ME 270 Spring 2015 Exam 3 NAME (Last, First): Please ...€¦ · ME 270 – Spring 2015 Exam 3 NAME (Last, First): _____ ME 270 Exam 3 – Spring 2015 Page 3 PROBLEM 1. Cont. 1C

ME 270 – Spring 2015 Exam 3 NAME (Last, First): ________________________________

ME 270 Exam 3 – Spring 2015 Page 1

Please review the following statement:

I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.

Signature: ______________________________________

INSTRUCTIONS

Begin each problem in the space provided on the examination sheets. If additional space is required, use the white lined paper provided to you.

Work on one side of each sheet only, with only one problem on a sheet.

Each problem is worth 20 points.

Please remember that for you to obtain maximum credit for a problem, it must be clearly presented, i.e.

The only authorized exam calculator is the TI-30IIS

The allowable exam time for Exam 1 is 70 minutes.

The coordinate system must be clearly identified.

Where appropriate, free body diagrams must be drawn. These should be drawn separately from the given figures.

Units must be clearly stated as part of the answer.

You must carefully delineate vector and scalar quantities.

If the solution does not follow a logical thought process, it will be assumed in error.

When handing in the test, please make sure that all sheets are in the correct sequential order and make sure that your name is at the top of every page that you wish to have graded.

Instructor’s Name and Section:

Sections: J Jones 9:30-10:20AM I Bilionis 1:30-2:20PM J Ackerman 3:30-4:20PM J Jones Distance Learning

Problem 1 __________

Problem 2 __________

Problem 3 __________

Total ______________



PROBLEM 1 (20 points).

Please show all work.

1A. A rigid beam (AC) is supported by thin circular rod (AB) and a square post (CD). The diameter of the rod is 20 mm and the square post is 20mm a side. Determine the axial stress in rod (AB) and post (CD) in MPa if the load on beam AC is P = 2000 N. Remember 1MPa = 1x106Pa (5 pts)

1B. A joint is fastened together with two bolts as shown.

a) If the failure shear stress for the bolts is

failτ = 800MPa , determine the minimum diameter in

mm of each bolt assuming a factor of safety FS of 2.

b) If the plates were glued together rather than bolted,

determine the maximum shear stress in MPa for the

loading shown. Neglect the factor of safety for part b.

AB = ________________________σ (2 pts)

CD = ________________________σ (3 pts)

mina) (d) = (3 pts)

maxb) ( ) = (2 pts)



PROBLEM 1. Cont.

1C. A shaft is made of a steel alloy having an allowable shear

stress of allowτ = 12 ksi. If the diameter of the shaft is 1.5in,

determine the maximum torque TA in that can be transmitted. What would be the maximum torque TB if a 1 inch diameter hole is bored through the shaft? Express the torques in units of in-lbs. Sketch the shear-stress distribution along a radial line for the hollow shaft. Remember 1ksi = 1000lbs/in2.

T = A (2 pts)

T = B (2 pts)

1 pt



PROBLEM 1. Cont.

1D. Given the shear-force and bending-moment diagrams provided below, sketch the equivalent

loading condition on the beam provided below. Make sure you indicate both the magnitude and the

direction of each load.

A C B



PROBLEM 2. (20 points)

GIVEN: Consider the beam ABCD with the given external loads shown in the figure below. The support at B is a roller and the support at C is a pin. The cross section of the beam is a rectangle with the cross section shown. The external loads are M = 4 kNm, F = 5 kN, and p0 = 2 kN/m.

FIND: a) Draw the free body diagram of the beam ABCD and calculate the reaction forces at the supports (3 pts).

By = _______________________________________________

Cx = _______________________________________________

Cy = _______________________________________________

(3 pts)



b) Draw the shear force and bending moment diagram of the beam ABCD. You must label the

shear force and bending moment values on the diagram at points A, B, C, and D to receive full

credit (8 pts). You may use the graphical method.

Labeled Shear and Bending Moment diagram (above) (8 pts)



c) Over what section is the beam in pure bending (1 pt)?

d) Calculate the maximum bending stress of the beam (σmax) in this section where pure bending

occurs. Recall the rectangular cross section of the beam shown on the right (8 pt).

c) Is the top of this pure bending section in compression or tension (1 pt)?

Section __________ (1 pt)

σmax = ____________________________________________ (7 pt)

Compression or Tension (circle one) (1 pt)



PROBLEM 3. (20 points)

GIVEN: Consider the beam AB shown in the figure. The support at A is a pin while the support at B is a rocker. Assume that the beam has a cross section area of A = 1in2, young modulus E = 29,000ksi, and shear modulus G = 10,900ksi.

FIND: a) Draw the free body diagram of the beam AB. If you choose to replace the distributed load with an equivalent force, make sure you show the magnitude and the point of application. (3 pts)

b) Compute the support reactions. (3 pts)

Ax = _______________________________________________

Ay = _______________________________________________

By = _______________________________________________

(3 pts)



c) Point C is located 6ft to the right of A (we copied the figure of the problem on the right for your convenience). Cut the beam at C and draw the free body diagram of the left part AC. Make sure you clearly show 1) the part of the distributed load that remains on top of AC and its magnitude; 2) the normal and shear forces at C; 3) the bending moment at C. Please follow the common conventions when drawing the directions of the internal forces and moment. (5 pts)

d) Compute the internal shear and normal forces as well as the bending moment at C. As in c), make sure you use the common sign conventions. (3 pts)

e) Compute the average shear stress τC and strain γC at C. (2 pts)

Normal force at C: NC = ____________________________________________

Shear force at C: VC = ____________________________________________

Bending moment at C: MC = ____________________________________________

(3 pts)



f) Consider a small (differential) material square located at C. The figure below shows a magnified version of the undeformed configuration of this material square. τC in this figure is the average shear stress applied on the right boundary of this material square. Draw arrows representing the average shear stress on the other sides of this square. Hint: The square must be in equilibrium. (1 pts)

g) The differential material square we considered in f) is under pure shear. Draw its deformed configuration below. Label on the deformed diagram the shear strain angle. (1 pts)

Average shear stress at C: τC = _________________________________________

Shear strain at at C: γC = ____________________________________________

(4 pts)

τC

Δx



ME 270 Exam 3 Equations

Normal Stress and Strain

σx =Fn

A

σx =−My

I

εx =σx

E=

∆L

L

εy = εz = − ϑεx

εx =−y

ρ

FS =σfail

σallow

Shear Stress and Strain

τ =Tρ

J

τ =V

A

τ = Gγ

G =E

2�1 + ϑ

γ =δs

Ls=

π

2− θ

I = y2dA

A

I =1

12bh3 Rectangle

I =π

4r4 Circle

IB = IO + AdOB2

J =π

2�ro

4 − ri4 Tube

Shear Force and Bending Moment

V�x = V�0 + p�ϵ dϵx

0

M�x = M�0 + V�ϵ dϵx

0



ME 270 Exam 3 Solution – Spring 2015

1a. ABσ = 3.18 MPa CDσ = - 2.50 MPa

1b. min

d = 7.98 mm max

τ = 13.3 MPa

1c. AT = 7948 in-lbs BT = 6378 in-lbs

1d. Sketch the equivalent loading condition on the beam provided below.

2a. yB = 1 kN xC = 0 kN yC = 8 kN

2b. Sketch shear force and bending moment on diagrams provided

2c. Section AB

2d. 6

maxσ = 6 x 10 Pa

2e. bCompression + M

3a. FBD

3b. xA = 0 kip yA = 9 kip yB = 18 kip

3c. FBD

3d. CN = 0 kip CV = 6 kip CM = 48 kip-ft

3e. CAvg shear stress at C: T = 6 ksi -4

CShear stress at C: γ = 5.5 x 10

3f. See sketch for arrows representing the avg shear stress applied on the right boundary of the

material square.

3g. See sketch for drawing of deformed configuration

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ME 270 Spring 2015 Exam 3 NAME (Last, First): Please ...€¦ · ME 270 – Spring 2015 Exam 3 NAME (Last, First): _____ ME 270 Exam 3 – Spring 2015 Page 3 PROBLEM 1. Cont. 1C