Page 2 of 15

x

y

B DC

2 ft 2 ft 2 ft

4 kips / ft2 kips / ft

20 kip ⋅ ft

x

x

V (x)

M (x)

H

R

a

b

ME 323 Examination # 2 Name ___________________________________

July 19, 2017

PROBLEM NO. 1 – 25 points max. Consider the semicircular cross-section beam (with R = 0.4 ft ) shown below with a roller support at C and a pin support at H. The beam is loaded with constant line loads between B and C, and between D and H, along with a concentrated couple at point D.

a) Draw the shear force and bending moment diagrams in the plot axes provided below. Provide sufficient detail in your analysis to justify your diagrams.

b) Determine the maximum magnitude normal stress in the beam. At which location does this stress exist (describe in terms of the x coordinate and the location on the cross section).

Please provide sufficient details on your work that will allow the grader to fairly assess your analysis.

Page 5 of 15

ME 323 Examination # 2 Name ___________________________________

July 19, 2017 PROBLEM NO. 2 – 30 points max.

Consider the propped cantilevered beam shown above with a linearly increasing line load, with the line load at end D having a value of p0 (force/length). Using the second-order integration technique from lecture, determine the reaction on the beam at C.

v(x)

B DC

d

p0

d

x

Page 9 of 15

ME 323 Examination # 2 Name ___________________________________

July 19, 2017 PROBLEM NO. 3 – 25 points max.

A “+” shaped structural member is made up on elements BC and DH (each of length 2L) welded together at their midpoints at O. Element DH is aligned with the z-axis, where as element BC is aligned with the x-axis. The member is built into a fixed wall at end B. A set of three forces in the y-direction act at ends D, H and C, as shown in the above figure.

a) Determine the components of the state of stress at point “a” at end B of the element. Show these components on the stress element “a” shown above.

b) Determine the components of the state of stress at point “b” at end B of the element. Show these components on the stress element “b” shown above.

Please provide sufficient details on your work that will allow the grader to fairly assess your analysis.

x

y

z

P

2P

d

C

D

H

L L

L

L

O

x

y

z

stresselementat“a”

x

y

z

stresselementat“b”

Ba

b

P

x

y

z

a

b

ProblemNo.1–28pointsA rectangular cross-section bar experiences a bending couple M at its ends andcarriesanaxialloadofP=1KN.Thestateofstressduetothisloadingisshownonthestressaxisbelow,withthisstresselementrotatedthroughanangleof θ = 30° from the x-axis. The shear component of stressτ at this orientation is unknown.Usingh=12cmandb=2cm:

a) DeterminethebendingcoupleMappliedtothebar.b) Determinetheshearstressτ atthisrotation.

ME 323: Mechanics of Materials Homework Set 11 Fall 2014 Semester Due: November 12, 2014 Name:_____________________ Section _____________ Problem 1 (10 Points): A rectangular cross-section bar experiences a bending couple M at its ends and carries an axial load P = 1000N . The state of stress at location B due to this loading is shown on the stress element below, with this stress element rotated through an angle of θ = 30° from the x-axis. The shear stress τ for this stress element orientation is unknown. Use the following cross section dimensions in your analysis: h = 12cm and b = 2cm .

a) Determine the bending couple M applied to the bar.

b) What is the shear stress τ acting on the stress element shown below?

MM

P Px

y

h

b

h / 4B

bar$cross$sec)on$

θ

x

y

′x′y

B

τ

τ

6MPa

6MPa

2MPa

2MPa

ProblemNo.2–29pointsThecrankshaftofaninternalcombustionengineisacteduponaloadingPatendD,

and isheld inequilibriumby reactionsat theoppositeendof the shaftKB that is

attachedtothetransmission.Attheinstantshown,sectionBCisalignedwiththey-

axis,andthelineofactionofPisata30°anglewithrespecttothez-axis,asshown

inthefigure.

Withthisloading,calculatethestatesofstressatlocations“a”and“b”onthecross

sectionofshaftKB.Showthecomponentsofthesestressstatesonthefacesofthe

stresselementsprovidedforpoints“a”and”b”.

Problem 8.2 (20 points)

The crankshaft of an internal combustion engine is rotating about the 𝑥 axis with constant angular velocity, and is therefore in static equilibrium. A section of the crankshaft KBCD is shown in Fig 8.2, when crank BC is aligned with the 𝑦 axis. At this instant, the force of the connecting rod 𝑃 acts in the 𝑦𝑧 plane at an angle of 30° from the 𝑧 axis at the center of shaft CD. The diameter of shaft KB is specified as 𝑑.

a) Calculate the reactions on the ‘positive 𝑥 face’ at K due to the applied load 𝑃.

b) Calculate the individual stress contributions due to bending, shear, and torsion at elements ‘a’ (0,0.5𝑑, 0) and ‘b’ (0,0,0.5𝑑). Indicate both direction and magnitude of each contributing stress and specify the corresponding axes.

c) Show the resultant stress contributions (state of stress) at ‘a’ and ‘b’ on their respective stress elements.

d) Plot the Mohr’s circle corresponding to each element, and calculate the principal stresses and the absolute maximum shear stress.

Use 𝐿 = 4𝑑 to express all stresses in b), c) and d) in terms of 𝑃/𝑑2.

ProblemNo.3–28pointsA simply-supported beam is subjected to the loadings shown below. Using thediscontinuityfunctionapproach,determinetheexpressionforthedeflectionv(x)forthesectionofthebeambetween3L<x<4L.

ME 323: Mechanics of Materials Homework Set 8 Fall 2015 Due: Wednesday, October 28

Problem 8.4 (10 points)

The simply supported beam AB is subjected to a uniformly distributed load w0 in AC, a concentrated load 𝑃𝑃0 at D, and a concentrated moment 𝑀𝑀0 at E.

(a) Set up the moment-deflection equation for the 2nd order integration method. (b) List the appropriate the boundary conditions and continuity conditions for the 2nd order

integration method. (c) Set up the load-deflection equation for the 4th order integration method. (d) List the appropriate the boundary conditions and continuity conditions for the 4th order

integration method. (e) Determine the deflection at C, D, and E using the superposition method.

A B

Fig. P.8.4

w0 N/m 𝑃𝑃0 N

𝑀𝑀0 N m

L m L m L m L m

C D E

y

x

PROBLEM NO. 1 Beam deflection using discontinuity functions.

B A

C

P

p0

a a

x

y,v

By

Ay P

p0

By

Ay P

p0a

Page 12 of 15

ME 323 Examination # 2 Name ___________________________________

July 19, 2017 PROBLEM NO. 4 - PART A – 5 points max.

The Mohr’s circle for the state of stress at a point in a structure experiencing plane stress is represented above. It is known that this state of stress has a maximum in-plane shear stress and an absolute maximum shear stress of

τ

max,in− plane= 20ksi and

τ

max,abs= 30ksi , respectively. For this state of stress:

a) Determine the normal and shear stress components related to the x-y axes.

b) Show these components of stress on the stress element provided above.

x − axis

30°

τ

x

y

σ

Page 2 of 12

ME 323 Examination #2 Name ___________________________________ July 22, 2016 PROBLEM NO. 1 – 30 points max.

A circular cross-section shaft (of diameter d and length 2L) is built into a fixed wall at end B. A rigid pulley is attached to the other end of the shaft. A cable (not shown) wrapped around the pulley applies a pair of tension forces of P and 2P on the pulley, as shown in the figure (note that these two forces both act in the positive z-direction). Let E and G represent the Young’s modulus and shear modulus, respectively, for the material of the shaft.

a) Determine the state of stress at points “a” and “b” on the shaft. Show these states of stress on the stress elements above.

b) Determine the principal stresses at points “a” and “b”. Use L / R = 3in your calculations.

x

y

z

P

2P

Ba

R

d

LL

C

bx

y

z

x

y

z

stresselementat“a” stresselementat“b”

Page 6 of 12

ME 323 Examination #2 Name ___________________________________ July 22, 2016 PROBLEM NO. 3 – 20 points max.

A cantilevered beam is loaded with concentrated couples and transverse force loadings. Although this loading is not shown below, a plot of the resulting internal bending moment M(x) is provided. Note that the bending moment curve M(x) is made up of straight-line segments.

a) Determine the shear force distribution for the beam and show this in the above plot for V(x).

b) Determine the loading on the beam and show this on the beam above.

c) Determine the location of the maximum tensile axial stress in the beam (giving both x and y coordinates). What is this maximum tensile value of axial stress?

x

V (x) (kips)

x (inches)

M (x) (kips ⋅ in)

x (inches)

0 5 10 15 20 25

2 in

4 in

y

z

beamcross-sec+on

20

4030

y

10

50

ME 323 Examination #3 Name ___________________________________ (Print) (Last) (First) December 18, 2014 Instructor ______________________________

Page 11 of 16

Problem 3 (9 points): The cross section is subject to loadings as shown.

Determine the state of stress at point A:

A) !! = !!

!! +!!!!!!

!!" = − !!!!!!

B) !! = !!

!! −!!!!!!

!!" = 0

C) !! = !!

!! +!!!!!!

!!" = 0 D) !! = !!

!! −!!!!!!

!!" =!!!!!!

Determine the state of stress at point B:

A) !! = !!

!! +!!!!!!

!!" = − !!!!!!

B) !! = !!

!! −!!!!!!

!!" =!!!!!!

C) !! = !!

!! +!!!!!!

!!" =!!!!!!

D) !! = !!

!! −!!!!!!

!!" = 0

x

y

z

Mz

My

Fx

Fy

A

B C

y

z

b

h

ME 323 Examination #3 Name ___________________________________ (Print) (Last) (First) December 18, 2014 Instructor ______________________________

Page 12 of 16

Determine the state of stress at point C:

A) !! = !!

!!!!" = 0 B)

!! = !!!!

!!" = − !!!!!!

C) !! = !!

!!!!" = 0 D)

!! = !!!!

!!" =!!!!!!