ME 270 Final Exam - PM F20 (P2-P3 Soln)...12 324 I I I DEAD 0.0833 at EYE fo 0096911 fIga o0.833 ftp.oogoga ggit fo3Fo.og6gja o.igg at 4103 N t VI zfok.NO Em 210.0505mi 20 ME 270 –

ME 270 – Fall 2020 Final Exam - PM NAME (Last, First):________________________________

ME 270 Final Exam – Fall 2020 – PM 1

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Purdue Honor Pledge – “As a Boilermaker pursuing academic excellence, I pledge to be honest and true in all that I do. Accountable together – We are Purdue.”

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Instructor’s Name and Section (circle your section)

J. Jones MWF 9:30AM-10:20AM M. Murphy TR 7:30AM-8:45AM J. Jones MWF 10:30AM-11:20AM M. Murphy TR 9:00AM-10:15AM L. Vasconcelos 10:30AM-11:20AM J. Jones Distance Learning S. Dyke MWF 11:30AM-12:20PM J. Jones Distance (Online Only) C. Krousgrill Online F. Semperlotti 1:30PM-2:20PM F. Semperlotti 4:30PM-5:20PM INSTRUCTIONS

Begin each problem in the space provided on the examination sheets. If additional space is required, please request additional paper from your instructor.

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

Each problem is worth 25 points.

Please remember that for you to obtain maximum credit for a problem, it must be clearly presented. Also, please make note of the following instructions.

x The allowable exam time for Exam 1 is 90 minutes. x All coordinate systems must be clearly identified. x Where appropriate, free body diagrams must be drawn. These should be drawn separately from the

given figures. x Units must be clearly stated as part of numerical answers. x You must carefully delineate vector and scalar quantities. x Please use a dark lead pencil or black pen for the exam. x If the solution does not follow a logical thought process, it will be assumed in error.

When submitting your exam on Gradescope, 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. Also, be sure to identify the page numbers for each problem before final submission on Gradescope. Do not include the cover page or the equation sheet with any of the problems.



Problem 1. (25 points) 1A. (7 points) GIVEN: The massless particle A can slide in the smooth inclined guide realized inside a block. The block is bolted to the ground. The particle A is connected to an elastic spring whose elastic stiffness constant is k=1000 N/m. At equilibrium, the spring is compressed by 0.01m (with respect to the unstretched condition). An external force of magnitude F is applied to the block at equilibrium.

FIND: a) The magnitude of the force F. b) The magnitude of the normal component N of the reaction force that the smooth guide applies to the particle. In the figure, !"is the unit vector indicating the direction normal to the slot.

$ = &' = ()))*+ ∗ ). )(+ = ()*

∑/! = ) /012(4)) − $ = ) / = "#$%('() = ((. 78*

∑/* = ) / −/9:0(4)) + * = ) * = /9:0(4)) = 7. <<*

|/| = ((. 78 [N] (3 pts)

* = 7. << [N] (4 pts)



1B. (7 points) GIVEN: A uniform rigid bar with smooth roller supports at points A and B is supported by a horizontal and vertical wall, and by the cable AC. The weight of the bar is 60 lbf.

FIND: a) Draw the free body diagram of the rigid bar on the schematic provided here below. (1 pts)

>?@(A) =87

B = CDE+, F-./- G = ). H88IJK = 8H. L4

b) Find the magnitude T of the tension in the cable AC at equilibrium. c) Find the reactions at A and B and express them in vector form.

∑/! = ) −M! + $9:0(B) = ) $ = 0!12#(3) = 4). (NOPQ

∑/* = ) R* + $012(B) −S = ) R* = S− $012(B) = (7. )TOPQ ∑U4 = ) LM! −S9:0(A) ∗

56 = ) M! = S9:0(A) ∗ 5' = 8)OPQ

|$| = 4). (N [lbf] (2 pts)

RVV = ())X + ((7. )T)Z [lbf] (2 pts)

MVV = (−8))X + ())Z [lbf] (2 pts)



1C. (4 points) GIVEN: The frame structure provided in the figure has smooth roller supports at A and C, a hinge support at D, and it is connected to a block of mass M via a cable wrapped around an ideal pulley. Neglect the weight of the frame elements.

The six figures below represent possible free body diagrams of individual elements of the frame. CIRCLE one or more free body diagrams that represent the correct set of forces at equilibrium. To indicate your selection, please circle the label “Case #”.



1D. (7 points) GIVEN: The truss structure depicted in the figure is hinged at joint A and has a roller constraint at joint I. The truss is loaded by the forces PD=6kN and PC=5kN. FIND: a) Identify any possible zero-force member.

Zero-force member(s): CE, BL, BG (3 pts) b) Using the method of sections, find the magnitude of the force in the element BF and indicate if the element is in tension or in compression (circle one in the box below). Show all your work including the free body diagram.

∑U0 = ) −8[7 − T[8 − /\ ∗ 9:0(B) ∗ ( − /\ ∗ 012(B) ∗ T = )

/\ = − -9":69112#(3):6#$%(3) =-19.007 (C)

∑/! = ) [7 + [8 − /\012(B) − M/012(B) = )

M/ = 9":9#:;<#$%(3)#$%(3) = 7. 7N (T)

|]^|= 5.59 [kN] (3 pts)

Circle one: Tension Compression (1 pts)



Problem 2. (25 points) 2A. (6 points) Using the method of integration, determine the second moment of area of the shape with respect to the x-axis. Give your answer in terms of 𝑎.

𝐼 = ______________ (6 pts)

2B. (7 points) For shaded regions below, 𝐼 and 𝐼 correspond to the second area moments about the x-axis and y-axis, respectively, and 𝑏 > 𝑐.

a) Choose the correct answer below. The subscripts 1 and 2 refer to areas 1 and 2, respectively. No

calculation needed.

(𝐼 )1 > (𝐼 ) (𝐼 )1 = (𝐼 ) (𝐼 )1 < (𝐼 ) (1.5 pt)

b) Choose the correct answer below. The subscripts 1 and 2 refer to areas 1 and 2, respectively. No calculation needed.

𝐼1

> 𝐼 𝐼1

= 𝐼 𝐼1

< 𝐼 (1.5 pt)

X

dy jIx fyDA foayzkfytdxy.VE y3

Ix atfoay ydy.at sdy a f fa

a

O

O



c) Find the second area moment about the x-axis for area 1 using the method of composite parts. Take 𝑟 = 𝑎/3 and 𝑏 = 𝑎/2 . Give your answer in terms of 𝑎.

(𝐼 )1 = _________________ (4 pts)

2C. (5 points): The pin shown below keeps the 30mm thick plate connected to the wall. The pin has 10mm x 10mm square cross section. A 4KN force is applied to the place. Determine the shear stress experienced by the pin.

|𝜏| = ______________ [MPa] (5 pts)

OYI I E Ii d Ai

f µ A a Az Tr togagl ag di da E g a1 X t

Iz Ig 0.009690I b

z190 0,0833a12

324

I I DFA I DEAD

0.0833 at EYE fo0096911 fIga

o0.833 a't ftp.oogoga ggitfo3F o.og6gja o.igg

0.236 at

4103 Nt VI zfok.NO Em 210.0505mi

20



2D. (7 points) GIVEN: The bar below is made up of two members (AB and BC) and it is attached to a wall at A. The members are connected by the rigid connector B. The dimension of the square cross sections of each member are shown below.

a) (2 points): Determine the magnitude of the internal axial force in member AB.

|𝐹 | = ______________ [KN] (2 pts)

b) (3 points): Determine the axial stress in member AB.

𝜎 = ______________ [MPa] (3 pts)

d) (2 points): If both members are made of the same aluminum alloy (yield stress of 𝜎 = 400 MPa), determine the safety factor (FS) of the bar.

FS= _________ (2 pts)

6000

OT F A B 2 0.27 10.110.10.03m2

A F 6000kW

O 60001102 200MPa0.03 m2

200 compression

FS Of TOO2 00

2

2



Problem 3. (25 points) GIVEN: A solid shaft attached to a fixed support at A is designed to resist the torque by the force couple at C. The system is in static equilibrium and the deformation is within the elastic regime of the materials composing the shaft. The weight of the shaft is negligible. Segment (1) has is made of a material with shear modulus 𝐺1 = 20 𝐺𝑃𝑎, while segments (2) and (3) are made of another material with shear modulus 𝐺 = 𝐺 = 60 𝐺𝑃𝑎.

FIND:

a) Determine the magnitude of the internal torque in the segments (1), (2) and (3).

|𝑻𝟏| = ______________ [N.m] (2 pts)

|𝑻𝟐| = ______________ [N.m] (2 pts)

|𝑻𝟑| = ______________ [N.m] (2 pts)

b) Determine the magnitude of maximum shear stress in the shaft and draw the shear stress profile as a function of the radial coordinate 𝜌 in the diagram below:

|𝝉𝒎𝒂𝒙| = ______________ [MPa] (6 pts)

𝜌

|𝜏|

(2 pts)

R 100mm

IT1 1200KN112mL TOOkN m

400 A400 KO K

Imanfp D T 127MDaJ

o.atJ I r

2

tmax 2ffmjx.sefkNjY254 MR

254



c) Determine the segment, radial position, and value of the maximum shear strain in the shaft.

Segment with the maximum strain: (1 pts)

Radial position of the maximum strain: (1 pts)

|𝜸𝒎𝒂𝒙| = ______________ (5 pts)

d) If the outer diameter of the solid shaft was increased, what would be the effect on the maximum shear stress in the shaft? No calculation needed.

|𝝉𝒎𝒂𝒙|: decrease stay the same increase (2 pt)

e) If the solid shaft was replaced by a hollow shaft with the same outer diameter, what would be the effect on the maximum shear stress in the shaft? No calculation needed.

|𝝉𝒎𝒂𝒙|: decrease stay the same increase (2 pt)

Extra page (if needed)

Since Ti Tz and Gi Ga the max shearstrain

takes place in section CH

E Gr254 254 MPa

Jae MPI20 GPa 201103Mpa

0.01272

4

0.0127D r 0 Im

0

O



PROBLEM 4 (25 points) GIVEN: Consider the cantilevered beam shown. The beam is loaded as shown, and is in equilibrium. One concentrated load is applied at point C (+ 500N), a constant distributed load is applied between points A and B (+ 500N/m), and an external moment is applied at point D (+ 500 N-m).

FIND: a) On the artwork provided below draw the free body diagram for the beam: (pts 3)

b) Determine the reactions at the fixed end, point A. Show your work and write in vector form.

!"" = [N] (2 pts)

$!"""""" = [N-m] (1 pts)

500N=

500NM

At MA y 4500NqbN 7It

Ax

Efx =0=A×C]E Fy=O= Ay +500N +500N → Ay=-IoooN#

g MA=D = - MA t 500N Gm ) t 500am ) tmg

MA=3000NTLO E t 1000J

3000 I.



c) On the axis provided below complete the shear force (5 pts) and bending moment diagrams (5 pts). Label the values of the shear force and bending moment at points A, B, C, and D. You may use the graphical method. Show your calculations below.

ON ON

IT- 1000N @ *500N

,SOON

3000M• NM pm

1500go

ON M500

#e#quadratic linear constant

✓ B= - 1000N t 250 (2) N = - 500N

Uc = - 500N t 500N = ON

Up = ON

MB = 3000 N m It §C- too ont x ) dx

=3 ooo Nm t- 1000 (2) t 5¥ ¥ = 1500 Nm

Mc = 1500 Nm t 6500NK2m ) = soo Nm

Mb = 560

Mm



d) Identify the segments(s) where the beam is under pure bending (circle your answer).

AB BC CD (3 pts)

e) The cross section of the beam has the shape of a trapezoid, as shown in the figure to the right. The centroid of this cross section is shown in the figure. The second moment of area of this section about the centroid is I = 0.2 cm4, and the dimensions above and below the centroid are a = 3 cm and b = 2 cm. Determine the maximum magnitude of the normal stress in the region of pure bending, provide the location of the point on the beam cross section where this occurs, and indicate if it is in tension or compression.

|('))*+| = [GPa] (3 pts)

Circle one: Tension Compression (3 pts)

Location: Top Neutral Axis Bottom Centroid (3 pts)

IoTma ,

=

lmlmoxf.mn,

ymax-

dI

=

( 500 Nm ) C. 03M )

⇒4= 7.5409km ' moment positive

= 2.56Pa top sickinconfusion

705

oO

Documents

ME 270 Final Exam - PM F20 (P2-P3 Soln)...12 324 I I I DEAD 0.0833 at EYE fo 0096911 fIga o0.833 ftp.oogoga ggit fo3Fo.og6gja o.igg at 4103 N t VI zfok.NO Em 210.0505mi 20 ME 270 –