ME 323: Mechanics of Materials

Fall 2020

Homework 3

Due Wednesday, Sept 16

Problem 3.1 (10 Points)

Rigid link CDHK is pinned to ground at H. Elastic rod (1), having a cross-sectional area of 2A, is con-

nected between end C of the link and ground pin B. Elastic rod (2), having a cross-sectional area of A, isconnected between D on the link and the ground at N. The Young’s moduli of (1) and (2) are E and 3E,

respectively. A downward load of P is applied at end K. For this problem, use the following: d = 180 mm,

A = 25 mm2, E = 70 GPa, and P = 500 N.

For this loading, determine:

(a) Draw a free-body diagram of the rigid link CDHK.

(b) The normal stress in rod (1) and rod (2). Hint: The rotation in the rigid bar CDHK is small.

(c) The rotation angle of link CDHK in radians.

1

ZAIE

9 ABE

ft NH ale H't Hy

K Z ptsd d p

bd 572

x

Equilibriumequations

tfEMH 0

F.cl tFz.d p.zd o

fe fz 2P eptsFor fi 21000

compatibilityequation8 82 epts

xEn Fe 3 5

solving andF 250N Fz 750N Iptcompression Tension

Normal Stress8 FAI

250I 925 106

Sfpd

82 152 7251 6

30 MpaHt

C

c so K

D O

H

82 2 1750 180 10 Ipt3 109425156

0.02571mm dtan Sf 002571180 82r 0 1.43 104rad

Ipt

0.00818370

ME 323: Mechanics of Materials

Fall 2020

Homework 3

Due Wednesday, Sept 16

Problem 3.2 (10 Points)

Consider the three-member truss shown. Members (1), (2) and (3) have cross-sectional areas of A, 2A and

2A, respectively, and are made up of materials having Young’s moduli of 2E, E and E, respectively. A

force P is applied at joint C of the truss, as shown. It is desired to determine the axial load carried by

each member. To this end:

(a) Draw a free-body diagram of joint C, and write down the corresponding equilibrium equations. Based

on these equations, is the truss determinate or indeterminate? Explain.

(b) Write down the force-elongation equation for each member.

(c) Write down the compatibility relations among the elongations of members (1), (2) and (3), and the

xy-components of displacement of joint C, (uC , vC).(d) Solve for the axial loads in the members. Also, determine the values of uC and vC . Leave your

answers in terms of the parameters defined here in the problem statement.

2

ZA E

2E A2

2pts

fz 2 FZc

HjkF3 A

fi 9 E3 13

l is53

joint K taEfx O

52 7 Fs PCE O d

Efy o

F Fs E Mfs o

2 equations Ic 23Unknowns Fir Es

indeterminate

b S F 6d sadZpts EXA EAT

82 56 3fzd_E ZA ZEA

S3 fEf r adcompatibility

81 20 q 3I d IptEA

Sz Uc Uc ZFE Ipt83 143 124

SED 4 3TEA Ksk

d from last part IptSfsdl Rfid 9 I

7 5 1 ioIE f3 fit 9

10

System of equations2 Fit Fz Sz 5 0

Et Eg 5 32 pFl t g4fz zp

4FF EBp g F3FE EzP 3

12 48 54 27BP Ffg Eg P go F3 FEO

Jgp 124 73 f325 806

andf 149 pZpts 403

FE 753 p806

Uc ZFE D UE 2259 P A1612

Vc 3FId D Vc 447 PIE A 403 EA

Ipt

ME 323: Mechanics of Materials

Fall 2020

Homework 3

Due Wednesday, Sept 16

Problem 3.3 (10 Points)

A rigid block with a weight of 80 ⇥ 103lbs is supported by posts A, B, and C. The block is attached to

the posts at all times. The Young’s modulus for posts A and B is EA = EB = 29⇥ 103ksi, and for post C

is EC = 14.6⇥ 103ksi. All posts have the same initial length before loading with a cross-sectional area of

8 in2. After heating post C, its temperature increased by 20

�F while the temperature of A and B is held

constant. The coe�cient of thermal expansion for post C is ↵C = 9.8⇥ 10�6 �

F�1. Determine the normal

stress in each post before and after heating post C.

3

Free BodyDiagram80 18lbsi Iptn n

e se s2ft 2ftFA Fc FB

Equilibriumequations

MEOFB 2fA 0iFA fB f ZptsEfy 0

2 80 18 0Before heatingcompatibility equation68 88

fLa_ IptEAAA

Le LA L and AEAA lksi

tooolbslin24FI o6 zqx o 0F 2fc a

solving and a

F 32 18 lbs 32 Kip IpeE 16 28 lbs 16 tipNormal stress

32A B aE 8 4KsiF 1 z Ksi Ipt

tfterheatingcompatibility equationb c f FSF

exDTk E 2 ptsLA L

EX fL44.618 8

98 10 294 eg8 5616 4.31037 1.9618 2b

6

Solving and b

F 22.81 103165 22 81Kip IpeE 34.38 10316 3438 tipNormal stress8A dB AE 22381 285 Ksi

F 34838 4.30 Ksi Ipt

ME 323: Mechanics of Materials

Fall 2020

Homework 3

Due Wednesday, Sept 16

Problem 3.4 (5 points)

Consider a bar made of two sections fixed at both ends. For section (1) let the length be L1, area A1,

Young’s modulus E1 and coe�cient of thermal expansion ↵1. The corresponding values for section (2) are

L2, A2, E2 and ↵2. It is known that L1 < L2, E1 > E2, A1 > A2 and ↵1 > ↵2. The bar is free of stress at

temperature T1. Let �1 and �2 represent the axial stresses in section 1 and 2, respectively, after the rise in

temperature. When the temperature is raised from T1 to T2 (T2 > T1):

1. Which of the following is a true statement about the stresses? (2.5 points)

(a) |�1| > |�2|(b) |�1| = |�2|(c) |�1| < |�2|(d) |�1| = |�2| = 0

2. If �1 is the change in length of section 1 and �2 is the change in length of section 2, which of the

following statements is true? (2.5 points)

(a) �1 = �2 = 0

(b) �1 + �2 = 0

(c) �1 = �2 6= 0

(d) �1 =E1E2�2

Briefly justify your answers!!

4

f EAJ FF Fz

i stressislargerinthecrosssectionwithlowerarea

I SE Sa t Sit82SE SA o Elongationatthesupport

shouldbeZero