Name (Print) (Last) (First)

ME 323 – Mechanics of Materials Final Examination – Summer 2021

August 4, 2021

Instructions:

This is an open-lecture book exam. Feel free to use any material in your lecture book that either came with the book, or was provided to you in class, or work that you added by hand. You are not allowed to use any other material.

Please remember that for you to obtain maximum credit for a problem, you must present your solution clearly. Accordingly,

§ coordinate systems must be clearly identified, § free body diagrams must be shown, § write down clarifying remarks, § state your assumptions, etc.

If your solution cannot be followed, it will be assumed that it is in error.

PLEASE DO NOT WRITE ON THE BACK SIDE OF THE EXAM PAPERS. When handing in the test, make sure that ALL SHEETS are in the correct sequential order.

Prob. 1 ______________________ Prob. 2 ______________________ Prob. 3 ______________________ Prob. 4 ______________________ Prob. 5 ______________________ Total ________________________

SOLUTION

ME 323 Final Examination Name ___________________________________ August 4, 2021 PROBLEM NO. 1 – 20 points max.

Consider the cantilevered beam having a square cross-section and with the loading shown below. Let

p0 =3F0 /a and M0 = F0a .

a) Construct the shear force/bending moment diagrams for this loading on the beam. Express all your answers in terms of, at most, F0 and a. Show all of your work leading up to these diagrams. Label the values of shear force and bending moments on these diagrams.

b) Calculate the shear and normal components of stress at point “k” on the beam cross-section at point D (x = 2a) on the beam. Express all your answers in terms of, at most, F0 , a and b. Show this state of stress on the stress element provided below.

B C D

x

y

F0 p0

a a a

M0

H

x

x

V x( )

M x( )

b

b

y

z O

beamcross-section

k

x

z

y

stresselementat“k”onthecross-sectionat

pointD

SOLUTION

Fo poore y

gMo

Dy Hy fo of D AtsEo

Ma D at Cpoa 302 Fo a 2

Mo O 2Fo

Dy ta Zzpoof 3Fo at Mo Foaf Foata Z 3ta a stoat Foa

Sz fo Foa

Fy Fo poor D 1Hy oZ

H p a Fo Dy3Ffa a Fo Fo EFo

BI r T

T 6 Fo

Mca Mco Fo a Fo acoTC Zai T poor Fo 3Ifa ZtoM at Mcas Mo Foa Foa o

M Catd Mcat ICEDd E FoaAl Za M told I Fo d to Foa Fofza Foa

DltTCzat Zai t Dy 2 Fo t Iz Fo I FoTt a TCZat I FoM Za M za Foz a O

b T M 1 EERIETE bis

3 Fogb 3

I O free surface

ME 323 Final Examination Name ___________________________________ August 4, 2021 PROBLEM NO. 2 – 20 points max. A beam having a constant cross-section with a second-order area moment of I and made up of a material with a Young’s modulus of E is supported by a roller at C and a fixed support at end D. A constant line load of p0 acts on the beam between C and D. The goal of this problem is to determine the vertical deflection of end B of the beam. To this end, please follow the steps below:

a) Draw a free body diagram (FBD) of the beam. From this FBD, write down the equilibrium equations for the beam.

b) If the problem is indeterminate, choose the redundant load(s) on the beam, and write the remaining reactions in terms of the redundant load(s).

c) Write down the strain energy of the beam.

v

B C D

p0

L L

x

SOLUTION

PolFd Mrs

Eno Most ODED GIL b

Fd 4 0 2

Fy Cy Dy Pol Ed O

Cy Dy

Choose Cy as the redundant load

Dy Pol Cy Edtab PI Cy L t Fal

Z

XM

EMA Mi Fax o M Eeo E I IB C D

Oth To Fd

2Cyd Midid X

ME 323 Final Examination Name ___________________________________

August 4, 2021 PROBLEM NO. 2 (continued)

d) Using Castigliano’s theorem, determine the reactions on the beam at C and D. Express your answers in terms of, at most, p0 and L.

e) Using Castigliano’s theorem, determine the vertical deflection of the beam at end B. Express your answer in terms of, at most, p0 , E, I and L.

SOLUTION

Fd Pox

TZ14k Felt pox Mz Cy XMz Eel I post Cy XdMz Mz

xCy Cy X

2142Fed L

Tt it 02 e Midst MEDX

o 8 In.tofdxtIMzEEdDea o

O E Ipo cyx2 dx Polat'S 3

Cy ZooPoleg

Dy pole Cy PolMp poor Cyl poll

0

Drs d IM.TTddxtIMzdIEIdDdTdlfd o Fd o

It pox Lt CyxL dXo

poL4 Icy is PIL6 E I

E t 8 9

ME 323 Final Examination Name ___________________________________ August 4, 2021

PROBLEM NO. 3 – 20 points max. Identical elastic members (1) and (2) (each having a length of L, cross-sectional area A and made up of a material having a Young’s modulus E and coefficient of thermal expansion of α ) are attached to a thin rigid ring of radius R. Member (1) is at a 45° angle with respect to the horizontal, and member (2) is horizontal. The temperature of member (1) is increased by an amount of ΔT , while the temperature of (2) is held constant. It is desired to find the axial stress in each member as a result of the temperature increase of member (1). To this end:

1. Equilibrium: Draw a free body diagram of the ring. Write down the appropriate equilibrium equation(s) for the ring.

2. Load/deformation equations: Write down the load/deformation equations for members (1) and (2).

1( )

B

C

L

H

2( )

D

K

L

1

1

OR

SOLUTION

Fz

MB Fz 2R tFfrzR O H

D Fz F ERF ER

Bx

By

2 e LATL

ez FIEA

ME 323 Final Examination Name ___________________________________ August 4, 2021

3. Compatibility equation: Write down the compatibility equation relating the elongations of members (1) and (2).

4. Solve: Determine the axial stress carried by each member.

5. Assessment: For each member, state whether it experiences tensile or compressive stress, and whether it experiences tensile or compressive strain.

SOLUTION

E FROez zeo

O ETI Zzz4 e tr ez

Fifa t ATL FE FELAF Tz Ee 2 DTEA

Solving F iz e xDTEA Fc Zz2 DEAFz F L TEA

J FLITE Tz 2 DTE

T E Tz compressiveEz Iyaz O compressive

E Feta LAT Zz L DT 25 152AT 70tensile

ME 323 Final Examination Name ___________________________________ August 4, 2021

PROBLEM NO. 4 – 20 points max. A circular shaft is mounted on a fixed wall at end B. A rigid plate is welded to the shaft at end C. Two loads, both of magnitude P, act on the plate at locations D and H, as shown. A torque T also acts on the plate, as shown in the figure. Ignore the weight of the shaft and plate. It is desired to know the maximum shear stress at point “a” on the cross-section K of the shaft. To this end:

a) Determine the resultants acting on the left side of the cut at the cross-section K of the shaft. Show these on the figure provided below.

b) Show the different components of stress at point “a”on the cross-section K. Use the figures provided on the next page.

c) Make of list of the components of stress from b), and provide the equations for these components as related to the resultants found in (a). Use the table provided on the next page.

d) Show the stress components on the stress element provided on the next page.

e) Determine the maximum shear stress at point “a” corresponding to T = 3PL. Express your answer in terms of, at most, P, L and d.

L

T

P

L

L

a

P

x

y

B

C

H

d

D

K

a

B

K

T

P

P

C

H D

K

SOLUTION

Mg M

f pgFx Fx y

FyEFX F t P o Fx P

Fy P Fy o Fy P

I Mx T My o tax T

E Mz Hz t PL t PL o My _2 PL

ME 323 Final Examination Name ___________________________________ August 4, 2021 PROBLEM NO. 4 (continued)

x

y

z

resultant stressat“a”

y

z

endview

a

sideviewy

a

x

y

a

x

y

a

x

y

a

x

SOLUTION

Tat F T FKA

Fy I 0

Mx ta ETz IpT

My TE MII

Iz

Tx Ti Tz T TzEa PL d

IIZ

Ixy Iz Td 3 PLIZIP Ip

III max abs V xz2i

p V ja hr JtZEdpPwlAcTCdlz2

I IT 144Ip Iz 2

4

ME 323 Final Examination Name ___________________________________ August 4, 2021 PROBLEM NO. 5 - PART A – 4 points max. A given state of plane stress in a ductile material with a yield strength of σY =175MPa has principal

stress components of σ P1 =180MPa and σ P2 =20MPa . TRUE or FALSE: The maximum shear stress theory predicts that the material will fail. TRUE or FALSE: The maximum distortional energy theory predicts that the material will fail.

Since the principal components of stress have the same sign, then: τ

max,abs=σ P1

2= 90 MPa

For MSS, failure occurs if :

τ

max,abs>σY2

= 87.5 MPa .

Therefore, failure is predicted by MSS. For MDE, failure occurs if :

σY <σ M = σ P1

2 −σ P1σ P2 +σ P22 = 1802 − 180( ) 20( ) + 202 = 170.9 MPa .

Therefore, failure is NOT predicted for MDE

ME 323 Final Examination Name ___________________________________ August 4, 2021 PROBLEM NO. 5 - PART B – 4 points max.

A thin-walled tank (having an inner radius of r and wall thickness t) constructed of a ductile material contains a gas with a pressure of p. A rigid cap of weight W = 2π pr2 rests on top of a seal on the tank (i.e., the cap is not attached to the tank). Ignore the weight of the tank. If the tank fails under this loading, determine on which plane will the failure occur (i.e., describe the orientation of this plane relative to the horizontal). With your answer, include a sketch of the Mohr’s circle for the stress state in the tank wall. From the FBD of the cap above, and with FT being the support force on the cap from the tank:

Fy∑ = −W +π pr2 + FT = 0 ⇒ FT =W −π pr2 = π pr2

With this, the hoop and axial components of stress in the tank are:

σ h =

prt

σ a = −

FTA

= − π pr2

2πrt= − pr

2t

Since σ h and σ a are principal components of stress, the Mohr’s circle for the state of stress in the tank wall is as shown below. From this, we see that the angle for the absolute maximum shear stress is 90°/2 = 45°.

rigid cap of weight W

t

r

g

W

p πr2( )+FT

W

p πr2( )+FT

x −axis

τ max,abs

90°

τ

σ

ME 323 Final Examination Name ___________________________________ August 4, 2021 PROBLEM NO. 5 - PART C – 4 points max.

A structure is made up of a horizontal rigid bar BC and a vertical elastic member CD, where CD has a circular cross section of diameter d and is made up of a material having a Young’s modulus of E. A distributed loading having a transverse force/length of p0 acts on bar BC. Ignore the weight of the

members. Determine the critical load value p0 that corresponds to the buckling of member CD using the Euler theory of buckling. From the FBD of BC:

M B∑ = − p0L( ) L

2⎛⎝⎜

⎞⎠⎟+ FC L = 0 ⇒ FC =

p0L2

Using the Euler theory for buckling:

FC( )cr= π 2 EI

Leff2 = π 2 Eπ d / 2( )4 / 4

7L / 10( )2⇒

p0( )crL

2= 25

784π 3 Ed4

L2 ⇒ p0( )cr= 25

392π 3 Ed4

L3

p0

Ld

LB

C

D

B y FC

p0L

FC

FC

ME 323 Final Examination Name ___________________________________ August 4, 2021 PROBLEM NO. 5 - PART D – 4 points max.

A truss is made up of members (1) and (2) as shown above, with each member being composed of a material with a Young’s modulus of E and thermal expansion coefficient of α , and having a cross-sectional area of A. With the members being initially unstressed and unstrained, the temperature of (1) is increased by an amount of ΔT and the temperature of (2) is decreased by an amount of 2ΔT .

a) Provide an argument supporting the claim that members (1) and (2) remain unstressed after the temperature change described above. HINT: Consider the free body diagram of joint C.

b) TRUE or FALSE: Members (1) and (2) have zero strain.

a) From the FBD of joint C:

Fy∑ = −F2sinθ = 0 ⇒ F2 = 0

Fx∑ = −F1 − F2cosθ = 0 ⇒ F1 = −F2cosθ = 0

b) Calculating strains:

ε1 =F1EA

+αΔT =αΔT ≠ 0

ε2 =F2EA

− 2αΔT = −2αΔT ≠ 0

B

L

1( )C

D

2( )

y

x

0.5L

F1

F2θ

ME 323 Final Examination Name ___________________________________ August 4, 2021 PROBLEM NO. 5 - PART

• Structure (a) is an L-shaped frame made up of elements (1) and (2), with all connections in the frame

being pin joints. Elements (1) are (2) are both made up on steel. Let σ 2( )a be the axial stress in element

(2) of structures (a). • Structure (b) is identical to structure (a), except here element (2) is made up of aluminum. Let

σ 2( )b be

the axial stress in element (2) of structure (b). Circle the answer below that most accurately describes the states of stress in element (2) for these two frames:

a) σ 2( )a < σ 2( )b

b) σ 2( )a = σ 2( )b

c) σ 2( )a > σ 2( )b

The structure is DETERMINANT. Therefore, reactions and internal resultants can be found using only equilibrium relations, and, as a result, do not depend on material properties. From this, we conclude that the stresses in structure (a) and (b) must be the same.

P

B

C

D

(1)

(2) L

L / 2 L / 2

steel

steel

P

B

C

D

(1)

(2) L

L / 2 L / 2

steel

aluminum

Structure(a) Structure(b)