New Chapter 19 Normal stress: axial loading homeworkweb.ics.purdue.edu/~krousgri/me270/wp-content/uploads/20... · 2012. 11. 5. · 19-6 Chapter 19: Normal stress: axial loading homework

Chapter 19: Normal stress: axial loading homework 19-1

Chapter 19

Normal stress: axial loadinghomework

19-2 Chapter 19: Normal stress: axial loading homework


Problem 14.1

Given: A rigid rectangular cross section strut weighing W is welded to a rigid triangularsupporting plate. The supporting plate is bolted to ground with three identical bolts B, Cand E, as shown. The weight of the plate and bolts are negligible compared to the weightof the strut. Couples My and Mz are applied in the y- and z-directions, respectively, at endD of the strut. Each bolt has a cross-sectional area of A.

Find: Determine the average normal stress in each of the three bolts. State whether eachbolt is in tension or in compression.

x

y

Mz M y

D

W

SIDE VIEW

strut

supporting plate

E

B

C

y

z

b / 2

b / 2

b / 3 2b / 3

O

TOP VIEW

supporting plate

strut

Use the following parameters in your analysis: W = 600lb, b = 1ft, My = Mz = 200ft · lband A = 0.005ft2.


Problem 14.2

Given: A truss is made up of links BD and CD. Links BD and CD have identical rectangularcross sections (with cross-sectional areas of A, and are made up of the same material (withYoung’s modulus E and Poisson’s ratio ν. A vertical load P acts at pin D on the truss.

Find:

• Determine the average normal stresses in links BD and CD.

• Determine the average strains �x, �y and �z in link CD.

L

0.8L

P

x

y

C

BD

Use the following parameters in your analysis: P = 1000lb, L = 5ft, A = 0.01ft2,E = 30× 106psi and ν = 0.3.


Problem 14.3

Given: The stone column (of total weight W and length L) shown has a square cross section.

Find: Consider a perpendicular cross section of the column at a position x along its length.

• Determine the axial load carried by the column as a function of x.

• Determine the normal stress, σave, across the cross section of the cut as a function ofx.

• Make a plot of σave vs. x.

bb

xL

Use the following parameters in your analysis: W = 75, 000N , b = 0.5m, L = 10m andA = 0.005ft2.


Problem 14.4

Given: An aluminum rod (having an outer diameter of d) is supported by a steel pipehaving an outside diameter of D and thickness t. The rod is subjected to an axial load of P .

Find: Determine the average normal stresses in the rod and pipe.

A B

d

D

t

rod

fixed wallpipe

P

Use the following parameters in your analysis: P = 20kips, D = 2in, d = 0.4in andt = 0.05in.


Problem 14.5

Given: Identical columns BD and AE are made up of steel pipes having outer diameters ofD and thicknesses t. These columns support a horizontal, rigid beam of length L. A linearlyvarying distributed loading is applied to the beam, with the loading having a values of p0(force/length) at end C and zero at end A.

Find: Determine the average normal stresses in the pipes. State whether these stresses arecompressive or tensile.

C

L2

L2

ED

AB

po

Use the following parameters in your analysis: p0 = 10kips/ft, L = 30ft, t = 0.3in andD = 8in.


Problem 14.6

Given: A rigid beam AB having a length of L is supported by two identical rods havingdiameters of D. This beam in turn supports a crate having a weight of W with its center ofmass at C. The weights of the beam and rods are to be considered negligible compared tothe crate.

Find: Determine the average normal stresses in the rods.

C

L2

GA B

L2

d

Use the following parameters in your analysis: L = 14ft, D = 0.3in, d = 3ft andW = 1200lb.


Problem 14.7

Given: The truss shown is made up of links DE, CE, CD and BC. All four links have thesame cross-sectional area A. A vertical load P is applied to joint D. The allowable stressin tension for each link is (σT )allow, and the allowable stress in compression for each link is(σC)allow.

Find: Using a factor of safety FS, determine the maximum allowable load P on the truss.

D

L

B

E

C

L

60° 60°

P

Use the following parameters in your analysis: A = 0.5in2, (σT )allow = 20ksi and (σC)allow =12ksi.


Problem 14.8

Given: A stepped, circular rod is made up on two sections joined by a rigid plate. Theupper section of the rod has a cross-sectional diameter of 2d, and the lower section of therod has a cross-sectional diameter of d. The lower section of the rod is embedded in soil.With an axial of P applied to the upper section of the rod, the soil produces a resistive,linearly-varying distributed axial load p(x) (force/length) on the lower section of the rod.The allowable tensile stress in either section of the rod is (σT )allow.

Find: Using a factor of safety FS, determine the maximum allowable load P on the truss.Assume that the rod does not slip on the soil.

x

p(x)

P

2d

Soil


Problem 14.9

Given: An L-shaped bracket CDE is used to support a crate of weight W and with thecenter of mass of the crate being at G. Rod BC connects end C of the bracket to groundwhere the rod has a cross-sectional area of A. The allowable tensile stress in rod BC is(σT )allow. The weights of the bracket and rod are negligible as compared to the weight ofthe crate.

Find: Using a factor of safety FS, determine the maximum allowable weight W of the crate.

L

θ

b

E BD

C

G

Use the following parameters in your analysis: θ = 30◦, L = 5ft, b = 2ft, A = 0.5in2,(σT )allow = 15ksi and FS = 1.4.

Chapter 20: Shear stress: direct shear and torsional loading homework 20-1

Chapter 20

Shear stress: direct shear andtorsional loading homework

20-2 Chapter 20: Shear stress: direct shear and torsional loading homework


Problem 15.1

Given: The rectangular element shown below experiences a state of pure shear τ . Thematerial of the element has a Young’s modulus E and a Poison’s ratio ν.

Find:

• Determine the shear strain in the element.

• Determine the horizontal displacement δ of the upper righthand corner of the element.

τ

τ

τ τb

b

δ δ

Use the following parameters in your analysis: b = 3in, E = 30 × 106psi, ν = 0.3 andτ = 10ksi.


Problem 15.2

Given: A horizontal bar is attached to a fixed bracket through two rubber pads onto theupper and lower inner surfaces of the bracket, as shown in the figure. The rubber has a shearmodulus of G. A horizontal load P acts on the free end of the bar.

Find:

• Determine the average shear stress in the rubber pads.

• Determine the average shear strain in the rubber pads.

b

b

b

b

P

P

b

w Bar

Bar

Bracket

BracketRubber Pad

Rub

ber

Pad

Rubber Pad

Top View

Side View

Use the following parameters in your analysis: P = 450N , b = 60mm, w = 100mm andG = 0.6MPa.


Problem 15.3

Given: A vibration isolation mount is made up of a steel rod bonded to the center of anannular rubber cylinder. The outer surface of the rubber cylinder is bonded to the innersurface of a cylindrical bracket. The bracket is rigidly attached to a fixed surface. A load Pacts on the upper end of the rod.

Find:

• Determine the average shear stress in the rubber where it is bonded to the rod.

• Determine the average shear stress in the rubber where it is bonded to the bracket.

d

3d

h

P

bracket

rubber

rod

Top View

Side View


Problem 15.4

Given: A slot is to be punched out of an aluminum plate of thickness t. The slot is ofa rectangular shape with semicircular ends, as shown. It is known that the average shearresistance is τpunch.

Find: Determine the required punching force P .

40mm

10mm

Use the following parameters in your analysis: t = 10mm and τpunch = 250MPa.


Problem 15.5

Given: A truss is made up of members BC, CE and DE. Members DE and CE are joinedby a two-sided pin connection at E, as shown in the figure, with DE and CE aligned witheach other. The circular pin at E has a diameter of d. A vertical load P acts at pin C of thetruss.

Find: Determine the average direct shear stress in pin C.

P

x

y

D

B

C

θ

E

d

pin E

Top view

of pin E


Problem 15.6

Given: A frame is made up of links BD and CE with the frame being loaded as shown inthe figure. The frame is attached to ground with two-sided pin connections at B and C. Thediameter of the circular pin at C is d.

Find: Determine the average direct shear stress in pin C.

B

P

E

C

D

P

L

2

L

2

L

2

L

2

pin C

bracketd

End view of

bracket at pin C

Side view


Problem 15.7

Given: A shaft is made up of a solid circular section between B and C (of radius do),and a hollow circular section between C and D (of outer radius d0 and inner radius di).Concentrated torques T1 and T2 are applied at locations C and D, respectively. End B ofthe shaft is built into a fixed support.

Find: Determine the maximum shear stress in the shaft and identify the location on theshaft where this maximum shear stress occurs.

L2

L2

y

x

B DC

T1

T2do di

Use the following parameters in your analysis: do = 50mm, di = 20mm, T1 = 4kN ·m andT1 = 6kN ·m.


Problem 15.8

Given: A solid circular shaft of radius R is built into a fixed wall at its left end. Aconcentrated torque T acts on the right end of the shaft. A constant distributed torque t0(torque/length) acts along the length of the shaft.

Find: Determine the maximum shear stress in the shaft and identify the location on theshaft where this maximum shear stress occurs.

L

xR

to

T

Use the following parameters in your analysis: T = 10kN ·m, R = 20mm, L = 750mmand t0 = 20N/mm.


Problem 15.9

Given: A torque T is applied to the front end of the driveshaft on a rear-wheel drive au-tomobile. With the driveshaft rotating with a rate of ω, the torque is known to deliver apower of P to the driveshaft. The driveshaft has a hollow cross section with an outer radiusof R and with a thickness of t. The allowable shear stress in the driveshaft is known to beτallow.

Find: Determine the minimum permissible thickness t of the driveshaft.

Note: Shaft power is given by P = Tω/550, where P is in horsepower, T is in ft · lb andω is in radians/sec.

T ω

Rear Wheels

DifferentialDrive Shaft

tR

Cross-section

of drive shaft

Use the following parameters in your analysis: P = 200horsepower, ω = 1250rev/min,R = 1.5in and τallow = 6ksi.

Chapter 21: Normal and shear stresses in beams homework 21-1

Chapter 21

Normal and shear stresses in beamshomework

21-2 Chapter 21: Normal and shear stresses in beams homework


Problem 16.1

Given: The beam shown is supported by a pin joint at C and a roller at E. Concentratedforces P1 and P2 act as shown in the figure. The material making up this beam has a Young’smodulus of E. Consider a perpendicular cut through the beam at location F.

Find:

• Show that a state of pure bending exists at this cut in the beam.

• Determine the maximum normal stress acting on this cut.

• What is the corresponding axial strain corresponding to the maximum tensile stresson this? Where on this cut does this maximum tensile strain occur on the cut?

L2

L2

xB

DC

P1d1 d2

P2

EF

h

b

Use the following parameters in your analysis: L = 6ft, b = 5in, h = 6in, d1 = 1ft,d2 = 2ft, P1 = 2000lb, P2 = 1000lb and E = 30× 106psi.


Problem 16.2

Given: The beam shown is supported by a pin joint at C and a roller at D. A concentratedforce P and a concentrated couple MB act on the as shown in the figure. The materialmaking up this beam has a Young’s modulus of E. The beam has a rectangular cross sectionas shown in the figure. The beam has a circular cross section as shown in the figure. Considera perpendicular cut through the beam at location E.

Find:



• What is the corresponding axial strain corresponding to the maximum tensile stresson this? Where on this cut does this maximum tensile strain occur on the cut?

L4

x

B E

MB

P

DC

L4

L4

L4

R

Use the following parameters in your analysis: L = 8ft, R = 3in, P = 2000lb, MB =2000ft · lb and E = 10× 106psi.


Problem 16.3

Given: A cantilevered beam is built into a fixed wall at end B. Three concentrated forcesact on the beam as shown. The material making up this beam has a Young’s modulus of E.The beam has a rectangular cross section as shown in the figure. Consider a perpendicularcut through the beam at location F.

Find:



• Make a sketch of the normal stress σ(y) over the depth of the beam at F. The x-axispasses through the neutral axis of the beam.

L2

x

B E

2P

DC F

L3

L3

L3

P P

y

y

z

b

h

Use the following parameters in your analysis: L = 6ft, h = 4in, b = 6in, P = 1000lb andE = 30× 106psi.


Problem 16.4

Given: An ”I-beam” has a cross section as shown below.

Find: Determine the second area moment of the cross section corresponding to the neutralaxis of the beam.

z

y

b1 b1b2

h1

h1

h2o

y

x

Neutral Axis


Problem 16.5

Given: A beam has a cross section made up of a rectangular beam welded to a hollow pipe.

Find:

• Determine the location of the neutral axis for the beam relative to the center of thepipe (that is, find the distance d).

• Determine the second area moment of the cross section corresponding to the neutralaxis of the beam.

y

x

Neutral Axis z

y

od

b2

Ri

Ro

b2

h


Problem 16.6

Given: A cantilevered beam is loaded as shown. The cross section of the beam is as alsoshown below.

Find:

• Determine the location of the neutral axis for the beam.

• Determine the second area moment of the cross section corresponding to the neutralaxis of the beam.

• Make a sketch of the normal stress in the beam σ(y) as a function of y correspondingto a perpendicular cut through the beam at location C.

x

Neutral Axis

L3

B EDC

P

P

L3

L3

z

y

o

b1 b1b2

h1

h2

Use the following parameters in your analysis: L = 1.2 m, P = 500 N, b1 = h2 = 10 cm,b2 = 30 cm, and h1 = 8 cm.

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New Chapter 19 Normal stress: axial loading homeworkweb.ics.purdue.edu/~krousgri/me270/wp-content/uploads/20... · 2012. 11. 5. · 19-6 Chapter 19: Normal stress: axial loading homework