UNIT - 1
SIMPLE STRESSES & STRAIN
1. A load of 5 KN is to be raised with the help of a steel wire. Find the diameter of steel wire, if the maximum stress is not to exceed 100 MN/m2.[ 7.28 mm]
2. A circular rod of diameter 20 m and 500 m long is subjected to tensile force of 45kN. The modulus of elasticity for steel may be taken as 200 KN/m2. Find stress, strain and elongation of bar due to applied load.[ 143.24 N/m2, 0.000716, 0.36]
3. Find the minimum diameter of the steel wire, which is used to raise a load of 4000N if the stress in the rod is not to exceed 95 MN/m2 . [d=7.32 mm]
4. Find the young modulus of a brass rod of diameter 25mm and of length 250mm which is subjected to a tensile load of 50kN when the extension of the rod is equal to 0.3mm. [E = 84.88 X 109N/m2]
5. A hollow cast-iron cylinder 4 m long, 300 mm outer diameter, and thickness of metal 50 mm is subjected to a central load on the top when standing straight. The stress produced is 75000 kN/m2. Assume Young's modulus for cast iron as 1.5 x 108 KN/m2 find
a. magnitude of the load, b. longitudinal strain produced and
C. Total decrease in length. [P = 2945.2 kN, 0.0005, 2 mm]
6. A steel wire 2 m long and 3 mm in diameter is extended by 0.75 mm when a weight P is suspended from the wire. If the same weight is suspended from a brass wire, 2.5 m long and 2 mm in diameter, it is elongated by 4.64 mm. determine the modulus of elasticity of brass if that of steel be 2.0 x 105 N/mm2. [Eb = 0.909 × 105 N/mm2]
7. The following data was recorded during tensile test made on a standard tensile test specimen:
Original diameter and gauge length =25 mm and 80 mm;
Minimum diameter at fracture =15 mm;
Distance between gauge points at fracture = 95 mm;
Load at yield point and at fracture = 50 kN and 65 kN;
Maximum load that specimen could take = 86 kN.
Make calculations for
(a) Yield strength, ultimate tensile strength and breaking strength
(b) Percentage elongation and percentage reduction in area after fracture
(c) Nominal and true stress and fracture.
[2.09 × 108, 12.8 × 104 kN/m2, 19.2 × 104 kN/m2,24.5%]
8. A conical bar tapers uniformly from a diameter of 4 cm to 1.5 cm in a length of 40 cm. If an axial force of 80 kN is applied at each end, determine the elongation of the bar. Take E = 2 × 105 N/mm2 . [0.3397]
9. A steel bar AB of uniform thickness 2 cm, tapers uniformly from 1.5 cm to 7.5 cm in a length of 50 cm. From first principles determine the elongation of plate; if an, axial tensile force of 100 kN is applied on it. [E = 2 x 105 N/mm2] [0.1386 mm]
10. A vertical bar fixed at the upper end, and of uniform strength carries an axial load of 12KN. The bar is 2.4m long having a weight per unit volume of 0.0001N/mm3. If the area of the bar at the lower end is 520mm2, find the area of the bar at the upper end. [525.44mm2]
11. A vertical rod of 4 m long is rigidly fixed at upper end and carries an axial tensile load of 50kN force. Calculate total extension of the bar if the rod topers uniformly from a diameter of 50 mm at top to 30 mm at bottom. Take density of material as 1 × 105 kg/m3 and E = 210 GN/m2.[ 0.8362mm]
12. An aerial copper wire (E =1 x 105 N/mm2) 40 m long has cross sectional area of 80 mm2 and weighs 0.6 N per meter run. If the wire is suspended vertically, calculate
(a) The elongation of wire due to self weight,
(b) The total elongation when a weight of 200 N is attached to its lower end, and
(c) The maximum weight which this wire can support at its lower end if the limiting value of stress is 65 N/mm2.
13. The bar shown in Fig is subjected to an axial pull of 150kN. Determine diameter of the middle portion if stress there is limited to 125N/mm2. Proceed to determine the length of this middle portion if total extension of the bar is specified as 0.15 mm. Take modulus of elasticity of bar material E = 2 × 105 N/mm2. [d2 = 39.1mm, L2 = 145.58 mm]
COMPOSITE BARS &THERMAL STRESS
Q.1: A beam weighing 50 N is held in horizontal position by three wires. The outer wires are of brass of 1.8 mm diameter and attached to each end of the beam. The central wire is of steel of 0.9 mm diameter and attached to the middle of the beam. The beam is rigid and the wires are of the same length and unstressed before the beam is attached. Determine the stress induced in each of the wire. Take E for brass as 80 GN/m2 and for steel as 200 GN/m2.[ σb = 7.49 N/mm2, σs = 18.5 N/mm2 ]
Q.2: Two copper rods and one steel rod lie in a vertical plane and together support a load of 50kN as shown in Figure. Each rod is 25 mm in diameter, length of steel rod is 3 m and length of each copper rod is 2m. If modulus of elasticity of steel is twice that of copper, make calculations for the stress induced in each rod. It may be presumed that each rod deforms by the same amount. [σc = 30.60 N/mm2, σs = 40.7 N/mm2]
3. Steel rod of length 15 m is at a temperature of 15˚C. Find the free expansion of length when the temperature is raised to 65˚C. Also find the temperature stress produced when (a) the expansion of rod is prevented (b) Rod is permitted to expand by 6mm. take E = 200 GPa, α = 12 X 10-6/˚C.
4. A copper plate measuring 60 X 30 mm is brazed to another 60X60 mm steel plate. If the combination is heated through 120˚C. Find (a) Stress produced (b) Shear force which tends to rupture the brazing (c) shear stress. Take αc =18.5 X 10-6 /˚C, αs =12 X 10-6 /˚C, Ec = 110Gpa, Es = 220 GPa.
5. A steel bar is placed between two copper bar each having the same area and length as steel bar at 20˚C. At this stage, they are rigidly connected together at both the ends when the temperature is raised to 320˚C, the length of the bar increases by 1.5mm. Determine the original length and final stresses in the bar. . Take αc =17.5 X 10-6 /˚C, αs =12 X 10-6 /˚C, Ec = 110Gpa, Es = 220 GPa.
6. A copper bar of 50mm diameter is placed within a steel tube of external diameter 75mm and internal diameter 50 mm of exactly the same length. The two pieces are rigidly fixed together by two pins 18 mm in diameter, one at each end passing through the bar and tube. Calculate the stresses induced in the copper bar, steel tube and pins if the temperature is raised to 50˚C. Take αc =17 X 10-6 /˚C, αs =11.5 X 10-6 /˚C, Ec = 105Gpa, Es = 210 GPa.
7. A composite bar of aluminum and steel is held between two supports. The bars are stress free at 40˚C. what will be the stresses in the two bar when the temperature is 20˚C if (a) support are non yielding (b) support came nearer to each other by 0.1 mm. it can be assumed that the change in temperature is uniform all along the length of bar. Take αc =23.5 X 10-6 /˚C, αs =11.7 X 10-6 /˚C, Ec = 74Gpa, Es = 210 GPa.
8. The steel bar of uniformly varying diameter is held between two unyielding supports at room temperature. What is the maximum stress induced in the bar if temperature rise by 30˚C. Take αs =12 X 10-6 /˚C, Es = 110Gpa.
9. In a laboratory, tensile test is conducted and young modulus of material is found to be 210GPa. On the same material torsion test is conducted and modulus of rigidity is found to be 78GPa. Find poission’s ratio and Bulk Modulus.
10. A rectangular bar of cross section 30 X 60 mm and length 200mm is restrained from expansion along its 30X200 mm sides by surrounding material. Find the change in dimensions and volume when a compressive force of 180 KN acts in axial direction. E = 200GPa, μ = 0.3. What are the changes if surrounding material can restrain only 50% of expansion on 30 X 200 mm side.
11. In a tension test specimen of diameter 25mm, 200mm gauge length stretched over 0.0975 mm under tensile load of 50KN. In a torsion test the same specimen twisted 0.025 rad over a length of 200mm when a torque of 0.4 KN-m was applied. Determine Poisson ratio and E.
12. A concrete cylinder of 50mm diameter and length 300 mm when subjected to axial compression load of 240KN resulted an increase of diameter by 0.127 mm and a decrease in length by 0.28 mm. calculate E and μ.
UNIT-II PRINCIPAL STRESS AND STRAIN
1. A rectangular bar of cross sectional area 10000 mm2 is subjected to an axial load of 20KN. Determine the normal and shear stresses on a section which is inclined at an angle of 30 deg with the normal cross section of the bar.
2. Find the diameter of a circular bar which is subjected to an axial pull of 160 kN, if the maximum allowable shear stress on any section id 65 N/mm2.
3. A rectangular bar of cross sectional area of 11000 mm2 is subjected to a tensile load P. the permissible shear and normal stresses on the oblique plane at 60 deg are given as 7 N/mm2 and 3.5 N/mm2 respectively. Determine the safe value of P.
4. Write short notes on Mohr’s Circle.
5. At a point in a stressed body, the principal stresses are 100MN/m2 (tensile) and 60MN/m2 (Compressive). Determine the normal stress and shear stress on a plane inclined at 50˚ to the axis of major principal stress. Also calculate maximum shear stress at the point. [Normal stress=33.9N.mm2, Shear stress=78.78N/mm2, Maximum Shear stress= 80N/mm2]
6. Direct stresses of 160N/mm2 tensile and 120 N/mm2 compressive exist on two perpendicular plate at a certain point in a body. They are also accompanied by shear stresses on these planes. The greatest principal stresses at the point due to this is 200N/mm2. Find
a. The magnitude of shearing stresses on the two planes.
b. The maximum shearing stresses at that point and location. [τxy = 113.14 N/mm2, θ=65˚, τmax = 180 N/mm2]
7. A rectangular block of material is subjected to a tensile stress of 100 N/mm2 on one plane and a compressive stress of 50 N/mm2 on a plane at right angles, together with the shear stresses of 60 N/mm2 on the faces. Find (a). The direction of principal stress (b). The magnitude of principal stress (c). The magnitude of greatest shear stresses (d). The location of planes containing maximum shear stresses. [θ=19˚ & 109˚, σ1 = 121 N/mm2, σ2 = 71 N/mm2, τmax = 96N/mm2, θ=25˚, 115˚]
8. At a point in a strained material, the principal tensile stress across two perpendicular planes is 80N/mm2 and 40 N/mm2. Determine normal stress, shear stress and resultant stress on a plane inclined at 20˚ with the major principal plane . Determine also the obliquity. What will be the intensity of stress which acting alone will produce the same maximum strain if Poisson’s ratio is ¼. [σn = 75.32 N/mm2, σr = 12.86 N/mm2, τ = 12.86N/mm2, σ = 70 N/mm2]
UNIT-IV: BENDING STRESS
Q.1: A simply supported cast iron square beam of 800-mm length and 15mm X 15mm in section fails on applying a load of 360 N at the mid span. Find the maximum UDL that can be applied safely to a 40 mm wide, 75 mm deep, and 1.6 mm long cantilever made of the same material. [3750 N/m]
Q.2: A timber beam of rectangular section of length 8m is simply supported. The beam carries a U.D.L. of 12kN/m run over the entire length and a point load of 10kN at 3m from the left support. If the depth is two times the width and the stress in the timber is not to exceed 8N/mm2, find the suitable dimension of the section. [b=275.5mm, d=551mm]
Q.3: Three beams have the same length, same allowable bending stress and are subjected to the same maximum BM. The cross section of beams is a circle, a square and a rectangle with depth twice the width. Find the ratio of weight of the circular and rectangular beams with respect to the square beam.
Q.4: A beam of symmetrical section, 400 mm deep and MI = 12X107 mm4, carries UDL of 10KN/m. Find maximum span of beam if the maximum stress is not to exceed 160N/mm2. With the same span, calculate maximum central point load if maximum stress is not to exceed as given above.
Q.5: A hollow circular bar used as a beam has its outside diameter thrice the inside diameter. It is subjected to a maximum bending moment of 60 KN-m. Determine the inside diameter of the beam if the permissible bending stress is limited to 120 MPa. [57.6 mm]
Q.6: A symmetrical section, 200 mm deep has a MI= 2.26X10-5 m4 about its neutral axis. Determine the longest span over which, when simply supported the beam would carry an UDL of 4 KN/m run without bending stress exceeding 125MN/m2. [7.76 m]
Q.7: A floor carries a load of 8 kN/m2 and is supported by joists that are 120 mm wide and 240 mm deep over a span of 6 m. determine the spacing centre to centre of the joists if the maximum allowable bending stress is 10 MPa. [320 mm]
Q.8: A flitched beam consists of a wooden joists 12 cm wide and 20 cm deep strengthened by a steel plate 1 cm thick and 18 cm deep, one on either side of the joists. If the stresses in wood and steel are not to exceed 7.5 MN/m2 and 127.5 MN/m2, find the moment of resistance of the beam. Take the modulus of elasticity of steel equal to twenty times that of wood. [19436.4 N-m]
Q.9: An I-Section beam has Flanges – 150 mm X 20 mm and Web – 300 mm X 10 mm. the maximum shear stress developed in the beam is 16.8 N/mm2. Find the shear force to which the beam is subjected. [50 kN]
Q.10: A beam of rectangular section is 100 mm wide and 200 mm deep. If the section is subjected to a maximum shear force of 10 KN, find the maximum shear stress. Also draw the shear stress distribution across the section. [0.75 N/mm2]
Q.11: The tension flange of cast iron I-section beam is 100 mm wide and 20 mm deep, the compression flange is 50 mm wide and 20 mm deep whereas the web is 100 mm X 20 mm. if a similar I-Section is welded on the top of it to form a symmetrical section, find the ratio of the moment of resistance of this section to that of the previous section assuming the allowable stress in tension and compression to the same. [4.495]
Q.12: A beam of rectangular section is 100 mm wide and 200 mm deep. If the section is subjected to a maximum shear force of 10 KN, find the maximum shear stress. Also draw the shear stress distribution across the section. [0.75 N/mm2]
Q.13: An I-Section beam has Flanges – 150 mm X 20 mm and Web – 300 mm X 10 mm. the maximum shear stress developed in the beam is 16.8 N/mm2. Find the shear force to which the beam is subjected. [50 kN]
Q.14: A cantilever is constructed from metal strip 25 mm deep throughout its length of 750 mm. Its width, however, varies uniformly from zero at the free end to 50 mm at the support. Determine the deflection of the free end of the cantilever if it carries uniformly distributed load of 300 N/m across its length. E = 200 GN/m2. [1.2 mm.]
SLOPE AND DEFLECTION OF BEAMS
Q.1: A cantilever of square section 200 mm X 200 mm, 2 m long, just fails in flexure when a load of 12 kN is placed at its free end. A beam of same material and having cross section 150 mm wide and 300 mm deep is simply supported over a span of 3 meter. Calculate the minimum concentrated load required to fail the beam. [54 kN]
Q.2: A cantilever beam 1.5 m long carries UDL over the entire length, find the deflection at the free end if the slope at the free end is 1.5˚. [RK-YB = 29.45 mm]
Q.3: A 2 m long cantilever made of steel tube of section 150 mm external diameter and 10 mm thick is loaded with W at end and 2W at 0.5 m left of end. If E = 200 GN/m2. (a) Calculate the value of W so that the maximum bending stress is 150 MN/m2. (b) The maximum deflection for the loading. [W= 4320 N]
Q.4: A fixed beam of 6 m span carries UDL of 5 KN/m run. Find the maximum deflection of the beam. Take E = 2 X 108 kN/m2 and I = 0.48 X 10-4 m4. [-1.76 mm]
Q.5: A 250 mm long cantilever beam of rectangular section 40 mm wide and 30 mm deep carries an UDL. Calculate the value of W if the the maximum deflection of cantilever is not to exceed 0.5 mm. Take E = 70 GN/m2 . [6.45kN/m]
Q.6: A steel girder of 6 m length acting as a beam carries a UDL load w N/m run throughout its length. If I = 30 X 10-6 m4 and depth 270 mm, calculate; (i) The magnitude of w so that the maximum stress developed in the beam section does not exceed 72 MN/m2. (ii) The slope and deflection under this load in the beam at a distance of 1.8 m from one end. Take E = 200 GN/m2 . [3555 N/m, θc = -0.173˚, Downward deflection = 8.13 mm]
Q.7: When a load of 20 kN is gradually applied at a certain point on a beam it produces a deflection of 13 mm and a maximum bending stress of 75 MN/m2. From what height can a load of 5 kN fall on to the beam at this point if the maximum bending stress is to be 150 MN/m2 ? [78 mm.]
ASSIGNMENT – V: TORSION OF SHAFT
Q.1: Derive the Torsion equation.
Q.2: 2250 kW power is to be transmitted at 1Hz speed. If the permissible shear stress is 80 N/mm2, determine the necessary diameter for a solid shaft of circular section. If a hollow circular section is used of internal diameter equal to ¾ of external diameter, calculate the saving in mass per meter length of shaft. [D=284mm, Matr. Saving=43.76%]
Q.3: A solid shaft of 200 mm diameter is proposed to be replaced by a hollow shaft of external diameter two times the internal diameter. If, the same power is to be transmitted at the same speed and at the same level of shear stress. Find the size of hollow shaft. [d=102.17mm & D=204.34mm]
Q.4: A maximum shear stress of 400 MPa is induced in a hollow shaft of 100 mm and 70 mm external and internal diameter respectively. What maximum shear stress will be developed in a solid shaft of the same weight, material and length subjected to same torque? [τs =835 MPa]
Q.5: A shaft 1.2 m long tapers uniformly from a radius of 40 mm to a radius of 60 mm. If the shaft transmits a torque 12 kN-m find (i) Angle of twist (ii) Maximum shear stress developed . Take C = 80 GN/m2.
Q.6: A hollow shaft subjected to pure torque attains a maximum shear stress τ. If the strain energy per unit volume is τ2/3c. Calculate the ratio of shaft.
Q.7: What must be length of 5 mm diameter Al, wire so that it can be twisted one complete revolution without exceeding shearing stress of 42 MPa. Take C = 27 GPa.
Q.8: A solid steel shaft has to transmit 75 KW at 200 rpm taking allowable shear stress 70 MPa find suitable diameter of shaft if the maximum torque transmitted on each revolution exceeds the mean torque by 30%. (69.7 mm)
Q.9: A hollow circular shaft is to transmit 300 kw at 80 rpm if the shear stress is not to exceed 60 MN/m2 and internal diameter 0.6 of external diameter. Find the external and internal diameter assuming that the maximum torque is 1.4 times the mean.
Q.10: a hollow circular shaft 20 mm thick transmits 294 KW at 200 rpm determines the diameter of shaft if shear strain due to torsion is not to exceed 8.6 X 10-4. Take C = 80 GPa.
UNIT – V COLUMN & STRUT
Q.1: State Euler’s theory for long columns. State and derive Euler’s Formula
Pcr = 𝜋2EI/ l2
Q.2: Discuss Rankine’s hypothesis of critical load for columns. How does it differ from Euler’s approach?
Q.3: Calculate the maximum value of the slenderness ratio of a mild steel column for which Euler’s formula is valid
300MN/ m2 E= 210 GN/ m2
Q.4: A material subjected to a simple tension test shows an elastic limit of 240 MN/m2. Calculate the factor of safety provided if the principal stresses set up in a complex two dimensional stress system are limited to 140 MN/m2 tensile and 45 MN/m2 compressive. The appropriate theories of failure on which your answer should be based are:
a. the maximum shear stress theory;
b. the maximum shear strain energy theory.
Q.5: A steel tube has a mean diameter of 100mm and a thickness of 3 mm. Calculate the torque which can be transmitted by the tube with a factor of safety of 2.25 if the criterion of failure is (a) maximum shear stress; (b) maximum strain energy; (c) maximum shear strain energy. The elastic limit of the steel in tension is 225 MN/m2 and Poisson’s ratio v is 0.3.
Q.6: A rectangular beam 300 mm deep is simply supported over a span of 5 m. What uniformly distributed load per meter the beam may carry? If the bending stress is not to exceed 130N/mm2. Take I = 8.5 × 106 mm4.
Q.7: A structure is composed of circular members of diameter d. At a certain position along one member the loading is found to consist of a shear force of 10 kN together with an axial tensile load of 20 kN. If the elastic limit in tension of the material of the members is 270 MN/m2 and there is a factor of safety of 4, estimate the magnitude of d required according to (a) the maximum principal stress theory, and (b) the maximum shear strain energy per unit volume theory. Poisson’s ratio v = 0.283.
Q.8: A thin cylindrical pressure vessel with closed ends is required to withstand an internal pressure of 4 MN/m2. The inside diameter of the vessel is to be 1000 mm and a factor of safety of 4 is required. A sample of the proposed material tested in simple tension gave a yield stress of 360 MN/m2. Find the thickness of the vessel, assuming the criterion of elastic failure to be (a) the maximum shear stress, (b) the shear strain energy. [11.1,9.62mm]
Q.9: What is Factor of safety (FOS)?
Q.10: What are the different theories of failures?
Q.11: Write the statement of maximum principal stress theory.
Q.12: Write the statement of maximum principal strain theory.
Q.13: Write the statement of maximum shear stress theory.
Q.14: Write the statement of maximum strain energy theory.