a uniform weight of 80 lb/ft. If the trolley supports a load of 1500 lb, determine the resultant internal loadings acting on the cross sections through points C and D.

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 in the beam at cross sections through points D and E. Point E is just to the right of the 3-kip load.

Determine the normal force, shear force, and moment at a section through point C. Take P = 8 kN.

The cable will fail when subjected to a tension of 2 kN. Determine the largest vertical load P the frame will support and calculate the internal normal force, shear force, and moment at the cross section through point C for this loading.

Determine the resultant internal loadings on the cross section through point C. Assume the reactions at the supports A and B are vertical.

Determine the resultant internal loadings on the cross section through 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 hoist and load weigh 300 lb, determine the resultant internal loadings in the crane on cross sections through points A , B , and C.

The forearm and biceps support the 2-kg load at A. If C can be assumed as a pin support, determine the resultant internal loadings acting on the cross section of the bone of the forearm at E. The biceps pulls on the bone along BD.

The serving tray T used on an airplane is supported on each side by an arm. The tray is pin connected to the arm at A, and at B there is a smooth pin. (The pin can move within the slot in the arms to permit folding the tray against the front passenger seat when not in use.) Determine the resultant internal loadings acting on the cross section of the arm through point C when the tray arm supports the loads shown.

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.

A 150-lb bucket is suspended from a cable on the wooden frame. Determine the resultant internal loadings on the cross section at D.

A 150-lb bucket is suspended from a cable on the wooden frame. Determine the resultant internal loadings acting on the cross section at E.

Determine resultant internal loadings acting on section a – a and section b – b . Each section passes through the centerline at point C.

The bolt shank is subjected to a tension of 80 lb. Determine the resultant internal loadings acting on the cross section at point C.

Determine the resultant internal loadings acting on the cross section through point C. Assume the reactions at the supports A and B are vertical.

Determine the resultant internal loadings acting on the cross section through point D. Assume the reactions at the supports A and B are vertical.

The forged steel clamp exerts a force of F = 900 N on the wooden block. Determine the resultant internal loadings acting on section a – a passing through point A.

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 internal resultant loadings acting on the cross section passing through the handle arm at D.

Solve Prob. 1–22 for the resultant internal loadings acting on the cross section passing through the handle arm at E and at a cross section of the short link BC.

Determine the resultant internal loadings acting on the cross section of the semicircular arch at C.

Determine the resultant internal loadings acting on the cross section through point B of the signpost. The post is fixed to the ground and a uniform pressure of \(7 \ \mathrm{lb} / \mathrm{ft}^{2}\) acts perpendicular to the face of the sign.

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 located at point C. The 300-N forces act in the -z direction and the 500-N forces act in the +x direction. The journal bearings at A and B exert only x and z components of force on the shaft.

The pipe assembly is subjected to a force of 600 N at B. Determine the resultant internal loadings acting on the cross section at C.

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 vertical plane, determine the resultant internal loadings acting on the cross section through point B. Hint : The distance from the centroid C of segment AB to point O is \(O C=[2 r \ \sin (\theta / 2)] / \theta\).

A differential element taken from a curved bar is shown in the figure. Show that \(d N / d \theta=V, \ d V / d \theta=-N\), \(d M / d \theta=-T\), 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 as shown. If the wheel is subjected to a normal force of 3 kN, determine the average shear stress developed in the pin. Neglect friction between the inner scaffold puller leg and the tube used on the wheel.

The lever is held to the fixed shaft using a tapered pin AB, which has a mean diameter of 6 mm. If a couple is applied to the lever, determine the average shear stress in the pin between the pin and lever.

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 built-up shaft consists of a pipe AB and solid rod BC. The pipe has an inner diameter of 20 mm and outer diameter of 28 mm. The rod has a diameter of 12 mm. Determine the average normal stress at points D and E and represent the stress on a volume element located at each of these points.

If the turnbuckle is subjected to an axial force of P = 900 lb, determine the average normal stress developed in section a – a and in each of the bolt shanks at B and C. Each bolt shank has a diameter of 0.5 in.

The average normal stresses developed in section a – a of the turnbuckle, and the bolts shanks at B and C, are not allowed to exceed 15 ksi and 45 ksi, respectively. Determine the maximum axial force P that can be applied to the turnbuckle. Each bolt shank has a diameter of 0.5 in.

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 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.

If the block is subjected to the centrally applied force of 600 kN, determine the averege normal stress in the material. Show the stress acting on a differential volume element of the material.

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 average shear stress developed in pin 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.

The 150-kg bucket is suspended from end E of the frame. Determine the average normal stress in the 6-mm diameter wire CF and the 15-mm diameter short strut BD.

The 150-kg bucket is suspended from end E of the frame. If the diameters of the pins at A and D are 6 mm and 10 mm, respectively, determine the average shear stress developed in these pins. Each pin is subjected to double shear.

The pedestal has a triangular cross section as shown. If it is subjected to a compressive force of 500 lb, specify the x and y coordinates for the location of point P ( x , y ), where the load must be applied on the cross section, so that the average normal stress is uniform. Compute the stress and sketch its distribution acting on the cross section at a location removed from the point of load application.

The 20-kg chandelier is suspended from the wall and ceiling using rods AB and BC , which have diameters of 3 mm and 4 mm, respectively. Determine the angle \(\theta\) so that the average normal stress in both rods is the same.

The chandelier is suspended from the wall and ceiling using rods AB and BC, which have diameters of 3 mm and 4 mm, respectively. If the average normal stress in both rods is not allowed to exceed 150 MPa, determine the largest mass of the chandelier that can be supported if \(\theta=45^{\circ}\).

The beam is supported by a pin at A and a short link BC. If P = 15 kN, determine the average shear stress developed in the pins at A, B, and C. All pins are in double shear as shown, and each has a diameter of 18 mm.

The joint is subjected to the axial member force of 6 kip. Determine the average normal stress acting on sections AB and BC. Assume the member is smooth and is 1.5-in. thick.

The driver of the sports car applies his rear brakes and causes the tires to slip. If the normal force on each rear tire is 400 lb and the coefficient of kinetic friction between the tires and the pavement is \(\mu_{k}=0.5\), determine the average shear stress developed by the friction force on the tires. Assume the rubber of the tires is flexible and each tire is filled with an air pressure of 32 psi.

During the tension test, the wooden specimen is subjected to an average normal stress of 2 ksi. Determine the axial force P applied to the specimen. Also, find the average shear stress developed along section a – a of the specimen.

If the joint is subjected to an axial force of P = 9 kN, determine the averege shear stress developed in each of the 6-mm diameter bolts between the plates and the members and along each of the four shaded shear planes.

The average shear stress in each of the 6-mm diameter bolts and along each of the four shaded shear planes is not allowed to exceed 80 MPa and 500 kPa, respectively. Determine the maximum axial force P that can be applied to the joint.

When the hand is holding the 5-lb stone, the humerus H, assumed to be smooth, exerts normal forces \(F_{C}\) and \(F_{A}\) on the radius C and ulna A, respectively, as shown. If the smallest cross-sectional area of the ligament at B is \(0.30 \ \mathrm{in}^{2}\), determine the greatest average tensile stress to which it is subjected.

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 developed 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 developed in this cable is the same as in the 10-mm diameter cable AC.

If the concrete pedestal has a specific weight of \(\gamma\), determine the average normal stress developed in the pedestal as a function of z.

The anchor bolt was pulled out of the concrete wall and the failure surface formed part of a frustum and cylinder. This indicates a shear failure occurred along the cylinder BC and tension failure along the frustum AB. If the shear and normal stresses along these surfaces have the magnitudes shown, determine the force P that must have been applied to the bolt.

The jib crane is pinned at A and supports a chain hoist that can travel along the bottom flange of the beam, \(1 \ \mathrm{ft} \leq x \leq 12 \ \mathrm{ft}\). If the hoist is rated to support a maximum of 1500 lb, determine the maximum average normal stress in the \(\frac{3}{4}-\mathrm{in}\). diameter tie rod BC and the maximum average shear stress in the \(\frac{5}{8}-\mathrm{in}\). -diamater pin at B.

If the shaft is subjected to an axial force of 5 kN, determine the bearing stress acting on the collar A.

If the 60-mm diameter shaft is subjected to an axial force of 5 kN, determine the average shear stress developed in the shear plane where the collar A and shaft are connected.

The crimping tool is used to crimp the end of the wire E. If a force of 20 lb is applied to the handles, determine the average shear stress in the pin at A. The pin is subjected to double shear and has a diameter of 0.2 in. Only a vertical force is exerted on the wire.

Solve Prob. 1–62 for pin B. The pin is subjected to double shear and has a diameter of 0.2 in.

A vertical force of P = 1500 N is applied to the bell crank. Determine the average normal stress developed in the 10-mm diameter rod CD, and the average shear stress developed in the 6-mm diameter pin B that is subjected to double shear.

Determine the maximum vertical force P that can be applied to the bell crank so that the average normal stress developed in the 10-mm diameter rod CD, and the average shear stress developed in the 6-mm diameter double sheared pin B not exceed 175 MPa and 75 MPa respectively.

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 \(\sigma=150 \ \mathrm{MPa}\) and \(\tau=60 \ \mathrm{MPa}\), respectively. Member CB has a square cross section of 25 mm on each side.

The pedestal in the shape of a frustum of a cone is made of concrete having a specific weight of \(150 \ \mathrm{lb} / \mathrm{ft}^{3}\). Determine the average normal stress acting in the pedestal at its base. Hint: The volume of a cone of radius r and height h is \(V=\frac{1}{2} \pi r^{2} h\).

The pedestal in the shape of a frustum of a cone is made of concrete having a specific weight of \(150 \ \mathrm{lb} / \mathrm{ft}^{3}\). Determine the average normal stress acting in the pedestal at its midheight, z = 4 ft. Hint: The volume of a cone of radius r and height h is \(V=\frac{1}{3} \pi r^{2} h\).

Member B is subjected to a compressive force of 800 lb. If A and B are both made of wood and are \(\frac{3}{8} \text { in }\). thick, determine to the nearest \(\frac{1}{4} \text { in }\). the smallest dimension h of the horizontal segment so that it does not fail in shear. The average 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 joint is fastened together using two bolts. Determine the required diameter of the bolts if the failure shear stress for the bolts is \(\tau_{\text {fail }}=350 \ \mathrm{MPa}\). Use a factor of safety for shear of F.S. = 2.5.

The truss is used to support the loading shown. Determine the required cross-sectional area of member BC if the allowable normal strees is \(\sigma_{\text {allow }}=24 \ \mathrm{ksi}\).

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 normal stress for the washer is \(\sigma_{\max }=60 \ \mathrm{ksi}\) and the maximum average shear stress is \(\tau_{\max }=21 \ \mathrm{ksi}\), determine the force F that must be applied to the bushing that will cause this to happen. The washer is \(\frac{1}{16} \mathrm{in}\). thick.

Member B is subjected to a compressive force of 600 lb. If A and B are both made of wood and are 1.5 in. thick, determine to the nearest \(\frac{1}{8} \text { in }\). the smallest dimension a of the support so that the average shear stress along the blue line does not exceed \(\tau_{\text {allow }}=50 \ \mathrm{psi}\). Neglect friction.

The hangers support the joist uniformly, so that it is assumed the four nails on each hanger carry an equal portion of the load. If the joist is subjected to the loading shown, determine the average shear stress in each nail of the hanger at ends A and B. Each nail has a diameter of 0.25 in. The hangers only support vertical loads.

The hangers support the joists uniformly, so that it is assumed the four nails on each hanger carry an equal portion of the load. Determine the smallest diameter of the nails at A and at B if the allowable stress for the nails is \(\tau_{\text {allow }}=4 \ \mathrm{ksi}\). The hangers only support vertical loads.

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 50-kg flowerpot is suspended from wires AB and BC. If the wires have a normal failure stress of \(\sigma_{\text {fail }}=350 \ \mathrm{MPa}\), determine the minimum diameter of each wire. Use a factor of safety of 2.5.

The 50-kg flowerpot is suspended from wires AB and BC which have diameters of 1.5 mm and 2 mm, respectively. If the wires have a normal failure stress of \(\sigma_{\text {fail }}=350 \ \mathrm{MPa}\), determine the factor of safety of each wire.

The thrust bearing consists of a circular collar A fixed to the shaft B. Determine the maximum axial force P that can be applied to the shaft so that it does not cause the shear stress along a cylindrical surface a or b to exceed an allowable shear stress of \(\tau_{\text {allow }}=170 \ \mathrm{MPa}\).

The steel pipe is supported on the circular base plate and concrete pedestal. If the normal failure stress for the steel is \(\left(\sigma_{\text {fail }}\right)_{\mathrm{st}}=350 \ \mathrm{MPa}\), determine the minimum thickness t of the pipe if it supports the force of 500 kN. Use a factor of safety against failure of 1.5. Also, find the minimum radius r of the base plate so that the minimum factor of safety against failure of the concrete due to bearing is 2.5. The failure bearing stress for concrete is \(\left(\sigma_{\text {fail }}\right)_{\mathrm{con}}=25 \ \mathrm{MPa}\).

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 \(\left(\sigma_{\text {fail }}\right)_{\mathrm{con}}=25 \ \mathrm{MPa}\), respectively.

The \(60 \mathrm{~mm} \times 60 \mathrm{~mm}\) oak post is supported on the pine block. If the allowable bearing stresses for these materials are \(\sigma_{\text {oak }}=43 \ \mathrm{MPa}\) and \(\sigma_{\text {pine }}=25 \ \mathrm{MPa}\), determine the greatest load P that can be supported. If a rigid bearing plate is used between these materials, determine its required area so that the maximum load P can be supported. What is this load?

The frame is subjected to the load of 4 kN which acts on member ABD at D. Determine the required diameter of the pins at D and C if the allowable shear stress for the material is \(\tau_{\text {allow }}=40 \ \mathrm{MPa}\). Pin C is subjected to double shear, whereas pin D is subjected to single shear.

The beam is made from southern pine and is supported by base plates resting on brick work. If the allowable bearing stresses for the materials are \(\left(\sigma_{\text {pine }}\right)_{\text {allow }}=2.81 \ \mathrm{ksi}\) and \(\left(\sigma_{\text {brick }}\right)_{\text {allow }}=6.70 \ \mathrm{ksi}\), determine the required length of the base plates at A and B to the nearest \(\frac{1}{4}\) inch in order to support the load shown. The plates are 3 in. wide.

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 \(\sigma_{\text {allow }}=150 \ \mathrm{MPa}\).

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 \(\sigma_{\text {allow }}=150 \ \mathrm{MPa}\).

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.

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 \(\sigma_{\text {fail }}=6 \ \mathrm{ksi}\), and 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 shear failure stress of \(\tau_{\text {fail }}=1.5 \ \mathrm{ksi}\). Use a factor of safety against failure of 2.

If the allowable bearing stress for the material under the supports at A and B is \(\left(\sigma_{b}\right)_{\text {allow }}=1.5 \ \mathrm{MPa}\), determine the size of square bearing plates \(A^{\prime}\) and \(B^{\prime}\) required to support the load. Dimension the plates to the nearest mm. The reactions at the supports are vertical. Take P = 100 kN.

If the allowable bearing stress for the material under the supports at A and B is \(\left(\sigma_{b}\right)_{\text {allow }}=1.5 \ \mathrm{MPa}\), determine the maximum load P that can be applied to the beam. The bearing plates \(A^{\prime}\) and \(B^{\prime}\) have square cross sections of \(150 \mathrm{~mm} \times 150 \mathrm{~mm}\) and \(250 \mathrm{~mm} \times 250 \mathrm{~mm}\), respectively.

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 constant 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.

The pin support A and roller support B of the bridge truss are supported on concrete abutments. If the bearing failure stress of the concrete is \(\left(\sigma_{\text {fail }}\right)_{b}=4 \ \mathrm{ksi}\), determine the required minimum dimension of the square bearing plates at C and D to the nearest \(\frac{1}{16} \text { in }\). Apply a factor of safety of 2 against failure.

The pin support A and roller support B of the bridge truss are supported on the concrete abutments. If the square bearing plates at C and D are \(21 \ \text { in. } \times 21 \ \text { in. }\)., and the bearing failure stress for concrete is \(\left(\sigma_{\text {fail }}\right)_{b}=4 \ \mathrm{ksi}\), determine the factor of safety against bearing failure for the concrete under each plate.

The beam AB is pin supported at A and supported by a cable BC. A separate cable CG is used to hold up the frame. If AB weighs 120 lb/ft and the column FC has a weight of 180 lb/ft, determine the resultant internal loadings acting on cross sections located at points D and E. Neglect the thickness of both the beam and column in the calculation.

The long bolt passes through the 30-mm-thick plate. If the force in the bolt shank is 8 kN, determine the average normal stress in the shank, the average shear stress along the cylindrical area of the plate defined by the section lines a – a, and the average shear stress in the bolt head along the cylindrical area defined by the section lines b – b.

To the nearest \(\frac{1}{16} \text { in }\)., determine the required thickness of member BC and the diameter of the pins at A and B if the allowable normal stress for member BC is \(\sigma_{\text {allow }}=29 \ \mathrm{ksi}\) and the allowable shear stress for the pins is \(\sigma_{\text {allow }}=10 \ \mathrm{ksi}\).

The circular punch B exerts a force of 2 kN on the top of the plate A. Determine the average shear stress in the plate due to this loading.

Determine the average punching shear stress the circular shaft creates in the metal plate through section AC and BD. Also, what is the bearing stress developed on the surface of the plate under the shaft?

The bearing pad consists of a 150 mm by 150 mm block of aluminum that supports a compressive load of 6 kN. Determine the average normal and shear stress acting on the plane through section a – a. Show the results on a differential volume element located on the plane.

The yoke-and-rod connection is subjected to a tensile force of 5 kN. Determine the average normal stress in each rod and the average shear stress in the pin A between the members.

The cable has a specific weight \(\gamma\) (weight/volume) and cross-sectional area A. If the sag s is small, so that its length is approximately L and its weight can be distributed uniformly along the horizontal axis, determine the average normal stress in the cable at its lowest point C.