Estatica_1

Contents

Nomenclature

8

1

Introduction

1

1.1

What is the mechanics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Mechanical behavior, constitutive laws . . . . . . . . . . . . . . . . . . . . . .

2

1.3

The geometry of motion and deformation, kinematics . . . . . . . . . . . . . .

2

1.4

Relation force-motion, the laws of mechanics, kinetics . . . . . . . . . . . . .

3

2

Scalars and Vectors

4

2.1

Adding vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

Relative position vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.3

Multiplying a vector by a scalar using components . . . . . . . . . . . . . . .

6

2.3.1

Magnitude of a vector using components . . . . . . . . . . . . . . . .

6

2.3.2

Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

The dot product of two vectors . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.4.1

Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.4

2.5

CONTENTS

2

2.5.1

Direction of a Cartesian Vector . . . . . . . . . . . . . . . . . . . . . .

9

2.5.2

Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.5.3

Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.5.4

Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.5.5

Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.5.6

Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.5.7

Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.5.8

Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.5.9

Exercise 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.5.10 Exercise 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.5.11 Exercise 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.5.12 Exercise 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.5.13 Exercise 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.5.14 Exercise 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.5.15 Exercise 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.5.16 Exercise 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.5.17 Exercise 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.5.18 Exercise 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.5.19 Exercise 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.5.20 Exercise 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.5.21 Exercise 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.5.22 Exercise 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

CONTENTS

3

3

2.5.23 Exercise 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.5.24 Exercise 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Equilibrium

22

3.1

Equations of Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3.2

Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.3

Types of Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.3.1

Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.3.2

Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.3.3

Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.3.4

Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.3.5

Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.3.6

Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.3.7

Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.3.8

Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.3.9

Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.4.1

Plane Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.4.2

Truss Connections and Supports . . . . . . . . . . . . . . . . . . . . .

40

3.4.3

Method of Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.4.4

Internal and External Redundancy . . . . . . . . . . . . . . . . . . . .

41

3.4.5

Method of Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.4

List of Figures

2.1

Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

Adding vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.3

Relative position of points A, B, and C . . . . . . . . . . . . . . . . . . . . . .

5

2.4

Exercise1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.5

Ejercise1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.6

Direction of a Cartesian Vector . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.7

Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.8

Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.9

Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.10 Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.11 Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.12 Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.13 Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.14 Exercise 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.15 Exercise 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.16 Exercise 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

LIST OF FIGURES

5

2.17 Exercise 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.18 Exercise 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.19 Exercise 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.20 Exercise 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.21 Exercise 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.22 Exercise 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.23 Exercise 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.24 Exercise 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.25 Exercise 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.26 Exercise 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.27 Exercise 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.28 Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3.1

3-D positive axis system (Right-Hand) . . . . . . . . . . . . . . . . . . . . . .

22

3.2

2-D positive axis system (Right-Hand) . . . . . . . . . . . . . . . . . . . . . .

23

3.3

Types of Supports

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.4

Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.5

Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.6

Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.7

Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.8

Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.9

Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

LIST OF FIGURES

6

3.10 Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.11 Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.12 Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.13 Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.14 Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.15 Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.16 Exercise 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.17 Exercise 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.18 Exercise 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.19 Exercise 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.20 Exercise 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.21 Exercise 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.22 Exercise 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.23 Exercise 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.24 Exercise 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.25 Exercise 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.26 Trusse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.27 Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.28 Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.29 Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.30 Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.31 Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

LIST OF FIGURES

7

3.32 Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.33 Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.34 Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.35 Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.36 Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.37 Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.38 Method of Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.39 Exercise 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.40 Exercise 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.41 Exercise 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.42 Exercise 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

List of Tables

Chapter 1 Introduction

1.1

What is the mechanics?

Mechanics can predict forces and motions by using the three pillars of the subject: I. models of physical behavior, II. geometry,and III. the basic mechanics balance laws. Mechanics is the study of force, deformation and motion, and the relations between them. We care about forces because we want to know how hard to push something to make it move or whether it will break when we push. We care about deformation and motion because we want things to move or not move in certain ways. For each special case of the general mechanics problem we need to identify the system(s) of interest, idealize the system(s), use classical (high school, Euclidean) geometry to describe the layout, deformation and motion, and finally use the laws of Newtonian mechanics. Those who want to know how machines, structures, plants, animals and planets hold together and move about need to know Newtonian mechanics.

1.2. MECHANICAL BEHAVIOR, CONSTITUTIVE LAWS

1.2

2

Mechanical behavior, constitutive laws

The Mechanical behavior of something is the description of how loads cause deformation (or vice versa). • Definition of force. Force can be defined by the amount of spring stretch it causes. Thus, most modern force measurement devices measure force indirectly by measuring the deformation it causes in a calibrated spring of some kind. • Deformation is often hard to see. Distortion in the presence of forces is easy to see or imagine in the flesh of squeezed fingertips, in chewing gum between teeth or when a piece of paper bends. But pieces of rock or metal have deformation that is essentially invisible and sometimes hard to imagine. Most structural materials deform less than one part per thousand with working loads. These small deformations, even though essentially invisible, are important because they are enough to break bones and collapse bridges. • Rigid-object mechanics. Mechanics calculations in which deformation has been neglected are called rigid-object (or rigid-body). The assumption of rigidity also simplifies the introduction of more general mechanics concepts. Thus, for understanding the steering dynamics of a car we might treat the car as a rigid object, whereas for crash analysis where rigidity is clearly a poor approximation, we might treat a car as highly deformable. • Contact behavior. Most constitutive models describe the material inside an object. But to solve a mechanics problem involving friction or collisions one also has to have a constitutive model for the contact interactions. The standard friction model (or idealization) is an example of a contact constitutive model.

1.3

The geometry of motion and deformation, kinematics

In mechanics we use classical Greek (Euclidean) geometry to describe the layout, deformation and large-motions of objects. • Deformation is defined by changes of lengths and angles between various pairs and triplets of points. We often need to understand deformation of individual parts to predict when they will break. • Motion is defined by the changes of the position of points in time.

1.4. RELATION FORCE-MOTION, THE LAWS OF MECHANICS, KINETICS

1.4

3

Relation force-motion, the laws of mechanics, kinetics

The same intuitive ‘force’ that causes deformation also causes motion, or more precisely, acceleration of mass. The relation between force and acceleration of mass makes up the third pillar holding up mechanics. We loosely call this Newton’s laws; synonyms include the laws of mechanics, momentum and energy balance and kinetics. 2 Force is related to deformation by material properties (elasticity, viscosity, etc.) and force is related to motion by the laws of mechanics. Kinetics and kinematics. It is easy to confuse these similar looking and sounding words. Kinematics concerns geometry with no mention of force, and kinetics concerns the relation of force to motion.

Chapter 2 Scalars and Vectors All physical quantities in engineering mechanics are measured using either scalars or vectors. Scalar. A scalar is any positive or negative physical quantity that can be completely specified by its magnitude. Vector. A vector is any physical quantity that requires both a magnitude and a direction. B Sense Magnitude

A

α

Direction

Figure 2.1: Vector

2.1

Adding vectors

~ and B ~ is defined by the tail of A ~ to the tip of B. ~ Tip to tail rule. The sum of two vectors A ~ is drawn first, as shown in fig. 2.3b. Putting Parallelogram rule. The same sum is achieved if B ~ and B ~ on the same picture draws a parallelogram as shown in fig. 2.2b. both ways of adding A Hence the tip to tail rule of vector addition is also called the parallelogram rule.

2.2. RELATIVE POSITION VECTORS

5

~ B

~ A C~

~ B

~ A

~ A

~ B

C~ (a) Tip to tail rule

(b) Parallelogram rule

Figure 2.2: Adding vectors

2.2

Relative position vectors

The position of point B relative to point A is represented by the vector ~rB/A (pronounced ”r of B relative to A”).An alternate notation for the relative position vector ~rB/A is ~rB/A = ~rA/B (pronounced ”r A B” or”r A to B”).

~rB/A = −~rA/B +~rC/B or ~rCA = ~rBA +~rCB

(2.1)

~ = ~rB/A and ~rAB = −~rBA rC/A

(2.2)

B

~rC/B

~rB/A

C ~rC/A A Figure 2.3: Relative position of points A, B, and C

2.3. MULTIPLYING A VECTOR BY A SCALAR USING COMPONENTS

2.3

6

Multiplying a vector by a scalar using components

~ is multiplied by c then so must be each of the component vectors. If A

h i ~ xyz = cA ~ x cA ~y c A ~z c A

(2.3a)

Matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. The number of columns of the 1st matrix must equal the number of rows of the 2nd matrix.       A x Bx C x  a aA x + bBx + cC x        M =  Ay By Cy  ∗ b =  aAy + bBy + cCy        Az Bz Cz c aAz + bBz + cCz

2.3.1

(2.4)

Magnitude of a vector using components

q A ~ xyz = A2x + A2y + A2z

(2.5a)

Exercise 1

2.3.2

7

Exercise 1

Finding a force vector given its magnitude and line of action: A string is pulled with a force F = 100N as shown in fig. 2.4 Write F~ as a vector.

Figure 2.4: Exercise1

Figure 2.5: Ejercise1

2.4. THE DOT PRODUCT OF TWO VECTORS

2.4

8

The dot product of two vectors

The dot product is used to project a vector in a given direction, to reduce a vector to components, to reduce vector equations to scalar equations, to define work and power, and to help solve geometry problems.

~ ~ ~ ∗ B ~ . cos(θAB) A . B = A

2.4.1

(2.6a)

Exercise 2

Finding angle between two vectors using dot product:

~ = 2i + 3 j A

2.5

~ = 2i − j B

(2.7a)

Cross product

The cross product a×b of two linearly independent vectors a and b is a vector that is perpendicular to both and therefore normal to the plane containing them.

A ~xB ~ = A ~ ∗ B ~ . sin(θAB)

(2.8a)

Direction of a Cartesian Vector

9

  # " # " #  i j k  " A1 A3 A1 A2   A2 A3 A x B = A1 A2 A3  = i− j+ k B1 B3 B1 B2   B2 B3 B1 B2 B3

2.5.1

(2.9)

Direction of a Cartesian Vector

cos (α) =

Ax A

cos (θ) =

Ay A

cos (γ) =

cos2 (α) +cos2 (θ) + cos2 (γ) = 1

Figure 2.6: Direction of a Cartesian Vector

Az A

(2.10a)

(2.11a)

Exercise 3

2.5.2

10

Exercise 3

Finding a vector normal to two given vectors:

~ = i−2j+k A

2.5.3

~ = 3 j + 2k B

(2.12a)

Exercise 4

A force F~ = 1i + 20 j[N] acts at point A of an object pinned at O as shown in the figure. The distance OA = 2m. Find the moment of the force about the pin at point O.

Figure 2.7: Exercise 4

Exercise 5

2.5.4

11

Exercise 5

A 2m x 2m square plate hangs from one of its corners as shown in the figure. At the diagonally opposite end, a force of 50N is applied by pulling on the string AB. Find the moment of the applied force about the center C of the plate using


2.5.5

Exercise 6

A string connects a particle A at (1m, 2m) to a support B at (3m, 5m). The tension in the string is 10N. There are other strings also holding the particle in place. What is the force of string AB on the particle?


Exercise 7

2.5.6

12

Exercise 7

A frictionless ramp connects A at (3m, 5m) to B at (12m, 17m). The ramp pushes a block with a force of 50N normal to the ramp surface. Express the force from the ramp as a vector F~ (ignore the other forces that also act on the block holding it in place).


2.5.7

Exercise 8

During a slam-dunk, a basketball player pulls on the hoop with a 250 lbf at point C of the ring as shown in the figure. Find the moment of the force about • The point of the ring attachment to the board (point B), and • the root of the pole, point O.


Exercise 9

2.5.8

13

Exercise 9

The screw eye in fig is subjected to two forces F1 and F2 . Determine the magnitude and direction of the resultant force.


2.5.9

Exercise 10

The force F = 450lb acts on the frame. Resolve this force into components acting along members AB and AC, and determine the magnitude of each component.


Exercise 11

2.5.10

14

Exercise 11

The device is used for surgical replacement of the knee joint. If the force acting along the leg is 360 N. Determine its components along the x’ and y axes.


2.5.11

Exercise 12

The contact point between the femur and tibia bones of the leg is at A. If a vrtical force of 175lb is applied at this point. Determine the components along the x and y component represents the normal force on the load-bearing region of the bones. Both the x and y components of this force cause synovial fluid to be squeeze out of the bearing space.


Exercise 13

2.5.12

15

Exercise 13

The top of an L-shaped bar, shown in the figure, is to be tied by strings AD and BD to the points A and B in the yz plane. Find the length of the strings AD and BD using vectors ~rAD and ~rBD


2.5.13

Exercise 14

Determine the magnitude and direction of the resultant force acting on the pipe assembly


Exercise 15

2.5.14

16

Exercise 15

Calcular el momento de ua fuerza de 250N, aplicada en el punto de la llave de grifo.


2.5.15

Exercise 16

Calcular el momento en el punto en el cual esta sometida una fuerza de 160KN.


Exercise 17

2.5.16

17

Exercise 17

Determine the n and t components of the force F which is exerted by the rod AB on the crank OA. Evaluate your general expression for F = 100N and 1) θ = 30◦ β = 10◦ and 2) θ = 15◦ β = 25◦


2.5.17

Exercise 18

The guy cables AB and AC are attached to the top of the transmission tower. The tension in cable AB is 8 kN. Determine the required tension T in cable AC such that the net effect of the two cable tensions is a downward force at point A. Determine the magnitude R of this downward force.


Exercise 19

2.5.18

18

Exercise 19

In the design of the robot to insert the small cylindrical part into a close-fitting circular hole, the robot arm must exert a 90-N force P on the part parallel to the axis of the hole as shown. Determine the components of the force which the part exerts on the robot along axes (a) parallel and perpendicular to the arm AB, and (b) parallel and perpendicular to the arm BC.


2.5.19

Exercise 20

Compute the moment of the 0.4-lb force about the pivot O of the wall-switch toggle.


Exercise 21

2.5.20

19

Exercise 21

The lower lumbar region A of the spine is the part of the spinal column most susceptible to abuse while resisting excessive bending caused by the moment about A of a force F. For given values of and F, b, and h, determine the angle which causes the most severe bending strain.


2.5.21

Exercise 22

Determine the combined moment about O due to the weight of the mailbox and the cross member AB. The mailbox weighs 4 lb and the uniform cross member weighs 10 lb. Both weights act at the geometric centers of the respective items.


Exercise 23

2.5.22

20

Exercise 23

Elements of the lower arm are shown in the figure. The weight of the forearm is 5 lb with mass center at G. Determine the combined moment about the elbow pivot O of the weights of the forearm and the sphere. What must the biceps tension force be so that the overall moment about O is zero?


2.5.23

Exercise 24

The directions of the two thrust vectors of an experimental aircraft can be independently changed from the conventional forward direction within limits. For the thrust configuration shown, determine the equivalent force–couple system at point O. Then replace this force–couple system by a single force and specify the point on the x-axis through which the line of action of this resultant passes. These results are vital to assessing design performance.


Exercise 25

2.5.24

21

Exercise 25

The pedal–chainwheel unit of a bicycle is shown in the figure. The left foot of the rider exerts the 40-lb force, while the use of toe clips allows the right foot to exert the nearly upward 20-lb force. Determine the equivalent force–couple system at point O.


Chapter 3 Equilibrium

3.1

Equations of Statics

For a 3-D body at rest the coordinate system used is the x-y-z Cartesian system, in which the definition of positive moments is given by the right hand rule that states that moments are positive is their sense is counterclockwise as shown in Fig. 1.1.

Figure 3.1: 3-D positive axis system (Right-Hand)

3.2. GOVERNING EQUATIONS

23

Figure 3.2: 2-D positive axis system (Right-Hand)

3.2

Governing Equations

Force equilibrium equation is given as:

X

Fx = 0

X

Fy = 0

X

Fz = 0

X

Mx = 0

X

My = 0

X

Mz = 0

(3.1)

(3.2)

3.3. TYPES OF SUPPORTS

3.3

24

Types of Supports

There are 6 commonly used types of supports which prevent a structure, or part of it, from accelerating when acted upon by external forces. In general, supports are there to keep a structure in equilibrium.

Figure 3.3: Types of Supports

Exercise 1

3.3.1

25

Exercise 1

The weight of the bicycle is 29 lb with center of gravity at G. Determine the normal forces at A and B when the bicycle is in equilibrium.


3.3.2

Exercise 2

A 120-lb crate rests on the 60-lb pickup tailgate. Calculate the tension T in each of the two restraining cables, one of which is shown. The centers of gravity are at G1 and G2. The crate is located midway between the two cables.


Exercise 3

3.3.3

26

Exercise 3

Determine the force magnitude P required to lift one end of the 250-kg crate with the lever dolly as shown. State any assumptions.


3.3.4

Exercise 4

Three cables are joined at the junction ring C. Determine the tensions in cables AC and BC caused by the weight of the 30-kg cylinder.


Exercise 5

3.3.5

27

Exercise 5

A woodcutter wishes to cause the tree trunk to fall uphill, even though the trunk is leaning downhill. With the aid of the winch W, what tension T in the cable will be required? The 1200-lb trunk has a center of gravity at G. The felling notch at O is sufficiently large so that the resisting moment there is negligible.


3.3.6

Exercise 6

A pipe P is being bent by the pipe bender as shown. If the hydraulic cylinder applies a force of magnitude kN to the pipe at C, determine the magnitude of the roller reactions at A and B.


Exercise 7

3.3.7

28

Exercise 7

The asymmetric simple truss is loaded as shown. Determine the reactions at A and D. Neglect the weight of the structure compared with the applied loads. Is knowledge of the size of the structure necessary?


3.3.8

Exercise 8

In a procedure to evaluate the strength of the triceps muscle, a person pushes down on a load cell with the palm of his hand as indicated in the figure. If the load-cell reading is 160 N, determine the vertical tensile force F generated by the triceps muscle. The mass of the lower arm is 1.5 kg with mass center at G. State any assumptions.


Exercise 9

3.3.9

29

Exercise 9

The portable floor crane in the automotive shop is lifting a 420-lb engine. For the position shown compute the magnitude of the force supported by the pin at C and the oil pressure p against the 3.20-in.-diameter piston of the hydraulic-cylinder unit AB.


Exercise 9

30

EVALUACION VECTORES MECANICA RACIONAL I

30 de Octubre 2015

El sistema ODA está atado a la tensión del cable AB igual a 2.4kN. Determine la tensión T como fuerza que actua en el miembro AD y AC (a lo largo de su l´ınea de acción).


Un barril está montado en el gancho del sistema con un peso de 100N. Al mismo tiempo actua un hidráulico en la dirección de B a C. Determine la fuerza que ejerce el hidráulico. El a´ ngulo es θ = 40◦


Exercise 9

31

El ensamble esta soportado por 3 cables como se muestra en la figura. El cable CD esta tensionado con una fuerza de 1.2KN. Exprese esa Fuerza de forma vectorial en coordenadas x,y, z and además en coordenadas x’,y’,z’


Exercise 9

32

Encuentre las reacciones en cada uno de los elementos

Figure 3.16: Exercise 10 Encuentre las reacciones en cada uno de los elementos


Exercise 9

33

The uniform I-beam has a mass of 60 kg per meter of its length. Determine the tension in the two supporting cables and the reaction at D.

Figure 3.18: Exercise 12 The light right-angle boom which supports the 400- kg cylinder is supported by three cables and a balland- socket joint at O attached to the vertical x-y surface. Determine the reactions at O and the cable tensions.


Exercise 9

34

The square steel plate has a mass of 1800 kg with mass center at its center G. Calculate the tension in each of the three cables with which the plate is lifted while remaining horizontal.


A 200-N force is applied to the handle of the hoist in the direction shown. The bearing A supports the thrust (force in the direction of the shaft axis), while bearing B supports only radial load (load normal to the shaft axis). Determine the mass m which can be supported and the total radial force exerted on the shaft by each bearing. Assume neither bearing to be capable of supporting a moment about a line normal to the shaft axis.


Exercise 9

35

The pin A, which connects the 200-kg steel beam with center of gravity at G to the vertical column, is welded both to the beam and to the column. To test the weld, the 80-kg man loads the beam by exerting a 300-N force on the rope which passes through a hole in the beam as shown. Calculate the torque (couple) M supported by the pin.


Exercise 9

36

The portable reel is used to wind up and store an air hose. The tension in the hose is 100 N and a vertical 200-N force is applied to the handle in order to steady the reel frame. Determine the minimum force P which must be applied perpendicular to the handle DE and the vertical components of the force reactions at the feet A, B, and C. The diameter of the coil of reeled hose is 300 mm, and the weight of the loaded reel and its frame may be neglected. Note that force P is perpendicular to OD. State any assumptions.


Exercise 9

37

A vertical force P on the foot pedal of the bell crank is required to produce a tension T of 400 N in the vertical control rod. Determine the corresponding bearing reactions at A and B.


Exercise 9

38

The basic features of a small backhoe are shown in the illustration. Member BE (complete with hydraulic cylinder CD and bucket-control links DF and DE) weighs 500 lb with mass center at G1 . The bucket and its load of clay weigh 350 lb with mass center at G2 . To disclose the operational design characteristics of the backhoe, determine and plot the force T in the hydraulic cylinder AB as a function of the angular position of member BE over the range0 < θ < 90 . For what value of is the force T equal to zero? Member OH is fixed for this exercise; note that its controlling hydraulic cylinder (hidden) extends from near point O to pin I. Similarly, the bucket-control hydraulic cylinder CD is held at a fixed length. assumptions.


3.4. STRUCTURES

3.4

39

Structures

We analyze the internal forces acting in several types of structures—namely, trusses, frames, and machines. In this treatment we consider only statically determinate structures, which do not have more supporting constraints than are necessary to maintain an equilibrium configuration. Thus, as we have already seen, the equations of equilibrium are adequate to determine all unknown reactions. The analysis of trusses, frames and machines, and beams under concentrated loads constitutes a straightforward application of the material developed in the previous two chapters. The basic procedure developed in Equilibrium for isolating a body by constucting a correct free-body diagram is essential for the analysis of statically determinate structures.

Figure 3.26: Trusse

3.4.1

Plane Trusses

A framework composed of members joined at their ends to form a rigid structure is called a truss. Bridges, roof supports, derricks, and other such structures are common examples of trusses. Structural members commonly used are I-beams, channels, angles, bars, and special shapes which are fastened together at their ends by welding, riveted connections, or large bolts or pins. When the members of the truss lie essentially in a single plane, the truss is called a plane truss.

Truss Connections and Supports

40

Figure 3.27: Trusses

3.4.2

Truss Connections and Supports

When welded or riveted connections are used to join structural members, we may usually assume that the connection is a pin joint if the centerlines of the members are concurrent at the joint as in Fig. 4/5. We also assume in the analysis of simple trusses that all external forces are applied at the pin connections. This condition is satisfied in most trusses. In bridge trusses the deck is usually laid on cross beams which are supported at the joints. For large trusses, a roller, rocker, or some kind of slip joint is used at one of the supports to provide for expansion and contraction due to temperature changes and for deformation from applied loads. Trusses and frames in which no such provision is made are statically indeterminate, . Examples of such joints. Two methods for the force analysis of simple trusses will be given. Each method will be explained for the simple truss The free-body diagram of the truss as a whole. The external reactions are usually determined first, by applying the equilibrium equations to the truss as a whole. Then the force analysis of the remainder of the truss is performed.

Method of Joints

3.4.3

41

Method of Joints

This method for finding the forces in the members of a truss consists of satisfying the conditions of equilibrium for the forces acting on the connecting pin of each joint. The method therefore deals with the equilibrium of concurrent forces, and only two independent equilibrium equations are involved.

3.4.4

Internal and External Redundancy

If a plane truss has more external supports than are necessary to ensure a stable equilibrium configuration, the truss as a whole is statically indeterminate, and the extra supports constitute external redundancy. If a truss has more internal members than are necessary to prevent collapse when the truss is removed from its supports, then the extra members constitute internal redundancy and the truss is again statically indeterminate. For a truss which is statically determinate externally, there is a definite relation between the number of its members and the number of its joints necessary for internal stability without redundancy. Because we can specify the equilibrium of each joint by two scalar force equations, there are in all 2j such equations for a truss with j joints. For the entire truss composed of m two-force members and having the maximum of three unknown support reactions, there are in all m + 3 unknowns (m tension or compression forces and three reactions). Thus, for any plane truss, the equationm + 3 = 2 j will be satisfied if the truss is statically determinate internally.


42

Determine the force in each member of the loaded truss. Explain why knowledge of the lengths of the members is unnecessary.


Determine the force in each member of the loaded truss. Identify any zero-force members by inspection.



43

Determine the force in each member of the loaded truss.


Determine the force in each member of the loaded truss.



44

Calculate the force in each member of the loaded truss. All triangles are isosceles..


Determine the force in each member of the loaded truss. Make use of the symmetry of the truss and of the loading.



45

Compute the force in each member of the loaded cantilever truss by the method of joints.

Figure 3.34: Exercise 7 Determine the forces in members BC and BG of the loaded truss.



46

The tower for a transmission line is modeled by the truss shown. For the loads of 1.8 kN applied in the vertical plane, compute the forces induced in members AB, DB, and CD.

Figure 3.36: Exercise 9 Determine the forces in members AB, BC, and BD of the loaded truss.


Method of Sections

3.4.5

47

Method of Sections

When analyzing plane trusses by the method of joints, we need only two of the three equilibrium equations because the procedures involve concurrent forces at each joint. We can take advantage of the third or moment equation of equilibrium by selecting an entire section of the truss for the free body in equilibrium under the action of a nonconcurrent system of forces. This method of sections has the basic advantage that the force in almost any desired member may be found directly from an analysis of a section which has cut that member.

Figure 3.38: Method of Section

Method of Sections

48

Calculate the forces induced in members KL, CL, and CB by the 20-ton load on the cantilever truss.

Figure 3.39: Exercise 10 Calculate the force in member DJ of the Howe roof truss illustrated. Neglect any horizontal components of force at the supports.


Method of Sections

49

The truss shown is composed of right triangles. The crossed members in the center two panels are slender tie rods incapable of supporting compression. Retain the two rods which are under tension and compute the magnitudes of their tensions. Also find the force in member MN.

Figure 3.41: Exercise 12 The hinged frames ACE and DFB are connected by two hinged bars, AB and CD, which cross without being connected. Compute the force in AB.


Estatica_1

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