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SEDC Handouts
TABLE OF CONTENTS
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1.0 SEDC-Engineering Mechanics – Statics---------------------------------------------------------2-17
2.0 SEDC-Engineering Mechanics – Dynamics--------------------------------------------------18-27
3.0 SEDC-Strength of Materials-----------------------------------------------------------------------28-39
4.0 SEDC-Static Determinacy and Influence Lines-----------------------------------------------40-41
5.0 SEDC-Theory of Structures - Cables and Arches-------------------------------------------42-43
6.0 SEDC-Approximate Analysis----------------------------------------------------------------------45-46
7.0 SEDC-Deflections------------------------------------------------------------------------------------46-47
8.0 SEDC-Analysis of Statically Indeterminate Structures--------------------- ----------------48-49
9.0 SEDC-Reinforced Concrete Design 1 - Allowable Stress Design ---------------------=-50-51
10.0 SEDC-Reinforced Concrete Design 2 - Flexure Analysis --------------------------------52-54
11.0 SEDC-Reinforced Concrete Design 3 - Shear and Torsion ------------------------------55-57
12.0 SEDC-Reinforced Concrete Design 4 - Analysis and Design of Slabs ----------------58-59
13.0 SEDC-Reinforced Concrete Design 5 - Analysis and Design of Columns -----------60-66
14.0 SEDC-Reinforced Concrete Design 6 - Bond, Anchorage and Development Length--67
15.0 SEDC-Steel Design 1 - Design Philosophies and Load Calculation -----------------------68
16.0 SEDC-Steel Design 2 - Tension Members ---------------------------------------------------69-71
17.0 SEDC-Steel Design 3 - Simple Connections ------------------------------------------------72-74
18.0 SEDC-Steel Design 4 - Eccentric Connections ---------------------------------------------75-76
19.0 SEDC-Steel Design 5 – Beams -----------------------------------------------------------------77-79
20.0 SEDC-Steel Design 6 - Bearing Plates and Column Base Plates-----------------------80-81
21.0 SEDC-Steel Design 7 - Compression Members--------------------------------------------82-84
22.0 SEDC-Steel Design 8 - Bi-axial Bending------------------------------------------------------85-86
23.0 SEDC-Timber Design-------------------------------------------------------------------------------87-88
24.0 SEDC-Seismic Analysis----------------------------------------------------------------------------89-90
25.0 SEDC-Wind Loading Analysis------------------------------------------------------------------------91
26.0 SEDC-Prestressed Concrete Design-----------------------------------------------------------92-93
27.0 SEDC-Foundation Design-------------------------------------------------------------------------94-96
28.0 SEDC-Construction--------------------------------------------------------------------------------97-100
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Effectiveness. Efficiency. Convenience
Engineering Mechanics - Statics
ENGR. HANS LAWRENCE E. DELA CRUZ
INSTRUCTION: Select the correct answer for each of the
following questions. Mark only one answer for each item by
shading the box corresponding to the letter of your choice on
the answer sheet provided. STRICTLY NO ERASURES
ALLOWED. Use pencil no. 2 only.
Resultant of a Force System
• Components and Resultant of Concurrent Force System
1.
Two forces act on the hook. Determine the magnitude
of the resultant force.
4.
Compute the magnitude of the component of F
directed along the axis of member AB.
a.
963.38 lb
c.
869.33 lb
b.
636.40 lb
d.
689.33 lb
5.
Compute the magnitude of the component of F
directed along the axis of member AC.
a.
963.38 lb
c.
869.33 lb
b.
636.40 lb
d.
689.33 lb
a.
b.
746.56 N
756.64 N
c.
d.
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Situation 3:
The vertical force F acts downward at A on the two
membered frame. Set F = 500 N
665.74 N
656.74 N
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Situation 1:
From the figure shown below,
Determine the magnitude of the component of F
directed along the axis of AB.
a.
366.03 N
c.
303.66 N
b.
448.29 N
d.
484.92 N
7.
Determine the magnitude of the component of F
directed along the axis of AC.
a.
366.03 N
c.
303.66 N
b.
448.29 N
d.
484.92 N
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6.
Compute the magnitude of the resultant force F R.
a.
11.98 N
c.
19.18 N
b.
11.89 N
d.
18.19 N
3.
Compute the direction of the resultant force measured
counterclockwise from the positive x-axis.
a.
57.63°
c.
157.63°
b.
357.63°
d.
257.63°
Situation 4:
The force acting on the gear tooth is F = 30 lb.
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2.
Situation 2:
The force F = 450 lb acts on the frame.
0905 315 5857
Calculate the component of the force acting along the
line a-a.
a.
45.96 lb
c.
30.64 lb
b.
26.95 lb
d.
40.42 lb
9.
Calculate the component of the force acting along the
line b-b.
a.
45.96 lb
c.
30.64 lb
b.
26.95 lb
d.
40.42 lb
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Situation 5:
The plate is subjected to the two forces at A and B as
shown in the figure:
If the resultant force of the two tugboats is 3 kN,
directed along the positive x axis, determine the
required direction 𝜃 of force FB.
a.
38.26°
c.
42.76°
b.
35.08°
d.
29.05°
16.
If the resultant force of the two tugboats is required to
be directed towards the positive x axis, and F B is to be
a minimum, determine the magnitude of FR.
a.
2.12 kN
c.
1.73 kN
b.
1.59 kN
d.
2.67 kN
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• Resultant of Three or more Concurrent Forces
17.
Determine the magnitude of the resultant force and its
direction 𝜃 measured counterclockwise from the
positive x axis.
Determine the magnitude of the resultant force.
a.
11.58 kN
c.
10.40 kN
b.
9.51 kN
d.
11.10 kN
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Situation 6:
If the magnitude of the resultant force is to be 500 N,
directed along the positive y axis,
15.
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11.
Determine the angle 𝜃 for connecting member A to the
plate so that the resultant force of FA and FB is directed
horizontally to the right.
a.
74.93°
c.
54.93°
b.
44.93°
d.
64.93°
If the resultant force of the two tugboats is 3 kN,
directed along the positive x axis, determine the
required magnitude of force FB.
a.
7.07 kN
c.
1.61 kN
b.
4.07 kN
d.
3.66 kN
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10.
14.
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a.
b.
c.
d.
Determine the magnitude of the force F.
a.
840 N
c.
777 N
b.
960 N
d.
667 N
13.
And its direction 𝜃.
a.
45.21°
b.
50.10°
R
12.
c.
d.
𝐹𝑅
𝐹𝑅
𝐹𝑅
𝐹𝑅
= 31.24 𝑘𝑁;
= 39.81 𝑘𝑁;
= 38.19 𝑘𝑁;
= 34.21 𝑘𝑁;
𝜃
𝜃
𝜃
𝜃
= 39.81°
= 31.24°
= 34.21°
= 38.18°
Situation 8:
From the figure shown below,
53.56°
40.39°
Situation 7:
A large vessel is being towed by two tugboats A and B.
18.
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Determine the magnitude of F1 so that the resultant
force is directed vertically upward and has a
magnitude of 800 N.
a.
297.04 N
c.
247.09 N
b.
279.40 N
d.
274.90 N
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ENGR. HANS LAWRENCE E. DELA CRUZ
19.
Determine the direction 𝜃 of F1 so that the resultant
force is directed vertically upward and has a
magnitude of 800 N.
a.
29.10°
c.
60.90°
b.
35.57°
d.
54.43°
Situation 11:
From the given pipe assembly as shown below,
• Resultant of Concurrent, 3-D Force System
Situation 9:
From the figure shown below,
Calculate the magnitude of the resultant force F R.
a.
680.0 lb
c.
753.7 lb
b.
282.8 lb
d.
160.0 lb
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24.
25.
Express the force as a Cartesian vector in N.
a.
265.17𝑖 − 459.28𝑗 + 530.33𝑘
b.
265.17𝑖 + 459.28𝑗 + 530.33𝑘
c.
265.17𝑖 − 459.28𝑗 − 530.33𝑘
d.
265.17𝑖 + 459.28𝑗 − 530.33𝑘
21.
Determine the coordinate direction angles of the force.
a.
69.30°, 52.24°, 45°
b.
69.30°, 127.76°, 135°
c.
69.30°, 52.24°, 135°
d.
69.30°, 127.76°, 45°
Calculate the coordinate direction angles of the
resultant force FR.
a.
64.46°, 22.04°, 102.26°
b.
25.54°, 67.96°, 77.74°
c.
154.46°, 22.04°, 12.26°
d.
25.54°, 67.96°, 102.26°
Situation 12:
If the resultant force acting on the bracket is directed
along the positive y axis,
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20.
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Situation 10:
Two forces act on the hook shown. The resultant force
FR acts along the positive y axis and has a magnitude of
800 N.
22.
23.
Compute the magnitude of F2.
a.
700 N
c.
b.
600 N
d.
27.
Determine coordinate direction angles of F so that 𝛽 <
90°.
a.
121.3°, 52.5°, 126.9°
b.
58.7°, 52.5°, 126.9°
c.
121.3°, 52.5°, 53.13°
d.
58.7°, 52.5°, 53.13°
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Determine the magnitude of the resultant force so that
𝛽 < 90°.
a.
589.13 N
c.
754.14 N
b.
276.14 N
d.
145.86 N
588 N
905 N
Compute the coordinate direction angle of F2 with
respect to x-axis.
a.
158.2°
c.
21.8°
b.
77.6°
d.
107.6°
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ENGR. HANS LAWRENCE E. DELA CRUZ
• Resultant of Parallel Force System
Situation 13:
The loading on the bookshelf is distributed as shown.
32.
Determine the length “b” of the uniform load.
a.
4.5 ft
c.
7.5 ft
b.
9 ft
d.
3.75 ft
33.
Determine the position “a” of the uniform load.
a.
8.5 ft
c.
1.5 ft
b.
7.5 ft
d.
9.75 ft
Situation 16:
The beam is loaded as shown.
Determine the magnitude of the equivalent resultant
force of the distributed loading.
a.
13.25 lb
c.
5.25 lb
b.
8.00 lb
d.
15.13 lb
29.
Determine its location measured from point O.
a.
0.34 ft
c.
3.09 ft
b.
2.41 ft
d.
3.13 ft
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28.
Situation 14:
The beam is subjected to distributed loading as shown.
Determine the magnitude of the resultant force.
a.
140 N
c.
180 N
b.
120 N
d.
160 N
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34.
Determine its location measured from A.
a.
3.20 m
c.
2.80 m
b.
3.00 m
d.
2.60 m
Situation 17:
The beam is loaded as shown in the figure.
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35.
Determine the magnitude of the equivalent resultant
force of the distributed loading.
a.
10.65 kips
c.
4.80 kips
b.
13.50 kips
d.
5.85 kips
31.
Determine its location measured from point B.
a.
0.48 ft
c.
12.48 ft
b.
1.24 ft
d.
0.26 ft
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30.
Situation 15:
The beam is subjected to the distributed loading. The
resultant force and couple moment acting about the
fixed support are required to be zero.
36.
Determine the magnitude of the resultant force.
a.
1.88 kips
c.
0.625 kips
b.
2.50 kips
d.
1.25 kips
37.
Determine its location measured from O.
a.
7.5 ft
c.
4.5 ft
b.
3 ft
d.
6 ft
• Resultant of Non-Concurrent Force System
Situation 18:
The wooden frame is subjected to coplanar forces as
shown below.
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Moment of a Force
• Moment of Force about a Point
43.
Determine the moment of the force about point O.
a.
b.
44.
Determine the magnitude of the resultant force.
a.
352.05 N
c.
356.09 N
b.
401.07 N
d.
392.32 N
39.
Determine where the resultant’s force line of action
intersects a vertical line along member AB, measured
from A.
a.
3.318 m
c.
0.318 m
b.
2.318 m
d.
1.318 m
40.
Determine where the resultant’s force line of action
intersects a horizontal line along member CB,
measured from end C.
a.
1.00 m
c.
0.25 m
b.
1.50 m
d.
1.75 m
c.
d.
460 N-m
445 N-m
Determine the moment of the force about point O.
Neglect the thickness of the member.
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38.
340 N-m
160 N-m
45.
16.25 N-m
13.91 N-m
c.
d.
11.25 N-m
17.30 N-m
Determine the moment of the force about point O.
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• Resultant of Parallel, 3-D Force System
Situation 19:
The building slab is subjected to four parallel column
loadings. Take F1 = 8 kN and F2 = 9 kN
a.
b.
a.
b.
1,660.60 N-m
1,606.06 N-m
c.
d.
1,066.06 N-m
1,060.66 N-m
Situation 20:
A force F having a magnitude of F = 100 N acts along
the diagonal of the parallelepiped.
41.
Determine the location of the resultant force measured
from the x-axis.
a.
10.82 m
c.
11.54 m
b.
11.31 m
d.
20.87 m
42.
Determine the location of the resultant force measured
from the y-axis.
a.
10.82 m
c.
11.54 m
b.
11.31 m
d.
20.87 m
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ENGR. HANS LAWRENCE E. DELA CRUZ
46.
47.
Express the force F as a Cartesian Vector (N) .
a.
−53.45𝑖 + 80.18𝑗 − 26.73𝑘
b.
−53.45𝑖 − 80.18𝑗 + 26.73𝑘
c.
53.45𝑖 + 80.18𝑗 + 26.73𝑘
d.
−53.45𝑖 + 80.18𝑗 + 26.73𝑘
a.
b.
243.11 lb-ft
343.11 lb-ft
c.
d.
443.11 lb-ft
143.11 lb-ft
• Moment of Force about a Specified Axis
Situation 22:
Three forces act at points A, B, and C as shown.
Determine the moment of F about point A (N-m).
a.
−16.04𝑖 − 32.07𝑘
b.
−16.04𝑗 + 32.07𝑘
c.
−16.04𝑗 − 32.07𝑘
d.
−16.04𝑖 + 32.07𝑘
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Situation 21:
The pipe assembly is subjected to the 80-N force.
Determine the resultant moment of the three forces
about the x-axis.
a.
80 lb-ft
c.
120 lb-ft
b.
230 lb-ft
d.
220 lb-ft
Determine the moment of force F about point A (N-m).
a.
10.61𝑖 + 13.09𝑗 + 29.19𝑘
b.
−5.39𝑖 + 13.09𝑗 − 11.38𝑘
c.
−5.39𝑖 + 13.09𝑗 + 11.38𝑘
d.
10.61𝑖 − 13.09𝑗 + 29.19𝑘
Determine the moment of force F about point B (N-m).
a.
10.61𝑖 + 13.09𝑗 + 29.19𝑘
b.
−5.39𝑖 + 13.09𝑗 − 11.38𝑘
c.
−5.39𝑖 + 13.09𝑗 + 11.38𝑘
d.
10.61𝑖 − 13.09𝑗 + 29.19𝑘
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49.
50.
52.
Determine the resultant moment of the three forces
about the y-axis.
a.
80 lb-ft
c.
120 lb-ft
b.
230 lb-ft
d.
220 lb-ft
53.
Determine the resultant moment of the three forces
about the z-axis.
a.
80 lb-ft
c.
120 lb-ft
b.
230 lb-ft
d.
220 lb-ft
54.
Determine the moment MAB produced by the force F.
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48.
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51.
Find the combined moment of the forces P and Q about
point O. The magnitudes of the forces are P = 80 lb and
Q = 60 lb.
a.
b.
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c.
d.
60.50 N-m
30.50 N-m
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70.50 N-m
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ENGR. HANS LAWRENCE E. DELA CRUZ
11.52𝑖 + 8.64𝑗 c.
11.52𝑖 − 8.64𝑘 d.
−11.52𝑖 + 8.64𝑗
−11.52𝑖 − 8.64𝑘
58.
Determine the tension in cables BA.
a.
403.24 N
c.
475.66 N
b.
420.43 N
d.
456.76 N
59.
Determine the tension in cables BC.
a.
403.24 N
c.
475.66 N
b.
420.43 N
d.
456.76 N
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• Moment of a Couple
56.
Determine the resultant couple moment of the three
couples acting on the plate.
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b.
Equilibrium and Reactions
• Equilibrium of Concurrent Force System
Situation 23:
A 60-kg cylinder is supported by cables BA and BC.
w
Determine the moment of this force F about an axis
extending between A and C. Express the result as a
Cartesian vector (lb-ft).
R
55.
Determine the magnitude and direction 𝜃 of F so that
the particle is in equilibrium.
57.
820 lb-ft
950 lb-ft
c.
d.
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a.
b.
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60.
3,650 lb-ft
1,100 lb-ft
Determine the magnitude of the couple moment acting
on the gear.
a.
b.
c.
d.
61.
a.
b.
163.92 N-m
43.92 N-m
c.
d.
25.10 N-m
81.96 N-m
0905 315 5857
= 3.69 𝑘𝑁, 𝜃 = 47.23°
= 4.96 𝑘𝑁, 𝜃 = 82.23°
= 3.96 𝑘𝑁, 𝜃 = 82.23°
= 4.69 𝑘𝑁, 𝜃 = 47.23°
If the mass of cylinder C is 40 kg, determine the mass
of cylinder A in order to hold the assembly in the
position shown.
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𝐹
𝐹
𝐹
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64.
a.
b.
c.
d.
30 kg
24.6 kg
The man attempts to pull down the tree using the cable
and small pulley arrangement shown. If the tension in
AB is 60 lb, determine the tension in cable CAD and the
angle 𝜃 which the cable makes at the pulley.
22.22 kg
11.11 kg
c.
d.
44.44 kg
33.33 kg
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a.
b.
• Equilibrium in Springs
65.
The spring has a stiffness of k = 800 N/m and an
unstretched length of 200 mm. Determine the force in
cables BC and BD when the spring is held in the
position shown.
𝑇 = 46.30 𝑙𝑏, 𝜃
𝑇 = 30.46 𝑙𝑏, 𝜃
𝑇 = 46.30 𝑙𝑏, 𝜃
𝑇 = 30.46 𝑙𝑏, 𝜃
= 30°
= 20°
= 20°
= 30°
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a.
b.
c.
d.
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62.
34.6 kg
20 kg
Determine the force P required to hold the 100-kg
weight in equilibrium.
a.
b.
c.
d.
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• Equilibrium in Pulley System
63.
Determine the force P required to hold the 100-lb
weight in equilibrium.
66.
a.
b.
100 lb
50 lb
c.
d.
0905 315 5857
= 171.4 𝑁, 𝐹𝐵𝐷
= 145.5 𝑁, 𝐹𝐵𝐷
= 154.5 𝑁, 𝐹𝐵𝐷
= 174.1 𝑁, 𝐹𝐵𝐷
= 145.5 𝑁
= 171.4 𝑁
= 174.1 𝑁
= 154.5 𝑁
Determine the mass of each of the two cylinders if they
cause a sag of s = 0.5 m when suspended from the
rings at A and B. Note that s = 0 when the cylinders are
removed.
12.5 lb
25 lb
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𝐹𝐵𝐶
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a.
b.
46.45 kg
4.73 kg
c.
d.
23.22 kg
2.37 kg
• Equilibrium in Three-Dimensional Force System
Situation 24:
A 900-kN load is supported by cables AB, AC, and AD.
72.
Which of the following most nearly gives the reaction
at B if the weight of the plate is neglected?
a.
208.33 kN
c.
787.04 kN
b.
231.48 kN
d.
833.33 kN
w
Which of the following most nearly gives the reaction
at A if the weight of the plate is neglected?
a.
231.48 kN
c.
833.33 kN
b.
208.33 kN
d.
787.04 kN
Determine the tension developed in cable AB.
a.
1,125 N
c.
506.25 N
b.
843.75 N
d.
725.75 N
ev
ie
67.
71.
73.
Determine the tension developed in cable AC.
a.
1,125 N
c.
506.25 N
b.
843.75 N
d.
725.75 N
69.
Determine the tension developed in cable AD.
a.
1,125 N
c.
506.25 N
b.
843.75 N
d.
725.75 N
70.
A tripod supports a load W as shown in the figure.
Determine the maximum load W that can be supported
by the tripod if each leg has a capacity of 15 kN.
74.
The uniform plate has a weight of 500 lb. Determine
the tension developed in cable C.
R
EO
C
EL
E
R
68.
Which of the following most nearly gives the reaction
at C if the plate has a thickness of 12 mm and a unit
weight of 77 kN/m3?
a.
787.82 kN
c.
209.12 kN
b.
834.12 kN
d.
232.27 kN
a.
b.
200 lb
350 lb
c.
d.
100 lb
250 lb
Analysis of Structures
• Method of Joints and Method of Section
Situation 26:
A roof truss is loaded as shown in the figure.
a.
b.
24.00 kN
34.99 kN
c.
d.
34.48 kN
36.00 kN
Situation 25:
The figure shows a 1.80-m diameter horizontal steel
plate supported by three posts at A, B, and C. A load P
= 1,250 kN is placed at a distance x = 0.50 m.
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ENGR. HANS LAWRENCE E. DELA CRUZ
75.
Determine the force in member AH.
a.
30.20 kN (T)
c.
30.20 kN (C)
b.
22.99 kN (C)
d.
22.99 kN (T)
81.
Compute the force in member CD.
a.
2.25 kips (T)
c.
5.25 kips (C)
b.
5.25 kips (T)
d.
2.25 kips (C)
76.
Determine the force in member CD.
a.
21.375 kN (T) c.
17.58 kN (T)
b.
17.58 kN (C)
d.
21.375 kN (C)
82.
Compute the force in member GK.
a.
2.70 kips (T)
c.
1.80 kips (T)
b.
2.70 kips (C)
d.
1.80 kips (C)
77.
Determine the force in member CG.
a.
7.50 kN (T)
c.
8.11 kN (T)
b.
8.11 kN (C)
d.
7.50 kN (C)
83.
Compute the force in member GJ.
a.
1.00 kips (C)
c.
b.
2.00 kips (T)
d.
Situation 28:
A transmission tower is subjected to 1,200 lb and
1,600 lb force at G and E, respectively.
R
ev
ie
w
Situation 26:
The truss is loaded by a 4-kN and 5-kN force applied at
B and A, respectively.
0.50 kips (C)
1.50 kips (T)
Compute the force in member AE.
a.
3.11 kN (C)
c.
8.85 kN (C)
b.
3.11 kN (T)
d.
8.85 kN (T)
79.
Compute the force in member CF.
a.
8.77 kN (T)
c.
8.77 kN (C)
b.
16.33 kN (T)
d.
16.33 kN (C)
84.
Determine the force in members DC.
a.
0.90 kips (T)
c.
1.90 kips (T)
b.
0.90 kips (C)
d.
1.90 kips (C)
80.
Compute the force in member BF.
a.
6.20 kN (C)
c.
2.20 kN (T)
b.
2.20 kN (C)
d.
6.20 kN (T)
85.
Determine the force in members HI.
a.
0.90 kips (T)
c.
1.90 kips (T)
b.
0.90 kips (C)
d.
1.90 kips (C)
Situation 27:
A K-truss supports the concentrated loads as shown.
(Use P = 3,000 lb and Q = 1,000 lb)
86.
Determine the force in members JI.
a.
0 kips
c.
0.42 kips (T)
b.
2.10 kips (C)
d.
0.42 kips (C)
R
EO
C
EL
E
78.
• Method of Members
Situation 29:
A frame is used to support the 300-lb load applied as
shown in the figure,
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ENGR. HANS LAWRENCE E. DELA CRUZ
87.
Determine the horizontal component of force at C
which member ABC exerts on member CEF.
a.
300 lb
c.
75 lb
b.
100 lb
d.
37.5 lb
88.
Determine the vertical component of force at C which
member ABC exerts on member CEF.
a.
300 lb
c.
75 lb
b.
100 lb
d.
37.5 lb
92.
Compute the magnitude of the pin reaction at A.
a.
410.8 N
c.
415.5 N
b.
400.0 N
d.
93.75 N
93.
Compute the magnitude of the pin reaction at C.
a.
93.75 N
c.
400.0 N
b.
415.5 N
d.
410.8 N
Cables
• Cables under Concentrated Loads
Situation 32:
A cable is subjected concentrated loads as shown in the
figure.
Determine the tension in segment AB.
a.
39.17 lb
c.
46.71 lb
b.
88.15 lb
d.
82.99 lb
95.
Determine the tension in segment BC.
a.
46.71 lb
c.
82.99 lb
b.
39.17 lb
d.
88.15 lb
96.
Determine the tension in segment CD.
a.
82.99 lb
c.
88.15 lb
b.
46.71 lb
d.
39.17 lb
97.
Determine the cable’s total length.
a.
20.16 ft
c.
22.68 ft
b.
19.08 ft
d.
21.19 ft
EL
Determine the force acting in the hydraulic cylinder
AB.
a.
2.17 kN
c.
9.23 kN
b.
7.01 kN
d.
8.97 kN
Determine the horizontal component of force at the
pin C.
a.
9.23 kN
c.
8.97 kN
b.
2.17 kN
d.
7.01 kN
91.
R
EO
90.
94.
C
89.
E
R
ev
ie
w
Situation 30:
The engine hoist is used to support the 200-kg
engine.
Determine the vertical component of force at the pin C.
a.
2.17 kN
c.
8.97 kN
b.
9.23 kN
d.
7.01 kN
Situation 31:
A folding table is subjected to a uniformly distributed
load as shown. Neglecting friction and the weights of
the members,
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Situation 33:
A cable is subjected concentrated loads as shown in the
figure.
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ENGR. HANS LAWRENCE E. DELA CRUZ
98.
Determine the tension in segment AB.
a.
10.20 kN
c.
11.84 kN
b.
9.44 kN
d.
13.57 kN
99.
Determine the tension in segment CD.
a.
10.20 kN
c.
11.84 kN
b.
9.44 kN
d.
13.57 kN
100. Compute the sag yB.
a.
5.68 m
b.
6.32 m
c.
d.
• Catenary Cable
Situation 36:
The span L and the sag H of the cable AB are 100 m and
10 m, respectively. If the cable weighs 50 N/m,
8.84 m
3.16 m
107. Determine the maximum tension in the cable.
a.
6.33 kN
c.
2.57 kN
b.
6.83 kN
d.
3.86 kN
• Parabolic Cable
Situation 34:
The cable is subjected to a uniform loading of
w = 250 N/m.
w
108. Determine the minimum tension in the cable.
a.
6.33 kN
c.
2.57 kN
b.
6.83 kN
d.
3.86 kN
ev
ie
109. Determine the total length of the cable.
a.
100.26 m
c.
102.62 m
b.
103.89 m
d.
101.98 m
E
EL
102. Determine the minimum tension in the cable.
a.
12.50 kN
c.
6.25 kN
b.
13.02 kN
d.
14.44 kN
Situation 37:
A cable is supported at A and B, where B is 10 m higher
than A. The cable weighs 50 N/m and the lowest point
is observed to be 5 m lower than A and 20 m
horizontally from A.
R
101. Determine the maximum tension in the cable.
a.
14.44 kN
c.
13.02 kN
b.
6.25 kN
d.
12.50 kN
EO
C
103. Determine the total length of the cable.
a.
50.86 m
c.
51.86 m
b.
50.42 m
d.
51.42 m
R
Situation 35:
The cable supports a girder which weighs 850 lb/ft.
110. Determine the horizontal component of the tension at
any point.
a.
1.90 kN
c.
2.29 kN
b.
2.79 kN
d.
2.04 kN
111. Determine the tension at B.
a.
1.90 kN
c.
b.
2.04 kN
d.
104. Determine the minimum tension in the cable.
a.
49.79 kips
c.
36.46 kips
b.
61.71 kips
d.
50.68 kips
105. Determine the reaction at support A.
a.
49.79 kips
c.
36.46 kips
b.
61.71 kips
d.
50.68 kips
106. Determine the total length of the cable.
a.
118.72 ft
c.
120.72 ft
b.
122.93 ft
d.
116.93 ft
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112. Determine the total length of the cable.
a.
58.88 m
c.
57.77 m
b.
58.79 m
d.
57.23 m
Friction
• Dry Friction
Situation 38:
The uniform 10-kg ladder in the figure rests against
the smooth wall at B, and the end A rests on the rough
horizontal plane for which the coefficient of static
friction is 𝜇𝑠 = 0.3.
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2.79 kN
2.29 kN
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ENGR. HANS LAWRENCE E. DELA CRUZ
118. Determine the frictional force developed between the
crate and the ground if P = 200 N.
a.
120.0 N
c.
130.1 N
b.
195.15 N
d.
200.0 N
113. Determine the angle of inclination 𝜃 of the ladder if the
ladder is on the verge of slipping.
a.
59.04°
c.
16.70°
b.
30.96°
d.
73.30°
w
119. Determine the frictional force developed between the
crate and the ground if P = 450 N.
a.
170.1 N
c.
270.0 N
b.
255.15 N
d.
195.15 N
EO
R
C
EL
E
Situation 39:
A box weighing 100 N is subjected to a horizontal force
P as shown in the figure. The coefficients of static and
kinetic friction between the box and the ground are
𝜇𝑠 = 0.4 and 𝜇𝑘 = 0.2.
120. Blocks A, B, and C have weights of 50 N, 25 N, and 15
N, respectively. Determine the smallest horizontal
force P that will cause impending motion. The
coefficient of static friction between A and B is 𝜇𝑠 =
0.3, between B and C, 𝜇𝑠 = 0.4, and between block C
and the ground, 𝜇𝑠 = 0.35.
ev
ie
114. Determine the normal reaction at B.
a.
58.86 N
c.
29.43 N
b.
49.05 N
d.
98.10 N
R
115. Determine the force P needed to cause impending
motion of the block.
a.
40 N
c.
20 N
b.
30 N
d.
60 N
a.
b.
61.5 N
15.0 N
c.
d.
45.0 N
46.5 N
Situation 41:
The crate has a mass of 50 kg and the coefficients of
static and kinetic friction between the crate and the
ground is 𝜇𝑠 = 0.25 and 𝜇𝑘 = 0.20.
116. If P = 30 N, determine the frictional force developed
between the box and the ground.
a.
30 N
c.
60 N
b.
40 N
d.
20 N
117. If P = 50 N, determine the frictional force developed
between the box and the ground.
a.
20 N
c.
40 N
b.
30 N
d.
60 N
Situation 40:
A 50-kg crate is subjected to an inclined force P as
shown in the figure. The coefficients of static and
kinetic friction between the crate and the ground are
𝜇𝑠 = 0.3 and 𝜇𝑘 = 0.2.
0905 315 5857
122. Determine the minimum value of force P required to
push the crate up the plane.
a.
474.27 N
c.
140.31 N
b.
431.07 N
d.
165.93 N
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121. Determine the minimum horizontal force P required to
hold the crate from sliding down the plane.
a.
431.07 N
c.
165.93 N
b.
474.27 N
d.
140.31 N
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ENGR. HANS LAWRENCE E. DELA CRUZ
• Belt Friction
127. The maximum tension that can be developed in the
cord shown in the figure is 500 N. If the pulley at A is
free to rotate and the coefficient of static friction at the
fixed drums B and C is 𝜇𝑠 = 0.25, determine the
largest mass of the cylinder that can be lifted by the
cord.
124. A uniform 50-kg plank is resting on friction surfaces at
A and B. The coefficients of static friction are shown in
the figure. If an 80-kg man starts walking from A
toward B, determine the distance x when the plank will
start to slide.
c.
d.
3.14 m
4.14 m
165.55 kg
15.69 kg
128. A force of P = 25 N is just sufficient to prevent the 20kg cylinder from descending. Determine the required
force P to begin lifting the cylinder. The rope passes
over a rough peg with two and half turns.
476.60 N
524.90 N
c.
d.
R
a.
b.
EO
C
EL
E
• Wedges
125. The two 200-kg blocks are pushed apart by the 15°
wedge of negligible weight. The angle of static friction
is 12° at all contact surfaces. Determine the force P
required to start the blocks moving.
c.
d.
ev
ie
1.14 m
2.14 m
28.28 kg
95.86 kg
R
a.
b.
a.
b.
w
123. Determine the minimum value of force P required to
keep it moving up the plane.
a.
140.31 N
c.
165.93 N
b.
431.07 N
d.
474.27 N
2,249.45 N
1,962.00 N
126. Determine the smallest force P needed to lift the 3000lb load. The coefficient of static friction between A and
C and between B and D is 𝜇𝑠 = 0.3, and between A and
B is 𝜇𝑠 = 0.4. Neglect the weight of each wedge.
a.
b.
4,054.36 lb
4,485.37 lb
c.
d.
3,868.17 lb
2,893.91 lb
0905 315 5857
1,539.78 N
3.19 N
c.
d.
196.20 N
25.00 N
129. A cable is attached to the 20-kg plate B, passes over a
fixed peg at C, and is attached to the block at A. Using
the coefficients of static friction shown, determine the
smallest mass of block A so that it will prevent sliding
motion of B down the plane.
a.
b.
3.84 kg
19.25 kg
c.
d.
2.22 kg
1.93 kg
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b.
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ENGR. HANS LAWRENCE E. DELA CRUZ
•
Centroids, Center of Gravity, and Moment of Inertia
• Centroid and Center of Gravity
Situation 42:
The composite figure has the dimension shown.
Dimensions are in millimeter.
Moment of Inertia, Parallel-Axis Theorem, and Radius of
Gyration
Situation 43:
The composite figure has the dimension shown.
Dimensions are in millimeter.
130. Determine the location 𝑥̅ of the centroid of the area
from the reference y-axis.
a.
66.60 mm
c.
46.60 mm
b.
56.60 mm
d.
36.60 mm
a.
b.
R
EO
C
w
ev
ie
R
EL
132. Locate the centroid (𝑥̅ , 𝑦̅) of the shaded area.
(1.18, 1.39)
(1.39, 1.18)
c.
d.
135. Calculate the moment of inertia with respect to its
centroidal y-axis.
a.
7.74𝑥106 𝑚𝑚4 c.
3.81𝑥106 𝑚𝑚4
6
4
b.
23.61𝑥10 𝑚𝑚 d.
55.38𝑥106 𝑚𝑚4
E
131. Determine the location 𝑦̅ of the centroid of the area
from the reference x-axis.
a.
338.44 mm
c.
308.44 mm
b.
328.44 mm
d.
358.44 mm
134. Calculate the moment of inertia with respect to its
centroidal x-axis.
a.
7.74𝑥106 𝑚𝑚4 c.
3.81𝑥106 𝑚𝑚4
6
4
b.
23.61𝑥10 𝑚𝑚 d.
55.38𝑥106 𝑚𝑚4
(-1.18, 1.39)
(-1.39, 1.18)
133. Determine the location 𝑦̅ of the centroid C of the beam
having the cross-sectional area shown.
136. Calculate the moment of inertia about the x-axis.
a.
7.74𝑥106 𝑚𝑚4 c.
3.81𝑥106 𝑚𝑚4
b.
23.61𝑥106 𝑚𝑚4 d.
55.38𝑥106 𝑚𝑚4
137. Calculate the moment of inertia about the y-axis.
a.
7.74𝑥106 𝑚𝑚4 c.
3.81𝑥106 𝑚𝑚4
6
4
b.
23.61𝑥10 𝑚𝑚 d.
55.38𝑥106 𝑚𝑚4
138. Calculate the radius of gyration about the x-axis.
a.
60.62 mm
c.
34.72 mm
b.
24.36 mm
d.
92.84 mm
139. Calculate the radius of gyration about the y-axis.
a.
60.62 mm
c.
34.72 mm
b.
24.36 mm
d.
92.84 mm
140. The centroid of the plane region is located at C. If the
area of the region is 2000 mm2 and its moment of
inertia about the x-axis is Ix = 40 × 106 mm4,
determine Iu.
a.
b.
a.
b.
79.69 mm
90.00 mm
c.
d.
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23.80𝑥106 𝑚𝑚4
33.60𝑥106 𝑚𝑚4
85.75 mm
83.16 mm
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46.40𝑥106 𝑚𝑚4 c.
91.20𝑥106 𝑚𝑚4 d.
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ENGR. HANS LAWRENCE E. DELA CRUZ
141. Determine the moment of inertia of the cross-sectional
area of the T-beam with respect to its centroidal x-axis.
142. Determine the location 𝑦̅ of the centroid of the area
from the reference x-axis.
a.
9.04 in.
c.
8.49 in.
b.
5.20 in.
d.
10.77 in.
143. Determine the moment of inertia with respect to its
centroidal x-axis.
a.
407.30 in4
c.
542.32 in4
b.
342.27 in4
d.
477.29 in4
27.0𝑥106 𝑚𝑚4 c.
8.78𝑥106 𝑚𝑚4 d.
a.
b.
144. Determine the moment of inertia with respect to its
centroidal y-axis.
a.
90.60 in4
c.
225.62 in4
4
b.
25.57 in
d.
160.59 in4
37.0𝑥106 𝑚𝑚4
7.88𝑥106 𝑚𝑚4
Situation 44:
A W14 × 34 section is joined to a C10 × 15.3 section to
form a structural member that has the cross section
shown. The American Institute of Steel Construction
Structural Steel Handbook lists the following crosssectional properties:
w
ev
ie
R
For C10 x 15.3
A = 4.48 in2
𝐼 𝑥̅ = 67.3 in4
̅ = 2.27 in4
𝐼𝑦
x = 0.634 in
“Stay positive,
Work hard,
Make it happen.”
R
EO
C
EL
E
For W14 x 34
A = 10 in2
𝐼 𝑥̅ = 340 in4
̅ 23.3 in4
𝐼 𝑦=
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Engineering Mechanics - Dynamics
ENGR. HANS LAWRENCE E. DELA CRUZ
INSTRUCTION: Select the correct answer for each of the
following questions. Mark only one answer for each item by
shading the box corresponding to the letter of your choice on
the answer sheet provided. STRICTLY NO ERASURES
ALLOWED. Use pencil no. 2 only.
Kinematics of Rectilinear Translation
• Rectilinear Motion with Constant Acceleration
Situation 1:
A truck traveling along a straight road at speed of 20
kph, increases its speed to 120 kph in 15 sec.
What is the constant acceleration?
a.
3.33 m/s2
c.
1.85 m/s2
2
b.
3.70 m/s
d.
6.67 m/s2
2.
Determine the total distance travelled.
a.
833.33 m
c.
291.67 m
b.
500.00 m
d.
458.33 m
Car B is traveling a distance d ahead of car A. Both
cars are traveling at 60 ft/s when the driver of B
suddenly applies the brakes, causing his car to
decelerate at 12 ft/s2. It takes the driver of car A 0.75
seconds to react (this is the normal reaction time for
drivers). When he applies his brakes, he decelerates at
15 ft/s2. Determine the minimum distance d between
the cars so as to avoid a collision.
a.
15.0 ft
c.
16.9 ft
b.
20.3 ft
d.
12.1 ft
ev
ie
11.
13.
EL
Find the distance traveled at constant speed.
a.
240 m
c.
640 m
b.
360 m
d.
160 m
5.
Find the total time of travel.
a.
40 sec
c.
b.
42 sec
d.
C
4.
EO
34 sec
36 sec
62.64 m/s
54.92 m/s
Determine the maximum height reached by the stone.
a.
135.48 m
c.
122.63 m
b.
160.38 m
d.
176.58 m
At what time did it travel until it reaches the highest
point?
a.
7 sec
c.
8 sec
b.
5 sec
d.
6 sec
Situation 6:
A stone is projected vertically upward from the top of
a 120-m high tower at a velocity of 36 m/s.
14.
Determine the maximum height attained by the stone
measured from the ground.
a.
164.930 m
c.
186.055 m
b.
44.930 m
d.
66.055 m
R
Situation 3:
An automobile is driven at 30 mph for 12 min, then at
40 mph for 20 min, and finally at 50 mph for 8 min.
Calculate the initial velocity.
a.
58.86 m/s
c.
b.
49.05 m/s
d.
R
12.
E
Find the distance covered during the acceleration.
a.
300 m
c.
100 m
b.
200 m
d.
400 m
10.
• Free Falling Bodies, Air Resistance Neglected
Situation 5:
A stone is thrown vertically upward and returns to
earth in 12 seconds.
Situation 2:
An automobile starting from rest speeds up to 40
meters per second with a constant acceleration of 4
m/s2, runs at this speed for a time, and finally comes to
rest with deceleration of 5 m/s2. If the total distance
travelled is 1000 meters.
3.
What is the total distance traveled by car A to overtake
car B?
a.
29.17 mi
c.
37.50 mi
b.
33.33 mi
d.
41.67 mi
w
1.
9.
6.
Calculate the average speed over the interval.
a.
37 mph
c.
40 mph
b.
39 mph
d.
38 mph
15.
How long will it take the stone to reach the top?
a.
11.581 sec
c.
3.670 sec
b.
9.512 sec
d.
9.829 sec
7.
How fast must the automobile move in the last 8 min
to obtain an average speed of 35 mph?
a.
20 mph
c.
15 mph
b.
17 mph
d.
12 mph
16.
How long will it take the stone to reach the ground?
a.
9.512 sec
c.
11.581 sec
b.
9.829 sec
d.
3.670 sec
17.
With what velocity will it hit the ground?
a.
60.419 m/s
c.
53.561 m/s
b.
50.108 m/s
d.
40.397 m/s
Situation 4:
Car A at a gasoline station stays there for 10 minutes
after a car B passes at an average speed of 40 mph.
8.
How long will it take car A moving at an average speed
of 50 mph to overtake car B?
a.
35 min
c.
45 min
b.
40 min
d.
50 min
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Situation 7:
A ball is dropped from the top of a tower 80 ft high at
the same instant that a second ball is thrown upward
from the ground with an initial velocity of 40 ft/s.
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18.
Determine the time elapsed when the two ball pass
each other.
a.
0.5 sec
c.
2 sec
b.
1 sec
d.
1.75 sec
19.
How high are the two balls from the ground when they
pass each other?
a.
16.0 ft
c.
15.6 ft
b.
23.9 ft
d.
20.7 ft
• Rectilinear Motion with Variable Acceleration
Situation 8:
A car moves in a straight line such that for a short time
its velocity is defined by 𝑣 = (3𝑡 2 + 2𝑡) m/s, where t
is in seconds.
28.
Determine the time t’ needed to stop the car.
a.
60 sec
c.
40 sec
b.
50 sec
d.
30 sec
20.
Determine the distance travelled when t = 3 sec.
a.
30 m
c.
36 m
b.
39 m
d.
33 m
29.
How far has the car travelled?
a.
3,000 m
c.
b.
2,500 m
d.
21.
Determine the acceleration when t = 3 sec.
a.
25 m/s2
c.
15 m/s2
b.
20 m/s2
d.
10 m/s2
Situation 11:
A car accelerates from initial velocity of 10 m/s. The
acceleration is increasing uniformly from zero to 8
m/s2 in 6 seconds. During the next 2 seconds, the car
decelerates at a constant rate of 2 m/s2.
R
30.
E
Situation 9:
The position of a particle along a straight-line path is
defined by 𝑠 = (𝑡 3 − 6𝑡 2 − 15𝑡 + 7) ft, where t is in
seconds.
ev
ie
w
2,000 m
1,500 m
Determine the particle’s instantaneous velocity when
t = 10 sec.
a.
225 ft/s
c.
135 ft/s
b.
105 ft/s
d.
165 ft/s
23.
Determine the particle’s acceleration when t = 10 sec.
a.
46 ft/s2
c.
48 ft/s2
2
b.
52 ft/s
d.
50 ft/s2
24.
Calculate the particle’s displacement when t = 10 sec.
a.
250 ft
c.
450 ft
b.
257 ft
d.
457 ft
25.
Determine the particle’s total distance travelled when
t = 10 sec.
a.
450 ft
c.
250 ft
b.
457 ft
d.
257 ft
26.
Calculate the particle’s average speed when t = 10 sec.
a.
45.0 ft/s
c.
25.7 ft/s
b.
25.0 ft/s
d.
45.7 ft/s
27.
Calculate particle’s average velocity when t = 10 sec.
a.
25.7 ft/s
c.
45.7 ft/s
b.
45.0 ft/s
d.
25.0 ft/s
31.
Determine the distance traveled after 8 seconds.
a.
128 m
c.
165 m
b.
140 m
d.
172 m
Situation 12:
The a-t curve for a particle having rectilinear motion is
shown in the figure. At t = 0, the velocity is 8 ft/s and
the particle is 60 ft to the left of the origin of
displacement.
R
EO
C
EL
22.
Determine the velocity after 8 seconds.
a.
30 m/s
c.
20 m/s
b.
34 m/s
d.
24 m/s
• Motion Curves
Situation 10:
The car starts from rest and travels along a straight
track such that it accelerates at 10 m/s2 for 10 seconds,
and then decelerates at 2 m/s2.
0905 315 5857
Determine the velocity when t = 4 sec.
a.
20 ft/s
c.
30 ft/s
b.
12 ft/s
d.
38 ft/s
33.
Determine the displacement when t = 7 sec.
a.
75 ft
c.
66 ft
b.
48 ft
d.
37 ft
34.
Determine the displacement when t = 13 sec.
a.
141 ft
c.
138 ft
b.
166 ft
d.
129 ft
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Kinematics of Curvilinear Translation
• Projectile Motion, Air Resistance Neglected
Situation 13:
A projectile is fired from the edge of a 150-m cliff with
an initial velocity of 180 m/s at an angle of 30° with the
horizontal. Neglect air resistance.
76.45 m
61.16 m
c.
d.
31.18 m
45.87 m
• Tangential and Normal Components of Acceleration
Situation 14:
The boat is traveling along the circular path with a
speed of 𝑣 = (0.0625𝑡 2 ) m/s, where t is in seconds.
Find the horizontal distance from the gun to the point
where the projectile strikes the ground.
a.
2,606 m
c.
2,835 m
b.
3,441 m
d.
3,100 m
40.
Determine the tangential acceleration when t = 10 s.
a.
0.98 m/s2
c.
1.59 m/s2
b.
1.25 m/s2
d.
0.87 m/s2
w
35.
a.
b.
Determine the total time of flight.
a.
18.19 sec
c.
22.08 sec
b.
19.89 sec
d.
16.72 sec
41.
Determine the normal acceleration when t = 10 s.
a.
1.25 m/s2
c.
0.87 m/s2
2
b.
0.98 m/s
d.
1.59 m/s2
37.
Determine the greatest elevation above the ground
reached by the projectile.
a.
448.12 m
c.
412.84 m
b.
652.48 m
d.
562.84 m
42.
Determine the magnitude of its acceleration when t =
10 s.
a.
0.87 m/s2
c.
1.59 m/s2
2
b.
1.25 m/s
d.
0.98 m/s2
38.
An airplane used to drop water on brushfires is flying
horizontally in a straight line at 180 mi/h at an altitude
of 300 ft. Determine the distance d at which the pilot
should release the water so that it will hit the fire at B.
Situation 15:
The automobile has a speed of 80 ft/s at point A and an
acceleration having a magnitude of 10 ft/s2, acting in
the direction shown.
R
EO
C
EL
E
R
ev
ie
36.
a.
b.
740 ft
1,340 ft
c.
d.
1,140 ft
940 ft
43.
39.
Determine the radius of curvature of the path at point
A.
Water is sprayed at an angle of 90° from the slope at
20 m/s. Determine the range R.
a.
b.
44.
739 ft
1,626 ft
c.
d.
1,280 ft
915 ft
Determine the tangential component of acceleration.
a.
5.77 ft/s2
c.
8.66 ft/s2
2
b.
11.55 ft/s
d.
5.00 ft/s2
Situation 16:
The racing car shown in the figure is traveling at 90
km/h when it enters the semicircular curve at A. The
driver increases the speed at a uniform rate, emerging
from the curve at C at 144 km/h.
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Determine the tangential acceleration when the car is
at B.
a.
1.65 m/s2
c.
1.55 m/s2
2
b.
11.23 m/s
d.
11.13 m/s2
47.
Determine the magnitude of the acceleration when the
car is at B.
a.
1.55 m/s2
c.
11.13 m/s2
b.
1.65 m/s2
d.
11.23 m/s2
c.
d.
60 ft/s down
48 ft/s down
In the pulley configuration shown, cylinder A has a
downward velocity of 0.3 m/s. Determine the velocity
of B.
w
Determine the normal acceleration when the car is at
B.
a.
11.23 m/s2
c.
11.13 m/s2
2
b.
1.55 m/s
d.
1.65 m/s2
51.
21 ft/s up
24 ft/s up
ev
ie
46.
a.
b.
R
45.
a.
b.
0.20 m/s down c.
0.20 m/s up
d.
0.45 m/s down
0.45 m/s up
Situation 18:
Block B is pulled downward at 4 ft/s and the speed is
decreasing at 2 ft/s2.
R
EO
C
EL
E
Situation 17:
The automobile is originally at rest at s = 0. The
tangential acceleration can be expressed as 𝑎𝑡 =
(0.05𝑡 2 ) ft/s2, where t is in seconds.
48.
Determine the magnitude of its velocity at s = 550 ft.
a.
80.56 ft/s
c.
90.32 ft/s
b.
235.24 ft/s
d.
115.42 ft/s
49.
Determine the magnitude of its acceleration at s = 550
ft.
a.
7.42 ft/s2
c.
18.17 ft/s2
2
b.
58.40 ft/s
d.
34.79 ft/s2
Kinematics of Particle Systems
• Absolute Dependent Motion Analysis
50.
If block A of the pulley system is moving downward at
6 ft/s while block C is moving down at 18 ft/s,
determine the velocity of block B.
0905 315 5857
Determine the velocity of block A at that instant.
a.
1 ft/s up
c.
2 ft/s down
b.
1 ft/s down
d.
2 ft/s up
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53.
Determine the acceleration of block A at that instant.
a.
0.5 ft/s2
c.
2.5 ft/s2
2
b.
2.0 ft/s
d.
1.0 ft/s2
• Relative Motion
54.
Two planes, A and B, are flying at the same altitude. If
their velocities are 𝑣𝐴 = 500 𝑘𝑝ℎ and 𝑣𝐵 = 700 𝑘𝑝ℎ
such that the angle between their straight-line courses
is θ = 60°, determine the velocity of plane B with
respect to plane A.
a.
b.
103.9 km/h
34.6 km/h
c.
d.
30.0 km/h
52.0 km/h
59.
c.
d.
1,200 kph
850 kph
R
606 kph
1,044 kph
What time is required to acquire a clockwise angular
velocity of 15 rad/s?
a.
1.67 sec
c.
2.00 sec
b.
1.33 sec
d.
1.00 sec
E
a.
b.
Determine the number of revolutions it must undergo
to acquire a clockwise angular velocity of 15 rad/s.
a.
3.316 rev
c.
20.833 rev
b.
2.301 rev
d.
14.458 rev
ev
ie
58.
w
Kinematics of Rotation
• Rotation with Constant Angular Acceleration
Situation 20:
A wheel has an initial clockwise angular velocity of 10
rad/s and a constant angular acceleration of 3 rad/s2.
Situation 21:
The disk is originally rotating at 𝜔𝑜 = 12 𝑟𝑎𝑑/𝑠.
If it is subjected to a constant angular acceleration of
𝛼 = 20 𝑟𝑎𝑑/𝑠 2,
R
EO
C
EL
Situation 19:
At the instant shown, the car at A is traveling at 10 m/s
around the curve while increasing its speed at 5 m/s2.
The car at B is traveling at 18.5 m/s along the
straightaway and increasing its speed at 2 m/s2.
55.
Determine the relative velocity of A with respect to B
at this instant.
a.
13.44 m/s
c.
16.67 m/s
b.
11.43 m/s
d.
7.07 m/s
56.
Determine the relative acceleration of A with respect
to B at this instant.
a.
4.32 m/s2
c.
0.83 m/s2
2
b.
4.24 m/s
d.
3.85 m/s2
57.
A passenger in an automobile observes that raindrops
make an angle of 30° with the horizontal as the auto
travels forward with a speed of 60 km/h. Compute the
terminal (constant) velocity 𝑣𝑟 of the rain if it is
assumed to fall vertically.
0905 315 5857
Determine the magnitude of the velocity of point A at
the instant t =2 s.
a.
48.0 m/s
c.
26.0 m/s
b.
24.0 m/s
d.
52.0 m/s
61.
Determine the magnitude of the tangential component
of acceleration of point A at the instant t =2 s.
a.
81.9 m/s2
c.
163.2 m/s2
2
b.
10.0 m/s
d.
1,352.0 m/s2
62.
Determine the magnitude of the normal component of
acceleration of point A at the instant t =2 s.
a.
163.2 m/s2
c.
10.0 m/s2
b.
1,352.0 m/s2
d.
81.9 m/s2
63.
Determine the magnitude of the velocity of point B
when the disk undergoes 2 revolutions.
a.
25.43 m/s
c.
47.66 m/s
b.
10.17 m/s
d.
16.76 m/s
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When the angular velocity of a 4-ft diameter pulley is
3 rad/s, the total acceleration of a point on its rim is 30
ft/s2. Determine the angular acceleration of the pulley
at this instant.
a.
6.0 rad/s2
c.
15.0 rad/s2
b.
7.5 rad/s2
d.
12.0 rad/s2
Kinetics of a Particle: Force and Acceleration
• Kinetics of Rectilinear Translation
Situation 25:
The 300-N block A in the figure is at rest on the
horizontal plane when the force P is applied at t = 0.
The coefficient kinetic friction is 0.2.
• Rotation with Variable Angular Acceleration
Situation 22:
A body rotates according to the relation 𝛼 = 3𝑡 2 + 4,
displacement being measured in radians and time in
seconds. If its initial velocity is 4 rad/s and the initial
angular displacement is zero,
Compute the angular displacement for the instant
when t = 3 sec.
a.
38.25 rad
c.
32.58 rad
b.
50.25 rad
d.
25.05 rad
Compute the value of angular velocity at the instant
when t = 4 sec.
a.
256 rad/s
c.
264 rad/s
b.
240 rad/s
d.
272 rad/s
Find the displacement of the block when t = 5 seconds.
a.
27.0 m
c.
23.2 m
b.
38.1 m
d.
46.3 m
75.
A car is traveling at 100 km/h along a straight, level
road when its brakes become locked. Determine the
stopping distance of the car knowing that the
coefficient of kinetic friction between the tires and the
road is 0.65.
a.
117.05 m
c.
33.1 m
b.
83.4 m
d.
60.5 m
76.
Compute the value of angular acceleration at the
instant when t = 4 sec.
a.
347 rad/s2
c.
324 rad/s2
b.
338 rad/s2
d.
312 rad/s2
EO
68.
74.
C
EL
67.
Find the velocity of the block when t = 5 seconds.
a.
9.28 m/s
c.
18.51 m/s
b.
10.81 m/s
d.
15.24 m/s
E
Situation 23:
The rotation of a pulley is defined by the relation 𝜃 =
2𝑡 4 − 30𝑡 2 + 6, where 𝜃 is measured in radians and t
in seconds.
73.
ev
ie
66.
Compute the angular velocity for the instant when t =
3 sec.
a.
35 rad/s
c.
43 rad/s
b.
31 rad/s
d.
39 rad/s
Find the acceleration of the block.
a.
1.856 m/s2
c.
3.048 m/s2
2
b.
2.161 m/s
d.
3.702 m/s2
R
65.
72.
w
64.
Determine the acceleration of the blocks when the
system is released. The coefficient of kinetic friction is
0.25, and the mass of block A is 50 kg while the mass
of block B is 25 kg. Neglect the mass of the pulleys and
cord.
R
Situation 24:
The angular acceleration of a pulley which will rotate
from rest is increased uniformly from zero to 12 rad/s2
for 4 seconds, and then uniformly decreased to 4
rad/s2 during the next 3 seconds.
69.
Determine the angular velocity at the end of 4 seconds.
a.
18 rad/s
c.
24 rad/s
b.
15 rad/s
d.
21 rad/s
70.
Determine the angular displacement at the end of 4
seconds.
a.
23 rad
c.
46 rad
b.
32 rad
d.
64 rad
71.
Determine the angular displacement at the end of 7
seconds.
a.
146 rad
c.
114 rad
b.
152 rad
d.
134 rad
0905 315 5857
77.
7.52 m/s2
2.67 m/s2
c.
d.
5.72 m/s2
6.27 m/s2
If the mass of block A is twice the mass of block B, find
the acceleration of A. Neglect the masses of the pulleys.
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8.41 m/s2
6.54 m/s2
c.
d.
• Kinetics of Curvilinear Translation
Situation 28:
A bob of weight W = 100 lb is moving with a constant
velocity of v = 8.03 ft/s in a horizontal plane at the end
of the chord of length L = 18 in. The string generates a
cone while in motion, thus the system is called a
conical pendulum.
4.20 m/s2
3.27 m/s2
Situation 26:
Figure shows a system consisting of three blocks
connected by an inextensible cable that runs around
four pulleys. The masses of blocks A, B, and C are 60
kg, 80 kg, and 20 kg, respectively. Neglect the mass of
the pulleys and cord.
4.91 m/s2
0 m/s2
Find the acceleration of block C.
a.
3.68 m/s2
c.
b.
0 m/s2
d.
4.91 m/s2
2.45 m/s2
86.
A weight concentrated at the end of a cord forms a
conical pendulum for which the period is 1 sec.
Determine the velocity v of the weight if the cord
rotates inclined at 30° with the vertical.
a.
0.109 m/s
c.
1.901 m/s
b.
1.109 m/s
d.
0.901 m/s
87.
A “swing ride” is shown in the figure. Calculate the
necessary angular velocity 𝜔 for the swings to assume
an angle 𝜃 = 35° with the vertical. Neglect the mass of
the cables and treat the chair and person as one
particle.
R
Find the acceleration of block B.
a.
2.45 m/s2
c.
b.
3.68 m/s2
d.
E
80.
Determine the tension in the supporting cord.
a.
187 lb
c.
202 lb
b.
133 lb
d.
289 lb
4.91 m/s2
3.68 m/s2
85.
EL
79.
84.
Find the acceleration of block A.
a.
0 m/s2
c.
b.
2.45 m/s2
d.
C
78.
Determine its inclination 𝜃 with respect to the vertical.
a.
37.67°
c.
57.67°
b.
47.67°
d.
67.67°
ev
ie
83.
w
a.
b.
R
EO
Situation 27:
The 125-kg concrete block A is released from rest in
the position shown and pulls the 200-kg log B up the
30° ramp. The coefficient of kinetic friction between
the log and the ramp is 0.5.
81.
Determine the acceleration of block A.
a.
1.78 m/s2
c.
1.52 m/s2
b.
0.89 m/s2
d.
0.76 m/s2
82.
Determine the velocity of log B when block A hits the
ground.
a.
2.30 m/s
c.
4.61 m/s
b.
3.26 m/s
d.
1.85 m/s
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a.
b.
6.465 rad/s
1.064 rad/s
c.
d.
2.338 rad/s
0.651 rad/s
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Determine the time required to complete one
revolution.
a.
0.992 sec
c.
1.103 sec
b.
1.157 sec
d.
2.116 sec
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Engineering Mechanics - Dynamics
ENGR. HANS LAWRENCE E. DELA CRUZ
88.
The flatbed railway car travels at the constant speed of
60 km/h around a curve of radius 55 m and bank angle
15°. Determine the smallest coefficient of static
friction between the crate and the car that would
prevent the crate of mass M from sliding.
a.
b.
0.85 m
1.35 m
c.
d.
0.93 m
1.25 m
Situation 30:
The coefficient of kinetic friction between the box and
the surface is 0.3. The box is released from rest at A.
0.315
0.217
c.
d.
0.221
0.257
w
a.
b.
94.
EL
Determine the minimum constant speed at which the
car can travel around the track without sliding down
the slope.
a.
13.17 m/s
c.
12.25 m/s
b.
9.91 m/s
d.
11.36 m/s
Compute the total distance travelled by the box if it
stops at D.
a.
10.0032 m
c.
11.0032 m
b.
9.0032 m
d.
8.0032 m
96.
The small body has a speed 𝑣𝐴 = 5 m/s at point A.
Neglecting friction, determine its speed 𝑣𝐵 at point
B after it has risen 0.8 m.
EO
R
Kinetics of a Particle: Work and Energy
• Work-Energy Method
91.
Calculate the velocity v of the 50-kg crate when it
reaches the bottom of the chute at B if it is given an
initial velocity of 4 m/s down the chute at A. The
coefficient of kinetic friction is 0.30.
Compute the velocity of the box at C if the distance
between B and C is 3m.
a.
2.43 m/s
c.
0.43 m/s
b.
4.43 m/s
d.
1.43 m/s
95.
C
90.
Compute the velocity of the box when it reaches B.
a.
8.54 m/s
c.
5.84 m/s
b.
8.45 m/s
d.
4.85 m/s
R
Determine the maximum constant speed at which the
car can travel without slipping up the slope.
a.
24.43 m/s
c.
18.38 m/s
b.
21.17 m/s
d.
22.70 m/s
93.
E
89.
ev
ie
Situation 29:
A sports car, having a mass of 1,700 kg, travels
horizontally along a 20° banked track which is circular
and has a radius of curvature of 100 m. The coefficient
of static friction between the tires and the road is 0.20.
a.
b.
97.
a.
b.
92.
2.17 m/s
7.01 m/s
c.
d.
3.15 m/s
8.97 m/s
2.18 m/s
2.79 m/s
c.
d.
4.76 m/s
3.05 m/s
The 8-kg block is moving with an initial speed of 5 m/s.
If the coefficient of kinetic friction between the block
and plane is 𝜇𝑘 = 0.25, determine the compression in
the spring when the block momentarily stops.
The crate, which has a mass of 100 kg, is subjected to
the action of the two forces. If it is originally at rest,
determine the distance it slides in order to attain a
speed of 6 m/s. The coefficient of kinetic friction
between the crate and the surface is 𝜇𝑘 = 0.2.
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ENGR. HANS LAWRENCE E. DELA CRUZ
a.
b.
c.
d.
0.557 m
0.612 m
100. Determine the velocity of the cart at time 𝑡 = 8 𝑠.
a.
6.33 m/s
c.
2.57 m/s
b.
1.42 m/s
d.
3.86 m/s
The 6-kg cylinder is released from rest in the position
shown and falls on the spring, which has been initially
pre-compressed 50 mm by the light strap and
restraining wires. If the stiffness of the spring is 4
kN/m, compute the additional deflection of the spring
produced by the falling cylinder before it rebounds.
101. Calculate the time t at which the cart velocity is zero.
a.
8.25 sec
c.
9.25 sec
b.
8.17 sec
d.
9.17 sec
Situation 32:
A particle with a mass of 0.75 kg has an initial velocity
of 𝑣 = 6 𝑚/𝑠. Forces F1 and F2 act on the particle and
their magnitudes change with time according to the
graphical schedule shown.
79.43 mm
89.17 mm
c.
d.
29.43 mm
39.17 mm
R
EO
C
EL
E
Kinetics of a Particle: Impulse and Momentum
• Linear Impulse-Momentum
99.
If the coefficient of kinetic friction between the 150 N
crate and the ground is 𝜇𝑘 = 0.2, determine the speed
of the crate when t = 4 s. The crate starts from rest and
is towed by the 100-N force.
R
a.
b.
ev
ie
w
98.
0.688 m
0.508 m
a.
b.
17.4 m/s
19.8 m/s
c.
d.
22.7 m/s
21.2 m/s
Situation 31:
The cart is moving down the incline with a velocity
𝑣0 = 20 𝑚/𝑠 at 𝑡 = 0, at which time the force P
begins to act as shown. After 5 seconds the force
continues at the 50-N level.
102. Determine the vertical component of the particle’s
velocity after 3 seconds.
a.
14.54 m/s
c.
6.11 m/s
b.
7.87 m/s
d.
9.20 m/s
103. Determine the resultant velocity of the particle after 3
seconds.
a.
15.77 m/s
c.
8.96 m/s
b.
13.68 m/s
d.
12.04 m/s
104. Determine the direction of the resultant velocity of the
particle from the x-axis after 3 seconds (positive
counterclockwise).
a.
131.76°
c.
106.25°
b.
112.80°
d.
154.44°
• Conservation of Momentum
105. The 15-Mg boxcar A is coasting to the right at 1.5 m/s
on the horizontal track when it encounters a 12-Mg
tank car B coasting at 0.75 m/s toward it. If the cars
collide and couple together, determine the speed of
both cars just after the coupling.
a.
0.5 𝑚/𝑠 ←
c.
0.5 𝑚/𝑠 →
b.
1.5 𝑚/𝑠 ←
d.
1.5 𝑚/𝑠 →
106. The spring is fixed to block A and block B is pressed
against the spring. If the spring is compressed s = 200
mm and then the blocks are released, determine their
velocity at the instant block B loses contact with the
spring. The masses of blocks A and B are 10 kg and 15
kg, respectively.
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ENGR. HANS LAWRENCE E. DELA CRUZ
111. Determine the velocity of the 50 kg block after the
impact.
a.
0.599 m/s
c.
0.799 m/s
b.
0.699 m/s
d.
0.899 m/s
a.
b.
c.
d.
𝑣𝐴
𝑣𝐴
𝑣𝐴
𝑣𝐴
=
=
=
=
2.31 𝑚/𝑠 →;
2.31 𝑚/𝑠 ←;
3.46 𝑚/𝑠 ←;
3.46 𝑚/𝑠 ←;
𝑣𝐵
𝑣𝐵
𝑣𝐵
𝑣𝐵
112. Compute the percentage of the energy lost.
a.
85.99 %
c.
89.95 %
b.
99.85 %
d.
59.98 %
= 3.46 𝑚/𝑠 →
= 2.31 𝑚/𝑠 →
= 2.31 𝑚/𝑠 →
= 3.46 𝑚/𝑠 →
Situation 35:
The two identical steel balls moving with initial
velocities 𝑣𝐴 and 𝑣𝐵 collide as shown. The coefficient
of restitution is 𝑒 = 0.7.
4.79 m/s
2.84 m/s
c.
d.
7.38 m/s
5.68 m/s
113. Determine the velocity of ball A just after the impact.
a.
8.54 m/s
c.
10.29 m/s
b.
3.22 m/s
d.
6.73 m/s
E
a.
b.
R
ev
ie
w
107. The 5-kg block A has an initial speed of 5 m/s as it
slides down the smooth ramp, after which it collides
with the stationary block B of mass 8 kg. If the two
blocks couple together after collision, determine their
common velocity immediately after collision.
EO
C
EL
• Impact and Coefficient of Restitution
Situation 33:
Object A which weighs 10kg and moving to the right
at a speed of 10 m/s, collided object B, which weighs
5 kg and moving to the left at 5 m/s.
R
108. If the coefficient of restitution, 𝑒 = 0, find the velocity
of A after impact.
a.
1 m/s
c.
5 m/s
b.
0 m/s
d.
15 m/s
109. If the coefficient of restitution, 𝑒 = 0.80, find the
velocity of A after impact.
a.
5 m/s
c.
0 m/s
b.
1 m/s
d.
15 m/s
110. If the coefficient of restitution, 𝑒 = 1, find the velocity
of A after impact.
a.
0 m/s
c.
5 m/s
b.
15 m/s
d.
1 m/s
114. Determine the velocity of ball B just after the impact.
a.
8.54 m/s
c.
10.29 m/s
b.
3.22 m/s
d.
6.73 m/s
115. Compute the percentage loss of system kinetic energy.
a.
22.2 %
c.
44.4 %
b.
33.3 %
d.
55.5 %
END OF HANDOUT
“Claim it.
You will soon become an
Engineer.”
Situation 34:
A 75 g bullet traveling at 600 m/s strikes and becomes
embedded in the 50 kg block, which is initially at rest.
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Effectiveness. Efficiency. Convenience
Strength of Materials
ENGR. HANS LAWRENCE E. DELA CRUZ
INSTRUCTION: Select the correct answer for each of the
following questions. Mark only one answer for each item by
shading the box corresponding to the letter of your choice on
the answer sheet provided. STRICTLY NO ERASURES
ALLOWED. Use pencil no. 2 only.
Simple Stresses
• Normal, Shearing, and Bearing Stress
1.
A hollow steel tube with an outside diameter of 200
mm is subjected to an axial tensile force of 400 kN as
shown in the figure. Determine the thickness of the
tube if the stress is limited to 130 MPa.
a.
b.
c.
d.
115.07 kips
38.16 kips
Figure shows a two-member truss supporting a block
of weight W. The cross-sectional areas of the members
are 800 mm2 for AB and 400 mm2 for AC. Determine
the maximum safe value of W if the working stresses
are 110 MPa for AB and 120 MPa for AC.
12.04 mm
10.04 mm
c.
d.
5.02 mm
6.02 mm
EL
E
Situation 1:
The bar ABCD in the figure consists of three cylindrical
steel segments with different lengths and crosssectional areas. Axial loads are applied as shown.
R
a.
b.
ev
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w
7.
64.80 kips
50.27 kips
C
Calculate the normal stress in segment AB.
a.
6,667 𝑝𝑠𝑖 (𝑇)
c.
3333 𝑝𝑠𝑖 (𝑇)
b.
4375 𝑝𝑠𝑖 (𝐶)
d.
2778 𝑝𝑠𝑖 (𝐶)
3.
Calculate the normal stress in segment BC.
a.
3333 𝑝𝑠𝑖 (𝑇)
c.
6,667 𝑝𝑠𝑖 (𝑇)
b.
2778 𝑝𝑠𝑖 (𝐶)
d.
4375 𝑝𝑠𝑖 (𝐶)
R
EO
2.
4.
Calculate the normal stress in segment CD.
a.
2778 𝑝𝑠𝑖 (𝐶)
c.
4375 𝑝𝑠𝑖 (𝐶)
b.
3333 𝑝𝑠𝑖 (𝑇)
d.
6,667 𝑝𝑠𝑖 (𝑇)
5.
The cross-sectional area of bar ABCD is 600 mm2.
Determine the maximum normal stress in the bar.
a.
b.
6.
a.
b.
33.33 MPa
41.67 MPa
c.
d.
61.71 kN
136.00 kN
Situation 2:
A billboard 3 m high x 4 m wide is supported on each
end by a pin jointed assembly (bracing not shown for
simplification). The total weight of the billboard is 30
kN.
Wind pressure, q
Wind pressure coefficient, c
= 1.70 kPa
= 1.00
8.
Determine the horizontal reaction at A.
a.
18.08 kN
c.
9.05 kN
b.
7.89 kN
d.
15.78 kN
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c.
d.
58.33 MPa
50.00 MPa
The column consists of a wooden post and a concrete
footing, separated by a steel bearing plate. Find the
maximum safe value of the axial load P if the working
stresses are 1000 psi for wood and 450 psi for
concrete.
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98.13 kN
173.33 kN
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ENGR. HANS LAWRENCE E. DELA CRUZ
9.
Determine the axial stress in member BC if its crosssectional dimension is 6 mm x 76 mm.
a.
91.62 MPa
c.
39.96 MPa
b.
45.81 MPa
d.
19.98 MPa
10.
a.
b.
349.78 kN
449.78 kN
c.
d.
549.78 kN
649.78 kN
Situation 5:
A circular hole is to be punched in a plate that has a
shear strength of 40 ksi (see figure). The working
compressive stress for the punch is 50 ksi.
If strut AB was replaced by a 16-mm diameter cable,
determine the normal stress in the cable.
a.
90.63 MPa
c.
45.31 MPa
b.
103.89 MPa
d.
207.79 MPa
Situation 3:
The 250 mm x 700 mm rectangular wood panel is
formed by gluing together two boards along the 30°
seam.
Compute the maximum thickness of the plate in which
a hole 2.5 in. in diameter can be punched.
a.
0.76 in.
c.
0.80 in.
b.
0.78 in.
d.
0.82 in.
17.
If the plate is 0.25 in. thick, determine the diameter of
the smallest hole that can be punched.
a.
0.80 in.
c.
1.20 in.
b.
0.60 in.
d.
1.60 in.
18.
The lap joint is connected by three 20-mm-diameter
rivets. Assuming that the axial load P = 50 kN is
distributed equally among the three rivets, find the
shear stress in a rivet.
R
ev
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w
16.
Determine the normal stress in the glued joint if P =
10 kN.
a.
42.9 kPa
c.
28.6 kPa
b.
49.5 kPa
d.
24.7 kPa
12.
Determine the shear stress in the glued joint if P = 10
kN.
a.
42.9 kPa
c.
28.6 kPa
b.
49.5 kPa
d.
24.7 kPa
EO
C
EL
E
11.
Situation 4:
A beam is loaded as shown in the figure.
R
a.
b.
19.
13.
Determine the shear stress in the 20-mm-diameter pin
at 𝐴.
a.
8.84 𝑀𝑃𝑎
c.
17.68 𝑀𝑃𝑎
b.
68.00 𝑀𝑃𝑎
d.
34.00 𝑀𝑃𝑎
14.
Determine the shear stress in the 30-mm-diameter pin
at 𝐵.
a.
8.84 𝑀𝑃𝑎
c.
17.68 𝑀𝑃𝑎
b.
68.00 𝑀𝑃𝑎
d.
34.00 𝑀𝑃𝑎
15.
What force is required to punch a 20-mm-diameter
hole in a plate that is 25 mm thick? The shear strength
of the plate is 350 MN/m2.
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c.
d.
53.05 MPa
159.15 MPa
The cylindrical steel column has an outer diameter of
4 in. and inner diameter of 3.5 in. The column is
separated from the concrete foundation by a square
bearing plate. The working compressive stress is
26,000 psi for the column, and the working bearing
stress is 1,200 psi for concrete. Find the largest force P
that can be applied to the column.
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15.38 MPa
18.18 MPa
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ENGR. HANS LAWRENCE E. DELA CRUZ
a.
b.
67.69 MPa
135.38 MPa
c.
d.
58.80 kips
76.58 kips
Situation 6:
The lap joint is connected by three 20-mm-diameter
rivets. Assume that the axial load P = 50 kN is
distributed equally among the three rivets.
22.
Assume that the axial load P applied to the lap joint is
distributed equally among the three 20-mm diameter
rivets. What is the maximum load P that can be applied
if the allowable stresses are 40 MPa for shear in rivets,
90 MPa for bearing between a plate and a rivet, and
120 MPa for tension in the plates?
Situation 8:
The tank shown in the figure is fabricated from 1/8-in
steel plate. The tank has an internal pressure of 125
psi.
w
Find the maximum average tensile stress in each plate.
a.
15.38 MPa
c.
53.05 MPa
b.
18.18 MPa
d.
33.33 MPa
A cylindrical pressure vessel is fabricated from steel
plating that has a thickness of 20 mm. The diameter of
the pressure vessel is 450 mm, and its length is 2.0 m.
Determine the maximum internal pressure that can be
applied if the longitudinal stress is limited to 140 MPa,
and the circumferential stress is limited to 60 MPa.
a.
24.89 MPa
c.
10.67 MPa
b.
5.33 MPa
d.
12.44 MPa
ev
ie
21.
26.
R
Find the bearing stress between a plate and a rivet.
a.
18.18 MPa
c.
33.33 MPa
b.
15.38 MPa
d.
53.05 MPa
Calculate the circumferential stress in the wall of the
cylinder.
a.
9,400 psi
c.
9,600 psi
b.
4,800 psi
d.
4,700 psi
a.
b.
R
EO
C
EL
E
20.
25.
330.0 kN
390.0 kN
c.
d.
135.0 kN
37.7 kN
• Thin-Walled Pressure Vessels
23.
A spherical shell with 70-in. outer diameter and 67-in.
inner diameter contains helium at a pressure of 1200
psi. Compute the stress in the shell.
a.
26,800 psi
c.
13,400 psi
b.
28,000 psi
d.
14,000 psi
Situation 7:
The cylindrical portion of the propane tank has an
outer diameter of 12 in. and a wall thickness of 0.125
in. The tank is pressurized to 200 psi.
24.
Calculate the longitudinal stress in the wall of the
cylinder.
a.
4,700 psi
c.
4,800 psi
b.
9,400 psi
d.
9,600 psi
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Calculate the maximum longitudinal stress developed
in the tank.
a.
9,000 psi
c.
4,500 psi
b.
21,000 psi
d.
6,566 psi
28.
Calculate the maximum hoop stress developed in the
tank.
a.
21,000 psi
c.
6,566 psi
b.
9,000 psi
d.
4,500 psi
Simple Strain
• Hooke’s Law: Axial Deformation
Situation 9:
During a stress-strain test, the unit of deformation at a
stress of 35 MN/m2 was observed to be 166.67 𝑥 10−6
m/m and at a stress of 140 MN/m2 it was
666.67 𝑥 10−6 m/m. If the proportional limit was 200
MN/m2.
29.
30.
What is the modulus of elasticity.
a.
210 GPa
c.
b.
200 GPa
d.
What is
MN/m2.
a.
b.
c.
d.
250 GPa
230 GPa
the strain corresponding to a stress of 80
380.95 𝑥 10−6 𝑚/𝑚
495.08 𝑥 10−6 𝑚/𝑚
280.95 𝑥 10−6 𝑚/𝑚
595.08 𝑥 10−6 𝑚/𝑚
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ENGR. HANS LAWRENCE E. DELA CRUZ
31.
• Statically Indeterminate Members
Situation 11:
The concrete post in the figure is reinforced axially
with four symmetrically placed steel bars, each of
cross-sectional area 900 mm2. The moduli of elasticity
are 200 GPa for steel and 14 GPa for concrete.
A 4-mm-diameter steel wire, 3.2 m long, carries an
axial tensile load P. Find the maximum safe value of P
if the allowable normal stress is 280 MPa and the
elongation of the wire is limited to 4 mm. Use E = 200
GPa.
a.
3.52 kN
c.
2.65 kN
b.
1.40 kN
d.
3.14 kN
Situation 10:
A steel rod has a length of 2.5 m and has an axial
rigidity of 60,000 kN. It is subjected to an axial tensile
force of 60 kN.
33.
Determine the stiffness of the rod.
a.
30,000 N/mm c.
24,000 N/mm
b.
3,000 N/mm
d.
2,400 N/mm
Determine the flexibility of the rod.
a.
1/2,400 mm/N c.
1/30,000 mm/N
b.
1/3,000 mm/N d.
1/24,000 mm/N
The steel propeller shaft ABCD carries the axial loads
shown in figure. Determine the change in the length of
the shaft caused by these loads. Use E = 29 x 10 6 psi
for steel.
38.
Compute the stress in steel when the 1000-kN axial
load is applied.
a.
11.09 MPa
c.
103.65 MPa
b.
55.91 MPa
d.
7.26 MPa
Situation 12:
A copper rod of length = 3 m, diameter = 300 mm and
modulus of elasticity = 120 GPa is inserted into an
aluminum tube. The aluminum tube has a length =
2,999.995 mm, wall thickness of 20 mm and modulus
of elasticity of 70 GPa.
c.
d.
−0.01358 𝑖𝑛.
0.01358 𝑖𝑛.
39.
Determine the deformation of each material due to an
axial compressive load P = 120 kN.
a.
𝛿𝑐 = 0.0371 𝑚𝑚; 𝛿𝑎 = 0.0321 𝑚𝑚
b.
𝛿𝑐 = 0.0321 𝑚𝑚; 𝛿𝑎 = 0.0371 𝑚𝑚
c.
𝛿𝑐 = 0.0414 𝑚𝑚; 𝛿𝑎 = 0.0364 𝑚𝑚
d.
𝛿𝑐 = 0.0364 𝑚𝑚; 𝛿𝑎 = 0.0414 𝑚𝑚
A bronze bar is fastened between a steel bar and an
aluminum bar as shown in the figure. Axial loads are
applied at the positions indicated. Find the largest
value of P that will not exceed an overall deformation
of 3.0 mm, or the following stresses: 140 MPa in the
steel, 120 MPa in the bronze, and 80 MPa in the
aluminum. Assume that the assembly is suitably
braced to prevent buckling. Use Est = 200 GPa, Eal = 70
GPa, and Ebr = 83 GPa.
40.
Determine the stress in each member.
a.
𝜎𝑐 = 0.85 𝑀𝑃𝑎; 𝜎𝑎 = 1.46𝑀𝑃𝑎
b.
𝜎𝑐 = 1.48 𝑀𝑃𝑎; 𝜎𝑎 = 0.75 𝑀𝑃𝑎
c.
𝜎𝑐 = 1.46 𝑀𝑃𝑎; 𝜎𝑎 = 0.85𝑀𝑃𝑎
d.
𝜎𝑐 = 0.75 𝑀𝑃𝑎; 𝜎𝑎 = 1.48 𝑀𝑃𝑎
41.
Determine the axial strain if no gap exist.
a.
1.213 𝑥 10−7
c.
1.213 𝑥 10−5
−6
b.
1.213 𝑥 10
d.
1.213 𝑥 10−4
42.
Before the 400-kN load is applied, the rigid platform
rests on two steel bars, each of cross-sectional area
1400 mm2, as shown in the figure. The cross-sectional
area of the aluminum bar is 2800 mm2. Compute the
stress in the aluminum bar after the 400-kN load is
applied. Use E = 200 GPa for steel and E = 70 GPa for
aluminum. Neglect the weight of the platform.
R
36.
−0.03185 𝑖𝑛.
0.03185 𝑖𝑛.
EO
a.
b.
C
EL
E
35.
Compute the stress in concrete when the 1000-kN
axial load is applied.
a.
55.91 MPa
c.
103.65 MPa
b.
11.09 MPa
d.
7.26 MPa
R
34.
37.
w
Determine deformation of the rod.
a.
2.5 mm
c.
25 mm
b.
2.0 mm
d.
20 mm
ev
ie
32.
a.
b.
42.73 kN
12.80 kN
c.
d.
67.20 kN
39.00 kN
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a.
b.
32.60 MPa
63.28 MPa
c.
d.
46.
Determine the stress in the rod when the temperature
rises to 70℃ if the wall yields by 0.50 mm.
a.
97.71 MPa (C) c.
79.17 MPa (C)
b.
79.17 MPa (T) d.
97.71 MPa (T)
47.
What is the maximum temperature such that no stress
will be developed in steel rod if the wall yields by 0.75
mm?
a.
62.22℃
c.
32.22℃
b.
52.22℃
d.
42.22℃
Situation 15:
The figure shows a homogeneous, rigid block weighing
12 kips that is supported by three symmetrically
placed rods. The lower ends of the rods were at the
same level before the block was attached. The
temperature of all bars increases by 100℉. Use the
following data:
16.30 MPa
126.56 MPa
w
Situation 13:
The figure shows a rigid bar that is supported by a pin
at A and two rods, one made of steel and the other of
bronze. Neglect the weight of the bar.
𝑨 (𝒊𝒏𝟐 )
0.75
1.50
106
29 x
12 x 106
𝜶 (/℉)
6.5 x 10-6
10.0 x 10-6
C
EL
E
R
ev
ie
Each steel rod
Bronze rod
𝑬 (𝒑𝒔𝒊)
43.
44.
600
200
300
83
EO
Bronze
Compute the stress in steel rod caused by the 50-kN
load.
a.
106.14 MPa
c.
191.82 MPa
b.
383.63 MPa
d.
53.07 MPa
Compute the stress in bronze rod caused by the 50-kN
load.
a.
53.07 MPa
c.
106.14 MPa
b.
191.82 MPa
d.
383.63 MPa
• Thermal Stresses
Situation 14:
A steel rod 3 m long is secured between two walls. The
rod is stress free at 20℃. The cross-sectional area of
the rod is 1000 mm2 and 𝛼 = 11.25 𝜇𝑚/𝑚℃.
45.
48.
Determine the stress in each of the steel rod after the
temperature has risen.
a.
11.6 ksi (C)
c.
3.6 ksi (T)
b.
3.6 ksi (C)
d.
11.6 ksi (T)
49.
Determine the stress in the bronze rod after the
temperature has risen.
a.
11.6 ksi (C)
c.
3.6 ksi (T)
b.
3.6 ksi (C)
d.
11.6 ksi (T)
R
Area (mm2)
E (GPa)
Steel
Determine the stress in the rod when the temperature
rises to 70℃ if the walls are rigid.
a.
115.2 MPa (T) c.
112.5 MPa (C)
b.
152.1 MPa (C) d.
121.5 MPa (T)
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50.
Determine the Poisson’s ratio of the material.
a.
0.30
c.
0.33
b.
0.25
d.
0.27
51.
Determine the Modulus of Elasticity of the material.
a.
35,000 ksi
c.
40,000 ksi
b.
30,000 ksi
d.
45,000 ksi
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• Poisson’s Ratio: Uniaxial, Biaxial and Triaxial Loading
Situation 16:
A rectangular steel block 1 in. wide, ½ in. deep and 4
in. long is subjected to an axial tensile load of 10.5 kips.
Measurements show the block to increase in length by
2.8 x 10-3 in. and to decrease in width by 0.21 x 10 -3 in.
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Situation 17:
The thin-walled tube is subjected to an axial force of
40 kN. The tube elongates 3 mm, and its circumference
decreases by 0.09 mm. The material behaves
elastically.
55.
Determine the minimum diameter of the solid shaft
that will not exceed the allowable angular
deformation.
a.
125 mm
c.
114 mm
b.
132 mm
d.
119 mm
56.
What is the maximum shearing stress developed in the
shaft?
a.
41.30 MPa
c.
5.60 MPa
b.
29.20 MPa
d.
17.10 MPa
Situation 19:
A hollow steel shaft 2,510 mm long must transmit
torque of 34 kN-m. The total angle of twist must not
exceed 3°. The maximum shearing stress must not
exceed 110 MPa. Use G = 83 GPa.
52.
Determine the modulus of elasticity, E.
a.
67.91 GPa
c.
76.19 GPa
b.
101.53 GPa
d.
103.15 GPa
53.
Determine the value of Poisson’s ratio.
a.
0.344
c.
0.333
b.
0.244
d.
0.222
54.
A rectangular steel block is 300 mm in the x direction,
200 mm in the y direction, and 150 mm in the z
direction. The block is subjected to a triaxial loading
consisting of three uniformly distributed forces as
follows: 250 kN tension in the x direction, 320 kN
compression in the y direction, and 180 kN tension in
the z direction. For steel, 𝜈 = 0.30 and E = 200,000
MPa. Determine the total strain in the x direction due
to the forces applied in the three directions.
ev
ie
E
4.17 x 10-5
4.78 x 10-5
c.
d.
3.55 x 10-5
5.68 x 10-5
Torsion
• Torsion in Circular Shafts
Situation 18:
The 6-m solid steel shaft is subjected to a torque of 12
kN-m. The allowable angle of twist of the shaft is 3°.
Use G = 83 GPa.
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59.
Determine the maximum required inside diameter.
a.
92.85 mm
c.
65.77 mm
b.
88.25 mm
d.
100.16 mm
Situation 20:
A steel shaft is to be manufactured either as a solid
circular shaft or as circular tube. The shaft is to
transmit a torque of 1,200 N-m without exceeding an
allowable shear stress of 50 MPa nor an allowable rate
of twist of 7°/m. Use G = 78 GPa for steel.
60.
Determine the required diameter of the solid shaft.
a.
16.83 mm
c.
49.63 mm
b.
24.81 mm
d.
33.65 mm
61.
Determine the required outer diameter of the hollow
shaft if the thickness of the shaft is specified as onetenth of the outer diameter.
a.
29.58 mm
c.
38.39 mm
b.
59.16 mm
d.
19.20 mm
62.
What power is transmitted by the shaft at 20 Hz?
a.
163.2 kW
c.
24.0 kW
b.
150.8 kW
d.
81.9 kW
63.
Four pulleys are attached to the 50-mm-diameter
aluminum shaft. If torques are applied to the pulleys as
shown in the figure, determine the angle of rotation of
pulley D relative to pulley A. Use G = 28 GPa for
aluminum.
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Determine the minimum outside diameter so that the
allowable shearing stress will not be exceeded.
a.
127.06 mm
c.
112.92 mm
b.
125.70 mm
d.
132.59 mm
R
58.
EL
C
EO
R
a.
b.
Determine the minimum required polar moment of
inertia.
a.
19,637,035.44 mm4
b.
13,028,193.04 mm4
c.
16,691,566.41 mm4
d.
10,809,638.58 mm4
w
57.
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ENGR. HANS LAWRENCE E. DELA CRUZ
a.
b.
2.56 kN-m
3.51 kN-m
c.
d.
2.64 kN-m
5.11 kN-m
• Flange-Bolt Couplings
68.
A flanged bolt coupling consists of ten 20-mmdiameter bolts spaced evenly around a bolt circle 400
mm in diameter. Determine the torque capacity of the
coupling if the allowable shearing stress in the bolts is
40 MPa.
−5.34°
−6.34°
c.
d.
−7.34°
−8.34°
• Statically Indeterminate Shafts
Situation 21:
The compound shaft shown in the figure is made of
bronze and steel and is fixed at both ends. It is
subjected to a concentrated torque at the junction. The
steel segment is 1 m long.
a.
b.
c.
d.
25.13 kN-m
31.25 kN-m
A flanged bolt coupling consists of eight 10-mmdiameter steel bolts on a bolt circle 400 mm in
diameter, and six 10-mm-diameter steel bolts on a
concentric bolt circle 300 mm in diameter, as shown in
the figure. What torque can be applied without
exceeding a shearing stress of 60 MPa in the bolts?
Determine the ratio of the length of the steel segment
to that of the bronze such that the materials will be
stressed to their limits.
a.
1.05
c.
1.30
b.
1.42
d.
1.19
a.
b.
EO
C
64.
EL
E
R
ev
ie
69.
24.72 kN-m
33.81 kN-m
w
a.
b.
Compute the torsional reaction at the right support if
the applied torque at the junction is maximized.
a.
4.97 kN-m
c.
1.96 kN-m
b.
2.98 kN-m
d.
3.47 kN-m
66.
Determine the angle of twist at the junction.
a.
3.90°
c.
5.59°
b.
2.21°
d.
3.35°
67.
The compound shaft is attached to a rigid wall at each
end. For the bronze segment AB, the diameter is 75
mm and G = 35 GPa. For the steel segment BC, the
diameter is 50 mm and G = 83 GPa. Given that a = 2 m
and b = 1.5 m, compute the largest torque T that can
be applied as shown in the figure if the maximum shear
stress is limited to 60 MPa in the bronze and 80 MPa in
the steel.
R
65.
10.72 kN-m
11.16 kN-m
70.
Determine the shearing stress caused by the applied
torque.
a.
170.74 MPa
c.
89.36 MPa
b.
78.36 MPa
d.
134.63 MPa
71.
Find the shear flow.
a.
156.73 N/mm
b.
341.48 N/mm
c.
d.
Determine the angle of twist.
a.
43.4°
c.
b.
13.4°
d.
178.73 N/mm
269.26 N/mm
22.4°
33.4°
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c.
d.
• Torsion in Thin-Walled Tubes: Shear Flow
Situation 22:
A tube 2 mm thick has a shape as shown in the figure
and is subjected to a torque of 600 N-m. Use G = 35
GPa and L = 3m.
72.
(074) 665 6774
18.11 kN-m
15.08 kN-m
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Situation 23:
The tube is subjected to a torque of 750 N-m.
79.
Determine the maximum bending moment that occur
in the span.
a.
80.0 kN-m
c.
30.0 kN-m
b.
37.3 kN-m
d.
20.0 kN-m
Situation 26:
The beam is loaded by two triangular loads as shown.
Take L = 5 m and use wo = 12 kN/m.
Find the average shear stress in the horizontal side of
the tube.
a.
8.7 MPa
c.
13.2 MPa
b.
15.6 MPa
d.
10.4 MPa
• Helical Spring
Situation 24:
A light helical spring is fabricated by wrapping wire ¾
in. in diameter around a forming cylinder 8 in. in
diameter. Use G = 12 x 10 6 psi.
Determine the axial force exerted by the spring
without exceeding the allowable shearing stress of 18
ksi.
a.
382 lb
c.
341 lb
b.
374 lb
d.
356 lb
76.
Determine the number of turns required to permit an
elongation of 4 in.
a.
13 turns
c.
10 turns
b.
14 turns
d.
11 turns
77.
Compute the maximum shearing stress developed in a
phosphor bronze spring having mean diameter of 200
mm and consisting of 24 turns of 20-mm diameter
wire when the spring is stretched 100 mm. Assume G
= 42 GPa and use Wahl’s Formula.
a.
23.10 MPa
c.
42.03 MPa
b.
31.89 MPa
d.
32.71 MPa
Determine the maximum shear that occur in the span.
a.
12 kN
c.
15 kN
b.
24 kN
d.
30 kN
81.
Determine the maximum bending moment that occur
in the span.
a.
12.5 kN-m
c.
11.5 kN-m
b.
15.0 kN-m
d.
16.5 kN-m
Situation 27:
The shear diagram for a beam is given below.
R
EO
C
EL
E
75.
80.
w
74.
ev
ie
Find the average shear stress in the vertical side of the
tube.
a.
8.7 MPa
c.
13.2 MPa
b.
15.6 MPa
d.
10.4 MPa
R
73.
Shear and Moment in Beams
• Shear-Moment Diagram
Situation 25:
The beam is loaded as shown in the figure.
78.
82.
Determine the maximum concentrated load applied on
the beam.
a.
40 kN
c.
25 kN
b.
10 kN
d.
50 kN
83.
Determine the maximum negative moment in the
beam.
a.
10 kN-m
c.
25 kN-m
b.
15 kN-m
d.
30 kN-m
84.
Determine the location of the point of contraflexure
from the right end of the beam.
a.
3.25 m
c.
0.25 m
b.
1.25 m
d.
4.75 m
Determine the maximum shear that occur in the span.
a.
80 kN
c.
40 kN
b.
120 kN
d.
20 kN
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Situation 28:
Refer in the figure. A barge shown in diagram carries
the following loads:
90.
Determine the distance of the resultant from the 10 kN
load.
a.
3m
c.
5m
b.
2m
d.
2.5 m
W1 = 290 kN/m
W2 = 580 kN/m
(For every strip along longitudinal direction)
91.
Calculate the maximum shear in the beam.
a.
42 kN
c.
45 kN
b.
48 kN
d.
36 kN
Dimensions are:
L1 = L3 = 3 m
L2 = 6 m
92.
Calculate the maximum moment in the beam.
a.
94.17 kN-m
c.
98.17 kN-m
b.
104.17 kN-m
d.
108.17 kN-m
86.
If shear is zero at 5 m from the left end, what is the
upward pressure in kN/m?
a.
96 kN/m
c.
58 kN/m
b.
192 kN/m
d.
174 kN/m
R
Find the total length, L (m), so that the upward
pressure is uniform, and the barge remains horizontal.
a.
15.0 m
c.
24.0 m
b.
21.0 m
d.
18.0 m
93.
If the upward pressure is 145 kN/m, what is the
resulting moment (kN-m) at first point of zero shear?
a.
2,088 kN-m
c.
1,566 kN-m
b.
1,305 kN-m
d.
3,915 kN-m
EO
C
87.
EL
E
85.
ev
ie
w
Stresses in Beams
• Bending/Flexural Stress
Situation 31:
The simply supported beam in the figure has a
rectangular cross section 120 mm wide and 200 mm
high.
R
• Moving Loads
Situation 29:
A truck with axle loads of 40 kN and 60 kN on a
wheelbase of 5 m rolls across a 10-m span.
88.
Compute the maximum bending moment.
a.
122.5 kN-m
c.
156.5 kN-m
b.
160.0 kN-m
d.
133.5 kN-m
89.
Compute the maximum shearing force.
a.
80 kN
c.
100 kN
b.
90 kN
d.
75 kN
Situation 30:
A truck and trailer combination having the axle loads
shown in the figure rolls across the simply supported
span of 12 m.
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Compute the bending stress at a point on section B that
is 25 mm below the top of the beam.
a.
8.70 MPa (C)
c.
7.80 MPa (C)
b.
8.70 MPa (T)
d.
7.80 MPa (T)
Situation 32:
Given a rectangular hollow box with outer dimension
85 mm by 125 mm and constant thickness of 10 mm
has an ultimate stress of 415 MPa. Use Modulus of
elasticity E = 110 GPa.
95.
Determine the bending moment M for which the factor
of safety will be 2.50.
a.
10. 32 kN-m
c.
11.02 kN-m
b.
20.09 kN-m
d.
50.23 kN-m
96.
Determine the corresponding radius of curvature of
the section.
a.
2.18 m
c.
4.76 m
b.
27.91 m
d.
41.42 m
97.
Determine the minimum height h of the beam shown
in the figure if the flexural stress is not to exceed 20
MPa.
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94.
Compute the maximum bending stress in the beam.
a.
35.0 MPa
c.
25.0 MPa
b.
30.0 MPa
d.
20.0 MPa
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a.
b.
137 mm
113 mm
c.
d.
155 mm
162 mm
Situation 36:
A wide flange beam carries a shear of 150 kN. The
section properties are shown in the table.
Situation 33:
A beam is loaded as shown in the figure. Its crosssectional properties are also given.
Flange width, bf
Flange thickness, t f
Overall depth, d
Web thickness, t w
= 200 mm
= 20 mm
= 300 mm
= 20 mm
106. Determine the shear stress on the web at the junction
of the flange and the web?
a.
20.94 MPa
c.
22.54 MPa
b.
24.66 MPa
d.
27.04 MPa
99.
Determine the maximum tensile stress.
a.
45 MPa
c.
30 MPa
b.
60 MPa
d.
90 MPa
107. Determine the maximum shear stress on the section?
a.
32.08 MPa
c.
34.65 MPa
b.
29.34 MPa
d.
37.87 MPa
108. Determine the percentage of the shear carried by the
web.
a.
94.18 %
c.
93.85 %
b.
95.30 %
d.
92.61 %
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Determine the maximum moment in the beam.
a.
9.25 kN-m
c.
11.25 kN-m
b.
12.25 kN-m
d.
10.25 kN-m
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Situation 34:
A beam is loaded as shown in the figure. Its crosssectional properties are also given.
Situation 37:
A glued-laminated (glulam) timber beam is made from
5 – 40 mm by 140 mm (finish dimensions) boards as
shown in the figure. The strength of the wood in
horizontal shear is 550 kPa (which takes into account
a factor of safety).
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100. Determine the maximum compressive stress.
a.
30 MPa
c.
45 MPa
b.
90 MPa
d.
60 MPa
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101. Determine the maximum tensile stress.
a.
25.00 MPa
c.
24.00 MPa
b.
16.25 MPa
d.
15.60 MPa
102. Determine the maximum compressive stress.
a.
24.00 MPa
c.
25.00 MPa
b.
15.60 MPa
d.
16.25 MPa
• Horizontal Shearing Stress
Situation 35:
A rectangular beam 100 mm x 200 mm carries a
uniformly distributed load of 5 kN/m over its simple
span of 5 m.
103. Determine the shear 1 m from the left support.
a.
7.50 kN
c.
12.50 kN
b.
8.75 kN
d.
5.00 kN
104. Determine the shear stress 1 m from the left support
at a point 30 mm from the top of the beam.
a.
0.19 MPa
c.
0.33 MPa
b.
0.29 MPa
d.
0.48 MPa
105. Determine the maximum shear stress in the beam.
a.
0.56 MPa
c.
0.94 MPa
b.
0.38 MPa
d.
0.66 MPa
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110. What is the required shear strength of the glue joints
(MPa) considering a factor of safety against failure of
the glue joints of FS = 3.0? Neglect thickness of the glue
joint.
a.
1.584 MPa
c.
0.176 MPa
b.
0.528 MPa
d.
4.752 MPa
Combined Stresses
• Combined Axial and Flexural Loads
Situation 38:
A solid circular pole has a height of 3 m and a diameter
of 250 mm. The pole carries a compressive load of 3 kN
at an eccentricity of 100 mm and a lateral force of 0.45
kN at its top. The unit weight of the pole is 22 kN/m3.
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109. What is the maximum allowable vertical shear force V
for the section?
a.
6.21 kN
c.
9.07 kN
b.
10.27 kN
d.
15.42 kN
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38
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REAL EXCELLENCE ONLINE CIVIL ENGINEERING REVIEW
Effectiveness. Efficiency. Convenience
Strength of Materials
ENGR. HANS LAWRENCE E. DELA CRUZ
111. Determine the maximum compressive stress at the
base of the pole.
a.
0.95 MPa
c.
1.14 MPa
b.
1.03 MPa
d.
1.20 MPa
112. Compute the maximum tensile stress at the base of the
pole.
a.
1.03 MPa
c.
1.20 MPa
b.
0.95 MPa
d.
1.14 MPa
119. Determine the maximum shear stress.
a.
87.5 MPa
c.
62.5 MPa
b.
150.0 MPa
d.
125.0 MPa
113. Determine the maximum shearing stress at the base of
the pole.
a.
9.17 kPa
c.
13.75 kPa
b.
11.38 kPa
d.
12.22 kPa
120. Determine the normal stress on the diagonal AB.
a.
31.6 MPa
c.
28.0 MPa
b.
50.0 MPa
d.
37.8 MPa
Situation 39:
To reduce interference, a link in a machine is designed
so that its cross-sectional area in the center section is
reduced by one-half, as shown in the figure. The
thickness of the link is 50 mm. Use P = 40 kN.
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121. Determine the shearing stress on the diagonal AB.
a.
31.6 MPa
c.
50.0 MPa
b.
37.8 MPa
d.
28.0 MPa
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114. Determine the maximum value of the normal stress
acting on section m-n.
a.
60 MPa
c.
40 MPa
b.
80 MPa
d.
20 MPa
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Situation 42:
The state of plane stress at a point with respect to the
xy-axes is shown in the figure.
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115. Determine the minimum value of the normal stress
acting on section m-n.
a.
0 MPa
c.
-40 MPa
b.
-60 MPa
d.
20 MPa
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• State of Stress at a Point (Plane Stress)
Situation 40:
A wooden beam whose length is 1 m is 100 mm wide
and 200 mm high. It carries a load of 100 kN as shown.
122. Determine the equivalent normal stress with respect
to the x’ y’-axes.
a.
𝜎𝑥′ = 17.86 𝑀𝑃𝑎; 𝜎𝑦′ = 72.14 𝑀𝑃𝑎
b.
𝜎𝑥′ = 72.14 𝑀𝑃𝑎; 𝜎𝑦′ = 17.86 𝑀𝑃𝑎
c.
𝜎𝑥′ = −72.14 𝑀𝑃𝑎; 𝜎𝑦′ = 17.86 𝑀𝑃𝑎
d.
𝜎𝑥′ = 17.86 𝑀𝑃𝑎; 𝜎𝑦′ = −72.14 𝑀𝑃𝑎
123. Determine the equivalent shear stress with respect to
the x’ y’-axes.
a.
32.99 MPa
c.
30.99 MPa
b.
33.99 MPa
d.
31.99 MPa
116. Determine the total tensile stress at A.
a.
34.98 MPa
c.
29.98 MPa
b.
32.48 MPa
d.
27.48 MPa
117. Determine the maximum principal stress at A.
a.
37.55 MPa
c.
39.85 MPa
b.
35.65 MPa
d.
42.16 MPa
Situation 43:
The state of stress at a point is the result of two
loadings. When acting separately, the loadings
produce the two states of stress shown in the figure. If
the two loadings are applied simultaneously:
118. Determine the maximum shearing stress at A.
a.
11.94 MPa
c.
18.16 MPa
b.
4.87 MPa
d.
16.18 MPa
Situation 41:
A small block is 40 mm long, 20 mm high, and 10 mm
thick. The block is subjected to the forces shown in the
figure.
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REAL EXCELLENCE ONLINE CIVIL ENGINEERING REVIEW
Effectiveness. Efficiency. Convenience
Strength of Materials
ENGR. HANS LAWRENCE E. DELA CRUZ
124. Determine the maximum principal stress for the
combined state of stress.
a.
39.5 MPa
c.
30.0MPa
b.
38.3 MPa
d.
32.4 MPa
125. Determine the principal planes for the combined state
of stress.
a.
22.5°
c.
45.0°
b.
13.3°
d.
26.6°
END OF HANDOUT
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“Don’t watch the clock.
Do what it does.
Keep going.”
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-Sam Levenson
Effectiveness. Efficiency. Convenience
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Page 12 of 12 | SEDC Handouts No. 03