Members AB and BC of the truss shown are made of the same

Two solid cylindrical rods AB and BC are welded together at B and loaded as shown. Knowing that \(d_{1}=30 \mathrm{~mm}\) and \(d_{2}=50 \mathrm{~mm}\), find the average normal stress at the midsection of (a) rod AB, (b) rod BC.

Two solid cylindrical rods AB and BC are welded together at B and loaded as shown. Knowing that the average normal stress must not exceed 150 MPa in either rod, determine the smallest allowable values of the diameters \(d_{1}\) and \(d_{2}\).

Two solid cylindrical rods AB and BC are welded together at B and loaded as shown. Knowing that P = 10 kips, find the average normal stress at the midsection of (a) rod AB, (b) rod BC.

Two solid cylindrical rods AB and BC are welded together at B and loaded as shown. Determine the magnitude of the force P for which the tensile stresses in rods AB and BC are equal.

A strain gage located at C on the surface of bone AB indicates that the average normal stress in the bone is 3.80 MPa when the bone is subjected to two 1200-N forces as shown. Assuming the cross section of the bone at C to be annular and knowing that its outer diameter is 25 mm, determine the inner diameter of the bone’s cross section at C.

Two brass rods AB and BC, each of uniform diameter, will be brazed together at B to form a nonuniform rod of total length 100 m that will be suspended from a support at A as shown. Knowing that the density of brass is \(8470 \mathrm{~kg} / \mathrm{m}^{3}\), determine (a) the length of rod AB for which the maximum normal stress in ABC is minimum, (b) the corresponding value of the maximum normal stress.

Each of the four vertical links has an 8 X 36-mm uniform rectangular cross section, and each of the four pins has a 16-mm diameter. Determine the maximum value of the average normal stress in the links connecting (a) points B and D, (b) points C and E.

Link AC has a uniform rectangular cross section \(\frac{1}{8}\) in. thick and 1 in. wide. Determine the normal stress in the central portion of the link.

Three forces, each of magnitude P = 4 kN, are applied to the structure shown. Determine the cross-sectional area of the uniform portion of rod BE for which the normal stress in that portion is +100 MPa.

Link BD consists of a single bar 1 in. wide and \(\frac{1}{2}\) in. thick. Knowing that each pin has a \(\frac{3}{8}\)-in. diameter, determine the maximum value of the average normal stress in link BD if \((a) \theta=0,(b) \theta=90^{\circ}\).

For the Pratt bridge truss and loading shown, determine the average normal stress in member BE, knowing that the cross-sectional area of that member is \(5.87 \mathrm{in}^{2}\).

The frame shown consists of four wooden members, ABC, DEF, BE, and CF. Knowing that each member has a \(2 \times 4 \text {-in. }\) rectangular cross section and that each pin has a \(\frac{1}{2} \text {-in. }\) diameter, determine the maximum value of the average normal stress (a) in member BE, (b) in member CF.

An aircraft tow bar is positioned by means of a single hydraulic cylinder connected by a 25-mm-diameter steel rod to two identical arm-and-wheel units DEF. The mass of the entire tow bar is 200 kg, and its center of gravity is located at G. For the position shown, determine the normal stress in the rod.

Two hydraulic cylinders are used to control the position of the robotic arm ABC. Knowing that the control rods attached at A and D each have a 20-mm diameter and happen to be parallel in the position shown, determine the average normal stress in (a) member AE, (b) member DG.

Determine the diameter of the largest circular hole that can be punched into a sheet of polystyrene 6 mm thick, knowing that the force exerted by the punch is 45 kN and that a 55-MPa average shearing stress is required to cause the material to fail.

Two wooden planks, each \(\frac{1}{2} \mathrm{in}\). thick and 9 in. wide, are joined by the dry mortise joint shown. Knowing that the wood used shears off along its grain when the average shearing stress reaches 1.20 ksi, determine the magnitude P of the axial load that will cause the joint to fail.

When the force P reached 1600 lb, the wooden specimen shown failed in shear along the surface indicated by the dashed line. Determine the average shearing stress along that surface at the time of failure.

A load P is applied to a steel rod supported as shown by an aluminum plate into which a 12-mm-diameter hole has been drilled. Knowing that the shearing stress must not exceed 180 MPa in the steel rod and 70 MPa in the aluminum plate, determine the largest load P that can be applied to the rod.

The axial force in the column supporting the timber beam shown is P = 20 kips. Determine the smallest allowable length L of the bearing plate if the bearing stress in the timber is not to exceed 400 psi.

Three wooden planks are fastened together by a series of bolts to form a column. The diameter of each bolt is 12 mm and the inner diameter of each washer is 16 mm, which is slightly larger than the diameter of the holes in the planks. Determine the smallest allowable outer diameter d of the washers, knowing that the average normal stress in the bolts is 36 MPa and that the bearing stress between the washers and the planks must not exceed 8.5 MPa.

A 40-kN axial load is applied to a short wooden post that is supported by a concrete footing resting on undisturbed soil. Determine (a) the maximum bearing stress on the concrete footing, (b) the size of the footing for which the average bearing stress in the soil is 145 kPa.

An axial load P is supported by a short \(\text { W8 } \times 40\) column of crosssectional area \(A=11.7 \mathrm{in}^{2}\) and is distributed to a concrete foundation by a square plate as shown. Knowing that the average normal stress in the column must not exceed 30 ksi and that the bearing stress on the concrete foundation must not exceed 3.0 ksi, determine the side a of the plate that will provide the most economical and safe design.

Link AB, of width \(b=2 \text { in. }\) and thickness \(t=\frac{1}{4} \mathrm{in}\)., is used to support the end of a horizontal beam. Knowing that the average normal stress in the link is -20 ksi and that the average shearing stress in each of the two pins is 12 ksi determine (a) the diameter d of the pins, (b) the average bearing stress in the link.

Determine the largest load P that can be applied at A when \(\theta=60^{\circ}\), knowing that the average shearing stress in the 10-mm-diameter pin at B must not exceed 120 MPa and that the average bearing stress in member AB and in the bracket at B must not exceed 90 MPa.

Knowing that \(\theta=40^{\circ}\) and P = 9 kN, determine (a) the smallest allowable diameter of the pin at B if the average shearing stress in the pin is not to exceed 120 MPa, (b) the corresponding average bearing stress in member AB at B, (c) the corresponding average bearing stress in each of the support brackets at B.

The hydraulic cylinder CF, which partially controls the position of rod DE, has been locked in the position shown. Member BD is 15 mm thick and is connected at C to the vertical rod by a 9-mm-diameter bolt. Knowing that P = 2 kN and \(\theta=75^{\circ}\), determine (a) the average shearing stress in the bolt, (b) the bearing stress at C in member BD.

For the assembly and loading of Prob. 1.7, determine (a) the average shearing stress in the pin at B, (b) the average bearing stress at B in member BD, (c) the average bearing stress at B in member ABC, knowing that this member has a 10 X 50-mm uniform rectangular cross section.

Two identical linkage-and-hydraulic-cylinder systems control the position of the forks of a fork-lift truck. The load supported by the one system shown is 1500 lb. Knowing that the thickness of member BD is\(\frac{5}{8} \mathrm{in} .\), determine (a) the average shearing stress in the \(\frac{1}{2} \text {-in. }\)-diameter pin at B, (b) the bearing stress at B in member BD.

Two wooden members of uniform rectangular cross section are joined by the simple glued scarf splice shown. Knowing that P = 11 kN, determine the normal and shearing stresses in the glued splice.

Two wooden members of uniform rectangular cross section are joined by the simple glued scarf splice shown. Knowing that the maximum allowable shearing stress in the glued splice is 620 kPa, determine (a) the largest load P that can be safely applied, (b) the corresponding tensile stress in the splice.

The 1.4-kip load P is supported by two wooden members of uniform cross section that are joined by the simple glued scarf splice shown. Determine the normal and shearing stresses in the glued splice.

Two wooden members of uniform cross section are joined by the simple scarf splice shown. Knowing that the maximum allowable tensile stress in the glued splice is 75 psi, determine (a) the largest load P that can be safely supported, (b) the corresponding shearing stress in the splice.

A centric load P is applied to the granite block shown. Knowing that the resulting maximum value of the shearing stress in the block is 2.5 ksi, determine (a) the magnitude of P, (b) the orientation of the surface on which the maximum shearing stress occurs, (c) the normal stress exerted on that surface, (d) the maximum value of the normal stress in the block.

A 240-kip load P is applied to the granite block shown. Determine the resulting maximum value of (a) the normal stress, (b) the shearing stress. Specify the orientation of the plane on which each of these maximum values occurs.

A steel pipe of 400-mm outer diameter is fabricated from 10-mm-thick plate by welding along a helix that forms an angle of \(20^{\circ}\) with a plane perpendicular to the axis of the pipe. Knowing that a 300-kN axial force P is applied to the pipe, determine the normal and shearing stresses in directions respectively normal and tangential to the weld.

A steel pipe of 400-mm outer diameter is fabricated from 10-mm-thick plate by welding along a helix that forms an angle of \(20^{\circ}\) with a plane perpendicular to the axis of the pipe. Knowing that the maximum allowable normal and shearing stresses in the directions respectively normal and tangential to the weld are \(\sigma=60\) MPa and \(\tau=36\) MPa, determine the magnitude P of the largest axial force that can be applied to the pipe.

A steel loop ABCD of length 5 ft and of \(\frac{3}{8} \text {-in. }\) diameter is placed as shown around a 1-in.-diameter aluminum rod AC. Cables BE and DF, each of \(\frac{1}{2} \text {-in. }\) diameter, are used to apply the load Q. Knowing that the ultimate strength of the steel used for the loop and the cables is 70 ksi, and that the ultimate strength of the aluminum used for the rod is 38 ksi, determine the largest load Q that can be applied if an overall factor of safety of 3 is desired.

Link BC is 6 mm thick, has a width \(w=25 \mathrm{~mm}\), and is made of a steel with a 480-MPa ultimate strength in tension. What is the factor of safety used if the structure shown was designed to support a 16-kN load P?

Link BC is 6 mm thick and is made of a steel with a 450-MPa ultimate strength in tension. What should be its width w if the structure shown is being designed to support a 20-kN load P with a factor of safety of 3?

Members AB and BC of the truss shown are made of the same alloy. It is known that a 20-mm-square bar of the same alloy was tested to failure and that an ultimate load of 120 kN was re-corded. If a factor of safety of 3.2 is to be achieved for both bars, determine the required cross-sectional area of (a) bar AB, (b) bar AC.

Members AB and BC of the truss shown are made of the same alloy. It is known that a 20-mm-square bar of the same alloy was tested to failure and that an ultimate load of 120 kN was recorded. If bar AB has a cross-sectional area of \(225 \mathrm{~mm}^{2}\), determine (a) the factor of safety for bar AB, (b) the cross-sectional area of bar ACif it is to have the same factor of safety as bar AB.

Link AB is to be made of a steel for which the ultimate normal stress is 65 ksi. Determine the cross-sectional area of AB for which the factor of safety will be 3.20. Assume that the link will be adequately reinforced around the pins at A and B.

Two wooden members are joined by plywood splice plates that are fully glued on the contact surfaces. Knowing that the clearance between the ends of the members is 6 mm and that the ultimate shearing stress in the glued joint is 2.5 MPa, determine the length L for which the factor of safety is 2.75 for the loading shown.

For the joint and loading of Prob. 1.43, determine the factor of safety when L = 180 mm.

Three \(\frac{3}{4} \text {-in. }\)-diameter steel bolts are to be used to attach the steel plate shown to a wooden beam. Knowing that the plate will support a load P = 24 kips and that the ultimate shearing stress for the steel used is 52 ksi, determine the factor of safety for this design.

Three steel bolts are to be used to attach the steel plate shown to a wooden beam. Knowing that the plate will support a load P = 28 kips, that the ultimate shearing stress for the steel used is 52 ksi, and that a factor of safety of 3.25 is desired, determine the required diameter of the bolts.

A load P is supported as shown by a steel pin that has been inserted in a short wooden member hanging from the ceiling. The ultimate strength of the wood used is 60 MPa in tension and 7.5 MPa in shear, while the ultimate strength of the steel is 145 MPa in shear. Knowing that b = 40 mm, c = 55 mm, and d = 12 mm, determine the load P if an overall factor of safety of 3.2 is desired.

For the support of Prob. 1.47, knowing that the diameter of the pin is d = 16 mm and that the magnitude of the load is P = 20 kN, determine (a) the factor of safety for the pin (b) the required values of b and c if the factor of safety for the wooden member is the same as that found in part a for the pin.

A steel plate \(\frac{1}{4} \mathrm{in} .\) thick is embedded in a concrete wall to anchor a high-strength cable as shown. The diameter of the hole in the plate is \(\frac{3}{4} \text { in. }\), the ultimate strength of the steel used is 36 ksi. and the ultimate bonding stress between plate and concrete is 300 psi. Knowing that a factor of safety of 3.60 is desired when P = 2.5 kips, determine (a) the required width a of the plate, (b) the minimum depth b to which a plate of that width should be embedded in the concrete slab. (Neglect the normal stresses between the concrete and the end of the plate.)

Determine the factor of safety for the cable anchor in Prob. 1.49 when P = 2.5 kips, knowing that a = 2 in. and b = 6 in.

Link AC is made of a steel with a 65-ksi ultimate normal stress and has a \(\frac{1}{4} \times \frac{1}{2}-\mathrm{in}\) uniform rectangular cross section. It is connected to a support at A and to member BCD at C by \(\frac{3}{4} \text {-in. }\)-diameter pins, while member BCD is connected to its support at B by a \(\frac{5}{16} \text {-in. }\)-diameter pin. All of the pins are made of a steel with a 25-ksi ultimate shearing stress and are in single shear. Knowing that a factor of safety of 3.25 is desired, determine the largest load P that can be applied at D. Note that link AC is not reinforced around the pin holes.

Solve Prob. 1.51, assuming that the structure has been redesigned to use \(\frac{5}{16}-\mathrm{in}\).-diameter pins at A and C as well as at B and that no other changes have been made.

Each of the two vertical links CF connecting the two horizontal members AD and EG has a \(10 \times 40-\mathrm{mm}\) uniform rectangular cross section and is made of a steel with an ultimate strength in tension of 400 MPa, while each of the pins at C and F has a 20-mm diameter and are made of a steel with an ultimate strength in shear of 150 MPa. Determine the overall factor of safety for the links CF and the pins connecting them to the horizontal members.

Solve Prob. 1.53, assuming that the pins at C and F have been replaced by pins with a 30-mm diameter.

In the structure shown, an 8-mm-diameter pin is used at A, and 12-mm-diameter pins are used at B and D. Knowing that the ultimate shearing stress is 100 MPa at all connections and that the ultimate normal stress is 250 MPa in each of the two links joining B and D, determine the allowable load P if an overall factor of safety of 3.0 is desired.

In an alternative design for the structure of Prob. 1.55, a pin of 10-mm-diameter is to be used at A. Assuming that all other specifications remain unchanged, determine the allowable load P if an overall factor of safety of 3.0 is desired.

A 40-kg platform is attached to the end B of a 50-kg wooden beam AB, which is supported as shown by a pin at A and by a slender steel rod BC with a 12-kN ultimate load. (a) Using the Load and Resistance Factor Design method with a resistance factor \(\phi=0.90\) and load factors \(\gamma_{D}=1.25 \text { and } \gamma_{L}=1.6\), determine the largest load that can be safely placed on the platform. (b) What is the corresponding conventional factor of safety for rod BC ?

The Load and Resistance Factor Design method is to be used to select the two cables that will raise and lower a platform supporting two window washers. The platform weighs 160 lb and each of the window washers is assumed to weigh 195 lb with equipment. Since these workers are free to move on the platform, 75% of their total weight and the weight of their equipment will be used as the design live load of each cable. (a) Assuming a resistance factor \(\phi=0.85\) and load factors \(\gamma_{D}=1.2) and \(\gamma_{L}=1.5\), determine the required minimum ultimate load of one cable. (b) What is the corresponding conventional factor of safety for the selected cables?

In the marine crane shown, link CD is known to have a uniform cross section of 50 X 150 mm. For the loading shown, determine the normal stress in the central portion of that link.

Two horizontal 5-kip forces are applied to pin B of the assembly shown. Knowing that a pin of 0.8-in. diameter is used at each connection, determine the maximum value of the average normal stress (a) in link AB, (b) in link BC.

For the assembly and loading of Prob. 1.60, determine (a) the average shearing stress in the pin at C, (b) the average bearing stress at C in member BC, (c) the average bearing stress at B in member BC.

Two steel plates are to be held together by means of 16-mmdiameter high-strength steel bolts fitting snugly inside cylindrical brass spacers. Knowing that the average normal stress must not exceed 200 MPa in the bolts and 130 MPa in the spacers, determine the outer diameter of the spacers that yields the most economical and safe design.

A couple M of magnitude 1500 N • m is applied to the crank of an engine. For the position shown, determine (a) the force P required to hold the engine system in equilibrium, (b) the average normal stress in the connecting rod BC, which has a \(\text { 450-mm }{ }^{2}\) uniform cross section.

Knowing that link DE is \(\frac{1}{8} \mathrm{in} .\) thick and 1 in. wide, determine the normal stress in the central portion of that link when \(\text { (a) } \theta=0,(b) \theta=90^{\circ}\).

A \(\frac{5}{8}-i n .\)-diameter steel rod AB is fitted to a round hole near end C of the wooden member CD. For the loading shown, determine (a) the maximum average normal stress in the wood, (b) the distance b for which the average shearing stress is 100 psi on the surfaces indicated by the dashed lines, (c) the average bearing stress on the wood.

In the steel structure shown, a 6-mm-diameter pin is used at C and 10-mm-diameter pins are used at B and D. The ultimate shearing stress is 150 MPa at all connections, and the ultimate normal stress is 400 MPa in link BD. Knowing that a factor of safety of 3.0 is desired, determine the largest load P that can be applied at A. Note that link BD is not reinforced around the pin holes.

Member ABC, which is supported by a pin and bracket at C and a cable BD, was designed to support the 16-kN load P as shown. Knowing that the ultimate load for cable BD is 100 kN, determine the factor of safety with respect to cable failure.

A force P is applied as shown to a steel reinforcing bar that has been embedded in a block of concrete. Determine the smallest length L for which the full allowable normal stress in the bar can be developed. Express the result in terms of the diameter d of the bar, the allowable normal stress \(\sigma_{\text {all }}\) in the steel, and the average allowable bond stress \(\tau_{\mathrm{all}}\) between the concrete and the cylindrical surface of the bar. (Neglect the normal stresses between the concrete and the end of the bar.)

The two portions of member AB are glued together along a plane forming an angle \(\boldsymbol{\theta}\) with the horizontal. Knowing that the ultimate stress for the glued joint is 2.5 ksi in tension and 1.3 ksi in shear, determine (a) the value of \(\boldsymbol{\theta}\) for which the factor of safety of the member is maximum, (b) the corresponding value of the factor of safety. (Hint: Equate the expressions obtained for the factors of safety with respect to the normal and shearing stresses.)

The two portions of member AB are glued together along a plane forming an angle \(\boldsymbol{\theta}\) with the horizontal. Knowing that the ultimate stress for the glued joint is 2.5 ksi in tension and 1.3 ksi in shear, determine the range of values of \(\boldsymbol{\theta}\) for which the factor of safety of the members is at least 3.0.

The following problems are designed to be solved with a computer. A solid steel rod consisting of n cylindrical elements welded together is subjected to the loading shown. The diameter of element i is denoted by \(d_{i}\) and the load applied to its lower end by \(\mathbf{P}_{i}\), with the magnitude \(\mathbf{P}_{i}\) of this load being assumed positive if \(\mathbf{P}_{i}\) is directed downward as shown and negative otherwise. (a) Write a computer program that can be used with either SI or U.S. customary units to determine the average stress in each element of the rod. (b) Use this program to solve Probs. 1.1 and 1.3.

The following problems are designed to be solved with a computer. A 20-kN load is applied as shown to the horizontal member ABC. Member ABC has a 10 X 50-mm uniform rectangular cross section and is supported by four vertical links, each of 8 X 36-mm uniform rectangular cross section. Each of the four pins at A, B, C, and D has the same diameter d and is in double shear. (a) Write a computer program to calculate for values of d from 10 to 30 mm, using 1-mm increments, (i) the maximum value of the average normal stress in the links connecting pins B and D, (ii) the average normal stress in the links connecting pins C and E, (iii) the average shearing stress in pin B, (iv) the average shearing stress in pin C, (v) the average bearing stress at B in member ABC, and (vi) the average bearing stress at C in member ABC. (b) Check your program by comparing the values obtained for d = 16 mm with the answers given for Probs. 1.7 and 1.27. (c) Use this program to find the permissible values of the diameter d of the pins, knowing that the allowable values of the normal, shearing, and bearing stresses for the steel used are, respectively, 150 MPa, 90 MPa, and 230 MPa. (d) Solve part c, assuming that the thickness of member ABC has been reduced from 10 to 8 mm.

The following problems are designed to be solved with a computer. Two horizontal 5-kip forces are applied to pin B of the assembly shown. Each of the three pins at A, B, and C has the same diameter d and is in double shear. (a) Write a computer program to calculate for values of d from 0.50 to 1.50 in., using 0.05-in. increments, (i) the maximum value of the average normal stress in member AB, (ii) the average normal stress in member BC, (iii) the average shearing stress in pin A, (iv) the average shearing stress in pin C, (v) the average bearing stress at A in member AB, (vi) the average bearing stress at C in member BC, and (vii) the average bearing stress at B in member BC. (b) Check your program by comparing the values obtained for d = 0.8 in. with the answers given for Probs. 1.60 and 1.61. (c) Use this program to find the permissible values of the diameter d of the pins, knowing that the allowable values of the normal, shearing, and bearing stresses for the steel used are, respectively, 22 ksi, 13 ksi, and 36 ksi. (d) Solve part c, assuming that a new design is being investigated in which the thickness and width of the two members are changed, respectively, from 0.5 to 0.3 in. and from 1.8 to 2.4 in.

The following problems are designed to be solved with a computer. A 4-kip force P forming an angle \(\alpha\) with the vertical is applied as shown to member ABC, which is supported by a pin and bracket at C and by a cable BD forming an angle \(\beta\) with the horizontal. (a) Knowing that the ultimate load of the cable is 25 kips, write a computer program to construct a table of the values of the factor of safety of the cable for values of \(\alpha\) and \(\beta\) from 0 to \(45^{\circ}\), using increments in \(\alpha\) and \(\beta\) corresponding to 0.1 increments in tan \(\alpha\) and tan \(\beta\). (b) Check that for any given value of \(\alpha\), the maximum value of the factor of safety is obtained for \(\beta=38.66^{\circ}\) and explain why. (c) Determine the smallest possible value of the factor of safety for \(\beta=38.66^{\circ}\), as well as the corresponding value of \(\apha\), and explain the result obtained.

The following problems are designed to be solved with a computer. A load P is supported as shown by two wooden members of uniform rectangular cross section that are joined by a simple glued scarf splice. (a) Denoting by \(\sigma_{U}\) and \(\tau_{U}\), respectively, the ultimate strength of the joint in tension and in shear, write a computer program which, for given values of a, b, P, \(\sigma_{U}\) and \(\tau_{U}\), expressed in either SI or U.S. customary units, and for values of \(\alpha\) from 5 to \(85^{\circ}\) at \(5^{\circ}\) intervals, can calculate (i) the normal stress in the joint, (ii) the shearing stress in the joint, (iii) the factor of safety relative to failure in tension, (iv) the factor of safety relative to failure in shear, and (v) the overall factor of safety for the glued joint. (b) Apply this program, using the dimensions and loading of the members of Probs. 1.29 and 1.31, knowing that \(\sigma_{U}=150 \mathrm{psi} \text { and } \tau_{U}=214 \mathrm{psi}\) for the glue used in Prob. 1.29 and that \(\sigma_{U}=1.26 \mathrm{MPa} \text { and } \tau_{U}=1.50 \mathrm{MPa}\) for the glue used in Prob. 1.31. (c) Verify in each of these two cases that the shearing stress is maximum for \(\alpha=45^{\circ}\).

The following problems are designed to be solved with a computer. Member ABC is supported by a pin and bracket at A, and by two links that are pin-connected to the member at B and to a fixed support at D. (a) Write a computer program to calculate the allowable load \(P_{\text {all }}\) for any given values of (i) the diameter \(d_{1}\) of the pin at A, (ii) the common diameter \(d_{2}\) of the pins at B and D, (iii) the ultimate normal stress \(\sigma_{U}\) in each of the two links, (iv) the ultimate shearing stress \(\tau_{U}\) in each of the three pins, and (v) the desired overall factor of safety F.S. (b) Your program should also indicate which of the following three stresses is critical: the normal stress in the links, the shearing stress in the pin at A, or the shearing stress in the pins at B and D. (c) Check your program by using the data of Probs. 1.55 and 1.56, respectively, and comparing the answers obtained for \(P_{\text {all }}\) with those given in the text. (d ) Use your program to determine the allowable load \(P_{\text {all }}\), as well as which of the stresses is critical, when \(d_{1}=d_{2}=15 \mathrm{~mm}, \sigma_{U}=110 \mathrm{MPa}\) for aluminum links, \(\tau_{U}=100 \mathrm{MPa}\) for steel pins, and \(F S .=3.2\).