strenght of materials(es-64)

(Mindanao State University-General Santos City, Philippines) BSEE-(G.R.F) students ‘04

Solved Problems in

Mechanics of Materials

ES-64 1. 12-7. Determine the equations of the elastic curve using the x1 and x2

coordinates. Specify the slope at A and the maximum deflection. EI is constant. 1. 12-10.The beam is made of two rods and is subjected to the concentrated

load P. Determine the slope at C. The moments of inertia of the rods are IAB and IAC, and the modulus of elasticity is E.

2. 12-13. Determine the elastic curve for the cantilevered beam, which is subjected to the couple moment Mo. Also compute the maximum slope and maximum deflection of the beam. EI is constant.

3. 12-15. Determine the deflection at the center of the beam and the slope at B. EI is constant.

4. 12-16. Determine the elastic curve for the simply supported beam, which is subjected to the couple moments Mo. Also, compute the maximum slope and the maximum deflection of the beam. EI is constant.

5. 12-19. Determine the equations of the elastic curve using the coordinates x1 and x2, and specify the slope at A. EI is constant.

6. 12-22.The floor beam of the airplane is subjected to the loading shown. Assuming that the fuselage exerts only vertical reactions on the ends of the beam, determine the maximum deflection of the beam. EI is constant.

7. 12-27. Determine the elastic curve for the simply supported beam using the x coordinates 0 ≤ x ≤ L/2. Also, determine the slope at A and the maximum deflection of the beam. EI is constant.

8. 12-34.The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant.

9. 12-35.The shaft is supported at A by a journal bearing that exerts only vertical reactions on the shaft, and at C by a thrust bearing that exerts horizontal and vertical reactions on the shaft. Determine the equation of the elastic curve. EI is constant.

10. 12-37.The shafts support the two pulley loads shown. Determine the equation of the elastic curve. The bearing at A and B exerts only vertical reactions on the shaft. EI is constant.

11. 12-38.The shafts support the two pulley loads shown. Determine the equation of the elastic curve. The bearing at A and B exerts only vertical reactions on the shaft. EI is constant.

14. 12-55. Determine the slope and deflection at B. EI is constant. 15. 12-61. Determine the maximum slope and the maximum deflection of the

beam. EI is constant. 16. 12-40.The beam is subjected to the load shown. Determine the equation of

the elastic curve. EI is constant. 17. 12-45.The beam is subjected to the load shown. Determine the equation of

the elastic curve. EI is constant. 18. 12-50. Determine the equation of the elastic curve. Specify the slope at A. EI

is constant. 19. 12-25.The beam is subjected to the linearly varying distributed load.

Determine the maximum slope of the beam. EI is constant. 20. 12-42.The beam is subjected to the load shown. Determine the equation of

the slope and elastic curve. EI is constant. 21. 12-45. The beam is subjected to the load shown. Determine the equation of

the elastic curve. EI is constant. 22. 12-46.The wooden beam is subjected to the load shown. Determine the

equation of the elastic curve. Specify the deflection at the end C. EW=1.6(103) ksi.

23. 12-63. Determine the deflection and slope at C. EI is constant. 24. 12-66. Determine the deflection at C and the slope of the beam at A, B, and

C. EI is constant. 25. 12-73. Determine the slope at B and deflection at C. EI is constant. 26. 12-86.The beam is subjected to the load shown. Determine the slope at B

and deflection at C. EI is constant.

1. 5-5.The copper pipe has an outer diameter of 2.50 in. and an inner diameter

of 2.30 in. If it is tightly secured to the wall at C and the three torques are applied to it as shown., determine the shear stress developed at points A and

B. These points lie on the pipe’s outer surface. Sketch the shear stress on the volume elements located at A and B.

2. 5-9. The assembly consists of two sections of galvanized steel pipe connected together using a reducing coupling at B. The smaller pipe has an outer diameter of 0.75 in. and an inner diameter of 0.68 in., whereas the larger pipe has an outer diameter of 1 in. and an inner diameter of 0.86 in. If the pipe is tightly secured to the wall at C, determine the maximum shear stress developed in each section of the pipe when the couple shown is applied to the handles of the wrench.

3. 5-10.The link acts as part of the elevator control for a small airplane. If the attached aluminum tube has an inner diameter of 25 mm, determine the maximum shear stress in the tube when the cable force of 600 N is applied to the cables. Also, sketch the shear stress distribution over the cross section.

4. 5-2.The solid shaft of radius r is subjected to a torque T. Determine the radius r of the inner cone of the shaft that resists one-half of the applied torque (T/2). Solve the problem two ways: (a) by using the torsion formula, (b) by finding the resultant of the shear stress distribution.

5. 5-3.The solid shaft of radius r is subjected to a torque T. Determine the radius r of the inner core of the shaft that resists one-quarter of the applied torque (T/4). Solve the problem two ways: (a) by using the torsion formula, (b) by finding the resultant of the shear stress distribution.

6. 5-13. A steel tube having an outer diameter of 2.5 in. is used to transmit 35 hp when turning at 2700 rev/min. Determine the inner diameter d of the tube to nearest 1/8 in. if the allowable shear stress is τallow = 10 ksi.

7. 5-17. The steel shaft has a diameter of 1 in. and is screwed into the wall using a wrench. Determine the maximum shear stress in the shaft if the couple forces have a magnitude of F = 30 lb.

8. 5-21. The 20-mm-diameter steel shafts are connected using a brass coupling. If the yield point for the steel is (τY)st = 100 MPa, determine the applied torque T necessary to cause the steel to yield. If d = 40 mm, determine the maximum shear stress in the brass. The coupling has an inner diameter of 20 mm.

9. 5-22. The coupling is used to connect the two shafts together. Assuming that the shear stress in the bolts is uniform, determine the number of bolts necessary to make the maximum shear in the shaft equal to the shear stress in the bolts. Each bolt has a diameter d.

10. 5-29. The shaft has a diameter of 80 mm and due to friction at its surface within the hole, it is subjected to a variable torque described by the function

25 xxt = N⋅m/m, where x is in meters. Determine the minimum torque T0

needed to overcome friction and cause it to twist. Also, determine the absolute maximum stress in the shaft.

11. 5-30. The solid shaft has a linear taper from rA at one end to rB at the other. Derive and equation that gives the maximum shear stress in the shaft at a location x along the shaft’s axis.

12. 5-32. The drive shaft AB of an automobile is made of a steel having an allowable shear stress of τallow = 8 ksi. If the outer diameter of the shaft is 2.5 in. and the engine delivers 200 hp to the shaft when it is turning at 1140 rev/min, determine the minimum required thickness of the shaft’s wall.

13. 5-33. The drive shaft AB of an automobile is to be designed as a thin walled tube. The engine delivers 150 hp when the shaft is turning at 1500 rev/min. Determine the minimum thickness of the shaft’s wall if the shaft’s outer diameter is 2.5 in. The material has an allowable shear stress τallow = 7 ksi.

14. 5-34. The drive shaft of a tractor is to be designed as a thin-walled tube. The engine delivers 200 hp when the shaft is turning at 1200 rev/min. Determine the minimum thickness of the wall of the shaft if the shaft’s outer diameter is 3 in. The material has an allowable shear stress of τallow = 7 ksi.

15. 5-35. A motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. If it is rotating at 200 rad/s. determine its largest inner diameter to the nearest 1/8 in. If the allowable shear stress for the material is τallow = 25 ksi.

16. 5-36. The drive shaft of a tractor is made of a steel tube having an allowable shear stress of τallow = 6 ksi. If the outer diameter is 3 in. and the engine delivers 175 hp to the shaft when it is turning at 1250 rev/min. determine the minimum required thickness of the shaft’s wall.

17. 5-37. A motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. and an inner diameter of 1.84 in. Determine the smallest angular velocity at which it can rotate if the allowable shear stress for the material is τallow = 25 ksi.

18. 5-38. The 0.75 in. diameter shaft for the electric motor develops 0.5 hp and runs at 1740 rev/min. Determine the torque produced and compute the maximum shear stress in the shaft. The shaft is supported by ball bearings at A and B.

19. 5-42. The motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. and an inner diameter of 1.84 in. Determine the smallest angular velocity at which it can rotate if the allowable shear stress for the material is τallow = 25 ksi.

20. 5-46. The solid shaft of radius c is subjected to a torque T at its ends. Show that the maximum shear strain developed in the shaft is γmax = Tc/JG. What

is the shear strain on an element located at point A, c/2 from the center of the shaft? Sketch the strain distortion of this element.

21. 5-74. A rod is made from two segments: AB is steel and BC is brass. It is fixed a its ends and subjected to a torque of T = 680 N⋅m. If the steel portion has a diameter of 30mm. determine the required diameter of the brass portion so the reactions at the walls will be the same. Gst = 75 GPa, Gbr = 39 GPa.

22. 5-78. The composite shaft consists of a mid-section that includes the 1-in.-diameter solid shaft and a tube that is welded to the rigid flanges at A and B. Neglect the thickness of the flanges and determine the angle of twist of end C of the shaft relative to end D. The shaft is subjected to a torque of 800 lb⋅ft. The material is A-36 steel.

The column is subjected to an axial force of 8 kN at its top. If the cross-sectional area has the dimensions shown in the figure, determine the average normal stress acting at section a-a. Show this distribution of stress acting over the area’s cross section.

( ) ( )[ ]

( )MPa

mmmmmmmmA

14010150102

The anchor shackle supports a cable force of 600 lb. If the pin has a diameter of 0.25 in., determine the average shear stress in the pin.

( )ksi

04909.02

:stressshear doubleFor

04909.0

The small block has a thickness of 5 mm. if the stress distribution at the support developed by the load varies as shown, determine the force F applied to the block, and the distance d to where it is applied.

( )( )( )( ) ( )( )[ ]( )( )

( )( )( ) ( ) ( )( )( ) ( ) ( )( )( ) ( )

dforSolving

3960000

000,960,3

601203

2512020

160120

151204060

256040

000,36

005.012.01060402

1005.006.01040

A 175-lb woman stands on a vinyl floor wearing stiletto high-heel shoes. If the heel has the dimensions shown, determine the average normal stress she exerts on the floor and compare it with the average normal stress developed when a man having the same weight is wearing flat-heel shoes. Assume the load is applied slowly, so that dynamic effect can be ignored. Also assume the entire is supported only by the heel of one shoe.

( )( ) ( )( )

inininA

6.01.0

4.25.0

The 50-lb lamp is supported by three steel rods connected by a ring at A. Determine which rod is subjected to the greater average normal stress and compute its value. Take θ=30°. The diameter of each rod is given in the figure.

( )psi

61.745

21.634

45sin30cos

30sin45cos

5045sin30sin

45cos30cos

=°+°

∴Rod AC is subjected to greater average normal stress at 745.61 psi.

The 50-lb lamp is supported by three steel rods connected by a ring at A. Determine which rod is subjected to the greater average normal stress and compute its value. Take θ=45°. The diameter of each rod is given in the figure. ∴Rod AC is subjected to greater average normal stress at 745.61 psi.

( )psi

24.500

5045sin45sin

45cos45cos

=°+°

The two steel members are joined together using a 60° scarf weld. Determine the average normal and average shear stress resisted in the plane of the weld.

( )MPa

62.4025.866

8025.866

100093.6

025.866

060cos60sin

30sin30cos8

060cos30cos8

60sin30sin

°+°+−

=°+°+−

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

( )TMPa

0973.113

5929.301

0973.113

5929.301

The plastic block is subjected to an axial compressive force of 600N. Assuming that the caps at the top and bottom distribute the load uniformly throughout the block. Determine the average normal and average shear stress acting along section a-a.

( )( )

( )( ) kPa

kPammmm

30cos10050

1000300

30cos10050

100062.519

62.519

030cos30sin30cos

30sin600

30cos30sin

=°−°°

The specimen failed tension test at an angle of 52° when the axial load was 19.80 kip. If the diameter of the specimen is 0.5 in., determine the average normal and average shear stress acting on the area of the inclined failure plane. Also, what is the average normal stress acting on the cross section when failure occurs?

When failure occurs no shear stress exists, since the shear force at the section is 0.

2492.0

19.12'

2492.0

2492.0'

52sin'

052cos52sin

38sin38cos8.19

052cos38cos8.19

52sin38sin

°+°+−

=°+°+−

A tension specimen having a cross-sectional area A is subjected to an axial force P. Determine the maximum average shear stress in the specimen and indicate the orientation θ of a section on which it occurs.

AAwhereA

45sin45cos

sincos

2sin2cos

2sin2cos0

2cos2sin

sincos

θθτ

θθθ

The joint is subjected to the axial member force of 5 kN. Determine the average normal stress acting on sections AB and BC. Assume the member is smooth and is 50 mm thick.

( )( )

100049.1

100008.4

30sin08.445sin5

030cos45cos5

=°−°

=°+°−

The joint is subjected to the axial member force of 6kip. Determine the average normal stress acting on section AB and BC. Assume the member is smooth and is 1.5in. thick. ΣFy = 0 6sin60º = FBCsin70º FBC = 5.53 kip ΣFx = 0 FAB = 6cos60º+FBCcos70º FAB = 4.89 kip For section AB: σAB=FAB/A = 4.89 kip 2(1.5 in2) σAB=1.63 ksi For section BC: σBC=FBC/A = 5.53 kip 4.5 in(1.5 in) σBC=0.819 ksi

Rod AB and BC have diameters of 4 mm and 6 mm, respectively. If the load of 8 kN is applied to the ring at B, determine the average normal stress in each rod if θ = 60°.

ΣFy = 0 -8 kN+FBCsin60º = 0

ΣFx = 0

- FBC = 9.24 kN FAB+FBCcos60º = 0

FAB = 4.62 kN

σAB=4.62 kN(10003) = 367.65 MPa π(4mm2) σBC=9.24 kN(10003) = 326.8 MPa π(9mm2)

Rods AB and BC have diameters of 4mm and 6mm respectively. If the vertical load of 8kN is applied to the ring at B, determine the angle θ of rod BC so that the average normal stress in each rod is equivalent. What is this stress?

Given: dAB=4mm dBC=6mm Find: θ & σ ΣFy = 0 -8 kN+FBCsinθ = 0 FBC=8 kN/sinθ ΣFx = 0 -FAB+FBCcosθ = 0 FAB = FBCcosθ = 8 kNcosθ/sinθ σBC= 8 kN (sinθ)π(0.006m/2)2 σAB= 8 kNcosθ πsinθ(0.004m/2)2 σBC=σAB 8 kN = 8 kNcosθ πsinθ(0.006m/2)2 πsinθ(0.004m/2)2 θ=cos-1(0.0042/0.0062) θ=63.61º σ= 8 kN

πsin63.61º(0.006m/2)2 =316 MPa

The bars of the truss each have a cross-sectional area of 1.25 in2. Determine the average normal stress in each member due to the loading P=8kip. State whether the stress is tensile or compressive.

ΣFx=0 Cx-Bx=0 Cx=Bx ΣFy=0 Cy+Dy-0.75-P=0 Cy+Dy=1.75 P At joint A ΣFy=0 AB(4/5)=P AB=(5/3)P (T) ΣFx=0 AB(4/5)=AE AE=(4/3)P (C) At joint B ΣFy=0 -AB(3/5)-BE+BD(3/5)=0 BD=()5/3[.75P+(3/5)(5/3)P] BD=(35/12)P (C) ΣFx=0 -AB(4/5)-BD(4/5)+BC=0 BC=(11/3)P (T) ΣMD=0 -3Cx+4(.75P)+8P=0 Cx=(11/3)P Dx=(11/3)P At joint E ΣFx=0 AE=ED ED=(4/3)P (C) ΣFy=0 BE=.75P (P)

σAB= (5/3)P 1.25 =(4/3)(8)=10.7 ksi (T) σBC=(11/3)P/1.25=(44/5)(8) = 23.5 ksi (T) σAE=(4/3)P/1.25=(16/15)(8) =8.53 ksi (C) σBE=.75P/1.25=(3/5)(8) =4.8 ksi (T) σED=(4/3)P/1.25=(16/15)(8) =8.53 ksi (C) σBD=(35/12)P/1.25=(7/3)(8) =18.7 ksi (C)

The uniform beam is supported by two rods AB and CD that have cross-sectional areas of 12 mm2 and 8 mm2, respectively. If d = 1 m, determine the average normal stress in each rod.

ΣMC = 0 3ΤAB = 12 kN.m TAB=4 kN ΣFy = 0 TCD = 6 kN-4 kN = 2 kN σAB= 4000 N 12(1m2/10002) σAB=333 MPa σCD= 2000 N 8(1m2/10002) =250 MPa

The uniform beam is supported by two rods AB and CD that have cross-sectional areas of 12 mm2 and 8 mm2, respectively. If d = 1 m, determine the average normal stress in each rod.

ΣMA = 0 6d = TCD(3) TCD = 2d ΣFy = 0 TAB = 6-2d σAB = σCD 6-2d = 2d 12 m2 8 m2 10002 10002 (6-2d)(8)(10002) = (12)(2d)(10002) 48 = 40d :ּd = 1.2m

The railcar dock light is supported by the 1/8 in diameter pin at A. If the lamp weighs 4lb, and the extension arm AB has a weight of 0.5 lb/ft, determine the average shear stress in the pin needed to support the lamp. Hint: The shear force is cost by the couple moment required for equilibrium at A.

0.5 lb.ft M Ay 4 lb 3 ft ΣFy = 0 Ay = 4+0.5(3) Ay = 5.5 lb ΣMA = 0 M = 4(3)+0.5(3)(1.5) M = 14.25 lb.ft[12 in/1ft] M = 171 lb.in T = V/A = M/1.25 πr2 = 11147.47 lb/in2 = 11.1 ksi

The two member frame is subjected to the distributed loading shown. Determine the intensity w of the largest uniform loading that can be applied to the frame w/o causing either the average normal stress or the average shear stress at section b-b to exceed σ = 15MPa and τ = 16MPa, respectively. Member CB has a square cross-section of 35mm on each side.

( ) ( )

78.777,21

53516875.1

22.222,27

53515875.1

1-81. The 60mm × 60mm oak post is supported on the pin block. If the allowable bearing stresses for this material are σ oak =43MPa and σ pine =25MPa, determine the greatest load P that can be supported. If a rigid bearing plate is used between this material, determine its required area so that the maximum load P that can be supported. What is this load?

σOAK=43 MPa σPINE=25 MPa σOAK=P/A [43 N/mm2][602 mm2]=P 154,800 N=P P=154.8 kN σPINE=P/A [25 N/mm2][602 mm2] 90000 N=P P=90 kN :ּ P=90 kN since this is the greatest load the pine block can told. Since there is a bearing plate between the oak block and the pine block, the maximum load is the allowable load by the pine block that is P≈155 kN AALLOW= PMAX/σOAK = 154.8[1000/1 kN] 25N/mm2[(1000mm)2/1 m2

=6.19x10-3 m

1-82. The join is fastened together using two bolts. Determine the required diameter of the bolts if the allowable shear stress for the bolts is τallow = 110 MPa. Assume each bolt supports an equal portion of the load.

Τallow = V/4A 110 MPa = 80 kN(1000) 4π(d2/4) d = √80(1000)/π(110) d = 15.2 mm

1-83. The lever is attached to the shaft A using a key that has a width d and length of 25mm. If the shaft is fixed and a vertical force of 200N is applied perpendicular to the handle, determine the dimension d if the allowable shear stress for the key is τ allow = 35MPa. ΣFy=0 Ay=200 N ΣMA =0 M=200(500 mm)[1 m/1000 mm] M=100 N.m V=M/20 mm=100 N.m/.02 m =5000N A=V/TALLOW dL=V/TALLOW d=5000 N/(25x25) N d=5.71 mm

500 mm

1-86. 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 inch. Determine the maximum P that can be applied to the member if the allowable shear stress for the bolts is τ allow =12ksi and the allowable average normal stress is σ allow = 20ksi.

ΣFy = 0 Vsin60º = Nsin30º V = Nsin30º sin60º ΣFx = 0 -P+Ncos30º+Vcos60º = 0 N = 0.866 P V = 0.5 P σ = 0.866 P 2π(0.3 in/2)2 Τ = 0.5 P/2π(0.3 in/2)2 If σ=20 ksi P = 3.26 kip Τ = 11.55 ksi (safe) If Τ=12 ksi P = 3.39 kip σ=20.8 ksi (fail) :ּP = 3.26 kip

1-88. The two steel wires AB and AC are used to support the load. If both wires have an allowable tensile stress of σallow = 200 MPa, determine the required diameter of each wire if the applied load is P = 5 kN.

ΣFx = 0 -FABsin60º+FAC(4/5) = 0

FAB = 4FBC/5sin60º

ΣFy = 0 -5+FABcos60º+FAC(3/5) = 0 FAC = 4.71 kN FAB = 4.35 kN σAB=4.35(10004) π(dAB)2 dAB=√4.35(10004)/π(200) dAB = 5.26mm σAC=4.71(10004) π(dAC)2 dAC = √4.71(10004)/π(200) = 5.48 mm

1-89. The two steel wires AB and AC are used to support the load. If both wires have an allowable tensile stress of σ allow = 180MPa, and wire AB has a diameter of 6mm and AC has a diameter of 4mm, determine the greatest force P that can be applied to the chain before one of the wires fails.

ΣFx = 0 -FABsin60º+FAC(4/5) = 0 FAB = 4FAC 5sin60º ΣFy = 0 -P+FABcos60º+FAC(3/5) = 0 FAC = 0.942 P FAB = 0.87 P σAB = 0.87 P (1000) π(9m2) σAC=0.942 P(1000) π(4m2) If σAB = 180 MPa

P = 5.85 kN 2-1. An air filled rubber ball that has a diameter of 6in. If the air pressure w/in it is increase until the ball’s diameter becomes 7in., determine the average normal strain in the robber.

−=∈=

2-2. A thin strip of rubber has an unstretched length of 15 in. If it is stretched around a pipe having an outer diameter of 5in., determine the average normal strain in the strip.

L0472.0

708.0==∈=

2-3. The rigid beam is supported by a pin at A and wires BD and CE. If the load P on the beam causes the end C to be displaced 10mm downward, determine the normal strain developed in wires CE and BD.

3 m 2 m 2 m

1007143.1

×=∈

×==∈

2-5. The wire AB is unstretched when θ = 45°. If the load is applied to the bar AC, w/c causes θ = 47°, determine the normal strain in the wire.

034290324.1

247sin

−∈=

2-7. The two wires are connected together at A. If the force P causes point A to be displaced horizontally 2mm, determine the normal strain developed in each wire.

30°30°

300 mm

310779.5

230cos300300

230cos300

150tan

−×∈=

°∈=

2-9. Part of a control linkage for an airplane consists of rigid member CBD and a flexible cable AB. If the force is applied to the end D of the member and causes a normal strain in the cable of 0.0035mm/mm, determine the displacement of D. Originally the cable is unstretched.

300 mm

400 mm

2-13. The triangular plate is fixed at its base, and its apex A is given a horizontal displacement of 5mm. Determine the average normal strain ∋x along the x axis.

A A'5 mm

800 mm

mmmm31043.4

80075.44cos

545cos800

45sin800tan

−×∈=

2-14 The triangular plate is fixed at its base, and its apex A is given a horizontal displacement of 5mm. Determine the average normal strain ∋x’ along the x’ axis.

A A'5 mm

800 mm

31084.8

45cos800

−×=∈

°==∈

2-15. The corners of the square plate are given the displacements indicated. Determine the average normal strains ∋x and ∋y along the x and y axes.

10 in.

10 in.10 in.2 in.

3 in. 3 in

102.10

3.1010

−=∈

2-21. A thin wire, lying along the x axis, is strained such that each point on the wire is displaced x = kx2 along the x axis. If k is constant, what is the normal strain at any point P along the wire?

−∈=

−=∈

∫−

2-22. The wire is subjected to a normal strain that is defined by ∋ - xe-x2 , where x is millimeters. If the wire has an initial length L, determine the increase in its length.

2xxe∈=

xdxedu

dxexdu

−=∆

2-25. The piece rubber is originally rectangular. Determine the average shear strain γxy if the corners B and D are subjected to the deformation shown by the dashed lines. Determine the average normal strain along the diagonal DB and side AD.

300 mm

400 mm

2-26. The piece of rubber is originally rectangular and subjected to the deformation shown by the dashed lines. Determine the average normal strain along the diagonal DB and side AD.

300 mm

400 mm

1000.8

500229.0cos

3002tan

1081.2

40043.0cos

4003tan

×=∈

−°=∈

×=∈

−°=∈

5006.496

188.89cos382.0cos

43.0cos

382.cos

43.0cos

−×−=∈

−=∈

2-28. The material distorts into the dashed position shown. Determine the average normal strain that occurs along the diagonals AD and CF.

( ) ( )( ) ( )

−=∈

−−+=

2655.143

40573990cos157258024.12580222

100 mm

( ) ( )mmx

5739.4

2-29. The non uniform loading causes a normal strain in the shaft that can be expressed as ∋ = kx2 , where k is a constant. Determine the displacement of the end B. Also, what is the average normal strain in the rod?

∫∫ =2kx

2-30. The non uniform loading causes a normal strain in the shaft that can be expresses as ∋ = k sin ((π/L)x), where k is a constant. Determine the displacement of the center C and the average normal strain in the entire rod.

L/2 L/2

( ) ( )

:STRAIN NORMAL

−=∆

2-31. The curved pipe has an original radius of 2ft. If it is heated non uniformly, so that the normal strain along its length is ∋ = 0.05cosθ, determine the increase in length of the pipe.

( )ftx

sin1.0

cos1.0

cos05.0

4-27. Determine the relative displacement of one end of the tapered plate with respect to the other end when it is subjected to an axial load P.

2212 dd

ydydhdxh

−=−

4-50. The three suspender bars are made of the same material and have equal cross-sectional areas A. Determine the average normal stress in each bar if the rigid beam ACE is subjected to the force P.

d/2 d/2 d

4-53. The 10mm diameter steel bolt is surrounded by a bronze sleeve. The outer diameter of this sleeve is 20mm, and its inner diameter is 10mm. If the bolt is subjected to a compressive force of P= 20kN. Determine the average normal stress in the steel and the bronze. Est = 200Gpa, Ebr = 100GPa.

( ) ( )( )( )

( )( )

( )( )MPa

PbPstkN

STSTST

1010012102441.4

102008103662.6

20103662.6102441.4

20103662.6

102441.4

1010001.002.0

4.01000

−×=×

−×=

×−=

−−

4-110. A 0.25in diameter steel rivet having a temperature of 1500°F is secured between two plates such that at this temperature it is 2in long and exerts a clamping force of 250lb between the plates. Determine the approximate clamping force between the plates when the rivet cools to 70°F. For the calculation, assume that the heads of the rivet and the plates are rigid. Take αst = 8(10-6)/°F, Est = 29(103)ksi. Is the result a conservative estimate of the actual answer? Why or why not?

An air filled rubber ball that has a diameter of 6in. If the air pressure w/in it is increase until the ball’s diameter becomes 7in., determine the average normal strain in the robber.

−=∈=

A thin strip of rubber has an unstretched length of 15 in. If it is stretched around a pipe having an outer diameter of 5in., determine the average normal strain in the strip.

L0472.0

708.0==∈=

The rigid beam is supported by a pin at A and wires BD and CE. If the load P on the beam causes the end C to be displaced 10mm downward, determine the normal strain developed in wires CE and BD.

3 m 2 m 2 m

1007143.1

×=∈

×==∈

The wire AB is unstretched when θ = 45°. If the load is applied to the bar AC, w/c causes θ = 47°, determine the normal strain in the wire.

034290324.1

247sin

−∈=

The two wires are connected together at A. If the force P causes point A to be displaced horizontally 2mm, determine the normal strain developed in each wire.

30°30°

300 mm

310779.5

230cos300300

230cos300

150tan

−×∈=

°∈=

Part of a control linkage for an airplane consists of rigid member CBD and a flexible cable AB. If the force is applied to the end D of the member and causes a normal strain in the cable of 0.0035mm/mm, determine the displacement of D. Originally the cable is unstretched.

For the wire:

300 mm

400 mm

600tan

5000035.0

( ) ( )( ) ( )( )

+−+=

4185.0

90cos2400000625.1753

90cos300400230040075.501 222

The triangular plate is fixed at its base, and its apex A is given a horizontal displacement of 5mm. Determine the average normal strain ∋x along the x axis.

A A'5 mm

800 mm

mmmm31043.4

80075.44cos

545cos800

45sin800tan

−×∈=

The triangular plate is fixed at its base, and its apex A is given a horizontal displacement of 5mm. Determine the average normal strain ∋x’ along the x’ axis.

A A'5 mm

800 mm

31084.8

45cos800

−×=∈

°==∈

The corners of the square plate are given the displacements indicated. Determine the average normal strains ∋x and ∋y along the x and y axes.

10 in.

10 in.10 in.2 in.

3 in. 3 in

102.10

3.1010

−=∈

A thin wire, lying along the x axis, is strained such that each point on the wire is displaced x = kx2 along the x axis. If k is constant, what is the normal strain at any point P along the wire?

−∈=

−=∈

∫−

The wire is subjected to a normal strain that is defined by ∋ - xe-x2 , where x is millimeters. If the wire has an initial length L, determine the increase in its length.

2xxe∈=

xdxedu

dxexdu

−=∆

The piece rubber is originally rectangular. Determine the average shear strain γxy if the corners B and D are subjected to the deformation shown by the dashed lines. Determine the average normal strain along the diagonal DB and side AD.

300 mm

400 mm

The piece of rubber is originally rectangular and subjected to the deformation shown by the dashed lines. Determine the average normal strain along the diagonal DB and side AD.

300 mm

400 mm

1000.8

500229.0cos

3002tan

1081.2

40043.0cos

4003tan

×=∈

−°=∈

×=∈

−°=∈

5006.496

188.89cos382.0cos

43.0cos

382.cos

43.0cos

−×−=∈

−=∈

The material distorts into the dashed position shown. Determine the average normal strain that occurs along the diagonals AD and CF.

( ) ( )( ) ( )

( )( ) ( )

24708.0

15850003.157

003.157

90cos801585021585080

9.12515850

03465.0

220252655.143

2655.143

40573990cos157258024.12580

−=∈

+−+=

−=∈

−−+=

100 mm

( ) ( )mmx

5739.4

The non uniform loading causes a normal strain in the shaft that can be expresses as ∋ = k sin ((π/L)x), where k is a constant. Determine the displacement of the center C and the average normal strain in the entire rod.

L/2 L/2

( ) ( )

:STRAIN NORMAL

−=∆

The curved pipe has an original radius of 2ft. If it is heated non uniformly, so that the normal strain along its length is ∋ = 0.05cosθ, determine the increase in length of the pipe.

( )ftx

sin1.0

cos1.0

cos05.0

Determine the relative displacement of one end of the tapered plate with respect to the other end when it is subjected to an axial load P.

ydydhdxh

−=−

2212 dd

The three suspender bars are made of the same material and have equal cross-sectional areas A. Determine the average normal stress in each bar if the rigid beam ACE is subjected to the force P.

d/2 d/2 d

( ) ( )

dPdFdF

EFCDAB

eqnFFF

eqnPFFF

EFCDAB

EFCDEFAB

−=−

− σσσσ

guFinatee

eqnPFF

guFinatee

EFCDAB

)5&4(sinlim

)3&2(sinlim

)3&1(sinlim

−=−

−=+−

guFsolve

)1(sin

)4(sin

−=+−

(Mindanao State University-General Santos City, Philippines) BSEE-(G.R.F) students

The 10mm diameter steel bolt is surrounded by a bronze sleeve. The outer diameter of this sleeve is 20mm, and its inner diameter is 10mm. If the bolt is subjected to a compressive force of P= 20kN. Determine the average normal stress in the steel and the bronze. Est = 200Gpa, Ebr = 100GPa.

( ) ( )( )( )

( )( )

( )( )MPa

PbPstkN

STSTST

1010012102441.4

102008103662.6

20103662.6102441.4

20103662.6

102441.4

1010001.002.0

4.01000

−×=×

−×=

×−=

−−

A 0.25in diameter steel rivet having a temperature of 1500°F is secured between two plates such that at this temperature it is 2in long and exerts a clamping force of 250lb between the plates. Determine the approximate clamping force between the plates when the rivet cools to 70°F. For the calculation, assume that the heads of the rivet and the plates are rigid. Take αst = 8(10-6)/°F, Est = 29(103)ksi. Is the result a conservative estimate of the actual answer? Why or why not?

( )( )( ) ( )

( ) ( )

25.0289.16

102925.04

21500702108

=−×

−=∆

The rubber block is subjected to an elongation of 0.03 in along the x axis, and its vertical faces are given a tilt so that °= 3.89θ .Determine the strains x∈ , x∈ , xyγ .

Take 5.0=rv

Finding x∈

Finding y∈

00375.0

0075.05.0

−=∈

∈−=∈ υ

Finding xyγ

0075.0

3 in.θ

( )( )radxy

0122.0

1803.8990

°−°=

The non uniform loading causes a normal strain in the shaft that can be expressed as ∋ = kx2 , where k is a constant. Determine the displacement of the end B. Also, what is the average normal strain in the rod?

∫∫ =2kx

5- 2. The solid shaft of radius r is subjected to a torque T. Determine the radius r’ of the inner core of the shaft that resists one-half of the applied torque (T/2).

by ratio and proportion:

7651.32cos'

841.0'

5-3. The solid shaft of radius r is subjected to a torque T. Determine the radius r’ of the inner core of the shaft that resists one-quarter of the applied torque (T/4).

45sin'

707.0'

ππτ

5-5. The copper pipe has an outer diameter of 2.50 in. and an inner diameter of 2.30 in. If it is tightly secured to the wall at C and the three torques are applied to it as shown, determine the shear stress developed at points A and B. These points lie on the pipe’s outer surface. Sketch the shear stress on the volume elements located at A and B. sol’n:

600 lb-ft

350 lb-ft

450 lb-ft

450 lb⋅ft = 5400 lb⋅in 350 lb⋅ft = 4200 lb⋅in 600 lb⋅ft = 7200 lb⋅in

5400 + 4200 – 7200 = 2400

( )( )( )

6.2758

3.25.2

5.2240016

7200 - 4200 = 3000

( )( )( )

3.3448

3.25.2

5.2300016

5-10. The link acts as part of the elevator control for a small airplane. If the attached aluminum tube has an inner diameter of 25 mm, determine the maximum shear stress in the tube when the cable force of 600 N is applied to the cables. Also, sketch the shear stress distribution over the cross section.

)10108(

)0125.0)(90(

)10108(

)075.0)(90(

)025.0035.0(32

)075.0)(600(2

5-13. A steel tube having an outer diameter of 2.5 in. is used to transmit 35 hp when turning at 2700 rev/min. Determine the inner diameter d of the tube to nearest 1/8 in. if the allowable shear stress is τallow = 10 ksi.

2.5 in.

Given:

/75.min/2700

/.250,193

/000,10

srevrevf

sftlbhp

allowτ

Solution:

.97.4084

)75.0(2

)]75.0(2[19250

)2)(25.1)(4084097()25.1(

)25.1)(97.4084(

2)25.1(

)25.1(2

)25.1(97.4084

ind 4.2= ; 5.08

5-17. The steel shaft has a diameter of 1 in. and is screwed into the wall using a wrench. Determine the maximum shear stress in the shaft if the couple forces have a magnitude of F = 30 lb. sol’n:

8in. 8in.

( )( )( )

72.7818

68.075.0

75.021016

πτ ( )

( )( )( )

121016

AB D = 0.75 in d = 0.68 in BC D = 1 in d = 0.86 in

( ) ( )inlbT

=+=∑210

815615

5-21. The 20-mm-diameter steel shafts are connected using a brass coupling. If the yield point for the steel is (τY)st = 100 MPa, determine the applied torque T necessary to cause the steel to yield. If d = 40 mm, determine the maximum shear stress in the brass. The coupling has an inner diameter of 20 mm.

To find the T; steel;

07.157

)010.0(

210100

To find the max.., τ; brass;

)01.002.0(

)02.0)(07.157(2

5-22. The coupling is used to connect the two shafts together. Assuming that the shear stress in the bolts is uniform, determine the number of bolts necessary to make the maximum shear in the shaft equal to the shear stress in the bolts. Each bolt has a diameter d. Soln:

5-29. The shaft has a diameter of 80 mm and due to friction at its surface within the hole, it is subjected to a variable torque described by the function

25 xxt = N⋅m/m, where x is in meters. Determine the minimum torque T0 needed

to overcome friction and cause it to twist. Also, determine the absolute maximum stress in the shaft.

25 xxet =

To 80mm

Solution: d=80mm=0.08 m

25 xxet =

πτ =

dxxeTx2

025∫=

dxxeT x

let u= 2x du=2x dx du/2=x dx x=0, u=0 x=2,u=4

25 dueT

05.12 ue=

[ ]15.12 4 −= e

NmT 670=

MPa66.6=τ

33 )1080(

)670(16−

πτ =

5-30. The solid shaft has a linear taper from Ar at one end to Br at the other. Derive

and equation that gives the maximum shear stress in the shaft at a location x along the shaft’s axis.

Solution:

1rrr B +=

r BA −=

xLrrr BA ))((1

−−=

xLrrrr BA

))(( −−+=

xLrrLrr BAB ))(( −−+

xLrrxLLrr BAB )()( −−−+

xLrxLLrr AB )()]([ −+−−

xLrxrr AB )()( −+

xLrxrr AB )( −+

πτ = ;

πτ =

AB −+=

5-32. The drive shaft AB of an automobile is made of a steel having an allowable shear stress of τallow = 8 ksi. If the outer diameter of the shaft is 2.5 in. and the engine delivers 200 hp to the shaft when it is turning at 1140 rev/min, determine the minimum required thickness of the shaft’s wall.

Given: psiksi 80008 ==τ

inD 5.2= slbfthp /000,100200 ==Ρ sec/19min/1140 revrevf ==

?=t Solutions:

)/19(2

/000,100

lbftT 66.837= or lbin89.051,10=

1644 dD

)5.2)(89.051,10(16000,8

44 dpsi

)000,8(

65.402075)5.2( 44

π=− d

998.1539

ind 19.2= Thickness:

DR = ;

19.2=r

25.1=R 1.1=r rRt −=

1.125.1

5-33. The drive shaft AB of an automobile is to be designed as a thin walled tube. The engine delivers 150 hp when the shaft is turning at 1500 rev/min. Determine the minimum thickness of the shaft’s wall if the shaft’s outer diameter is 2.5 in. The material has an allowable shear stress τallow = 7 ksi.

000,77

/25min/500,1

/000,75150

psiksi

srevrevf

slbfthp

Solution

)/25(2

/000,75

58.729,5

46.477

5968.27

42.1039

)000,7(

183,22939

)5.2)(58.5729(16000,7

Thickness:

15.125.1

5-34. The drive shaft of a tractor is to be designed as a thin-walled tube. The engine delivers 200 hp when the shaft is turning at 1200 rev/min. Determine the minimum thickness of the wall of the shaft if the shaft’s outer diameter is 3 in. The material has an allowable shear stress of τallow = 7 ksi. Given:

srevrevf

sftlbhp

/20min/1200

/.100000200

psi7000= ?max =t

Solution:

slbftT

3.9549

)/20(2

/100000

84.2081

)7000(

4.45836681

)3)(3.9549(167000

Max Thickness:

5-35. A motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. If it is rotating at 200 rad/s. determine its largest inner diameter to the nearest 1/8 in. If the allowable shear stress for the material is τallow = 25 ksi.

( )( )( )( )

( )ind

11.616

2150001625000

250000

lbfthp

slbfthp

/250000/500

/25000

5-36. The drive shaft of a tractor is made of a steel tube having an allowable shear stress of τallow = 6 ksi. If the outer diameter is 3 in. and the engine delivers 175 hp to the shaft when it is turning at 1250 rev/min. determine the minimum required thickness of the shaft’s wall.

5-37. A motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. and an inner diameter of 1.84 in. Determine the smallest angular velocity at which it can rotate if the allowable shear stress for the material is τallow = 25 ksi.

5-38. The 0.75 in. diameter shaft for the electric motor develops 0.5 hp and runs at 1740 rev/min. Determine the torque produced and compute the maximum shear stress in the shaft. The shaft is supported by ball bearings at A and B.

5-42. The motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. and an inner diameter of 1.84 in. Determine the smallest angular velocity at which it can rotate if the allowable shear stress for the material is τallow = 25 ksi.

5-46. The solid shaft of radius c is subjected to a torque T at its ends. Show that the maximum shear strain developed in the shaft is γmax = Tc/JG. What is the shear strain on an element located at point A, c/2 from the center of the shaft? Sketch the strain distortion of this element.

5-74. A rod is made from two segments: AB is steel and BC is brass. It is fixed at its ends and subjected to a torque of T = 680 N⋅m. If the steel portion has a diameter of 30mm. determine the required diameter of the brass portion so the reactions at the walls will be the same. Gst = 75 GPa, Gbr = 39 GPa. sol’n: θAB = θBC

680 N⋅m 1.60 m

0.75 m

BCAB JG

( )( )( ) ( )( )

( )( )

( )( )94

103932

6.1680

107532

75.0680

dππ=

( )( )( )( )( )3975.0

75030.06.1 44 =d

( )4 46103.3 md −=

md 0427.0=

mmd 7.42=

5-78. The composite shaft consists of a mid section that includes the 1-in-diameter solid shaft and a tube that is welded to the rigid flanges at A and B. Neglect the thickness of the flanges and determine the angle of twist of end C of the shaft relative to end D. The shaft is subjected to a torque of 100-lb.ft. The material is A-36 steel.

The solid shaft has a diameter of 0.75 in. If it is subjected to a torque shown, determine the maximum shear stress developed in regions BC and DE of the shaft. The bearings at A and B allow free rotation of the shaft.

( )( )( )

75.032

12375.025

75.032

12358.035

The solid shaft has a diameter of 0.75 in. If it is subjected to a torque shown, determine the maximum shear stress developed in regions CD and EF of the shaft. The bearings at A and F allow free rotation of the shaft.

( )( )( )

75.032

12375.015

5-49. The splined ends and gears attached to the A-36 steel shaft are subjected to the torque shown. Determine the angle of twist of end B with respect to the end A. The shaft has a diameter of 40 mm.

5-58. The engine of the helicopter is delivering 600 hp to he rotor shaft AB when the blade is rotating at 1200 rev/min. Determine the nearest 1/8 in. diameter of the shaft AB if the allowable shear stress is τ allow= 8 ksi and the vibrations limit the angle of twist of the shaft to 0.05 rad. The shaft is 2 ft. long and made from 1.2 steel. 5-59.The engine of the helicopter is delivering 600 hp to he rotor shaft AB when the blade is rotating at 1200 rev/min. Determine the nearest 1/8 in. diameter of the shaft AB if the allowable shear stress is τ allow= 10.5 ksi and the vibrations limit the angle of twist of the shaft to 0.05 rad. The shaft is 2 ft. long and made from 1.2 steel. 5-6. the solid 1.25 in diameter shaft is used to transmit the torques applied to the gears. If it is supported by smooth bearings at A and B, which do not resist torque, determine the shear stress developed in the shaft at points C and D. Indicate the shear stress on volume elements located at these points. 5-7. The shaft has an outer diameter of 1.25 in and an inner diameter of 1 in. If it is subjected to the applied torques as shown, determine the absolute maximum shear stress developed in the shaft. The smooth bearings at A and B do not resist torque.

5-50.The splined ends and gears attached to the A-36 steel shaft are subjected to the torques shown. Determine the angle of twist of gear C with respect to gear D. The shaft has a diameter of 40 mm.

6-1. Draw the shear and moment diagrams for the shaft. The bearings at A and B exerts only vertical reactions on the shaft.

( ) ( )[ ]

5.75.7

5.3124

.8.05.3125.024

25.024

5.3124

05.1248.0

250 mm 800 mm

-24 kN

7.5 kN

-6 kN⋅m

6-2. The load binder is used to support a load. If the force applied to the handle is 50lb, determine the tensions T1 and T2 in each end of the chain and then draw the shear and moment diagrams for the arm ABC.

200200

12 in. 3 in.

200 lb

-50 lb

-600 lb⋅in

5- 2. The solid shaft of radius r is subjected to a torque T. Determine the radius r’ of the inner core of the shaft that resists one-half of the applied torque (T/2).

5-3. The solid shaft of radius r is subjected to a torque T. Determine the radius r’ of the inner core of the shaft that resists one-quarter of the applied torque (T/4). 5-5. The copper pipe has an outer diameter of 2.50 in. and an inner diameter of 2.30 in. If it is tightly secured to the wall at C and the three torques are applied to it as shown, determine the shear stress developed at points A and B. These points lie on the pipe’s outer surface. Sketch the shear stress on the volume elements located at A and B.

5-10. The link acts as part of the elevator control for a small airplane. If the attached aluminum tube has an inner diameter of 25 mm, determine the maximum shear stress in the tube when the cable force of 600 N is applied to the cables. Also, sketch the shear stress distribution over the cross section.

600 lb-ft

350 lb-ft

450 lb-ft

5-13. A steel tube having an outer diameter of 2.5 in. is used to transmit 35 hp when turning at 2700 rev/min. Determine the inner diameter d of the tube to nearest 1/8 in. if the allowable shear stress is τallow = 10 ksi.

2.5 in.

5-17. The steel shaft has a diameter of 1 in. and is screwed into the wall using a wrench. Determine the maximum shear stress in the shaft if the couple forces have a magnitude of F = 30 lb.

8in. 8in.

5-21. The 20-mm-diameter steel shafts are connected using a brass coupling. If the yield point for the steel is (τY)st = 100 MPa, determine the applied torque T necessary to cause the steel to yield. If d = 40 mm, determine the maximum shear stress in the brass. The coupling has an inner diameter of 20 mm. 5-22. The coupling is used to connect the two shafts together. Assuming that the shear stress in the bolts is uniform, determine the number of bolts necessary to make the maximum shear in the shaft equal to the shear stress in the bolts. Each bolt has a diameter d. 5-29. The shaft has a diameter of 80 mm and due to friction at its surface within the hole, it is subjected to a variable torque described by the function

25 xxet =

To 80mm

5-30. The solid shaft has a linear taper from Ar at one end to Br at the other. Derive

and equation that gives the maximum shear stress in the shaft at a location x along the shaft’s axis.

5-35. A motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. If it is rotating at 200 rad/s. determine its largest inner diameter to the nearest 1/8 in. If the allowable shear stress for the material is τallow = 25 ksi. 5-36. The drive shaft of a tractor is made of a steel tube having an allowable shear stress of τallow = 6 ksi. If the outer diameter is 3 in. and the engine delivers 175 hp to the shaft when it is turning at 1250 rev/min. determine the minimum required thickness of the shaft’s wall.

5-37. A motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. and an inner diameter of 1.84 in. Determine the smallest angular velocity at which it can rotate if the allowable shear stress for the material is τallow = 25 ksi. 5-38. The 0.75 in. diameter shaft for the electric motor develops 0.5 hp and runs at 1740 rev/min. Determine the torque produced and compute the maximum shear stress in the shaft. The shaft is supported by ball bearings at A and B. 5-42. The motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. and an inner diameter of 1.84 in. Determine the smallest angular velocity at which it can rotate if the allowable shear stress for the material is τallow = 25 ksi.

5-74. A rod is made from two segments: AB is steel and BC is brass. It is fixed at its ends and subjected to a torque of T = 680 N⋅m. If the steel portion has a diameter of 30mm. determine the required diameter of the brass portion so the reactions at the walls will be the same. Gst = 75 GPa, Gbr = 39 GPa. 5-78. The composite shaft consists of a mid section that includes the 1-in-diameter solid shaft and a tube that is welded to the rigid flanges at A and B. Neglect the thickness of the flanges and determine the angle of twist of end C of the shaft relative to end D. The shaft is subjected to a torque of 100-lb.ft. The material is A-36 steel.

680 N⋅m 1.60 m

0.75 m

6-3. Draw the shear and moment diagrams for the shaft. The bearings at A and D exert only vertical reactions on the shaft. The loading is applied to the pulleys at B and C and E.

6-4. Draw the shear and moment diagrams for the beam.

( ) ( ) ( )

( )( )

( )( ) ( )

( ) ( ) ( )

( ) ( )0

.}1276.142

2711047806124.82{

.24.420

1511035804924.82

.16.1196

20803424.82

.36.1151

.1424.82

76.107

76.1423511080

76.142

613534110148049

−−=Σ

−=Σ

−++=

14 in. 20 in. 15 in. 12 in.

80 lb 110 lb 35 lb

82.4 lb

2.24 lb

107.76 lb

1151.36 lb.ft

1196 lb.ft

-420.24 lb.ft

( )( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

4282122162204

4282122164

4282124

:symmetry By

−−−−=Σ

−−−=

−−=Σ

−=Σ

ftkipM

4 ft 4 ft 4 ft 4 ft 4 ft

2 kip 2 kip 2 kip 2 kip

-2 kip

-4 kip

16 kip.ft

24 kip.ft

16 kip.ft

6-5. Draw the shear and moment diagrams for the rod. It is supported by a pin at A and a smooth plate at B. The plate slides w/in the groove and so it cannot support a vertical force, although it can support a moment.

( ) ( )

215415

−=Σ

4 m 2 m

30 kN.m

60 kN.m

6-6. Draw the shear and moment diagrams for the shaft. The bearings at A and B exert only vertical reactions on the shaft. Also, express the shear and moment in the shaft as a function of x w/in the region 125mm<x<725mm.

( ) ( )

( ) ( ) ( )075.02500675.08008.0625.815

.328.111

6.0800725.0625.815

.953.101

125.0625.815

37.1484

625.8151500800

625.815

751500675800800

−−=Σ

−=Σ

125 mm

815.625 N

1500 N

600 mm 75 mm

15.625 N

-1484.375 N

101953 N.m

111328 N.m

( )( ) mNxM

.100625.15

125.0800625.815

625.16

725mm x 125mm @

−−=

( ) ( )

1558210

+−=Σ

−−−=Σ

10 kN 8 kN

2 m 3 m

15 kN.m

-39 kN.m-15 kN.m

-75 kN.m

6-8. Draw the shear and moment for the pipe. The end screw is subjected to a horizontal force of 5kN. Hint: the reaction at the pin C must be replaced by equivalent loadings at point B on the axis of the pipe.

( )( )

4.04.0

8.054.0

⋅+⋅−=

⋅−=

mkNmkNM

400 mm

80 mmA

-0.4 kN.m

6-9. Draw the shear and moment diagrams for the beam. Hint: the 20kip load must be replaced by equivalent loadings at point C on the axis of the beam.

( ) ( )

433.333.13

20415867.11

467.11

01567.11

33.333.367.11

33.367.11

67.1138.3

20815122041512

−−=Σ

+−==

+=−=

=Σ=Σ

kipVkipV

kipRkipR

6-10. The engine crane is used to support the engine, which has a weight of 1200 lb. Draw the shear and moment diagrams of the boom ABC when it is in the horizontal position shown.

60006000

12001200

32002000

12002000

520005

⋅−−=Σ

⋅−=

−=Σ

3 ft 5 ft

1200 lb

6-12. Draw the shear and moment diagrams for the compound beam w/c is pin connected at B. It is supported by a pin at A and a fix wall at C.

( ) ( )

4818614

kipkipV

( )( )

⋅−⋅=

⋅−=

ftkipftkipM

ftkipM

8 kip6 kip

4 ft 6 ft 4 ft 4 ft

6-15. Draw the shear and moment diagrams for the beam. Also, determine the shear and moment in the beam as a function of x, where 3ft<x<15ft.

( )( )( )

SimilarBy

−∆=

=Σ↑

50778.67

50778.82

167.13

167.13125.1

167.13

6125.15012

( ) ( )

25.46667.1775.0

75.65.475.05.39167.13

9675.05.39167.13

5.13167.13

5.17667.17

35.1167.13

−+−=

−+−−=

+−−−=

−−−=

−−=

50 kip.ft

1.5 kip/ft

3 ft 12 ft

13.167

-4.833

7.79 kip.ft

-50 kip.ft

( ) ( )

( )( ) ( )( )

( )( )

( )( ) ( )( )0

4880012880051200

4880051200

48800128800

88008800

−−=Σ

−=Σ

64006400

800 lb/ft

6-17. The 50lb man sits in the center of the boat, w/c has a uniform width and weight per linear foot of 3lb/ft. Determine the maximum bending moment exerted on the boat. Assume that the water exerts a uniform distributed load upward on the bottom of the boat.

175.075

( ) ( )

5.735.713

150 lb

7.5 ft 7.5 ft

-75 lb

281.25 lb.ft

2°2°

6-18. The footing supports the load transmitted by the two columns. Draw the shear and moment diagrams for the footing if the reaction of soil pressure on the footing is assumed to be uniform.

141424

kipkipV

ftkipw

kipkipw

14 kip 14 kip

12 ft 6 ft6 ft

21 kip.ft

-7 kip

21 kip.ft

-7 kip

A B CD

( )( )

( )[ ]

5.05.0

5.055.2

305.27

305.125210

⋅−=

⋅+−=

⋅−=

−=−=

ftkipM

30 kip.ft2 kip/ft

5 ft 5 ft 5 ft

-10 kip

-0.5 kip

1°-25 kip.ft

-27.5 kip.ft

2.5 kip.ft

6-21. Draw the shear and moment diagrams for the beam and determine the shear and moment in the beam as a function of x, where 4ft<x<10ft. by symmetry:

( ) ( )

7510503200

41504450200

1501050

4150450

−+−=

−−−+−=

−−=

150 lb/ft

4 ft 6 ft 4 ft

200 lb.ft200 lb.ft

200 lb.ft

-200 lb.ft -200 lb.ft

-450 lb

450 lb

2°2°

6-23. The T-beam is subjected to the loading shown. Draw the shear and moment diagrams for the beam.

( ) ( ) ( )( )

ftkipM

lbRlbRR

⋅−=

+−−=

−−−=

600100200067.35660

61002000

33.233

9181001862000

67.3566;3800

181002000

6-24. The beam is bolted or pinned at A and rests on a bearing pad at B that exerts a uniform distributed loading on the beam over its 2ft length. Draw the shear and moment diagrams for the beam if it supports a uniform loading of 2kip/ft.

( ) ( )( )

( )( )( )[ ]

( ) ( )( )[ ]

( ) ( )( ) ( )( )0

124682118

582102

+−=Σ

⋅−=Σ

⋅+=Σ

⋅=Σ

ftkipM

ftkipw

6424max

LLWLLWM

⋅−⋅=Σ

LWLWLW

LWLWLR

LLWLLWLR

⋅−+⋅=

⋅⋅=

−=Σ

⋅=⋅

0244.0

11785.00934.0

232222

WLLLWM

⋅−

−=Σ

similarby

LWLWRV

LWLWLWV

LWLWVR

LWLWLW

LLWLLLWLLLWLR

−===

−=−=

−−===

126696633

+−=Σ

−=Σ

−−=

LLWLLLWLLLWL

LLWLWM

LWLWLW

LLWLLLWLLLWM

LLWLLWM

OOOMAX

6-32. The ski supports the 180lb weight of the man. If the snow loading on its bottom surface is trapezoidal as shown, determine the intensity w, and then draw the shear and moment diagrams for the ski.

( ) ( )

( )( ) ( )

3090105

15.140

018090

118040

−=Σ

+−=Σ

=−+=

( ) ( ) ( ) ( )

( ) 05.43

5.11275.168

75.1685.43

5.1125.112

5.1120

5.1125.112

:5.112

75.84375.168

25.45.4

symmetrybykNR

50 kN/m50 kN/m

4.5 m 4.5 m

112.5 kN

-112.5 kN

168.75 kN.m

6-35. The smooth pin is supported by two leaves A and B and subjected to a compressive load of 0.4kN/m caused by bar C. Determine the intensity of the distributed load wo of the leaves on the pin and draw the shear and moment diagrams for the pin.

( ) ( )[ ]

( )( )

( )[ ]

00026.0

012.000008.0

00008.002.03

5.1012.0

06.04.0012.0

02.05.12

06.04.02.02

2.6x10-3 kN⋅m

60 mm 20 mm20 mm

0.4 kN/m

0.012 kN

0.012 kN30 mm

( )( )

( )( )[ ]

90001800

1500121625

05.063009000

106251625

300010625

127500

30001500

3 kip/ft

1500 lb.ftA

12 ft 6 ft

-18000

6-37. The compound beam consists of two segments that are pinned together at B. Draw the shear and moment diagrams if it support the distributed loading shown.

x?0;V @

( )( ) ( )( )( )

⋅−=

−−=

15.1312

18 kN/m

12 kN/m

-45 kN

-54 kN.m

6-39. Draw the shear and moment diagrams for the beam and determine the shear and moment as a function of x.

( )( ) ( )( )( )

( ) ( ) ( )

( ) ( )

6005009

1003100200

200200

100500

200600200200

13200200

120032

1200300900

132002

15.132006

−+−=

−−−−=

−−+−=

−−−−−=

700700

3.690;0

600;200

(Mindanao State University-General Santos City, Philippines) BSEE-(G.R.F) students

6-43. A member having the dimensions shown is to be used to resist an internal bending moment of M = 2 kip⋅ft. Determine the maximum stress in the member if the moment is applied (a) about the z axis, (b) about the y axis. Sketch the stress distribution for each case. a) about z axis

b) about y axis

6 in. x

kiplbinkip

)/1000(/167.0

)12)(6(2

)12)(6)(12

kiplbinkip

)/1000)(/333.0(

)12)(3)(2(

)6)(12)(1(

6-45. A member has the triangular cross section shown. Determine the largest internal moment M that can be applied to the cross section without exceeding allowable tensile and compressive stresses of (σallow)t = 22 ksi and (σallow)s = 15 ksi, respectively.

We used the smaller value

2 in. 2 in.

4 in. 4 in.

inkipM

)32)(4((22

inkipM

)32)(4((15

ftkipM

ftinkipM

6-46. A member has the triangular cross section shown. If a moment of M = 800 lb⋅ft is applied to the cross section, determine the maximum tensile and compressive bending stresses in the member. Also, sketch a three dimensional view of the stress distribution acting over the cross section.

2 in. 2 in.

4 in. 4 in.

( )( )

31.29600

)32)(4(

( )( )

15.19600

6-49. A beam has the cross section shown. If it is made of steel that has an allowable stress of σallow =24 ksi, determine the largest internal moment the beam can resist if the moment is applied (a) about the z axis, (b) about the y axis. a) about z-axis

b) about y-axis

0.25 in.

( )( ) ( )( )( ) ( )( )

( ) ( )

ftkipM

inkipM

.7.249

25.03/8125.3324

8125.33

6.025.0125.325.06.0

25.06.02

( )( ) ( )( )

ftkipM

inkipM

.06.72

25.06.0

6.025.02

6-50. The beam is subjected to a moment of M = 40 kN⋅m. Determine the bending stress acting at points A and B. sketch the results on a volume element acting at each of these points.

50 mm 50 mm

M = 40 kN⋅m

( )( ) ( )( )

( )( )( )

)10(51.1

1000025.0400

)10(5.1

1000075.0400

)10(51.1

05.005.02

15.005.0

6-73. Determine the smallest allowable diameter of the shaft which is subjected to the concentrated forces. The sleeve bearings at A and B support only vertical forces, and the allowable bending stress is σallow = 160 MPa.

0.8 1.2 0.6

600 400

( ) ( ) ( )

( )( )( )

0156.0

)10(160

8.06008.14002.1

6.040026002.1

6-81. The beam is subjected to the load P at its center. Determine the placement a of the supports so that the absolute maximum bending stress in the beam is as large as possible. What is this stress?

For symmetrical loading

≤≤

=Σ↑+

( )( )

maxmax

b L/2 L/2

6-89. The steel beam has the cross-sectional area shown. If w = 5 kip⋅ft, determine the absolute maximum bending stress in the beam.

( ) ( )

( )( ) ( )( ) ( )( )

( )( )( )

3.512160

103.015.53.08

30.082

24044024

=Σ↑+

8 ft 8 ft 8 ft

0.30 in.

10 in.

0.30 in.

6-90. The beam has a rectangular cross section as shown. Determine the largest load P that can be supported on its overhanging ends so that the bending stress does not exceed σmax = 10 MPa.

( )( )

)10(953.1

125.05.1)10(10

)10(953.1

25.05.0

35.15.1

=Σ↑+

1.5 m 1.5 m 1.5 m

250 mm

150 mm

6-91. The beam has the rectangular cross section shown. If P =12 kN, determine that absolute maximum bending stress in the beam. Sketch the stress distribution acting over the cross section.

( ) ( ) ( )

( )( )

)10(953125.1

125.018000

)10(953125.1

125.018

)10(953125.1

25.015.0

3125.1125.1

=Σ↑+

1.5 m 1.5 m 1.5 m

250 mm

150 mm

6-93. A log that is 2 ft in diameter is to be cut into a rectangular section for use as a simply supported beam. If the allowable bending stress for the wood is σallow = 8 ksi, determine the largest load P that can be supported if the width of the beam is b = 8 in.

8 ft 8 ft

6-102. The member has a square cross section and is subjected to a resultant moment of M = 850 N⋅m as shown. Determine that bending stress at each corner and sketch the stress distribution produced by M. Set θ = 45°.

125 mm

250 mm

M = 850 N⋅m

6-103. The member has a square cross section and is subjected to a resultant moment of M = 850 N⋅m as shown. Determine that bending stress at each corner and sketch the stress distribution produced by M. Set θ = 30°.

125 mm

250 mm

M = 850 N⋅m

6-105. The T-beam is subjected to a moment of M = 150 kip⋅in. directed as shown. Determine the maximum bending stress in the beam and the orientation of the neutral axis. The location y of the centroid, C, must be determined.

6-185. Determine the bending stress distribution in the beam at section a-a. sketch the distribution in three dimension acting over the cross section.

80 N 80 N

400 mm 400 mm300 mm 300 mm

100 mm

15 mm 15 mm

6-51. The aluminum machine part is subjected to a moment of M = 75 N⋅m. Determine the bending stress created at points B and C on the cross section. Sketch the results on a volume element located at each of these points.

B 10 mm

M = 75 N⋅m

10 mm 10 mm

6-53. A beam is constructed from four pieces of wood, glued together as shown. If the moment acting on the cross section is M = 450 N⋅m, determine the resultant force that bending stress produces on the top board A and on the side board B.

200 mm

M = 40 kN⋅m

6-47. The beam is made from three boards nailed together as shown. If the moment acting on the cross section is M = 600 N⋅m, determine the maximum bending stress in the beam. Sketch a three-dimensional view of the stress distribution acting over the cross section.

M = 600 N⋅m

200 mm

6-106. If the internal moment acting on the cross section of the strut has a magnitude of M = 800 N⋅m and is directed as shown, determine the bending stress at points A and B. The location z of the centroid C of the strut’s cross-sectional area must be determined. Also, specify the orientation of the neutral axis.

M = 800 N⋅m

200 mm

150 mm

6-71. The axle of the freight car is subjected to wheel loadings of 20 kip. If it is supported by two journal bearings at C and D, determine that maximum bending stress developed at the center of the axle, where the diameter is 5.5 in.

10 in.

20 kip

A B DC

10 in.60 in.

20 kip

6-74. Determine the absolute maximum bending stress in the 1.5-in.-diameter shaft which is subjected to the concentrated forces. The sleeve bearings at A and B support only vertical forces.

18 in.

15 in.

400 lb

300 lb

6-75. Determine the smallest allowable diameter of the shaft which is subjected to the concentrated forces. The sleeve bearings at A and B support only vertical forces, and the allowable bending stress is σallow = 22 ksi.

18 in.

15 in.

400 lb

300 lb

6-72. Determine the absolute maximum bending stress in the 30-mm-diameter shaft which is subjected to the concentrated forces. The sleeve bearings at A and B support only vertical forces.

0.8 m 1.2 m 0.6 m

600 N 400 N

6-79. The steel shaft has a circular cross section with a diameter of 2 in. It is supported on smooth journal bearings A and B, which exert only vertical reactions on the shaft. Determine the absolute maximum bending stress if it is subjected to the pulley loadings shown.

20 in. 20 in. 20 in. 20 in.

500 lb 300 lb 500 lb

The solid shaft of radius r is subjected to a torque T. Determine the radius r’ of the inner core of the shaft that resists one-half of the applied torque (T/2).

by ratio and proportion:

7651.32cos'

841.0'

The solid shaft of radius r is subjected to a torque T. Determine the radius r’ of the inner core of the shaft that resists one-quarter of the applied torque (T/4).

45sin'

707.0'

ππτ

The copper pipe has an outer diameter of 2.50 in. and an inner diameter of 2.30 in. If it is tightly secured to the wall at C and the three torques are applied to it as shown, determine the shear stress developed at points A and B. These points lie on the pipe’s outer surface. Sketch the shear stress on the volume elements located at A and B. sol’n:

600 lb-ft

350 lb-ft

450 lb-ft

450 lb⋅ft = 5400 lb⋅in 350 lb⋅ft = 4200 lb⋅in 600 lb⋅ft = 7200 lb⋅in

5400 + 4200 – 7200 = 2400

( )( )( )

6.2758

3.25.2

5.2240016

7200 - 4200 = 3000

( )( )( )

3.3448

3.25.2

5.2300016

The link acts as part of the elevator control for a small airplane. If the attached aluminum tube has an inner diameter of 25 mm, determine the maximum shear stress in the tube when the cable force of 600 N is applied to the cables. Also, sketch the shear stress distribution over the cross section.

)10108(

)0125.0)(90(

)10108(

)075.0)(90(

)025.0035.0(32

)075.0)(600(2

A steel tube having an outer diameter of 2.5 in. is used to transmit 35 hp when turning at 2700 rev/min. Determine the inner diameter d of the tube to nearest 1/8 in. if the allowable shear stress is τallow = 10 ksi.

2.5 in.

Given:

/75.min/2700

/.250,193

/000,10

srevrevf

sftlbhp

allowτ

Solution:

.97.4084

)75.0(2

)]75.0(2[19250

)2)(25.1)(4084097()25.1(

)25.1)(97.4084(

2)25.1(

)25.1(2

)25.1(97.4084

ind 4.2= ; 5.08

The steel shaft has a diameter of 1 in. and is screwed into the wall using a wrench. Determine the maximum shear stress in the shaft if the couple forces have a magnitude of F = 30 lb. sol’n:

8in. 8in.

( )( )( )

72.7818

68.075.0

75.021016

πτ ( )

( )( )( )

121016

AB D = 0.75 in d = 0.68 in BC D = 1 in d = 0.86 in

( ) ( )inlbT

=+=∑210

815615

The 20-mm-diameter steel shafts are connected using a brass coupling. If the yield point for the steel is (τY)st = 100 MPa, determine the applied torque T necessary to cause the steel to yield. If d = 40 mm, determine the maximum shear stress in the brass. The coupling has an inner diameter of 20 mm.

To find the T; steel;

07.157

)010.0(

210100

To find the max.., τ; brass;

)01.002.0(

)02.0)(07.157(2

The coupling is used to connect the two shafts together. Assuming that the shear stress in the bolts is uniform, determine the number of bolts necessary to make the maximum shear in the shaft equal to the shear stress in the bolts. Each bolt has a diameter d. Soln:

The shaft has a diameter of 80 mm and due to friction at its surface within the hole, it is subjected to a variable torque described by the function

25 xxet =

To 80mm

Solution: d=80mm=0.08 m

25 xxet =

πτ =

dxxeTx2

025∫=

dxxeT x

let u= 2x du=2x dx du/2=x dx x=0, u=0 x=2,u=4

25 dueT

05.12 ue=

[ ]15.12 4 −= e

NmT 670=

MPa66.6=τ

33 )1080(

)670(16−

πτ =

The solid shaft has a linear taper from Ar at one end to Br at the other. Derive and

equation that gives the maximum shear stress in the shaft at a location x along the shaft’s axis.

Solution:

1rrr B +=

r BA −=

xLrrr BA ))((1

−−=

xLrrrr BA

))(( −−+=

xLrrLrr BAB ))(( −−+

xLrrxLLrr BAB )()( −−−+

xLrxLLrr AB )()]([ −+−−

xLrxrr AB )()( −+

xLrxrr AB )( −+

πτ = ;

πτ =

AB −+=

The drive shaft AB of an automobile is made of a steel having an allowable shear stress of τallow = 8 ksi. If the outer diameter of the shaft is 2.5 in. and the engine delivers 200 hp to the shaft when it is turning at 1140 rev/min, determine the minimum required thickness of the shaft’s wall.

Given: psiksi 80008 ==τ

inD 5.2= slbfthp /000,100200 ==Ρ sec/19min/1140 revrevf ==

?=t Solutions:

)/19(2

/000,100

lbftT 66.837= or lbin89.051,10=

1644 dD

)5.2)(89.051,10(16000,8

44 dpsi

)000,8(

65.402075)5.2( 44

π=− d

998.1539

ind 19.2= Thickness:

DR = ;

19.2=r

25.1=R 1.1=r rRt −=

1.125.1

The drive shaft AB of an automobile is to be designed as a thin walled tube. The engine delivers 150 hp when the shaft is turning at 1500 rev/min. Determine the minimum thickness of the shaft’s wall if the shaft’s outer diameter is 2.5 in. The material has an allowable shear stress τallow = 7 ksi.

000,77

/25min/500,1

/000,75150

psiksi

srevrevf

slbfthp

Solution

)/25(2

/000,75

58.729,5

46.477

5968.27

42.1039

)000,7(

183,22939

)5.2)(58.5729(16000,7

Thickness:

15.125.1

The drive shaft of a tractor is to be designed as a thin-walled tube. The engine delivers 200 hp when the shaft is turning at 1200 rev/min. Determine the minimum thickness of the wall of the shaft if the shaft’s outer diameter is 3 in. The material has an allowable shear stress of τallow = 7 ksi. Given:

srevrevf

sftlbhp

/20min/1200

/.100000200

psi7000= ?max =t

Solution:

slbftT

3.9549

)/20(2

/100000

84.2081

)7000(

4.45836681

)3)(3.9549(167000

Max Thickness:

3925.12

A motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. If it is rotating at 200 rad/s. determine its largest inner diameter to the nearest 1/8 in. If the allowable shear stress for the material is τallow = 25 ksi.

( )( )( )( )

( )ind

11.616

2150001625000

250000

lbfthp

slbfthp

/250000/500

/25000

The drive shaft of a tractor is made of a steel tube having an allowable shear stress of τallow = 6 ksi. If the outer diameter is 3 in. and the engine delivers 175 hp to the shaft when it is turning at 1250 rev/min. determine the minimum required thickness of the shaft’s wall.

( )( )( )( )

thickness

87.32085639

5.24.8021166000

.4.8021

.45.668

833.202

( )( )( )( )

/17.483

5.42.132216

A motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. and an inner diameter of 1.84 in. Determine the smallest angular velocity at which it can rotate if the allowable shear stress for the material is τallow = 25 ksi.

( )( )

inlbTT

2.11137

.2.11137;84.12

21625000

The 0.75 in. diameter shaft for the electric motor develops 0.5 hp and runs at 1740 rev/min. Determine the torque produced and compute the maximum shear stress in the shaft. The shaft is supported by ball bearings at A and B.

( )( )

12.1816

.12.18

The motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. and an inner diameter of 1.84 in. Determine the smallest angular velocity at which it can rotate if the allowable shear stress for the material is τallow = 25 ksi.

( )( )

( ) ( )( )

/4.296

22.11137

84.122500103

The solid shaft of radius c is subjected to a torque T at its ends. Show that the maximum shear strain developed in the shaft is γmax = Tc/JG. What is the shear strain on an element located at point A, c/2 from the center of the shaft? Sketch the strain distortion of this element.

A rod is made from two segments: AB is steel and BC is brass. It is fixed at its ends and subjected to a torque of T = 680 N⋅m. If the steel portion has a diameter of 30mm. determine the required diameter of the brass portion so the reactions at the walls will be the same. Gst = 75 GPa, Gbr = 39 GPa. sol’n: θAB = θBC

680 N⋅m 1.60 m

0.75 m

BCAB JG

( )( )( ) ( )( )

( )( )

( )( )94

103932

6.1680

107532

75.0680

dππ=

( )( )( )( )( )3975.0

75030.06.1 44 =d

( )4 46103.3 md −=

md 0427.0=

mmd 7.42=

The composite shaft consists of a mid section that includes the 1-in-diameter solid shaft and a tube that is welded to the rigid flanges at A and B. Neglect the thickness of the flanges and determine the angle of twist of end C of the shaft relative to end D. The shaft is subjected to a torque of 100-lb.ft. The material is A-36 steel.

5-14. The solid shaft has a diameter of 0.75 in. If it is subjected to a torque shown, determine the maximum shear stress developed in regions BC and DE of the shaft. The bearings at A and B allow free rotation of the shaft.

( )( )( )

( )( )( )( )

12375.040

12375.035

5-15. The solid shaft has a diameter of 0.75 in. If it is subjected to a torque shown, determine the maximum shear stress developed in regions CD and EF of the shaft. The bearings at A and F allow free rotation of the shaft.

( )( )( )

12375.015

5-49. The splined ends and gears attached to the A-36 steel shaft are subjected to the torque shown. Determine the angle of twist of end B with respect to the end A. The shaft has a diameter of 40 mm.

( )( )( ) ( )( )

( )°=

0578.0

3.57010079813.0

107502.0

23.0300

107502.0

24.0200

107502.0

2.05.0400

949494/

πππθ

5-58. The engine of the helicopter is delivering 600 hp to he rotor shaft AB when the blade is rotating at 1200 rev/min. Determine the nearest 1/8 in. diameter of the shaft AB if the allowable shear stress is τ allow= 8 ksi and the vibrations limit the angle of twist of the shaft to 0.05 rad. The shaft is 2 ft. long and made from 1.2 steel.

5-59.The engine of the helicopter is delivering 600 hp to he rotor shaft AB when the blade is rotating at 1200 rev/min. Determine the nearest 1/8 in. diameter of the

( )( )

16.0;6.2

2864816

.28648

.3.2387

300000

shaft AB if the allowable shear stress is τ allow= 10.5 ksi and the vibrations limit the angle of twist of the shaft to 0.05 rad. The shaft is 2 ft. long and made from 1.2 steel.

( )( )

14.0;4.2

2864816

.28648

300000

5-6. the solid 1.25 in diameter shaft is used to transmit the torques applied to the gears. If it is supported by smooth bearings at A and B, which do not resist torque, determine the shear stress developed in the shaft at points C and D. Indicate the shear stress on volume elements located at these points.

( )( )( )

32625.0600

3216251500

5-7. The shaft has an outer diameter of 1.25 in and an inner diameter of 1 in. If it is subjected to the applied torques as shown, determine the absolute maximum shear stress developed in the shaft. The smooth bearings at A and B do not resist torque.

( )( )( )( )

inlbTdD

25.1150016

.1500;16

max44max

5-50.The splined ends and gears attached to the A-36 steel shaft are subjected to the torques shown. Determine the angle of twist of gear C with respect to gear D. The shaft has a diameter of 40 mm.

( )( )( ) ( )( )

107502.0

24.0200

107502.0

25.0400

/)10(75

ππθ

The simply supported shaft has a moment of inertia of 2I for given region BC and a moment of inertia I for regions AB and CD. Determine the maximum deflection

of the beam due to the load P.

128128

1536512192

384512

333333

PLPLPLT

PLPLPLPLPLPLT

+++++=

L/4 L/4 L/4 L/4

I I 2I

Determine the equation of the elastic curve for the beam using the x coordinate that is valid for 0 ≤ x ≤ L/2. Specify the slope at A and the beam’s maximum deflection. EI is constant.

Determine the equations of the elastic curve using the x1 and x2 coordinates.

Specify the slope at A and the maximum deflection. EI is constant.

The beam is made of two rods and is subjected to the concentrated load P. Determine the slope at C. The moments of inertia of the rods are IAB and IAC, and the modulus of elasticity is E. A

Determine the elastic curve for the cantilevered beam, which is subjected to the couple moment Mo. Also compute the maximum slope and maximum deflection of the beam. EI is constant.

CxMEIy

Determine the deflection at the center of the beam and the slope at B. EI is constant.

−+−=

+++−

+++−=

[ ]xLLxxLEI

−+−=

−+−

Determine the elastic curve for the simply supported beam, which is subjected to the couple moments Mo. Also, compute the maximum slope and the maximum deflection of the beam. EI is constant.

Determine the equations of the elastic curve using the coordinates x1 and x2, and specify the slope at A. EI is constant.

belowindicatessignnegative

The floor beam of the airplane is subjected to the loading shown. Assuming that the fuselage exerts only vertical reactions on the ends of the beam, determine the maximum deflection of the beam. EI is constant.

80 lb/ft

8 ft 2 ft 2 ft

( ) ( )( )

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

147204

147202

40160'

2401040320''

101080

688012

640880

ftlbEI

xxxxEIy

CxCxxxEIy

CxxxEIy

xxxEIy

−−−−1

++−−−1

+−−−+=

−−−+=

−−−

−−+=

.77.18

ftkipEI

The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant.

8 ft 4 ft 4 ft

3 kip/ft

5 kip-ft 5 kip-ft

( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) 343432

.13624128

1212312

ftkipxxxxxx

Cxxxxxx

CxCxxxxx

CxxxxxEIy

+−−+−+−−−+−=

++−+−+−−−+−=

+−+−+−−−+−=

−−+−+

−−−−+−=

The shaft is supported at A by a journal bearing that exerts only vertical reactions on the shaft, and at C by a thrust bearing that exerts horizontal and vertical reactions on the shaft. Determine the equation of the elastic curve. EI is constant.

( ) ( )( )

( )( )( )

( )( )

−++−=

++−+

−++−=

−+−=

PbxEIy

CaxbaP

PbxEIy

axRxRM

The shafts support the two pulley loads shown. Determine the equation of the

elastic curve. The bearing at A and B exerts only vertical reactions on the shaft. EI

is constant.

20 in. 20 in. 20 in.

40 lb 60 lb

( ) ( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( )[ ] 3333

.40064033.182067.667.11

40064033.182067.667.1

402067.64067.10

4033.182067.667.1

40110204010''

402040

6060204040

inlbxxxxEI

xxxxEIy

CxCxxxEIy

xxxEIy

xRxxRM

+−+−−−=

+−−=

++−+−−−=

−−

−+−−−=

The shafts support the two pulley loads shown. Determine the equation of the elastic curve. The bearing at A and B exerts only vertical reactions on the shaft. EI is constant.

12 in. 24 in. 24 in.

50 lb 80 lb

( ) ( ) ( )

( ) ( )

( ) ( )[ ] 3333

.576016803633.13121.1733.81

576016803633.13121.1733.8

60950400

12144000

3633.13121.1733.8

125.102

3680125.10250

2480481250

inlbxxxxEI

xxxxEIy

eqnequating

CxCxxxEIy

−+−−−+−=

++−=

++−−−+−=

−−

−−−+−=

9 ft. 15 ft.

6 kip/ft

( ) ( )( ) ( ) ( )( )

( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )2.24924180

4.7724

1.99180

194.77

156692

05.71561593

CxCxxx

xxxxxxxEIy

xxyxRxxyM

++−+−+−

++−+−+−=

+−+−+−=

−+−+−=

−+−+

−−+−+

−−+

( ) ( )

−+−+−=

55.26365.256

55.2636

eqnequating

Determine the slope and deflection at C. EI is constant.

B 15 ft.

30 ft.

15 kip

Determine the slope and deflection at B. EI is constant.

Determine the maximum slope and the maximum deflection of the beam. EI is constant.

LMLMEIy

LxMxMEIy

CxMEIy

4 kip/fit

4 kip 2 kip

8 ft. 8 ft.

1.5 m 3 m

( ) ( ) ( )

( ) ( )( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) 3333

.905.673

5.45.675.43

0,5.4@

105.4105.110'

205.4205.120

205.45.1

5.1205.4203

mkNxxxxEI

CxxxxEIy

CxCxxxEIy

CxxxEIy

xxRxRM

−+−−+−=

++−+=

++−−+−=

+−+=

++−−+−=

+−−+−=

−−+−=

Determine the equation of the elastic curve. Specify the slope at A. EI is constant.

The beam is subjected to the linearly varying distributed load. Determine the maximum slope of the beam. EI is constant.

( ) ( )

( )360

xwLxwEIy

xxyxRM

proportionandratioby

( ) ( )

LwLwLwEIy

−−=

The beam is subjected to the load shown. Determine the equation of the slope and elastic curve. EI is constant. A

3 kN/m

15 kN⋅m

5 m 3 m

( ) ( )( )

( )( )( )

( ) ( ) ( )

( ) ( )

−+−+−=

−+−+−

+−+−=

−−+−+

125.38

5575.1

125.32

5525.5

225.21

5575.00

5575.1

5525.5

225.2'

155.253)5(

0155.2535

The wooden beam is subjected to the load shown. Determine the equation of the elastic curve. Specify the deflection at the end C. EW=1.6(103) ksi.

0.8 kip/ft 1.5 kip

x 6 in.

12 in.

( ) ( ) ( )( ) ( )

( )( )

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )

( )( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ).765.0

106.1,18@

9.8;091080

98.0905.0

0;0,0@

194.094.5

8.03.0''

194.094.5

04.598.02

1185.19

CxCxxxxx

Cxxxxx

xxxxEIy

proportionandratioby

xyxxyxxM

==+−−

+−−

−+−+−+

−−=

−−+−+−+

−−=

=+−−

Determine the deflection and slope at C. EI is constant.

Determine the deflection at C and the slope of the beam at A, B, and C. EI is constant.

8 kN⋅m

( ) ( )

xCslope

xBslope

xAslope

CxCxxEIy

CxxEIy

kNRkNR

4040,40'

1616,16'

=−=−=

+−+−=

++−=

++−+−=

+−+−=

xxxEIy

xCdeflection

+−+−=

Determine the slope at B and deflection at C. EI is constant.

a a a a

The beam is subjected to the load shown. Determine the slope at B and deflection at C. EI is constant.

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

axCdeflection

axBslope

xwaaxwwxaxwwax

waaxwwxaxwwax

CxCaxwwxaxwwax

Caxwwxaxwwax

waxEIy

wxaxaxwxRM

2424246

22424246

=∆−=

−−−+=

−−−−−

+−−+=

++−−−−

+−−−−

−−−−+=

−−−−

−−+=

wawaaR

Determine the elastic curve for the simply supported beam the x coordinate 0 ≤x ≤ L/2. Also, determine the slope at A, and the maximum deflection of the beam. EI is constant.

282120

:PROPORTION AND RATIO BY

CxCLxw

LwLwLR

LRLLLwLLw

LwLwRR

+++−=

++−=

LLwLLwL

oooMAX

224260

Draw the shear and moment diagrams for the shaft. The bearings at A and B exerts only vertical reactions on the shaft.

( ) ( )[ ]

5.75.7

5.3124

.8.05.3125.024

25.024

5.3124

05.1248.0

250 mm 800 mm

-24 kN

7.5 kN

-6 kN⋅m

The load binder is used to support a load. If the force applied to the handle is 50lb, determine the tensions T1 and T2 in each end of the chain and then draw the shear and moment diagrams for the arm ABC.

200200

12 in. 3 in.

200 lb

-50 lb

-600 lb⋅in

Draw the shear and moment diagrams for the shaft. The bearings at A and D exert only vertical reactions on the shaft. The loading is applied to the pulleys at B and C and E.

Draw the shear and moment diagrams for the beam.

( ) ( ) ( )

( )( )

( )( ) ( )

( ) ( ) ( )

( ) ( )0

.}1276.142

2711047806124.82{

.24.420

1511035804924.82

.16.1196

20803424.82

.36.1151

.1424.82

76.107

76.1423511080

76.142

613534110148049

−−=Σ

−=Σ

−++=

14 in. 20 in. 15 in. 12 in.

80 lb 110 lb 35 lb

82.4 lb

2.24 lb

107.76 lb

1151.36 lb.ft

1196 lb.ft

-420.24 lb.ft

( )( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

4282122162204

4282122164

4282124

:symmetry By

−−−−=Σ

−−−=

−−=Σ

−=Σ

ftkipM

4 ft 4 ft 4 ft 4 ft 4 ft

2 kip 2 kip 2 kip 2 kip

-2 kip

-4 kip

16 kip.ft

24 kip.ft

16 kip.ft

Draw the shear and moment diagrams for the rod. It is supported by a pin at A and a smooth plate at B. The plate slides w/in the groove and so it cannot support a vertical force, although it can support a moment.

( ) ( )

215415

−=Σ

4 m 2 m

30 kN.m

60 kN.m

Draw the shear and moment diagrams for the shaft. The bearings at A and B exert only vertical reactions on the shaft. Also, express the shear and moment in the shaft as a function of x w/in the region 125mm<x<725mm.

125 mm

815.625 N

1500 N

600 mm 75 mm

15.625 N

-1484.375 N

101953 N.m

111328 N.m

( ) ( )

( ) ( ) ( )075.02500675.08008.0625.815

.328.111

6.0800725.0625.815

.953.101

125.0625.815

37.1484

625.8151500800

625.815

751500675800800

−−=Σ

−=Σ

( )( ) mNxM

.100625.15

125.0800625.815

625.16

725mm x 125mm @

−−=

( ) ( )

1558210

+−=Σ

−−−=Σ

10 kN 8 kN

2 m 3 m

15 kN.m

-39 kN.m-15 kN.m

-75 kN.m

Draw the shear and moment for the pipe. The end screw is subjected to a horizontal force of 5kN. Hint: the reaction at the pin C must be replaced by equivalent loadings at point B on the axis of the pipe.

( )( )

4.04.0

8.054.0

⋅+⋅−=

⋅−=

mkNmkNM

400 mm

80 mmA

-0.4 kN.m

The engine crane is used to support the engine, which has a weight of 1200 lb. Draw the shear and moment diagrams of the boom ABC when it is in the horizontal position shown.

60006000

12001200

32002000

12002000

520005

⋅−−=Σ

⋅−=

−=Σ

3 ft 5 ft

1200 lb

Draw the shear and moment diagrams for the compound beam w/c is pin connected at B. It is supported by a pin at A and a fix wall at C.

( ) ( )

4818614

kipkipV

( )( )

⋅−⋅=

⋅−=

ftkipftkipM

ftkipM

8 kip6 kip

4 ft 6 ft 4 ft 4 ft

Draw the shear and moment diagrams for the beam. Also, determine the shear and moment in the beam as a function of x, where 3ft<x<15ft.

( )( )( )

SimilarBy

−∆=

=Σ↑

50778.67

50778.82

167.13

167.13125.1

167.13

6125.15012

( ) ( )

25.46667.1775.0

75.65.475.05.39167.13

9675.05.39167.13

5.13167.13

5.17667.17

35.1167.13

−+−=

−+−−=

+−−−=

−−−=

−−=

50 kip.ft

1.5 kip/ft

3 ft 12 ft

13.167

-4.833

7.79 kip.ft

-50 kip.ft

( ) ( )

( )( ) ( )( )

( )( )

( )( ) ( )( )0

4880012880051200

4880051200

48800128800

88008800

−−=Σ

−=Σ

64006400

800 lb/ft

The 50lb man sits in the center of the boat, w/c has a uniform width and weight per linear foot of 3lb/ft. Determine the maximum bending moment exerted on the boat. Assume that the water exerts a uniform distributed load upward on the bottom of the boat.

175.075

( ) ( )

5.735.713

150 lb

7.5 ft 7.5 ft

-75 lb

281.25 lb.ft

2°2°

The footing supports the load transmitted by the two columns. Draw the shear and moment diagrams for the footing if the reaction of soil pressure on the footing is assumed to be uniform.

141424

kipkipV

ftkipw

kipkipw

14 kip 14 kip

12 ft 6 ft6 ft

21 kip.ft

-7 kip

21 kip.ft

-7 kip

A B CD

( )( )

( )[ ]

5.05.0

5.055.2

305.27

305.125210

⋅−=

⋅+−=

⋅−=

−=−=

ftkipM

30 kip.ft2 kip/ft

5 ft 5 ft 5 ft

-10 kip

-0.5 kip

1°-25 kip.ft

-27.5 kip.ft

2.5 kip.ft

Draw the shear and moment diagrams for the beam and determine the shear and moment in the beam as a function of x, where 4ft<x<10ft. by symmetry:

( ) ( )

7510503200

41504450200

1501050

4150450

−+−=

−−−+−=

−−=

150 lb/ft

4 ft 6 ft 4 ft

200 lb.ft200 lb.ft

200 lb.ft

-200 lb.ft -200 lb.ft

-450 lb

450 lb

2°2°

( ) ( ) ( ) ( )

( ) 05.43

5.11275.168

75.1685.43

5.1125.112

5.1120

5.1125.112

:5.112

75.84375.168

25.45.4

symmetrybykNR

50 kN/m50 kN/m

4.5 m 4.5 m

112.5 kN

-112.5 kN

168.75 kN.m

The smooth pin is supported by two leaves A and B and subjected to a compressive load of 0.4kN/m caused by bar C. Determine the intensity of the distributed load wo of the leaves on the pin and draw the shear and moment diagrams for the pin.

( ) ( )[ ]

( )( )

( )[ ]

00026.0

012.000008.0

00008.002.03

5.1012.0

06.04.0012.0

02.05.12

06.04.02.02

2.6x10-3 kN⋅m

60 mm 20 mm20 mm

0.4 kN/m

0.012 kN

0.012 kN30 mm

( )( )

( )( )[ ]

90001800

1500121625

05.063009000

106251625

300010625

127500

30001500

3 kip/ft

1500 lb.ftA

12 ft 6 ft

-18000

Draw the shear and moment diagrams for the beam and determine the shear and moment as a function of x.

( )( ) ( )( )( )

( ) ( ) ( )

( ) ( )

6005009

1003100200

200200

100500

200600200200

13200200

120032

1200300900

132002

15.132006

−+−=

−−−−=

−−+−=

−−−−−=

18 kN/m

12 kN/m

-45 kN

-54 kN.m

A member having the dimensions shown is to be used to resist an internal bending moment of M = 2 kip⋅ft. Determine the maximum stress in the member if the moment is applied (a) about the z axis, (b) about the y axis. Sketch the stress distribution for each case. a) about z axis

b) about y axis

6 in. x

kiplbinkip

)/1000(/167.0

)12)(6(2

)12)(6)(12

kiplbinkip

)/1000)(/333.0(

)12)(3)(2(

)6)(12)(1(

A member has the triangular cross section shown. Determine the largest internal moment M that can be applied to the cross section without exceeding allowable tensile and compressive stresses of (σallow)t = 22 ksi and (σallow)s = 15 ksi, respectively.

2 in. 2 in.

4 in. 4 in.

inkipM

)32)(4((22

inkipM

)32)(4((15

ftkipM

ftinkipM

A member has the triangular cross section shown. If a moment of M = 800 lb⋅ft is applied to the cross section, determine the maximum tensile and compressive bending stresses in the member. Also, sketch a three dimensional view of the stress distribution acting over the cross section.

2 in. 2 in.

4 in. 4 in.

( )( )

31.29600

)32)(4(

( )( )

15.19600

A beam has the cross section shown. If it is made of steel that has an allowable stress of σallow =24 ksi, determine the largest internal moment the beam can resist if the moment is applied (a) about the z axis, (b) about the y axis. a) about z-axis

b) about y-axis

0.25 in.

( )( ) ( )( )( ) ( )( )

( ) ( )

ftkipM

inkipM

.7.249

25.03/8125.3324

8125.33

6.025.0125.325.06.0

25.06.02

( )( ) ( )( )

ftkipM

inkipM

.06.72

25.06.0

6.025.02

The beam is subjected to a moment of M = 40 kN⋅m. Determine the bending stress acting at points A and B. sketch the results on a volume element acting at each of these points.

( )( ) ( )( )

( )( )( )

)10(51.1

1000025.0400

)10(5.1

1000075.0400

)10(51.1

05.005.02

15.005.0

50 mm 50 mm

M = 40 kN⋅m

Determine the smallest allowable diameter of the shaft which is subjected to the concentrated forces. The sleeve bearings at A and B support only vertical forces, and the allowable bending stress is σallow = 160 MPa.

0.8 1.2 0.6

600 400

( ) ( ) ( )

( )( )( )

0156.0

)10(160

8.06008.14002.1

6.040026002.1

The beam is subjected to the load P at its center. Determine the placement a of the supports so that the absolute maximum bending stress in the beam is as large as possible. What is this stress?

≤≤

=Σ↑+

( )( )

maxmax

b L/2 L/2

The beam has a rectangular cross section as shown. Determine the largest load P that can be supported on its overhanging ends so that the bending stress does not exceed σmax = 10 MPa.

( )( )

)10(953.1

125.05.1)10(10

)10(953.1

25.05.0

35.15.1

=Σ↑+

1.5 m 1.5 m 1.5 m

250 mm

150 mm

The beam has the rectangular cross section shown. If P =12 kN, determine that absolute maximum bending stress in the beam. Sketch the stress distribution acting over the cross section.

( ) ( ) ( )

( )( )

)10(953125.1

125.018000

)10(953125.1

125.018

)10(953125.1

25.015.0

3125.1125.1

=Σ↑+

1.5 m 1.5 m 1.5 m

250 mm

150 mm

A log that is 2 ft in diameter is to be cut into a rectangular section for use as a simply supported beam. If the allowable bending stress for the wood is σallow = 8 ksi, determine the largest load P that can be supported if the width of the beam is b = 8 in.

8 ft 8 ft

( )( )

75.562

75.5696

The member has a square cross section and is subjected to a resultant moment of M = 850 N⋅m as shown. Determine that bending stress at each corner and sketch the stress distribution produced by M. Set θ = 45°.

125 mm

250 mm

M = 850 N⋅m

( )( ) ( )

( )( )

102552.3

125.004.601

102552.3

125.004.601

102552.3

125.004.601

102552.3

125.004.601

102552.3

04.601125.0

102552.3

125.004.601

102552.3

125.004.601

102552.3

125.004.601

102552.325.025.012

04.60145sin850

04.60145cos850

+−−

−−=

−−

125 mm

250 mm

M = 850 N⋅m

( )( ) ( )

( )( )

102552.3

125.012.736

102552.3

125.0425

102552.3

125.012.736

102552.3

125.0425

102552.3

125.012.736

102552.3

125.0425

102552.3

125.012.736

102552.3

125.0425

102552.325.025.012

42530sin850

12.73630cos850

+−−

−−=

−−

The T-beam is subjected to a moment of M = 150 kip⋅in. directed as shown. Determine the maximum bending stress in the beam and the orientation of the neutral axis. The location y of the centroid, C, must be determined.

6in. 6in.

150 kip.in

( )( )( ) ( )( )( )[ ]( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )( )

( ) ( )

inkipM

0178.2

333.293

19.129

33.333

333.293

69.129

333.333

33.333

7921221212

1472882

33.293

1229284

7560cos150

9038.12960sin150

−−−=

⋅−=−=

( ) ( )

°−=

−−=

302.33

30tan33.293

333.333

tantan

The aluminum machine part is subjected to a moment of M = 75 N⋅m. Determine the bending stress created at points B and C on the cross section. Sketch the results on a volume element located at each of these points.

( )( ) ( )( )( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )

( )( )( )

103633.0

0075.075

103633.0

0175.075

103633.0

0175.003.004.001.012

12)005.00175.0(01.008.001.008.0

0175.0

01.004.008.001.0

03.001.004.0201.008.0005.0

−−=

M = 75 N⋅m

10 mm 10 mm

A beam is constructed from four pieces of wood, glued together as shown. If the moment acting on the cross section is M = 450 N⋅m, determine the resultant force that bending stress produces on the top board A and on the side board B.

200 mm

M = 40 kN⋅m

( )( ) ( )( )( ) ( )( )

( )( )

( )( )( )

015.024.034.410334

10316.1

12.04500

10316.1

24.0015.012

1211.002.02.002.02.0

The beam is made from three boards nailed together as shown. If the moment acting on the cross section is M = 600 N⋅m, determine the maximum bending stress in the beam. Sketch a three-dimensional view of the stress distribution acting over the cross section.

M = 600 N⋅m

200 mm

( )( )( ) ( )( )( )( )( ) ( )( )

( )( ) ( )( )( )

( )( )

1053.24

11875.0600

1053.34

05625.01.015.002.015.002.012

0125.005625.0025.024.0025.024.012

11875.005625.0175.0

05625.0

02.015.0224.0025.0

02.015.01.0224.0025.00125.0

If the internal moment acting on the cross section of the strut has a magnitude of M = 800 N⋅m and is directed as shown, determine the bending stress at points A and B. The location z of the centroid C of the strut’s cross-sectional area must be determined. Also, specify the orientation of the neutral axis.

M = 800 N⋅m

200 mm

150 mm

( )( )( ) ( )( )( )( )( )( ) ( )( )( )

( )( ) ( )( )( )

( )( )

( )( )( )

( )( )

°−=

−−=

−−+

−−

30tan103374.16

1018869.0tan

tantan

103374.16

0366.082.692

1011869.0

2.0400

103374.16

1134.082.692

101886.0

2.0400

1018869.0

4.0012.012

194.0012.0138.0012.0138.012

103374.16

0366.0081.0012.04.0012.04.012

1134.00366.015.0

138.0012.024.0012.0

138.0012.0081.024.0012.0006.0

The beam has a rectangular cross section as shown. Determine the largest load P that can be subjected to the concentrated forces. The sleeve bearings at A and B support only vertical forces.

0.8 m 1.2 m 0.6 m

600 N 400 N

( )( )

A member having the dimensions shown is to be used to resist an internal bending moment of M = 2 kip⋅ft. Determine the maximum stress in the member if the moment is applied (a) about the z axis, (b) about the y axis. Sketch the stress distribution for each case. a) about z axis

b) about y axis

6 in. x

kiplbinkip

)/1000(/167.0

)12)(6(2

)12)(6)(12

kiplbinkip

)/1000)(/333.0(

)12)(3)(2(

)6)(12)(1(

A member has the triangular cross section shown. Determine the largest internal moment M that can be applied to the cross section without exceeding allowable tensile and compressive stresses of (σallow)t = 22 ksi and (σallow)s = 15 ksi, respectively.

2 in. 2 in.

4 in. 4 in.

inkipM

)32)(4((22

inkipM

)32)(4((15

ftkipM

ftinkipM

A member has the triangular cross section shown. If a moment of M = 800 lb⋅ft is applied to the cross section, determine the maximum tensile and compressive bending stresses in the member. Also, sketch a three dimensional view of the stress distribution acting over the cross section.

2 in. 2 in.

4 in. 4 in.

( )( )

31.29600

)32)(4(

( )( )

15.19600

A beam has the cross section shown. If it is made of steel that has an allowable stress of σallow =24 ksi, determine the largest internal moment the beam can resist if the moment is applied (a) about the z axis, (b) about the y axis. a) about z-axis

b) about y-axis

0.25 in.

( )( ) ( )( )( ) ( )( )

( ) ( )

ftkipM

inkipM

.7.249

25.03/8125.3324

8125.33

6.025.0125.325.06.0

25.06.02

( )( ) ( )( )

ftkipM

inkipM

.06.72

25.06.0

6.025.02

The beam is subjected to a moment of M = 40 kN⋅m. Determine the bending stress acting at points A and B. sketch the results on a volume element acting at each of these points.

50 mm 50 mm

M = 40 kN⋅m

( )( ) ( )( )

( )( )( )

)10(51.1

1000025.0400

)10(5.1

1000075.0400

)10(51.1

05.005.02

15.005.0

Determine the smallest allowable diameter of the shaft which is subjected to the concentrated forces. The sleeve bearings at A and B support only vertical forces, and the allowable bending stress is σallow = 160 MPa.

0.8 1.2 0.6

600 400

( ) ( ) ( )

( )( )( )

0156.0

)10(160

8.06008.14002.1

6.040026002.1

The beam is subjected to the load P at its center. Determine the placement a of the supports so that the absolute maximum bending stress in the beam is as large as possible. What is this stress?

≤≤

=Σ↑+

( )( )

maxmax

b L/2 L/2

The steel beam has the cross-sectional area shown. If w = 5 kip⋅ft, determine the absolute maximum bending stress in the beam.

( ) ( )

( )( ) ( )( ) ( )( )

( )( )( )

3.512160

103.015.53.08

30.082

24044024

=Σ↑+

8 ft8 ft8 ft

0.3 in

8 in 0.30 in

0.30 in

160 kip.ft

-40 kip

40 kip

The beam has a rectangular cross section as shown. Determine the largest load P that can be supported on its overhanging ends so that the bending stress does not exceed σmax = 10 MPa.

( )( )

)10(953.1

125.05.1)10(10

)10(953.1

25.05.0

35.15.1

=Σ↑+

1.5 m 1.5 m 1.5 m

250 mm

150 mm

The beam has the rectangular cross section shown. If P =12 kN, determine that absolute maximum bending stress in the beam. Sketch the stress distribution acting over the cross section.

( ) ( ) ( )

( )( )

)10(953125.1

125.018000

)10(953125.1

125.018

)10(953125.1

25.015.0

3125.1125.1

=Σ↑+

1.5 m 1.5 m 1.5 m

250 mm

150 mm

. A log that is 2 ft in diameter is to be cut into a rectangular section for use as a simply supported beam. If the allowable bending stress for the wood is σallow = 8 ksi, determine the largest load P that can be supported if the width of the beam is b = 8 in.

8 ft 8 ft

( )( )

75.562

75.5696

125 mm

250 mm

M = 850 N⋅m

( )( ) ( )

( )( )

102552.3

125.004.601

102552.3

125.004.601

102552.3

125.004.601

102552.3

125.004.601

102552.3

04.601125.0

102552.3

125.004.601

102552.3

125.004.601

102552.3

125.004.601

102552.325.025.012

04.60145sin850

04.60145cos850

+−−

−−=

−−

125 mm

250 mm

M = 850 N⋅m

( )( ) ( )

( )( )

102552.3

125.012.736

102552.3

125.0425

102552.3

125.012.736

102552.3

125.0425

102552.3

125.012.736

102552.3

125.0425

102552.3

125.012.736

102552.3

125.0425

102552.325.025.012

42530sin850

12.73630cos850

+−−

−−=

−−

The T-beam is subjected to a moment of M = 150 kip⋅in. directed as shown. Determine the maximum bending stress in the beam and the orientation of the neutral axis. The location y of the centroid, C, must be determined.

( )( )( ) ( )( )( )[ ]( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )( )

( ) ( )

inkipM

0178.2

333.293

19.129

33.333

333.293

69.129

333.333

33.333

7921221212

1472882

33.293

1229284

7560cos150

9038.12960sin150

−−−=

⋅−=−=

( ) ( )

°−=

−−=

302.33

30tan33.293

333.333

tantan

6-185. Determine the bending stress distribution in the beam at section a-a. Sketch the distribution in three dimension acting over the cross section.

( )( ) ( )( )

( )( )

050.032

1052.2

15.075.0

1.015.0

The aluminum machine part is subjected to a moment of M = 75 N⋅m. Determine the bending stress created at points B and C on the cross section. Sketch the results on a volume element located at each of these points.

M = 75 N⋅m

10 mm 10 mm

( )( ) ( )( )( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )

( )( )( )

103633.0

0075.075

103633.0

0175.075

103633.0

0175.003.004.001.012

12)005.00175.0(01.008.001.008.0

0175.0

01.004.008.001.0

03.001.004.0201.008.0005.0

−−=

A beam is constructed from four pieces of wood, glued together as shown. If the moment acting on the cross section is M = 450 N⋅m, determine the resultant force that bending stress produces on the top board A and on the side board B.

200 mm

M = 40 kN⋅m

( )( ) ( )( )( ) ( )( )

( )( )

( )( )( )

015.024.034.410334

10316.1

12.04500

10316.1

24.0015.012

1211.002.02.002.02.0

The beam is made from three boards nailed together as shown. If the moment acting on the cross section is M = 600 N⋅m, determine the maximum bending stress in the beam. Sketch a three-dimensional view of the stress distribution acting over the cross section.

M = 600 N⋅m

200 mm

( )( )( ) ( )( )( )( )( ) ( )( )

( )( ) ( )( )( )

( )( )

1053.24

11875.0600

1053.34

05625.01.015.002.015.002.012

0125.005625.0025.024.0025.024.012

11875.005625.0175.0

05625.0

02.015.0224.0025.0

02.015.01.0224.0025.00125.0

If the internal moment acting on the cross section of the strut has a magnitude of M = 800 N⋅m and is directed as shown, determine the bending stress at points A and B. The location z of the centroid C of the strut’s cross-sectional area must be determined. Also, specify the orientation of the neutral axis.

M = 800 N⋅m

200 mm

150 mm

( )( )( ) ( )( )( )( )( )( ) ( )( )( )

( )( ) ( )( )( )

( )( )

( )( )( )

( )( )

°−=

−−=

−−+

−−

30tan103374.16

1018869.0tan

tantan

103374.16

0366.082.692

1011869.0

2.0400

103374.16

1134.082.692

101886.0

2.0400

1018869.0

4.0012.012

194.0012.0138.0012.0138.012

103374.16

0366.0081.0012.04.0012.04.012

1134.00366.015.0

138.0012.024.0012.0

138.0012.0081.024.0012.0006.0

The axle of the freight car is subjected to wheel loadings of 20 kip. If it is supported by two journal bearings at C and D, determine that maximum bending stress developed at the center of the axle, where the diameter is 5.5 in.

( ) ( )

( )( )( )

202020

80207020

6-74. Determine the absolute maximum bending stress in the 1.5-in.-diameter shaft which is subjected to the concentrated forces. The sleeve bearings at A and B support only vertical forces.

( ) ( ) ( )

( )( )( )Ksi

5.7450064

90300400

153001840030

6-75. Determine the smallest allowable diameter of the shaft which is subjected to the concentrated forces. The sleeve bearings at A and B support only vertical forces, and the allowable bending stress is σallow = 22 ksi.

Given:

Solution:

( ) ( ) ( )

90300400

153001840030

144000

45003222

ππσ

Determine the absolute maximum bending stress in the 30-mm-diameter shaft which is subjected to the concentrated forces. The sleeve bearings at A and B support only vertical forces.

0.8 m 1.2 m 0.6 m

600 N 400 N

( )( )

6-79. The steel shaft has a circular cross section with a diameter of 2 in. It is supported on smooth journal bearings A and B, which exert only vertical reactions on the shaft. Determine the absolute maximum bending stress if it is subjected to the pulley loadings shown.

( ) ( ) ( ) ( )

( )( )ksi

160004

20500403006080080

20500403006050080

ππσ

Determine the absolute maximum bending stress in the 20mm diameter pin. The smooth pin is supported by two leaves A and B and subjected to a compressive load of 0.4kN/m caused by bar C. Determine the intensity of the distributed load wo of the leaves on the pin and draw the shear and moment diagrams for the pin.

( )( )

10853.7

01.000026.0

10853.7

2.6x10-3 kN⋅m

60 mm 20 mm20 mm

0.4 kN/m

If the beam ABC has square cross-section 6in by 6in, determine the absolute bending stress in the beam. The 20kip load must be replaced by equivalent loadings at point C on the axis of the beam.

( )( )

12368.46

6-67. If the crane boom ABC has a rectangular cross-section with the base of 2in a height of 3in, determine the absolute maximum bending stress in the boom. The engine crane is used to support the engine, which has a weight of 1200 lb.

( )( )

kipftlbM

1000.6000

6-86. The simply supported beam is made from four ¾ in diameter rod w/c are bounded as shown determine the maximum bending stress in the beam due to the loading shown.

( ) ( ) ( )

( ) ( )

inlbftlb

192032

.1920.160

28088010

ππσ

6-48. The beam is made from three boards nailed together as shown. If the moment acting on the cross-section is M=600N.m, determine the resultant force bending stress produces on the top board.

( )( )( )NF

kPastressAve

1000/240255.598

5.5982

252945.

5-59. The beam is subjected to a moment M=30lb.ft. Determine the bending stress at points A and B. Also, sketch a three dimensional view of the stress distribution acting over the entire cross-sectional area.

(Mindanao State University-General Santos City, Philippines) BSEE-(G.R.F) students 2004

( )( ) ( )( )( )( )( )( ) ( )( )( )

( ) ( )( ) ( )( )( )

( )( )

977.32

4.01230

35.214

4.141230

4.12312

329.02

135.0212

2135.025.012

strenght of materials(es-64)

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