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CMGT 340 STATICS Chapter 1

Fundamental Concepts and Principles

Mechanics State of rest or motion of bodies under the action of forces

Statics Equilibrium of bodies under the action of balanced forces

Dynamics Motion of bodies under the action of unbalanced forces

Strength of Material Relationship among the external forces applied to the bodies, the resulting stresses

(intensity of internal forces), and deformation (change of size or shape). The

determination of the proper size and strength of structural members to satisfy

strength and deformation requirements are also important topics.

Statics

All bodies are assumed to be perfectly rigid.

A rigid body is a solid in which the distance between any two points remains the same.

The Nature of a Force

Any effect that may change the state of rest or motion of a body

Represents the action of one body on another

Applied by either direct physical contact between bodies or by remote action (gravitational, electrical,

magnetic)

Characteristics of a Force

A force can be completely defined by a Magnitude, a Direction, and a Point of Application

Magnitude Direction Point of Application

Scalar Value (Magnitude)

Number

U.S. units of pound (lb)

S.I. units of Newton (N)

Vector Value (Direction)

Angle, Ө (Degrees)

Direction indicated by a line with an

ARROWHEAD

Line of Action

Point at which the force is

exerted

100 lb (Magnitude)

Line of Action

Direction

Point of Application

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Scalar and Vector Quantities

Scalar Quantities:

Length

Area

Volume

Speed

Mass

Time

Added algebraically

Vector Quantities:

Force

Moment

Displacement

Velocity

Acceleration

Vector Quantities Have:

Magnitude

Direction

Line of Action

Point of Application (sometimes)

Added geometrically

Types of Forces

Distributed Concentrated

External and Internal Forces

The applied forces, F1, F2, and F3, and the reactions at the supports, Ax, Ay, and Ey are external forces

acting on the truss.

The internal forces are developed in the truss members due to the applied loads and reactions. These

forces hold the truss together.

F

A

B C

D

E Ax

Ay Ey

F1 F2

F3

W = F/L

L

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Types of Force Systems

Concurrent Coplanar Force System – The lines of action of all of the forces pass through a common

point and lie in the same plane.

Nonconcurrent Coplanar Force System – The lines of action of all the forces in the system lie in the

same plane but do not pass through a common point.

W

P

R

A B

RA

P Q

RB

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Spatial Force System – The lines of action of all of the forces in the system do not lie in the same

plane. Spatial force systems can be concurrent or nonconcurrent.

Concurrent Spatial Force System Nonconcurrent Spatial Force System

Newton’s Laws

First Law – A particle remains at rest or continues to move along a straight line with a constant velocity

if the force acting on it is zero.

Second Law – If the force acting on a particle is not zero, the particle accelerates (changes velocity with

respect to time) in the direction of the force, and the magnitude of the acceleration (the rate of change of

velocity per unit time) is proportional to the magnitude of the force.

If F is applied to a particle of mass, m, this law may be expressed mathematically as:

F = ma

where F = the force acting on the particle

m = mass of the particle

a = the acceleration of the particle caused by the force

The second law is the basis for the study of DYNAMICS

Third Law – The forces of action and reaction between interactive bodies always have the same

magnitudes and opposite directions.

F1 F2

F3

v Equilibrium

F a

Accelerated Motion

F F

Force of A on B

A B Force of B on A

Action - Reaction

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The Principle of Transmissibility

The point of application of a force acting on a rigid body may be placed anywhere along its line of

action without altering the conditions of equilibrium or motion of the rigid body.

Note: The internal effect of a force on a body is dependent on its point of application. The principle of

transmissibility does not apply if our concern is the internal force or deformation in the part labeled BC.

Systems of Units

U.S. Customary Units

Three base units:

Length: foot (ft)

Force: pound (lb)

Time: second (s)

SI Units

Three base units:

Length: meter (m)

Mass: kilogram (kg)

Time: second (s)

m

= F/a F = ma

1 slug = (1 lb)/(1 ft/s2) = 1 lb · s2/ft 1 N = (1 kg) · (m/s2) = 1 kg · m/s2

W = mg W = mg

= 1 slug x 32.2 ft/s2 = (1 kg) (9.81 m/s2)

= 1 lb · s2/ft x 32.2 ft/s2 = 9.81 kg · m/s2

= 32.2 lb = 9.81 N

Unit Conversion

Unit reduction – changing units within a system

Unit conversion – changing units from one system to another

Useful Conversion Factors:

1 ft = 0.3048 m

1 slug = 14.59 kg

1 lb = 4.448 N

F F

F

A A

F

B B

C

See Table 1-2, pg. 12 (textbook)

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Dimensional Analysis

■ When doing statics problems, you'll often be required to determine the numerical value and the units

of a variable in an equation.

■ A useful method for determining the units of a variable in an equation.

■ Another use of dimensional analysis is in checking the correctness of an equation which you have

derived after some algebraic manipulation.

Note:

Most physical quantities can be expressed in terms of combinations of five basic dimensions.

These are: mass (M), length (L), time (T), electrical current (I), and temperature (K). These five

dimensions have been chosen as being basic because they are easy to measure in experiments.

Dimensions aren't the same as units. For example, the physical quantity, speed, may be measured in

units of meters per second, miles per hour etc.; but regardless of the units used, speed is always a length

divided by a time, so we say that the dimensions of speed are length divided by time, or simply L/T.

Similarly, the dimensions of area are L2 since area can always be calculated as a length times a length.

For example, although the area of a circle is conventionally written as πr2, we could write it as

πr (which is a length) × r (another length).

Examples of Dimensions:

Quantity Dimensions Corresponding US

Units

Corresponding SI

Units

Volume L x L x L = L3 ft3 m3 (cubic meters)

Acceleration (velocity/time) L / T2 ft/s2 m/s2

Density (mass/volume) M / L3 slug/ft3 kg/m3

Force (mass x acceleration) M x L / T2 slug ft/s2 = lb kg · m/s2 = N

Work (in 1-D, force × distance) M x L2 / T2 lb · ft N · m

Power (work / time) M x L2 / T lb · ft/s N · m/s

Consistency of Units in an Equation

When solving statics problems students should form the habit of carrying units with all quantities when

substituting into an equation and making sure that the result is in the correct units.

Example 1.

A car is traveling from Chico, Ca to Sacramento, Ca a total distance of 90 miles. If the trip took 75

minutes, what was the vehicles speed in mph?

Solution.

90 mi

75 min

60 min

1 hr =

72 mi

hr Speed of vehicle = x

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Example 2.

Solve the equation mv = Ft for the velocity v and find its value if the force F = 2.56 N, the time

t = 16.8 s, and the mass m = 1.89 kg.

Solution.

Rules for Numerical Computations

Accuracy = number of significant digits

Precision = decimal position of the least significant digit i.e. 1.35 nearest hundredth

Rule 1 When approximate numbers are multiplied or divided, the result is expressed with the

same accuracy as the least accurate number.

Rule 2 When approximate numbers are added or subtracted, the result is expressed with the same

precision as the least precise number.

Units of Measure

Whatever you measure, you have to use Units.

Units are a critical part of EVERY equation.

How tall am I?

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5 inches

5 feet

5 meters

5 kilometers

5 furlongs

All are units of length

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Example of typical units:

Time Angles Force

seconds degrees Pounds (lb)

minutes radians Kip

hours Newton (N)

days

weeks

months

years

eons

Question: If a 72-inch man stands on a 6-foot scaffolding, how tall are they together?

Answer:

X = 72 + 6

X = 72 inches + 6 feet

Why not the first equation?

No idea if the numbers are in the same units or even what the units are.

Why not the second equation?

The two figures are given in different units.

In order to add the two numbers, you have to convert one of them to the units of the other.

Then, when you have the two numbers in the same units, you can add them.

In order to perform this conversion, you need a conversion factor.

If you want the final result to end up in the units of feet, then you will need to convert the man’s height

in inches to feet.

i.e. How many inches make up a foot?

Example 1. Convert 55 mi/hr to m/s

Solution.

Imagine trying to do this problem without dimensional analysis!

How could you possibly catch a mistake?

Conclusion: Units are a critical part of describing every measurement.

Which equation is clearer?

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Example 2. The specific weight (weight per unit volume) of concrete is 150 lb/ft3. What is its equivalent

value in KN/m3?

Solution.

Example 3. Convert the force exerted by a 112 lb person to SI units.

Solution.

Example 4. Convert a moment of force of 12 KN-m to U.S. customary units of kip-ft.

Solution.

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General Mathematical Principles

The mathematical concepts provided by this review should be thoroughly mastered for success in

solving problems in mechanics.

Right Triangles - Three-sided closed figure that has a right angle (angle equals 90º)

Sum of the interior angles of a triangle is 180º

A + B + 90º = 180º

Pythagorean Theorem

Trigonometry of Right Triangles

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Oblique Triangle – none of the interior angles is equal to 90º

Sum of the interior angles of a triangle is 180º

A + B + C = 180º

Trigonometry of Oblique Triangles

Law of Sines

Law of Cosines

Law of Tangents

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Methods for Solving Simultaneous Equations:

1. Method of Elimination by Substitution

2. Method of Elimination by Addition and Subtraction

3. Cramer’s Rule for Two Equations

Example 1.

Solve the following equations for x and y using the Method of Elimination by Substitution.

-15x + 21y = 12

-02x + 03y = 17

Solution.

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Example 2.

Solve the following equations for x and y using the Method of Elimination by Addition and

Subtraction.

-15x + 21y = 12

-02x + 03y = 17

Solution.

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Matrix Math

Matrix – array of numbers arranged in row and column format.

Systems of equations of any order can be expressed in Matrix format.

Square Matrix – Number of rows equals the number of columns.

-15x + 21y = 12

-02x + 03y = 17

-15 21 x =

12

-2 3 y 17

Matrix Form

Determinant of a Matrix

The determinant of a matrix is a number.

Determinant of a matrix of order 2:

A = a1 b1

a2 b2

Cramer’s Rule for Two Equations

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Example 3.

Solve the following equations for x and y using the Cramer’s Rule.

-15x + 21y = 12

-2x + 3y = 17

Solution.