# Engineering Statics/Introduction

Building large structures, such as these in Hong Kong, can only be designed by applying the principles of static (and dynamic) mechanics.

# 1 Introduction to Statics

## Mechanics

Sir Isaac Newton developed his three laws of motion, which form the basis of mechanical analysis.

Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment. There are few principles in mechanics, but they have wide applications in engineering. These principles form the basis for advanced research in vibrations, stability and strength of structures, fluid dynamics, and so on. Thus, a thorough understanding of mechanics is essential to progress in these fields of research, or to simply become a good engineer.

Mechanics is the oldest physical science. The main theory of mechanics in antiquity was Aristotelian mechanics. In the Middle Ages, Aristotle's theories were criticized and modified by a number of figures, beginning with John Philoponus in the 6th century. A central problem was that of projectile motion, which was discussed by Hipparchus and Philoponus. This led to the development of the theory of impetus by 14th century French Jean Buridan, which developed into the modern theories of inertia, velocity, acceleration and momentum. This work and others was developed in 14th century England by the Oxford Calculators such as Thomas Bradwardine, who studied and formulated various laws regarding falling bodies. On the question of a body subject to a constant (uniform) force, the 12th century Jewish-Muslim Hibat Allah Abu'l-Barakat al-Baghdaadi stated that constant force imparts constant acceleration, while the main properties are uniformly accelerated motion (as of falling bodies) was worked out by the 14th century Oxford Calculators. Two central figures in the early modern age are Galileo Galilei and Isaac Newton. Galileo's final statement of his mechanics, particularly of falling bodies, is his Two New Sciences (1638). Newton's 1687 Philosophiæ Naturalis Principia Mathematica provided a detailed mathematical account of mechanics, using the newly developed mathematics of calculus and providing the basis of Newtonian mechanics.

In this book, we will not give a mostly theoretical study — as we would in a physics book; the content will be balanced with the applications of mechanical principles. The principles of mechanics are best described mathematically, and thus we shall begin by describing the mathematical principles of statics.

branches of mechanics (green) and their relative place in the Island of "Physics of Mechanics" and "Mechanical engineering"

The topic of mechanics is divided in two parts: statics and dynamics; then it is further divided into topics we will not cover in this book.

## Basic Concepts

Definitions for
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Two-dimensional space is described with two coordinates (${\displaystyle x}$  and ${\displaystyle y}$ ), while three-dimensional space (physical reality) is described in three coordinates (${\displaystyle x}$ , ${\displaystyle y}$ , and ${\displaystyle z}$ ).
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of object (not related to analysis of statics problems).
Mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent: inertial mass, active gravitational mass, and passive gravitational mass. The latter being dealt with frequently in statics.
Force is any influence that causes an object to undergo a change in speed, a change in direction, or in a change in shape. Force can also be described by intuitive concepts such as a push or pull that can cause an object with mass to change its velocity, i.e. accelerate. A force has both magnitude and direction, which is a vector quantity (discussed in Chapter 2).
Particle is a small localized object to which can be ascribed several physical properties such as volume or mass. Whether objects can be considered particles depends on the scale of the context; if an object's own size is small or negligible, or if geometrical properties and structure are irrelevant, then it can be considered a particle.
Rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it. If the deformations on the material are negligible in affecting the solution, then a suitable problem model may be made using rigid bodies.

## Scalars and Vectors

There are two types of quantities in physics: scalars and vectors. Scalars have only a magnitude, e.g. time, volume, speed, energy, mass, and density. Vector quantities are described by both a magnitude and direction; examples include displacement, velocity, acceleration, force, moment, and momentum. Speed is velocity with its direction component.

Types of vectors
Free vectors have magnitude and direction, as any vector, but the initial point is not important, i.e. a vector floating in space. A free vector is convenient, as we'll see, in describing displacement of a moving body.
Sliding vectors have a line of action in space but no particular point of application. An example of this is a force – no matter the point of application of a force on a body, the result is the same.
Fixed vectors has a specific point of application. This type of vector becomes important when describing forces acting on non-rigid (deformable) bodies.

### Conventions for Diagrams and Equations

Figure 1.1: Vector drawing and labeling conventions

A vector quantity, ${\displaystyle {\textbf {V}}}$  is drawn as in Figure 1.1, a line segment with an arrow head to indicate direction. The length of the line segment indicates the vector magnitude, ${\displaystyle |V|}$  (printed as ${\displaystyle V}$ ), using a convenient scale, e.g. 1 centimeter is equal to 10 newtons.

Since boldening letters is difficult/time-consuming for handwritten work, a line or arrow over the letter as in ${\displaystyle {\overline {V}}}$  or ${\displaystyle {\vec {V}}}$ . Others might use a caret symbol, or underline ${\displaystyle {\underline {V}}}$ . Maintaining the mathematical distinction between vector and scalar is very important in arriving at the correct answer, so use what you're comfortable with.

### Using Vectors

A vector's direction may be described by an angle, given from a known origin and line of reference, as in Figure 1.1. Since magnitude is always positive, ${\displaystyle -{\textbf {V}}}$  has the same length as ${\displaystyle {\textbf {V}}}$  but in the opposite direction (${\displaystyle 180^{\circ }}$ ).

(a)
(b)
(c)
Figure 1.2: Parallelogram law of combination

Vectors obey the parallelogram law of combination. This law states that the free vectors, ${\displaystyle \mathbf {V_{1}} }$  and ${\displaystyle \mathbf {V_{2}} }$ , can be replaced by ${\displaystyle \mathbf {V} }$ , which is the diagonal of the parallelogram formed by the two free vectors as its sides, as in Figure 1.2a. This is known as the vector sum, which can be expressed mathematically as below.

${\displaystyle \mathbf {V} =\mathbf {V_{1}} +\mathbf {V_{2}} }$


Notationally, the vector sum is indicated by the addition sign between bold-faced vectors. This should not be confused with the scalar sum of the two vectors, ${\displaystyle V_{1}+V_{2}}$ . Because of the geometry of the parallelogram, clearly ${\displaystyle V\neq V_{1}+V_{2}}$ .

Alternatively, the vectors ${\displaystyle \mathbf {V_{1}} }$  and ${\displaystyle \mathbf {V_{2}} }$ , treated again as free vectors, may be summed using the triangle rule, as in Figure 1.2c, to result in the vector sum ${\displaystyle \mathbf {V} }$ . Use the method you feel most comfortable with — they are equivalent. Vector sums obey the commutative property, i.e. ${\displaystyle \mathbf {V_{1}} +\mathbf {V_{2}} =\mathbf {V_{2}} +\mathbf {V_{1}} }$ .

Figure 1.3: Vector subtraction methods

The vector difference of our two example vectors, ${\displaystyle \mathbf {V_{2}} -\mathbf {V_{1}} }$ , can be obtained using either the parallelogram law or the triangle law (see Figure 1.3) by adding ${\displaystyle -\mathbf {V_{1}} }$  and ${\displaystyle \mathbf {V_{2}} }$ , to obtain the vector difference ${\displaystyle \mathbf {V^{\prime }} }$  The equation which represents vector subtraction is as follows:

${\displaystyle \mathbf {V^{\prime }} =\mathbf {V_{2}} -\mathbf {V_{1}} }$


The foregoing vector operations may be applied to more than two vectors by successively adding or subtracting in pairs. The free vectors involved in these equations are called the vector components. To illustrate this, see Figure 1.4a: the vectors ${\displaystyle \mathbf {V_{1}} }$  and ${\displaystyle \mathbf {V_{2}} }$  form the component vectors of ${\displaystyle \mathbf {V} }$  in the direction of ${\displaystyle 1}$  and ${\displaystyle 2}$ , respectively. Typically we use rectangular components to take advantage of convenient trigonometric functions, as shown in Figure 1.4b. As another mathematical convenience, the Cartesian coordinate system is used to describe physical systems, as evidenced use of the ${\displaystyle x}$  and ${\displaystyle y}$  as subscripts and coordinates. Figure 1.4c illustrates the use of an alternate rectangular coordinate system, which may be treated as relative coordinates to ${\displaystyle x}$  and ${\displaystyle y}$ . Relative coordinate systems will become useful when you encounter complex mechanics problems.

(a)
(b)
(c)
Figure 1.4: Parallelogram law of combination

A vector can be separated into its magnitude and direction. Determining the vector's direction can readily be done by using the inverse tangent function as in the equation below; ${\displaystyle \theta }$  is taken to be the direction, when measured from the positive x-axis in the Cartesian coordinate system.

${\displaystyle \theta =\tan ^{-1}{\frac {V_{y}}{V_{x}}}}$


Taking ${\displaystyle V}$  as the vector magnitude, a vector may be described as its magnitude multiplied by its unit vector. The value of ${\displaystyle \mathbf {V} }$  does not change as a result, because the unit vector indicates the proper direction and has a magnitude of one.

${\displaystyle {\textbf {V}}=V{\textbf {n}}}$


In order to describe a vector in 2 or 3 (orthogonal) dimensions, we use the unit vectors, ${\displaystyle \mathbf {i} }$ , ${\displaystyle \mathbf {j} }$ , and ${\displaystyle \mathbf {k} }$  to indicate a vector's direction towards ${\displaystyle x}$ , ${\displaystyle y}$ , and ${\displaystyle z}$ , respectively (as seen in Figure 1.5).

${\displaystyle {\textbf {V}}=V_{x}{\textbf {i}}+V_{y}{\textbf {j}}+V_{z}{\textbf {k}}}$

Figure 1.5

## Newton's Laws

Newton's laws of motion are three physical laws that form the basic laws for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces.

1. Law 1: The velocity of a body remains constant unless the body is acted upon by an external force.
2. Law 2: The acceleration ${\displaystyle {\textbf {a}}}$  of a body is parallel and directly proportional to the vector sum of forces, ${\displaystyle {\textbf {F}}}$ , and inversely proportional to the mass ${\displaystyle m}$ .
3. Law 3: The mutual forces of action and reaction between two bodies are equal, opposite and collinear.

These laws are easily verifiable by accurate physical measurements – as you might have done in high school physics. Newton's second law is most useful in the analysis of dynamics, since it describes a direct mathematical relationship for quantities we can measure (and others we cannot). As applied to a particle mass (having no dimension), ${\displaystyle m}$ , Newton's second law may be stated as

${\displaystyle {\textbf {F}}=m{\textbf {a}}}$  (Eqn 1.1)

where ${\displaystyle {\textbf {F}}}$  is the vector sum of forces acting on the particle mass and ${\displaystyle {\textbf {a}}}$  is the resulting acceleration of the particle mass. Equation 1.1 is a vector equation because the direction of ${\displaystyle {\textbf {F}}}$  must be colinear with that of ${\displaystyle {\textbf {a}}}$ , and the magnitude of the vector sum of forces, ${\displaystyle |{\textbf {F}}|}$ , must equal the other side's, ${\displaystyle |m{\textbf {a}}|}$ .

Restating Newton's first law, a body stays at rest unless acted upon by an unbalanced sum of forces. This statement is a consequence of the velocity remaining the same in this situation (a body with a velocity of ${\displaystyle 0}$  is at rest). Therefore, the first law is of prime importance in our analysis of statics. This law is a consequence of Newton's second law, that is, the acceleration is zero when the force vector sum is zero.

Newton's third law is essential to our solution method of statics problems. When a force is acted on a body, the resulting force on the body is equal, and opposite in sign and direction. This law holds for all forces, whether they are variable or constant, from all sources for all time instants. For systems with gravity present, two bodies contacting each other will have equal and opposite forces unless their tangent of contact is colinear with the acceleration of gravity.

During the analysis of systems of bodies and forces, it is critical to notate the pairs of forces so as to avoid becoming confused. To simplify your analysis, it is best to isolate the system, considering one body with the forces acting on it.

## Units

In mechanics we use four fundamental quantities called dimensions: length, mass, force, and time. The units used to measure these quantities must be consistent with each other in the equations we use, e.g. Newton's second law, Eq. 1.1; that is, you may not mix SI and U.S. Customary units in the same equation. There are a number of unit systems, but this text will use the ones commonly used in science and engineering. See the table below for the four fundamental dimensions and their units and symbols.

QUANTITY DIMENSIONAL SYMBOL SI UNITS U.S. CUSTOMARY UNITS
UNIT SYMBOL UNIT SYMBOL
Mass M Base units ${\displaystyle {\Bigg \{}}$  kilogram kg slug
Length L meter m Base units ${\displaystyle {\Bigg \{}}$  foot ft
Time T second s second sec
Force F newton N pound lb

### SI Units

The International System of Units (abbreviated SI from French: Système international d'unités) is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. SI is the world's most widely used system of measurement, which is used both in everyday commerce and in science. The system has been nearly globally adopted with the United States being the only industrialised nation that does not mainly use the metric system in its commercial and standards activities. As shown in the table, the SI base units are kilogram (kg) for mass, meter (m) for length, and second (s) for time. The unit for force, the newton (N), is derived from the base units by Eq. 1.1.

${\displaystyle {\text{N}}={\text{kg}}\cdot {\text{m}}/{\text{s}}^{2}}$


Thus, 1 newton is the force required to accelerate a 1 kg mass to 1 m/s2.

Consider a mass, ${\displaystyle m}$ , which is free-falling toward the earth. Simply speaking, this mass only has the force of gravity acting on it; therefore it is falling with the acceleration of gravity, ${\displaystyle g}$ . This gravitational force is called the weight ${\displaystyle W}$  of the body, and is found with Eq. 1.1:

${\displaystyle W{\text{ (N)}}=m{\text{ (kg)}}\times g{\text{ (m}}/{\text{s}}^{2}{\text{)}}}$


### U.S. Customary Units

United States customary units, also called the foot-pound-second (FPS) system, are a system of measurements commonly used in the United States. Many U.S. units are virtually identical to their imperial counterparts, but the U.S. customary system developed from English units used in the British Empire before the system of imperial units was standardized in 1824. Several numerical differences from the imperial system are present. Engineers must be able to work in SI and U.S. Customary Units, and both systems are used freely in this text.

Similar to the derivation of the newton from the metric base units, we can derive the U.S. Customary unit of mass, the slug, using Eq. 1.1.

${\displaystyle {\text{slug}}={\frac {{\text{lb}}\cdot {\text{ft}}}{{\text{sec}}^{2}}}}$

Therefore, 1 slug is the mass which is accelerated to 1 ft/sec2 when acted on by a force of 1 lb. The slug may also be expressed in terms of the mass's gravitational force, or weight, using Eq. 1.1:

${\displaystyle m{\text{(slugs)}}={\frac {W{\text{ (lb)}}}{g{\text{ (ft/sec}}^{2}{\text{)}}}}}$

In U.S. units the pound is also used to denote mass, especially to specify thermal properties of liquids and gases. The abbreviations are then lbm for pound-mass and lbf for pound-force. In this text, we will use the pound (lb) only for force. Other useful units of force in engineering and science is the kilopound (kip) which is equal to 1000 lbs, and the ton which is equal to 2000 lbs.

The International System of Units is an absolute system because its base units are not dependent on the environment. Conversely, the U.S. system (FPS) is a gravitational system because its base quantity force is defined as the gravitational force (weight) of a standard mass at sea level, 45° latitude. A standard pound is also the force required to accelerate a one-pound mass to 32.1740 ft/sec2.

### Primary Standards

International standards have been established for the measurements of mass, length, and time.

The international prototype kilogram (IPK) is the kilogram. The IPK, which is roughly the size of a golf ball, is made of a platinum–iridium alloy.

#### Mass

The kilogram (kg) is the base unit of mass in the SI and is defined as being equal to the mass of the International Prototype Kilogram (IPK), which is almost exactly equal to the mass of one liter of water. An accurate copy of the IPK is kept at the National Institute of Standards and Technology (NIST) in the United States. The physical standards do not have constant mass; no plausible mechanism has been proposed to explain either a steady decrease in the mass of the IPK, or an increase in that of its replicas dispersed throughout the world. The long-term solution to this problem, however, is to liberate the SI system’s dependency on the IPK by developing a practical realization of the kilogram that can be reproduced in different laboratories by following a written specification. The international pound, used in both the Imperial system and U.S. customary units, is defined as exactly 0.45359237 kg, making one kilogram approximately equal to 2.2046 pounds.

#### Length

The meter (m) is the base unit of length in the SI. Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole (at sea level), its definition has been periodically refined to reflect growing knowledge of metrology. Since 1983, it has been defined as the length of the path travelled by light in vacuum during a time interval of ${\displaystyle ^{1}/_{299,792,458}}$  of a second.

#### Time

The second (s) is the base unit of time and in SI and other systems. Between 1000 (when al-Biruni used seconds) and 1960 the second was defined as 1/86,400 of a mean solar day (that definition still applies in some astronomical and legal contexts). Between 1960 and 1967, it was defined in terms of the period of the Earth's orbit around the Sun, but it is now defined more precisely in atomic terms. Since 1967, the second has been defined to be: the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

#### Gravity

Standard gravity, or standard acceleration due to free fall, ${\displaystyle g}$ , is the nominal acceleration of an object in a vacuum near the surface of the Earth. The value of ${\displaystyle g}$  is a nominal midrange value on Earth, originally based on the acceleration of a body in free fall at sea level at a geodetic latitude of 45°. Although the actual acceleration of free fall on Earth varies according to location, the above standard figure is always used for metrological purposes. It is defined as

SI Units
${\displaystyle 9.80665{\text{ m/s}}^{2}}$ , or ${\displaystyle \approx 35.30394{\text{ (km/h)/s}}}$ .
U.S. Units
${\displaystyle 32.1740{\text{ ft/s}}^{2}}$  or ${\displaystyle \approx 21.937{\text{ mph/s}}}$ .

For most engineering work, the accuracy of these standards is more than is necessary.

## Law of Gravitation

Often in mechanics we need to calculate the weight of a body. We can compute this force with Newton's law of gravitation:

${\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}}$  (Eqn 1.2)

${\displaystyle {\begin{array}{rcl}{\text{where }}F&=&{\text{the force between the masses}}\\G&=&{\text{the gravitational constant }}(=6.67384\times 10^{-11}{\frac {{\text{ m}}^{3}}{{\text{kg}}\cdot {\text{s}}^{2}}})\\m_{1},m_{2}&=&{\text{two particle masses}}\\r&=&{\text{the distance between the centers of the masses}}\end{array}}}$

Of course, the mutual force obeys Newton's laws of action and reaction. These forces act along ${\displaystyle r}$ , as shown in Fig. 1.7.