Strength of materials: Solved exercises

In the most varied mechanical projects, the calculation of the strength of materials is practically an indispensable item. Whether dimensioning an axis or structure, resistance calculations are key! Check out some examples of solved calculations below.. A good guide to basic strength applications.

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Trouble	Figure	Statement
01.016		Determine normal strength, the shear force and moment in the section passing through point C. Use P = 8 kN.
01.033		The column is subjected to an axial force of 8 kN on its top. Assuming the cross section has the dimensions shown in the figure, determine the average normal voltage acting on section a-a. Show this stress distribution acting on the cross-sectional area.
01.036		the lamp of 50 lbf is supported by two steel rods coupled by an A-ring. Determine which of the rods is subject to the highest average normal stress and calculate its value. suppose that ? = 60º. The diameter of each rod is given in the figure..
01.037		the lamp of 50 lbf is supported by two steel rods coupled by an A-ring. Determine which of the rods is subject to the highest average normal stress and calculate its value. suppose that ? = 45º. The diameter of each rod is given in the figure..
01.038		the lamp of 50 lbf is supported by two steel rods coupled by an A-ring. Determine the angle of the orientation of ? of AC, such that the mean normal voltage at rod AC is twice the mean normal voltage at rod AB. What is the intensity of this tension on each rod? The diameter of each rod is shown in the figure..
01.053		The plastic block is subjected to an axial compressive force of 600 N. Assuming the top and bottom covers distribute the load evenly across the block, determine the average normal and shear stresses along section a-a.
01.056		The joint is subjected to the force of 6 kip do elemento axial. Determine the average normal voltage acting on sections AB and BC. Assume the element is flat and has 1,5 inch thick.
01.060		The truss bars have a cross-sectional area of 1,25 pol2. Determine the average normal voltage in each element due to load P = 8 kip. Indicate whether the tension is tension or compression.
01.061		The truss bars have a cross-sectional area of 1,25 pol2. Assuming that the maximum average normal voltage at each bar does not exceed 20 ksi, determine the maximum magnitude P of the loads applied to the truss.
01.079		the eye (figure on the side) is used to support a load of 5 kip. Determine its diameter d, approx. 1/8 pol, and the required thickness h, so that the washer does not penetrate or shear the support. The normal allowable tension of the bolt is ?adm = 21 ksi, and the allowable shear stress of the support material is ?adm = 5 ksi.
01.080		The lap joint of the wooden element A of a truss is subjected to a compressive force of 5 kN. Determine the required diameter d of steel rod C and height h of element B if the allowable normal stress of the steel is (?adm)steel = 157 MPa and the allowable normal stress of wood is (?adm)mad = 2 MPa. Element B has 50 mm thick.
01.112		The two aluminum rods support the vertical load P = 20 kN. Determine their required diameters if the allowable tensile stress for aluminum is ?adm = 150 MPa.
02.005		The rigid beam is supported by an A-pin and BD and CE wires. If the maximum allowable normal deformation in each wire is ?max = 0,002 mm/mm, what will be the maximum vertical displacement caused by the load P on the wires?
02.008		Two Bars are used to support a load. Without her, the length of AB is 5 pol, AC's is 8 pol, and the A-ring has coordinates (0,0). If the charge P acts on the ring at A, the normal deformation in AB becomes ?AB = 0,02 in/in and the normal deformation in AC becomes ?AC = 0,035 half / half. Determine the position coordinates of the ring due to load.
02.009		Two bars are used to support a P load.. Without her, the length of AB is 5 pol, AC's is 8 pol, and the A-ring has coordinates (0,0). If a load P is applied to the ring at A, so that it moves to the coordinate position (0,25 pol, 0,73 pol), what will be the normal deformation in each bar?
02.013		The rectangular plate is subjected to the deformation shown by the dashed line. Determine the mean shear deformation ?xy of the plate.
02.015		The rectangular plate is subjected to the deformation shown by the dashed line. Determine normal deformations ?x, ?Y, ?x’, ?Y'.
02.017		The plastic part is originally rectangular. Determine shear deformation ?xy at corners A and B if the plastic distorts as shown by the dashed lines.
02.018		The plastic part is originally rectangular. Determine shear deformation ?xy at corners D and C if the plastic distorts as shown by the dashed lines.
02.019		The plastic part is originally rectangular. Determine the mean normal deformation that occurs along diagonals AC and DB.
02.024		the square deforms, going to the position shown by the dashed lines. Determine the shear deformation in each of corners A and C. The DB side remains horizontal.
02.025		The block is deformed, going to the position shown by the dashed lines. Determine the mean normal deformation along straight AB.
02.028		The AB elastic has an unstretched length of 1 late. If it is attached at B and attached to the surface at point A’, determine the average normal deformation of the elastic. The surface is defined by the function y=(x2) late, where x is given standing.
03.002		Data for a stress-strain test of a ceramic are given in the table. The curve is linear between the origin and the first point. Build the diagram and determine the modulus of elasticity and modulus of resilience.
03.003		Data for a stress-strain test of a ceramic are given in the table. The curve is linear between the origin and the first point. Construct the diagram and determine the approximate modulus of toughness if the breaking strength is 53,4 ksi.
03.004		Data for a stress-strain test of a ceramic are given in the table. The curve is linear between the origin and the first point. Build the diagram and determine the modulus of elasticity and modulus of resilience.
03.018		AB and AC steel wires support the mass of 200 kg. Assuming the normal allowable voltage for them is ?adm = 130 MPa, determine the required diameter for each wire. Furthermore, what will be the new length of wire AB after the load is applied? Assume the undeformation length of AB to be 750 mm. Eaço = 200 GPa.
03.024		The plastic rod is made of Kevlar 49 and has a diameter of 10 mm. Assuming an axial load of 80 kN, determine changes in its length and diameter.
04.006		The set consists of an A-36 steel CB rod and a 6061-T6 aluminum BA rod., each with a diameter of 1 pol. If the rod is subjected to an axial load P1 = 12 kip in A and P2 = 18 kip on connection B, determine the displacement of the connection and end A. The length of each segment without elongation is shown in the figure.. Disregard the size of the connections at B and C and assume they are rigid.
04.007		The set consists of an A-36 steel CB rod and a 6061-T6 aluminum BA rod., each with a diameter of 1 pol. Determine the applied loads P1 and P2 if A moves 0,08 in to the right and B moves 0,02 inch to the left when loads are applied. The length of each segment without elongation is shown in the figure.. Disregard the size of the connections at B and C and assume they are rigid.
04.042		The concrete column is reinforced with four steel bars, each with a diameter of 18 mm. Determine the average stress of concrete and steel if the column is subjected to an axial load of 800 kN. Eaço = 200 GPa e Ec = 25 GPa.
04.043		The column shown in the figure is made of high strength concrete (Ec=29 GPa) and four A36 steel reinforcing bars. If the column is subjected to an axial load of 800 kN, determine the diameter required for each bar so that one-quarter of the load is supported by steel and three-quarters by concrete.
05.001		One shaft is made of alloy steel with allowable shear stress of ?adm = 12 ksi. Assuming the shaft diameter is 1,5 pol, determine the maximum torque T that can be transmitted. What would the maximum torque T’ be if a hole were drilled 1 inch in diameter along axis? Plot the shear-stress distribution along a radial straight line in each case.
05.005		the massive axis of 30 mm diameter is used to transmit the torques applied to the gears.. Determine the shear stress developed at points C and D of the shaft. Indicate the shear stress on the volume elements located at these points.
05.006		The assembly consists of two segments of galvanized steel pipe coupled by a reduction in B. The smaller tube has an external diameter of 0,75 inch and inner diameter of 0,68 pol, while the larger tube has an external diameter of 1 inch and inner diameter of 0,86 pol. Assuming the tube is securely attached to the wall at C, determine the maximum shear stress developed in each pipe section when the conjugate shown is applied to the switch handle.
05.010		The solid shaft has a diameter of 0,75 pol. Assuming it is subjected to the torques shown, determine the maximum shear stress developed in the CD and EF regions. Bearings in A and F allow free rotation of the shaft.
05.025		The gear motor develops 1/10 hp when running a 300 rev / min. Assuming the shaft has a diameter of ½ in., determine the maximum shear stress developed therein.
05.026		The gear motor develops 1/10 hp when running a 300 rev / min. Assuming that the allowable shear stress for the shaft is ?adm = 4 ksi, determine the smallest shaft diameter that can be used to approximate 1/8 pol.
05.030		The pump operates with an engine that has power of 85 W. Assuming the impeller at B is turning a 150 rev / min, determine the maximum shear stress developed in A, located on the transmission shaft that has 20 mm in diameter.
05.031		A steel tube with an outside diameter of d1 = 2,5 pole transmits 35 hp when running a 2700 rev / min. Determine the inner diameter d2 of the tube, approx. 1/8 pol, if the allowable shear stress is ?max = 10 ksi.
05.043		A shaft is subjected to a torque T. Compare the effectiveness of the tube shown in the figure with that of a solid-section shaft of radius c. For this, calculate the percentage increase in torsional stress and torsional angle per unit of tube length relative to the values ​​of the solid section axis.
05.046		The A-36 steel shaft is made up of AB and CD tubes and a solid BC part.. It rests on plain bearings that allow it to rotate freely. If the ends are subject to torques of 85 N.m, what is the torsional angle of end A relative to end D? The tubes have an outside diameter of 30 mm and inner diameter of 20 mm. The solid part has a diameter of 40 mm.
05.047		The A-36 steel shaft is made up of AB and CD tubes and a solid BC part.. It rests on plain bearings that allow it to rotate freely. If ends A and D are subject to torques of 85 N.m, what is the torsional angle of the end B of the solid part in relation to the end C? The tubes have an outside diameter of 30 mm and inner diameter of 20 mm. The solid part has a diameter of 40 mm.
05.049		Gears coupled to the ASTM-304 stainless steel shaft are subjected to the torques shown. Determine the torsional angle of gear C relative to gear B. The shaft has a diameter of 1,5 pol.
05.054		The A-36 steel shaft has 3 m in length and outside diameter of 50 mm. Requires you to broadcast 35 kW of power from engine E to generator G. Determine the smallest angular velocity that the shaft can have if the maximum allowable twist is 1°. Adopt the transverse modulus of elasticity equal to 75 GPa.
05.058		Both axles are made of A-36 steel. Each has a diameter of 1 pol, and they are supported by bearings in A, B e C, which allows free rotation. Assuming support D is fixed, determine the torsional angle of end B when torques are applied to the assembly as shown.
05.059		Both axles are made of A-36 steel. Each has a diameter of 1 pol, and they are supported by bearings in A, B e C, which allows free rotation. Assuming support D is fixed, determine the torsional angle of end A when torques are applied to the assembly as shown.
06.001		Draw the shear force and moment diagrams for the shaft. Bearings in A and B exert only vertical reactions on the shaft.
06.002		The axle is subjected to the loads caused by the belts that pass over the two pulleys. Draw the shear force and moment diagrams. Bearings in A and B exert only vertical reactions on the shaft.
06.003		The three traffic lights have, each one, mass of 10 kg and the overhanging tube AB has a mass of 1,5 kg/m. Draw the shear force and moment diagrams for the tube. Dismiss the plate mass.
06.005		The reinforced concrete abutment is used to support the spars of a bridge platform. Draw your shear force and moment diagrams when it is subjected to the stringer loads shown. Assume that columns A and B exert only vertical reactions on the encounter.
06.006		Draw the shear force and moment diagrams for the shaft. The bearings in A and B only exert vertical reactions on it. Also express shear force and moment as a function of x in the region 125 mm < x < 725 mm.
06.032		Draw the shear force and moment diagrams for the wooden beam and determine the shear force and moment across the beam as a function of x.
06.042		Two solutions were proposed for the design of a beam. Determine which one will withstand an M = moment 150 kN.m with the lowest normal bending stress. what is this tension? With what percentage is it more efficient?
06.047		The aluminum machine part is subject to a moment M = 75 N.m. Determine the normal bending stress at points B and C of the cross section. Plot the results on a volume element located at each of these points.
06.048		The aluminum machine part is subject to a moment M = 75 N.m. Determine the maximum normal bending tension and compression stresses in the part.
06.055		The beam is subject to a moment of 15 chicken.pes. Determine the net force that the stress produces on the upper A and lower B flanges. Also calculate the maximum stress developed in the beam.
06.068		The cross section of a beam is subject to a moment of 12 kip . feet. Determine the net force that tension produces on the table (6 pol × 1 pol). Also calculate the maximum stress developed in this cross section of the beam.
06.071		Determine the normal absolute maximum bending stress on the axis of 30 mm in diameter that is subjected to concentrated forces. Bushings on A and B supports only withstand vertical forces.
06.072		Determine the smallest allowable diameter of the shaft subjected to concentrated forces. The bushings on supports A and B support only vertical forces and the allowable bending stress is ?adm = 160 MPa.
06.073		The beam has rectangular cross section as shown.. Determine the greatest load P that can be supported at its overhanging ends, so that the normal bending stress in the beam does not exceed ?adm = 10MPa.
06.077		The beam is subjected to the loading shown. Determine the required cross-sectional dimension a if the bending stress of the material is ?adm = 150 MPa.
06.079		Determine the intensity of the maximum load P that can be applied to the beam, assuming it is made of material with allowable bending stress (?adm)c = 16 ksi in compression and (?adm)t = 18 ksi in traction.
07.005		If the T beam is subjected to vertical shear V = 10 kip, what will be the maximum shear stress developed in it? Also calculate the shear stress jump at the AB tab-core junction. Draw the shear stress intensity variation across the cross section. Show that IEN = 532.04 in. 4.
07.015		Determine the maximum shear stress on the shaft with circular cross section of radius r and subject to shear force V. Express the answer in terms of cross-sectional area A.
07.017		Determine the greatest forces P at the ends that the element can withstand, assuming that the allowable shear stress is ?adm = 10 ksi. The supports at A and B exert only vertical reactions on the beam.
07.021		The supports at A and B exert vertical reactions on the wooden beam. Assuming that the allowable shear stress is ?adm = 400 psi, determine the intensity of the largest distributed load w that can be applied on the beam.
12.005		Determine the elastic line equations of the beam using coordinates x1 and x2. Specify slope at A and maximum deflection. Consider constant EI.
12.030		The shaft supports the loads of the three pulleys shown.. Determine the deflection at its center and its slope at A and B. The bearings only exert vertical reactions on it and EI is constant.
12.049		The rod consists of two axes for which the moment of inertia of AB is I and of BC is 2I. Determine the maximum slope and deflection of the rod due to loading. The modulus of elasticity is E.
13.005		The plane link is made of A-36 steel. (E = 29000 hp). Determine the smallest diameter of the rod, approx. 1/16 pol, that will bear the load of 4 kip unbuckled. The ends are secured by pins.
13.016		The L-2 tool steel link used in a forging machine is attached to the forks by pins at the ends.. Determine the maximum load P it can support without buckling. Use a safety factor for F.S buckling. = 1,75. Observe, in the left figure, that the ends are pinned for buckling and, on the right, that the ends are set.