fall 2019 problem 3.1 (10 points): a rigid bar is
TRANSCRIPT
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ME 323: Mechanics of Materials Homework Set 3
Fall 2019
Problem 3.1 (10 points): A rigid bar is supported by three elastic Aluminum alloy (Youngโs
Modulus E) posts with identical cylindrical cross-section of area A. The center post is slightly
shorter than the two outer posts by an amount ๐ฟ, as shown in Fig. 3.1. A load P acts downward
directly over the center of the middle post.
A = 200 mm2; L = 2 m; E = 72 GPa.
a) What is the width ๐ฟ of the gap closed by a load P = 24 kN?
b) If the gap was ๐ฟ = 1 mm, what is the new length of each bar when a load P = 40 kN is
applied?
Fig. 3.1
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Solution
Part a)
Free Body Diagram (FBD)
Looking at the forces on the rigid bar in Fig. 1A
Fig. 1A
Equilibrium equations
Force balance along the y-axis is:
ฮฃ๐น๐ฆ = 0 => โ๐ + 2๐น = 0
=> ๐น =1
2๐
[1.1]
The gap ๐ฟ must be such that the force applied ๐ = 24 ๐๐ would just close it without applying
any force on the post (2).
Therefore, only the forces applied by posts (1) act upon the rigid bar as shown in the FBD in fig.
๐1 =๐น๐ฟ1
๐ธ๐ด=
๐๐ฟ1
2๐ธ๐ด=
24 โ 103 โ 2
2 โ 72 โ 109 โ 200 โ 10โ6
=> ๐1 = 1.67 ๐๐
[1.2]
Part b)
There are many ways one can try to solve this part but the method which is highly encouraged to
be followed is using compatibility equations as depicted in Part b.2) below
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Part b.1)
First, it should be assessed if the rigid bar would deform post (2) when a load ๐ = 40 ๐๐ is
applied while a gap ๐ฟ = 1 ๐๐ is present between the rigid bar and end of the post (2).
So, using the same FBD in Fig. 1A, the change in length can be calculated using the Eqn. 1.2:
๐1 =๐น๐ฟ1
๐ธ๐ด=
๐๐ฟ1
2๐ธ๐ด=
40 โ 103 โ 2
2 โ 72 โ 109 โ 200 โ 10โ6
=> ๐1 = 2.78 ๐๐
[1.3]
The change in length is definitely higher than 1 mm, hence the rigid post not only makes contact
with the post (2) but also contracts it. Hence, the new FBD can be drawn after the rigid bar
makes contact with the post (2) as in Fig. 1B:
Fig. 1B
Equilibrium of the forces can be written as:
ฮฃ๐น๐ฅ = 0 โน 2๐น1 + ๐น2 = ๐ [1.4]
Since, the posts are all made of same materials ๐น1 = ๐น2 = ๐น =1
3๐
The change in length of the post (2) can be calculated as follows:
๐2 =๐น๐ฟ2
๐ธ๐ด=
๐๐ฟ2
3๐ธ๐ด=
40 โ 103 โ (2 โ 0.001)
3 โ 72 โ 109 โ 200 โ 10โ6
=> ๐2 = 1.85 ๐๐
[1.5]
Hence, the final lengths of all the three posts is calculated as:
๐ฟ๐๐๐๐๐ = ๐ฟ๐๐๐๐ก๐๐๐ โ ๐ฟ โ ๐2 = 2 โ 0.001 โ 0.00185
=> ๐ฟ๐๐๐๐๐ = 1997.15 ๐๐
[1.6]
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Part b.2) โ The best way to solve this part
The change in length is definitely higher than 1 mm, hence the rigid post not only makes contact
with the post (2) but also contracts it. Hence, the new FBD can be drawn after the rigid bar
makes contact with the post (2) as in Fig. 1B:
Fig. 1B
Equilibrium of the forces can be written as:
ฮฃ๐น๐ฅ = 0 โน 2๐น1 + ๐น2 = ๐
โน ๐น2 = ๐ โ 2๐น1
[1.4]
The compatibility conditions can be obtained from the figure below:
Fig. 1C
๐2 + ๐ฟ = ๐1 [1.3]
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โ๐น2๐ฟ2
๐ด๐ธ+ ๐ฟ = โ
๐น1๐ฟ1
๐ด๐ธ
โน(๐ โ 2๐น1)๐ฟ2
๐ด๐ธโ ๐ฟ =
๐น1๐ฟ1
๐ด๐ธ
โน ๐น1 =๐๐ฟ2 โ ๐ฟ๐ด๐ธ
๐ฟ1 + 2๐ฟ2=
40 โ 103 โ 1.999 โ 0.001 โ 200 โ 10โ6 โ 72 โ 10โ9
2 + 2 โ 1.999
โน ๐น1 = 10.93 ๐๐
[1.4]
๐1 =๐น1๐ฟ1
๐ธ๐ด=
10.93 โ 103 โ 2
200 โ 10โ6 โ 72 โ 109= 1.52 ๐๐
[1.5]
๐ฟ๐๐๐๐๐ = ๐ฟ1 โ ๐1 = 2 โ 0.00152 = 1998.47 ๐๐ [1.6]
Part b.3)
The change in length is definitely higher than 1 mm, hence the rigid post not only makes contact
with the post (2) but also contracts it. Hence, the new FBD can be drawn after the rigid bar
makes contact with the post (2) as in Fig. 1B:
Fig. 1B
The force required to close the gap ๐ฟ as seen from the FBD in Fig. 1B can be calculated as:
๐น1 =๐ฟ๐ธ๐ด
๐ฟ1= 0.001 โ 72 โ 109 โ 200 โ
10โ6
2= 7.2 ๐๐
[1.3]
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The force acting on the post (2) can be calculated as follows:
๐น2 =๐ โ 2๐น1
3=
40 โ 2 โ 7.2
3= 8.53 ๐๐
[1.4]
๐2 =๐น2๐ฟ2
๐ธ๐ด=
8.53 โ 103 โ (2 โ 0.001)
72 โ 109 โ 200 โ 10โ6= 1.18 ๐๐
[1.5]
๐ฟ๐๐๐๐๐ = ๐ฟ โ ๐ฟ โ ๐2 = 2000 โ 1 โ 1.18 = 1.9978 ๐๐ [1.6]
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Problem 3.2 (10 points): A three-segment rod that is initially stress-free is attached to rigid
supports at ends A, B, and D. Elastic segments (1) and (2) are joined by the left hand side of rigid
connector C, whereas elastic segment (3) is joined by the right hand side. Segment (1) has solid
circular cross-section with diameter d. Segments (2) and (3) have circular tube cross-sections.
Segment (2) has an inner diameter 2d and an outer diameter 3d. Segment (3) has an inner diameter
d and an outer diameter 2d. Segment (1) has a length of 2L, whereas segments (2) and (3) have
lengths of L. Each segment has a Youngโs modulus of E. An external load 2P is applied on the
connector C.
a) Determine the axial stresses induced in segments (1), (2), and (3).
b) Determine the displacement of the connector C.
Express your results in terms of P, d, L, E and ฯ.
Fig. 3.2
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Solution
Free Body Diagram (FBD)
From the FBD of the connector C
Fig. 2A
Equilibrium equations
Force balance is shown below along x-axis:
ฮฃ๐น๐ฅ = 0 => ๐น3 โ ๐น1 โ ๐น2 โ 2๐ = 0
[2.1]
There are 3 unknowns and only 1 equation, hence itโs an indeterminate problem.
Force-elongation equations for each of the three rods:
๐1 =๐น1๐ฟ1
๐ธ๐ด=
2๐น1๐ฟ
๐ธ (๐๐2
4)
=8๐น1๐ฟ
๐ธ๐๐2
๐2 =๐น2๐ฟ2
๐ธ๐ด=
๐น2๐ฟ
๐ธ (5๐๐2
4)
=4๐น2๐ฟ
5๐ธ๐๐2
๐3 =๐น3๐ฟ3
๐ธ๐ด=
๐น3๐ฟ
๐ธ (3๐๐2
4)
=4๐น3๐ฟ
3๐ธ๐๐2
[2.2]
Compatibility Equations:
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๐ข๐ด + ๐1 = ๐ข๐
๐ข๐ต + ๐2 = ๐ข๐
๐ข๐ + ๐3 = ๐ข๐ท
[2.3]
where, ๐ข๐ด = ๐ข๐ต = ๐ข๐ท = 0 as these walls are immovable. Therefore:
๐1 = ๐2
๐1 = โ๐3
๐1 = ๐2 = โ๐3
[2.4]
Using the Eqns in 2.4, Eqns 2.2 can be re-written as follows:
8๐น1๐ฟ
๐ธ๐๐2=
4๐น2๐ฟ
5๐ธ๐๐2 โน ๐น2 = 10๐น1
8๐น1๐ฟ
๐ธ๐๐2=
โ4๐น3๐ฟ
3๐ธ๐๐2 โน ๐น3 = โ6๐น1
[2.5]
Sub. ๐น2 and ๐น3 in Eqn. 2.1 to get:
๐น1 = โ2
17๐
๐น2 = โ20
17๐
๐น3 = +12
17๐
[2.6]
Part b)
The displacement of the connector C would be equal to the elongation of one of the bars
(according to Eqn. 2.4). Therefore,
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๐ข๐ = ๐1
๐ข๐ =8๐น1๐ฟ
๐ธ๐๐2= โ
16๐๐ฟ
17๐ธ๐๐2
[2.7]
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Problem 3.3 (10 points): Three elastic members form the pin-jointed truss in Fig. 3.3. The joint
at A is constrained by a slider block to move only in the horizontal direction (i.e., vA = 0). All
members have the same cross-sectional area A = 525 mm2 and a modulus of elasticity E = 150
GPa. The pin-joints B, C and D are colinear.
a) Determine the member forces F1, F2 and F3 which result in moving the joint A to the right
by 13 mm (i.e., uA = 13 mm).
b) Determine the horizontal load P.
c) Determine the vertical reaction between the slider block at A and the track.
Fig. 3.3
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Solution
Free Body Diagram (FBD)
The FBD at point A is drawn below:
Fig. 3A
Force balance equations:
๐ด๐น๐ฅ = 0 => โ3
5๐น1 โ ๐น2 โ
1
2๐น3 + ๐ = 0
[3.1]
๐ด๐น๐ฆ = 0 => ๐ด๐ฆ +4
5๐น1 โ
โ3
2๐น3 = 0
[3.2]
There are 5 unknowns and only 2 equations, hence this is an indeterminate problem.
Part a)
Force-elongation equations:
๐น1 =๐ด๐ธ
๐ฟ1๐1
๐น2 =๐ด๐ธ
๐ฟ2๐2
๐น3 =๐ด๐ธ
๐ฟ3๐3
[3.3]
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Elongations of the elastic members (1), (2) and (3) can be represented along with the motion of
pin joint A as:
Fig. 3B
From the Fig. 3B, there is no vertical displacement and hence the elongations can be represented
only in the horizontal displacement of the joint A as:
๐1 =3
5๐ข๐ด โน ๐น1 =
3๐ด๐ธ
5๐ฟ1๐ข๐ด
๐2 = ๐ข๐ด โน ๐น2 =๐ด๐ธ
๐ฟ2๐ข๐ด
๐3 =1
2๐ข๐ด โน ๐น3 =
๐ด๐ธ
2๐ฟ3๐ข๐ด
[3.4]
[OR] โ It can be obtained using:
๐1 = ๐ข๐ด๐๐๐ ๐1 + ๐ฃ๐ด๐ ๐๐๐1; ๐1 = 306.87๐
๐2 = ๐ข๐ด๐๐๐ ๐2 + ๐ฃ๐ด๐ ๐๐๐2; ๐2 = 0๐
๐3 = ๐ข๐ด๐๐๐ ๐3 + ๐ฃ๐ด๐ ๐๐๐3; ๐3 = 60๐
[3.4]
๐ฟ1 = 5 ๐; ๐ฟ2 = 3 ๐; ๐ฟ3 = 6 ๐; ๐ด = 525 ๐๐2; ๐ธ = 150 ๐บ๐๐
Substituting the above values in Eqn. 3.4, gives:
๐น1 = 122.85 ๐๐ [3.5]
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๐น2 = 341.25 ๐๐
๐น3 = 85.31 ๐๐
Part b)
Substitute the above force values in Eqn. 3.1 to calculate the value of ๐
๐ =3
5๐น1 + ๐น2 +
1
2๐น3 = 457.615 ๐๐
[3.3]
Part c)
Substitute the above force values in Eqn. 3.2 to calculate ๐ด๐ฆ
๐ด๐ฆ =โ3
2๐น3 โ
4
5๐น1 = โ24.4 ๐๐
[3.4]
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Problem 3.4 (10 points): The 50 ft long steel rails on a train track are laid with a small gap between
them to allow for thermal expansion.
(a) Determine the required gap so that the rails just touch one another when the temperature is
increased from T1 = -20 ยฐF to T2 = 120 ยฐF.
Hint: since all rails expand equally, each rail needs to effectively expand by ๐ฟ.
(b) Using the gap determined in (a), what would be the axial force in the rails if the temperature
were to rise to T3 = 140ยฐF? The cross-sectional area of each rail is 5.10 in2.
Hint: since all rails expand equally, study a rail of length 50 ft at T1 between two rigid walls
located at 50 ft + ๐ฟ from each other.
The coefficient of thermal expansion ๐ผ = 6.60ร10-6 /ยฐF, and Youngโs modulus E = 29ร103 ksi.
Fig. 3.4
Solution
Part a)
All the rails equally have a gap of ๐ฟ and when the temperature increases, all of them expand
equally by ๐ฟ
2 on both the sides, so a total of ๐ฟ increase can be considered for a single rail.
When is gap is just closed, there wouldnโt be any force acting on the rail. Therefore,
๐ฟ = ๐ = ๐ผฮ๐๐ฟ = 6.6 โ 10โ6 โ (120 โ (โ20)) โ 50 = 0.554 ๐๐ = 0.0462 ๐๐ก
[4.1]
Part b)
After the gap is filled, any temperature increase would result in experience of stress in the rail as
there isnโt any more free space to expand into. Therefore, the expansion can be expressed as:
๐ =๐๐ฟ
๐ธ๐ด+ ๐ผฮ๐๐ฟ
๐ = 0 here and hence the above equation can be re-written as:
๐ = โ๐ผฮ๐๐ธ๐ด = 6.6 โ 10โ6 โ (140 โ 120) โ 29 โ 106 โ 5.1 = 19.522 ๐๐๐๐
[4.2]