ME 270 Fall 2013 Final Exam NAME (Last, First): … · ME 270 – Fall 2013 Final Exam NAME (Last, First): _____ 1(c) Bar OA is loaded with a single force as shown and is held in

ME 270 – Fall 2013 Final Exam NAME (Last, First): ________________________________ 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 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. Silvers 8:30-9:30am B. Hylton 2:30-3:20pm J. Jones 11:30am-12:20pm J. Seipel 12:30-1:20pm M. Murphy 9:00-10:15am E. Nauman 9:30-10:20 am K. Li 1:30-2:20pm J. Jones Distance Learning

Problem 1 __________

Problem 2 __________

Problem 3 __________

Problem 4 __________

Problem 5 __________

Total ______________

ME 270 – Fall 2013 Final Exam NAME (Last, First): ________________________________ PROBLEM 1 (20 points) – Prob. 1 questions are all or nothing.

1(a) An overhead lamp is held in static equilibrium by cable AC and horizontal spring AB. Given the spring constant is k=300 N/m and the net deformation of the spring is 0.453m, determine the magnitude of the tension in cable AC and the weight of the lamp. (4pts)

1(b) Bar ABC is loaded at C with one force and one couple as shown. Determine the equivalent force-couple system at A. Express your solution in vector form. (Hint –

This is not a static equilibrium problem.) (4pts)

AC

Lamp

(2 pts)T =

W = (2 pts)

eq A

eq A

(F ) = (2 pts)

(M ) = (2 pts)

ME 270 – Fall 2013 Final Exam NAME (Last, First): ________________________________ 1(c) Bar OA is loaded with a single force as shown and is held in static equilibrium by a built-in support at O. Determine the reactions at O due to this loading. (4pts)

1(d) The fink truss is loaded as shown. Identify all zero-force members by placing a zero over that member. (4pts)

A

O

O

F = (2 pts)

M = (2 pts)

ME 270 – Fall 2013 Final Exam NAME (Last, First): ________________________________ 1(e) The 15° wedge shown is being removed by force P. On the figures provided, complete the free body diagram of the crate and wedge. Write the x and y-equilibrium equations for the wedge. Assume friction exists on all interfaces (A, B and C). Leave the friction forces generic (i.e., fA, fB, fC) DO NOT SOLVE THE EQUATIONS. (4pts)

1 pt

1 pt

x

y

F = 0 = (1 pt)

F = 0 =

(1 pt)

C

ME 270 – Fall 2013 Final Exam NAME (Last, First): ________________________________ PROBLEM 2 (20 points)

Given: Rod AB is loaded at its midpoint with a vertical

force CP 72 lb . The weight of the rod is negligible. It

is held in place at A by a ball and socket joint. At B, the rod rests against a smooth wall in the x-z plane, and is held in place by a cable (BD) which is parallel to the x-axis. Find:

a) On the figure provided, draw a complete free body diagram. (5 pts)

b) Write the position vector from A to B, and the position vector from A to C. (3 pts)

c) Determine the tension in cable BD, and the force from the smooth wall at B. Give your answers in vector form. (8 pts)

d) Determine the reactions from the ball and socket joint at A. Give your answer(s) in vector form. (4 pts)

Free Body Diagram

ME 270 – Fall 2013 Final Exam NAME (Last, First): ________________________________


Name (last, first)

Problem #3 Fall 2013 Final

a.

b. Ax=

Ay= Fy= .

c. |VA|= τA= .

d. γA=

e. Load BD tension or compression (circle one)

f. σBD

g. εx in member BD

12 ft

6 ft

3 ft

4 ft

80 kip

A

B

C

D

F

80 kip

Ax

B

C

D

F

80 kip

A

B

C

D

F

B D

C

The structure is subjected to a 80kip (80,000 lb) load as shown. The

mass of the members is negligible when compared to applied load.

BD is a two-force member. A support pin at Joint A makes contact

with the joint on two sides (double shear pin).

The modulus of elasticity (E) is 10x106 psi (10x10

3ksi) for all

members, including the pin at A.

Poisson’s Ratio for all members, including the pin at A.

The cross-sectional area of member BD is 0.5 in2.

The joint at A is supported on two sides by a pin with a cross-

sectional area of 0.5 in2.

Please place your answers in the box provided. Remember units! Coordinate axis is provided for this problem.

ALL steps of your work must be shown to earn credit.

a. The free-body diagrams for the entire structure is provided,

please complete the free-body diagram for the exploded

structure (3 points) on the figure provided.

b. Determine the reactions at A and F (3 points)

c. Determine the magnitude of the shear force |VA| and the shear

stress τA on the support pin at A (3 points).

d. Determine the shear strain on the Pin at A (3 Points)

e. Determine the load carried by member BD and circle

whether it is in tension or compression (3 points)

f. Determine the axial stress σBD in member BD (3 points)

g. Determine the axial strain εx in BD (2 points)

Ay Fy

Ax Ay

A

Fy

y

x

Joint A


ME 270 – Fall 2013 Final Exam NAME (Last, First): ________________________________ PROBLEM 4 (20 points)

4(a) A cantilevered structure is loaded as shown, with F = 100 N. Calculate the internal forces and moments, in vector form, at cut A. A coordinate system is provided on the figure. (4 pts)

4(b) A leather worker is punching a square hole in a leather strap. The punch cross-section is 5mm by 5mm. The leather strap is 2 mm thick. If the leather worker applies a 5 N force to the punch, calculate the average shear stress in the leather. (3 pts)

FBD

ME 270 – Fall 2013 Final Exam NAME (Last, First): ________________________________ 4(c) A cantilevered shaft is loaded as shown. Section AB is a solid aluminum shaft with diameter d. Section BC is a hollow steel shaft with an outer diameter d and wall thickness 0.1*d. Let d = 6 inches and T = 10 ft-kips. Calculate the polar moment of area and maximum torsional stress in the hollow region of the shaft (BC). Then, using the provided values, calculate the maximum torsional strain in the shaft. (6 pts)

AB BC

Polar Moment of Area (J) 127.23 in4 _________________

Maximum Torsional Stress ( ) 2.83 ksi _________________

Modulus of Rigidity (G) 3.8 x 103 ksi 11.2 x 103 ksi

Torsional Strain ( ) __________________ _________________

ME 270 – Fall 2013 Final Exam NAME (Last, First): ________________________________ 4(d) Setup but do not integrate the equation for the 2nd moment of area of the cross-section shown below, rotating about the given axis, x. (3 pts)

4(e) Using the method of composite parts, calculate the 2nd moment of area of the cross section shown below, rotating about the given axis, x. Both cross-bars have the same length and thickness. (4 pts)

ME 270 – Fall 2013 Final Exam NAME (Last, First): ________________________________ ME 270 –Fall 2013

Problem 5. (20 pts).

5a. The beam given below is supported by a pin and roller. Determine the reactions at points A and D and write them in the space provided below. Note that the beam cross section is rectangular, with a base of 200 mm and a height of 400 mm.

(Draw Free Body Diagrams used for part 5a here):

(2 pts) Reaction at point A (WRITE ANSWER HERE): _______________________________

(2 pts) Reaction at point D (WRITE ANSWER HERE): ________________________________

4 kN

1m

2m

1m

5 kN/m

4 kN-m

A B

C D


Problem 5 Continued

5b. For the beam given in part a above, draw the shear force and bending moment diagram in the space provided. (9 pts).

4 kN

Shear

Force

V(x)

Bending

Moment

M(x)

1m 2m 1m

5 kN/m

4 kN-m

A B

C

D


Problem 5 Continued

5c. Locate the point where pure bending occurs as the distance from pt A. (2 pts)

Location of pure bending measured from A (WRITE ANSWER HERE):______________________

5d. What is the maximum magnitude of bending moment that occurs in the beam. (2 pts).

Maximum bending moment (WRITE ANSWER HERE) : _______________________________

5e. For the location found in part c, determine the maximum axial stress due to bending. (3 pts).

Maximum axial stress (WRITE ANSWER HERE): _________________________________




ME 270 Final Exam Equations Fall 2013

Normal Stress and Strain

σx =Fn

A

σx(y) =−My

I

εx =σx

E=∆L

L

εy = εz = − ϑεx

εx(y) =−y

ρ

FS =σfail

σallow

Shear Stress and Strain

τ =V

A

τ(ρ) =Tρ

J

τ = Gγ

G =E

2 1 + ϑ

γ =δs

Ls=π

2− θ

For a rectangular cross-

section,

τ(y) =6V

Ah2

h2

4− y2

τmax =3V

2A

Second Area Moment

I = y2dA

A

I =1

12bh3 Rectangle

I =π

4r4 Circle

IB = IO + AdOB2

Polar Area Moment

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

Buoyancy

BF gV

Fluid Statics

p gh

eq avgF p Lw

Belt Friction

L

S

Te

T

Distributed Loads

L

eq 0F w x dx

L

eq 0xF x w x dx

Centroids

cx dAx

dA

cy dAy

dA

ci ii

ii

x A

xA

ci ii

ii

y A

yA

In 3D,

ci ii

ii

x V

xV

Centers of Mass

cmx dAx

dA

cmy dAy

dA

cmi i ii

i ii

x A

xA

cmi i ii

i ii

y A

yA


Fall 2013 ME 270 Final Answers

1A. ACT = 157 N LampW = 78.5 N

1B. eq A(F ) = 25 i + 43.3j lb eq A(M ) = 520 lb-ft

1C. OF = -80 i + 60j N OM = 460k N-m

1D. Zero Force Members: BP, CP, CO, CN, DN, EM, ON, & NE.

1E. o o

A B BXF = 0 = -P + f + f cos 15 - N sin 15

o o

A B ByF = 0 = -N + N cos 15 + f sin 15

2A. Free Body Diagram

2B. ABr = 4 i - 7j + 9k ft ACr = 2 i - 35j + 4.5k ft

2C. N = 28 lb j T = -16 lb i

2D. AF = 16 i - 28j + 72k lb

3A. Free Body Diagram

3B. xA = -80 kip yA = -60 kip yF = 60 kip

3C. AV = 100 kip A = 100 ksi

3D. Aγ = 0.026 in/in

3E. Load BD 120 kip Tension

3F. BD = 240 ksi

3G. x in member BD is 0.024 in/in

4A. F = (160 i + 120j) N M = (360k) N-m

4B. AVG = 0.125 MPa

4C. maxγ = 0.000745


4D.

3 43 - x 4 - y

4 2 3 24 3x 0 0

0 0

3I = y dydx or (3 - x) dxdy

4

4E. 4

xI = 1365.6 in

5A. FBD Reaction at point A: 9j kN Reaction at point D: 5j kN

5B. Shear force bending moment diagram

5C. Location of pure bending measured from A: x = 2m or 2m from pt A

5D. Maximum bending moment: 11.5 kN-m

5E. Maximum axial stress: 2.156 MPa

Documents

ME 270 Fall 2013 Final Exam NAME (Last, First): … · ME 270 – Fall 2013 Final Exam NAME (Last, First): _____ 1(c) Bar OA is loaded with a single force as shown and is held in