Solving Practical Engineering Mechanics Problems: Statics

iii

Synthesis Lectures on Mechanical Engineering

Synthesis Lectures on Mechanical Engineering series publishes 60–150 page publications pertaining to this diverse discipline of mechanical engineering. The series presents Lectures written for an audience of researchers, industry engineers, undergraduate and graduate students. Additional Synthesis series will be developed covering key areas within mechanical engineering.

Solving Practical Engineering Mechanics Problems: StaticsSayavur I. BakhtiyarovOctober 2017

Resistance Spot Welding: Fundamentals and Applications for the Automotive IndustryMenachem Kimchi and David H. PhillipsOctober 2017

Unmanned Aircraft Design: Review of FundamentalsMohammad SadraeySeptember 2017

Introduction to Refrigeration and Air Conditioning Systems: Theory and ApplicationsAllan KirkpatrickSeptember 2017

MEMS Barometers Toward Vertical Position Detection: Background Theory, System Prototyping, and Measure-ment AnalysisDimosthenis E. BolanakisMay 2017

Vehicle Suspension System Technology and DesignAvesta Goodarzi, Amir KhajepourMay 2017

Engineering Finite Element AnalysisRamana M. PidapartiMay 2017

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Solving Practical Engineering Mechanics Problems: StaticsSayavur I. Bakhtiyarovwww.morganclaypool.com

ISBN: 9781681731889 printISBN: 9781681731896 ebook

DOI 10.2200/S00799ED1V01Y201709MEC008

A Publication in the Morgan & Claypool Publishers seriesSYNTHESIS LECTURES ON MECHANICAL ENGINEERING, #8

Series ISSN: 2573-3168 Print 2573-3176 Electronic

Solving Practical Engineering Mechanics Problems: StaticsSayavur I. BakhtiyarovNew Mexico Institute of Mining and Technology

SYNTHESIS LECTURES ON MECHANICAL ENGINEERING #8

M&C MORGAN & CLAYPOOL PUBLISHERS

vi

ABSTRACTEngineering mechanics is one of the fundamental branches of science that is important in the education of profes-sional engineers of any major. Most of the basic engineering courses, such as mechanics of materials, fluid and gas mechanics, machine design, mechatronics, acoustics, vibrations, etc. are based on engineering mechanics courses. In order to absorb the materials of engineering mechanics, it is not enough to consume just theoretical laws and the-orems—a student also must develop an ability to solve practical problems. Therefore, it is necessary to solve many problems independently. This book is a part of a four-book series designed to supplement the engineering mechanics courses. This series instructs and applies the principles required to solve practical engineering problems in the follow-ing branches of mechanics: statics, kinematics, dynamics, and advanced kinetics. Each book contains between 6 and 8 topics on its specific branch and each topic features 30 problems to be assigned as homework, tests, and/or midterm/final exams with the consent of the instructor. A solution of one similar sample problem from each topic is provided.

This first book contains seven topics of statics, the branch of mechanics concerned with the analysis of forces acting on construction systems without an acceleration (a state of the static equilibrium). The book targets the un-dergraduate students of the sophomore/junior level majoring in science and engineering. The author welcomes all feedback/comments from the reader. Please feel free to contact him at sayavur.bakhtiyarov@nmt.edu.

KEYWORDSforce, moment, torque, gravity, equilibrium, center of gravity

vii

Contents Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Topic S-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Determination of the Reaction Forces of Supports for Rigid Body . . . . . . . . . . . . . . . . . . . . . 11.2 Sample Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Topic S-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Application of the Method of Joints to Find Unknown Forces in a Plane (2D) Truss Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Sample Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 Determining Reaction Forces by Analytical Method . . . . . . . . . . . . . . . . . . . . . . . . 322.3.2 Determining the Forces in the Truss Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Topic S-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1 Determination of Reaction Forces of Supports for Composite Stud . . . . . . . . . . . . . . . . . . . . 373.2 Sample Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Topic S-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1 Determination of Reaction Forces of Supports for Composite Construction (System of Two Bodies) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Sample Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Topic S-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.1 Determination of Reaction Forces of Supports for Composite Construction (System of Three Bodies) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 Sample Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6 Topic S-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.1 Determination of reaction forces of rods supporting rectangular plate . . . . . . . . . . . . . . . . . . 856.2 Sample Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7 Topic S-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.1 Determination of Center of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.2 Sample Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Author Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

ix

AcknowledgmentsThe author acknowledges that this work is essentially a translation and a revision of selected problems provided by Professor A. A. Yablonski (Collection of Problems for Course Projects in Theoretical Mechanics, 2nd ed., Vischaya Shkola Publishers, 1972, in Russian). The author intended to introduce this unique work to western academia, which is the product of material covered by him in many classes over a period of four decades in a number of universities and colleges.

1

CHAPTER 1

Topic S-1

1.1 DETERMINATION OF THE REACTION FORCES OF SUPPORTS FOR RIGID BODY

Find reaction forces in supports of the given construction systems schematically shown in Figures 1.1‒1.30. The sizes are in meters and the loads are shown in the table below.

Problem #G P M q α

kN kN m kN m Degrees1 10 5 20 1 302 12 8 10 4 603 8 4 5 2 604 14 - 8 3 305 - 6 7 1 456 - 10 4 2 607 - 6 5 1 458 16 7 6 2 609 6 6 4 2 3010 10 8 9 1 3011 - 4 7 0.5 4512 10 6 8 - 4513 12 10 6 2 3014 10 6 10 1 4515 4 4 4 2 6016 20 10 - 2 4517 25 5 - 0.5 4518 20 10 10 - 3019 - 4 8 1 4520 - 10 6 0.5 4521 - 8 7 0.5 3022 - 10 8 1 3023 - 7 10 2 3024 - 6 7 1.5 3025 - 14 20 0.5 4526 - 16 14 1 3027 5 4 8 2.5 4528 - 10 7 3 3029 - 6 8 1 1530 15 10 14 0 30

2 1. PROBLEM S-1

2 2

G

M

q

B

P

αA

Figure 1.1.

2 3

G

M

q

B

P

αA

1

α

D

C

Figure 1.2.

3

2 2

G

M

q

B

P

αA

Figure 1.3.

2

G

Mq

B

α

A

31

Figure 1.4.

1.1 DETERMINATION OF THE REACTION FORCES OF SUPPORTS FOR RIGID BODY

4 1. PROBLEM S-1

M4

2

q

A

α

C B

p

D

Figure 1.5.

3

M

qP

α

A

1

2

B

Figure 1.6.

5

2

M

qP

α

A

2

B

α

Figure 1.7.

3

M q

P

α

G

1

B

4

D

α

F

A

E

C

Figure 1.8.

1.1 DETERMINATION OF THE REACTION FORCES OF SUPPORTS FOR RIGID BODY

6 1. PROBLEM S-1

1

G

M

q

B

Pα

A

1 2

α

Figure 1.9.

2

Mq

A

1

EB

1

2α

2.5

1.5

Pα

C

D

G

Figure 1.10.

Mq

A

B

4

2αP

α

3

Figure 1.11.

7

BP

αMG

C

A

Figure 1.12.

M

qP

α

G

1 B

D

F

A

EC 1 1 1F

α

Figure 1.13.

1.1 DETERMINATION OF THE REACTION FORCES OF SUPPORTS FOR RIGID BODY

8 1. PROBLEM S-1

α

11

q3

PG

2

M A

Figure 1.14.

q

P

1

A

2

α

G

4

M

Figure 1.15.

9

2 2

G

q

B

Pα

A

2

Figure 1.16.

1.5

G

q BPα

A

2

4

Figure 1.17.

GB

P

α

Aαα

GM

Figure 1.18.

1.1 DETERMINATION OF THE REACTION FORCES OF SUPPORTS FOR RIGID BODY

10 1. PROBLEM S-1

α3

21 M

A

B

q2q

P

Figure 1.19.

4

q

C

P

α

A

3

2

B

α

M

Figure 1.20.

11

α

αP

2

A4

B

4 qM

Figure 1.21.

2

q

Pα

A

2

B

α

M

2

Figure 1.22.

1.1 DETERMINATION OF THE REACTION FORCES OF SUPPORTS FOR RIGID BODY

12 1. PROBLEM S-1

q

P

αA

2

M CB

Figure 1.23.

α

3

M

2α

B

A

2

P

q

Figure 1.24.

13

M

α

P

22

2

A

B

q

Figure 1.25.

M

αP

33

2

A

B

2q

q2 2Figure 1.26.

1.1 DETERMINATION OF THE REACTION FORCES OF SUPPORTS FOR RIGID BODY

14 1. PROBLEM S-1

Figure 1.27.

M

A

2

2

2

B

α

q

P

Figure 1.28.

15

M

α

α1

2

2

P2α

A

q

B

Figure 1.29.

A

M

B

α

αP

G

Figure 1.30.

1.2 SAMPLE PROBLEMDefine the reaction forces in support A and rod CD of the construction schematically shown in Figure 1.31.

G = 10 kN; P = 5 kN; M = 8 kN m; q = 0.5 kN/m; a = 30°. The sizes are in meters.

1.2 SAMPLE PROBLEM

16 1. PROBLEM S-1

1 2

G

M

q

B

P

αA

2 1

C

D

Figure 1.31.

1.3 SOLUTIONLet’s consider the equilibrium of the forces applied to the stud AB. Remove the supports: pin support at the point A, the member CD, and the string attached to point B. The actions of the support are replaced by the appropriate reac-tion forces (Figure 1.32). Because the direction of the reaction force of the pin A is unknown, we need to determine its components XA and YA. We also show the reaction SCD in member CD and the reaction S in the string, which is by magnitude equal to P. Evenly distributed load of the intensity q is replaced by the concentrated force Q, which is equal: Q = 2 ∙ q = 2 ∙ 0.5=1 kN and applied in the center of gravity of the distributed forces.

2G

y

S

B

Qα

A

1

xC

1 2

XA

YA

90° SCDM

Figure 1.32.

17

Three equilibrium equations can be written for the plane (2D) system of forces applied to the system:

∑ MiA = 0; ‒Q ∙ 1 ‒ G ∙ 3 + SCD ∙ 4 sin 30° ‒ M + S ∙ 6 = 0 (1.1)

∑ Xi = 0;XA ‒ SCD cos 30° = 0 (1.2)

∑ Yi = YA ‒ Q ‒ G + SCD cos 60° + S = 0 (1.3)

From Equation (1.1):

SCD = Q ∙ 1 + G ∙ 3 + M ‒ S ∙ 6 = 1 ∙ 1 + 10 ∙ 3 + 8 ‒ 5 ∙ 6 = 4.5 kN. 4 sin 30° 4 ∙ 0.5

From Equation (1.2):

XA = SCD cos 30° = 4.5 ∙ 0.866 = 3.90 kN.

From Equation (1.3):

YA = Q + G ‒ SCD cos 60° ‒ S = 1 + 10 ‒ 4.5 ∙ 0.5 ‒ 5 = 3.75 kN.

The values of XA, YA, and SCD are positive. It means the assumed directions of these forces coincide with their true directions.

1.3 SOLUTION