MD Module 1 Stress [PDF]

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MODULE I. STRESSES DEFINITIONS AND USEFU INFORMATION  STRENGTH - The ability of a material to withstand load without failure.  STRESS - It is the force or load applied to a material per unit area.  STRAIN - It is the amount by which a body changes (shorten or lengthen) due to the application of load divided by the original length. It is the percent elongation or compression of material due to application of load.



Ultimate Strength



Figure 1. STRESS-STRAIN CURVE



STRESS



Plastic Limit



Yield Point Proportional Limit Modulus line



STRAIN     







PROPORTIONAL LIMIT - It is the stress at which the stress-strain curve deviates from a straight line; it is the point on the stress-strain curve at which it begins to deviates from the straight line relationship ELASTIC LIMIT - It is the maximum stress at which a material may be subjected without causing permanent deformation YIELD POINT - It is the stress at which there is a marked increase in elongation without an increase in load; it is a point on the stress-strain curve at which a sudden increase in strain without additional load ULTIMATE STRENGTH - It is the highest point on the stress-strain curve; It is the maximum load divided by the original area before straining occurs; It is the stress that would cause failure; Also known as TENSILE STRENGTH MODULUS OF ELASTICITY (YOUNG’S MODULUS, E) - It is the ratio of the unit stress to unit strain within the PROPRTIONAL LIMIT. It is the proportionality constant of a material in tension or compression below the PROPORTIONALITY LIMIT on the stress-strain curve at which 6 stress is proportional to strain. E = 30 x 10 psi (STEEL) MODULUS OF ELASTICITY IN SHEAR (MODULUS OF RIGIDITY, G) - It is the ratio of the unit stress to unit strain with in the PROPORTIONAL 6 LIMIT in shear. G = 11.5 x 10 psi (STEEL)



G   



E 21   



POISSON’s RATIO () - It is the ratio of the lateral strain to longitudinal strain of a material subjected to uniform longitudinal stress within the PROPORTIONAL LIMIT. It is the ratio of the contraction to the extension when an element is loaded with a longitudinal tensile force. ENDURANCE LIMIT or FATIGUE LIMIT (SE, SN) - Maximum stress that will not cause failure when the force is reversed indefinitely. FACTOR OF SAFETY or DESIGN FACTOR (FS) - It is the factor by which the design stress is safe.



SIMPLE STRESSES AND STRAIN 1.



Tensile Stress



st  Where,



2.



F A



A



A



 2 D 4



st = tensile stress, psi, kPa, Mpa 2 2 A = cross sectional area, in , m



Compressive Stress



sc  Where,



F A



A



A



 2 D 4



sc = compressive stress, psi, kPa, Mpa 2 2 A = cross sectional area, in , m



F



F = tensile force, lb, kN D = diameter, in., m



F = compressive force, lb, kN D = diameter, in., m



1



F



3.



Shearing Stress



F ss  As Where, 4.



As



F



 A s  d2 4



sc = shearing stress, psi, kPa, Mpa 2 2 A = shearing area, in , m



F



F = shearing force, lb, kN d = diameter, in., m



d F



Bearing Stress



sb  Where,



F Ab



A b  LD



L



sb = bearing stress, psi, kPa, Mpa F = applied force, lb, kN D = diameter, in., m



2



Ab = bearing area, in , m L = length, in., m



2



D 5.



Torsional Stress



ss  Where,



Tc 16T  J  D3



ss = torsional stress, psi, kPa, Mpa c = distance of the farthest fiber from neutral axis, in., m



T = torque or twisting moment, in-lb, kN-m



D 2 D4 J , for solid shaft 32



4



c



J = polar moment of inertia, in. , m



J



D = shaft diameter, in., m Di = inside diameter of a hollow shaft, in., m



4



 Do4  Di4 , for hollow shaft 32











Do = outside diameter of a hollow shaft, in., m



T



6.



Bending or Flexural Stress



s Where,



Mc 32M  I D3



s = bending stress, psi, kPa, Mpa M = bending moment, in-lb, kN-m



I



c = distance of farthest fiber from the neutral axis, in., m 4 4 I = Moment of inertia about the neutral axis, in. , m



bh3 D4 , for rectangular cross section I  , for circular section 12 64 F



F/2



F/2 7.



Strain and Elongation o o



A



 Strain, Strain  L F Stress, Stress  A



L 2







F



o



Modulus of Elasticity (Young Modulus),



o



Elongation,  



Where,



8.



E



F 



A  Stress Strain L



FL L  s  AE E



 = elongation (or shortening), in., m, mm F = force, lb, kN s = stress, psi, kPa



L = length, m, in. 2 2 A = cross sectional area, in. , m



Thermal Elongation and stress



   kE  t 2  t1  L 



Thermal elongation,   kL  t   kL  t 2  t1  Where,



Stress, s  E 



o



o



 = elongation or shortening, in., m, mm k = coefficient of thermal expansion, m/m- C, ft/ft- F o o o o t1 = initial temperature, C, F t2 = final temperature, C, F



F



COMBINED STRESSES 9.



P



Combined Axial and Flexural Stresses



P Mc  A I



s o



For solid shaft, Where,



P



s



F/2



4P 32M  D2 D3



s = combined axial and flexural stress M = bending moment



10. Maximum Shear Stress Induced by External Tension: ss max 



F/2



P = axial load D = shaft diameter



1 2



2



s2  4s2s 



s 2  2   ss  



Where,



o



s = induced bending stress ss = induced shearing stress ssmzx = maximum shearing stress Induced stressed are those tensile, compressive, and shear stresses induced within a body by application of external forces and/or torques onto the body.



11. Maximum Normal Stress Induced by External Tension and Shearing Loads: s max  12. Variable Stresses, Where,



s  2



2



s 2  2   ss  



1 sm sa   N s y sn



N = factor of safety



sm 



smax  smin 2



sy = yield stress



sn = endurance stress



sa 



sa = variable component stress (alternating stress)



smax  smin 2



sm = mean stress



13. Equivalent Twisting Moment and Equivalent Bending Moment







Equivalent Twisting Moment: Te 



M2  T2







Equivalent Bending Moment: M e 



M  Te M   2



M2  T2 2



PRACTICE PROBLEMS (STRESSES)



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Prob. # 1] In the LRT project, steel railroad rails of 10 m long are to be installed. If lowest temperature considered is 16 C, and a maximum o -6 o temperature of 36 C is designed for, assuming coefficient of thermal expansion of steel to be 11.6 x 10 m/m- C and modulus of elasticity of steel to be 207 000 MPa. Determine the clearance between rails such that the adjoining rail will just touch at maximum design temperature. a) 2.32 mm b) 3.22 mm c) 3.43 mm d) 4.33 mm o Prob. # 2] In prob. # 1, determine the induced stress in the rails if the maximum temperature is 40 C. a) 96.048 kPa b) 9.6048 Mpa c) 960.48 kPa d) 98.604 kPa 6 Prob. # 3] An engine part is being tested with a load of 30 000 lb. The allowable tensile stress is 10 000 psi, modulus of elasticity of 40 x 10 psi. If the original length of specification is 42 inches with elongation not exceeding 0.0015 inch, what diameter of the specimen is required? a) 15.7 inches b) 5.71 inches c) 5.17 inches d) 5.51 inches Prob. # 3] Calculate the compressive stress of a signboard support with a load of 2000 lb. A hollow cylinder is used with an outside diameter of 6 inches and thickness of 0.75 inch.



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a) 161.68 psi b) 168.16 psi c) 166.18 psia d) 186.16 psi Prob. # 4] A thrust bearing has an inside diameter of 0.5 inch and an outside diameter of 3 inches. For an allowable bearing pressure of 9 psi, how much axial load ca it sustain? a) 618.5 lb b) 615.8 lb c) 681.5 lb d) 685.1 lb Prob. # 5] What force is required to punch a 0.5-inch hole on a 3/8-inch thick plate if the ultimate shear strength of the plate is 42 000 psi? a) 24 740 lb b) 27 440 lb c) 24 470 lb d) 20 744 lb Prob. # 6] A steel rod on bridge must be made to withstand a pull of 5000 lb. Find the diameter of the rod assuming a factor of safety of 5 and ultimate stress of 64 000 psi. (ME Brd Problem, 1996) a) 0.75 inch b) 0.84 inch c) 0.71 inch d) 0.79 inch Prob. # 7] If the ultimate shear strength of steel plate is 42 000 psi, what force is necessary to punch a 0.75 inch diameter hole in a 0.675 inch plate? (Brd Prob., 196) a) 63 000 lb b) 61 850 lb c) 68 080 lb d) 66 800 lb Prob. # 8] What force is required to punch a 0.5 inch hole on a 3/8 inch thick plate if the ultimate shear strength on the plate is 42 000 psi? (Brd Prob. 1995) a) 24 940 lb b) 24 620 lb c) 24 960 lb d) 24 740 lb o Prob. # 9] A 2.5 inches diameter by 2 inches long journal bearing is to carry a 5500 lb load at 3600 rpm using SAE 40 lube at 200 F through a single hole at 25 psi. Compute the bearing pressure. (Brd Prob, Oct 1995) a) 1100 psi b) 900 psi c) 1000 psi d) 950 psi Prob. # 10] A test specimen is under tension. The load is 20,000 lb, modulus of elasticity is 30 million psi, and original length of specimen is 40 in. What is the required cross-section, in square inches, if the resulting elongation must not be greater than 0.001 inch? a. 22.3 b. 26.6 c. 32.2 d. 62.6 Prob. # 11] A square bar of wrought iron, 2 inches on each side, is raised to a temperature of 100 deg F above its normal. If held so that it -6 6 cannot expand, what stress will be induced in it? k = 6.8 x 10 per deg F & E = 30 x 10 psi a. 17,400 psi b. 18,400 psi c. 19,400 psi d. 20,400 psi Prob. # 12] In problem no. 11, what is the force necessary to prevent expansion? a. 91,600 lbs b. 81,600 lbs c. 20,900 lbs d. 11,995 lbs Prob. # 13] A short cylindrical cast iron post supports a compressive load of 20 tons (40 kips). If the factor of safety is taken equal to 10, find the diameter of the post. Ultimate stress for compression is 80,000 psi. Neglect slenderness ratio and no buckling. a. 2.25 in. b. 2.52 in. c. 3.25 in. d. 3.52 in. Prob. # 14] A steel wire 10 m. long, hanging vertically, supports a tensile load of 2000N. Neglecting the weight of the wire, determine the required diameter if the total elongation is not to exceed 5 mm. Assume modulus of elasticity to be 200 GPa. a. 4.27 mm. b. 4.72 mm. c. 5.05 mm. d. 5.50 mm. Prob. # 15] A hollow iron pipe to be designed as a column has an outside diameter of 240 mm and is subjected to a force of 80 KN. Find the pipe thickness if the compressive stress is limited to 16 MPa. a. 5.85 mm b. 6.85 mm c. 7.85 mm d. 8.85 mm Prob. # 16] It is desired to check the design of a 2-in. medium steel shaft subjected to a turning moment of 40,000 in-lb.. Determine the factor of safety used in the design if ultimate stress is 50,000 psi. a. 1.96 b. 3.14 c. 1.60 d. 4.92 Prob. # 17] Compute the diameter (in inches) of a SAE 1030 steel shaft to transmit 8 hp at 90 rpm. The torsional deflection allowable is 0.03 degree per foot length. a. 3 1/8 b. 3 ¼ c. 3 5/8 d. 3 7/8 Prob. # 18] A uniform beam 12 meters long is fixed at one end. It has a uniform weight of 50 kg/m along its length. A load of 20 kgs. is suspended on the beam 4 m from the free end. The moment at the fixed end is a. 3760 kg-m b. 0.0 kg-m c. 60 kg-m d. 4800 kg-m Prob. # 19] An air cylinder has a bore of 25 mm and is operated with shop air at a pressure of 90 psi. Find the push force exerted by the piston rod in N. a) 304.7 N b) 340.7 N c) 247.0 N d. 307.4 N Prob. # 20] What pressure is required to punch a hole 2 inches in diameter through a 0.25 inch steel plate? (ME Brd Prob. April 1995) a) 10 tons b) 20 tons c) 30 tons d) 40 tons Prob. # 21] What force is required to punch a 0.5-inch hole on a 3/8-inch thick plate if the ultimate shear strength of the plate is 42 000 psi? (Brd Prob. Oct. 1995) a) 24 940 lb b) 24 620 lb c) 24 960 lb d) 24 740 lb Prob. # 22] what is the modulus of elasticity if the stress is 44 000 psi and a unit strain of 0.00105? (ME Brd Oct. 1995) 6 6 6 6 a) 41.905 x 10 b) 42.300 x 10 c) 41.200 x 10 d) 43.101 x 10 Prob. # 23] What modulus of elasticity in tension is required to obtain a unit deformation of 0.00105 m/m from a load producing a unit tensile stress of 44 000 psi? 6 6 6 6 a) 42.300 x 10 psi b) 42.202 x 10 psi c) 43.101 x 10 psi d) 41.905 x 10 psi Prob. # 24] If the ultimate shear strength of a steel plate is 42 000 psi, what force is necessary to punch a 0.75-inch diameter hole in a 0.635inch thick plate? (Brd Prob. April 1996) a) 63 008 lb b) 68 080 lb c) 61 850 lb d) 66 800 lb



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Prob. # 25] A steel tie rod on bridge must be made to withstand a pull of 5 000 lb. Find the diameter of the rod assuming a factor of safety of 5 and ultimate stress of 64 000 psi. (Brd Prob. April 1996) a) 0.75 in b) 0.71 inch c) 0.89 inch d) 0.79 inch 3 Prob. # 26] A steel weighing 480 lb/ft has an ultimate strength of 80 000 psi. What is the maximum length of steel rod that could be hung vertically from its upper end without rupturing? a) 2 000 ft b) 18 000 ft c) 24 000 ft d) 166.67 ft 6 Prob. # 27] An engine part is being tested with a load of 30 000 lb. The allowable tensile stress is 10 000 psi, modulus of elasticity of 40 x 10 psi. If the original length of specimen is 42 inches with elongation not exceeding 0.0015 inch, what diameter of the specimen is required? a) 4.2 inches b) 3.0 inches c) 2.5 inches d) 5.17 inches Prob. # 28] What force is required to punch a half inch hole on a 0.25-inch thick plate if the ultimate shearing stress of the plate is 44 000 psi? a) 25 000 lb b) 15 000 lb c) 20 000 lb d) 17 279 lb 2 Prob. # 29] A steel rod 75 inches long with a cross sectional area of 0.25 in is held vertically firm at one end while a load of 2500 lb is suspended from the other end. If the rod stretches 0.025 inch, find the modulus of elasticity of the steel. 6 6 6 6 a) 12 x 10 psi b) 11.5 x 10 psi c) 30 x 10 psi d) 27 x 10 psi Prob. # 30] What load in N must be applpied to a 25 mm round steel bar 2.5 m long (E = 207 Gpa) to stretch the bar 1.3 mm? a) 42 000 N b) 52 840 N c) 53 000 N d) 60 000 N 2 Prob. # 31] What load which cause a total deformation of 0.036 inch of a steel rack that has a cross sectional area of 4 in and a length of 6 ft. a) 55 000 lb b) 40 000 lb c) 60 000 lb c) 50 000 lb o



Prob. # 32] A railroad track is laid at a temperature of 15 F with gaps of 0.01 feet between the ends of the rails. The rails are 33 ft long. If they o -6 o are prevented from buckling, what stress will result from a temperature of 110 F? (k = 6.5 x 10 per F) a) 10 000 psi b) 8 530 psi c) 9 450 psi c) 9 980 psi Prob. # 33] A thrust washer has an inside diameter of 0.5 inch and an outside diameter of 3 inches. For an allowable bearing pressure of 90 psi, how much axial load can it sustain? a) 618.5 lb b) 537.2 lb c) 702.1 lb c) 871.2 lb



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