1

CMGT 350 FALL 2020Chapter 1 - Fundamental Concepts and PrinciplesReading: Chapter 1, Pgs. 3-31

1-1Introduction to Mechanics

Mechanics - branch of physical sciences that deals with the state of rest or motion of bodies under the action of forces.

Mechanics1. Statics2. Dynamics3. Strength of Materials

StaticsEquilibrium of bodies under the action of balanced forces.

DynamicsMotion of bodies under the action of unbalanced forces.

Strength of MaterialsRelationships 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 sizes of structural members to satisfy strength and deformation requirements are also important topics.

Externally Applied Forces

A B

F2F1

Pin Support Roller Support

Beam

2

AssumptionsStatics and DynamicsAll bodies are assumed to be perfectly rigid.

A rigid body is a solid in which the distance between any two points in the body remain unchanged.

Strength of MaterialsDeformation of structural members becomes very important because the concerns are the strength and stiffness of structural or machine members.

Strength and stiffness are directly or indirectly related to the deformation, even if the deformation is very small.

1-2The Nature of a Force

A force is any affect that may change the state of rest or motion of a body. The existence of a force can be observed by the effects that the force produces.

Force Applied by Direct Contact Force Applied by Remote Action

Rigid Body(Earth Marble)

Not Rigid Body(Earth Foam Stress Ball)

d

<d

d

The distance (d) between the two points shown remains unchanged regardless of the force applied to the surface. The force causes no deformation on the rigid body.

3

F1 = 200 lb

F2 = 150 lb

Characteristics of a ForceA force can be defined completely by:1. Magnitude - number with proper units2. Direction - line of action with an arrowhead3. Point of Application - point at which the force is exerted

1-3Scalar and Vector Quantities

Scalar QuantitiesDescribed completely by a magnitude.

Examples:Value U.S. Customary Units S.I. UnitsLength feet (ft) meter (m)Area ft x ft = ft2 m x m = m2Volume ft x ft x ft = ft3 m x m x m = m3Speed length / time = ft/s = fps length / time = m/sMass slug (lb x s2/ft) Kilogram (kg)Time Sec (s) Sec (s)

Vector QuantitiesCharacterized by its magnitude (scalar), Direction (arrowhead/line of action), and point of application.

Vectors do no add like scalar quantities.Vector quantities must be added geometrically, not algebraically.

Methods Used to Add Vectors:Parallelogram LawTriangle RuleRectangular ComponentsGraphically

Examples of Vector QuantitiesForceMomentDisplacementVelocityAcceleration

100 lb (magnitude)A

Point of Application

Direction (arrowhead)Line of Action

4

Wall

Floor

F = 200 lb

Ladder

Ɵ

A

1-4Types of Forces

Forces can be classified into the following types: Distributed and Concentrated Forces

Distributed force is exerted on a line, over an area, or throughout an entire volume.Example:

Concentrated force is an idealization in which a force is assumed to act at a point.Example:A 200 lb person standing on a ladder at point A

Equivalent concentrated force for a distributed load w is the load intensity (force/length)c = centroid of the shape

b

A B

Uniformly Distributed Load Triangular Distributed Load

ww

b

A B

Uniformly Distributed Load Triangular Distributed Load

F = w x b F = 1/2(w x b)

w

b

w

b

C C

5

External and Internal Forces

External Force - force exerted on the body by another body.

Internal Force - structures formed by several connected components, the forces holding the component parts together are internal forces within the structure.

Example.Truss - basic structure made up of triangles

External Forces to the truss:

P and Q Applied Forces (loads)

AX , Ay , By "Reactions" at the supports

Internal Forces are developed in the truss members due to the applied loads and the reactions. These internal forces are responsible for holding the truss together.

A B

C

D

Q

P

Ax

Ay By

Pin Support Roller Support

6

1-5Types of Force Systems

Concurrent Coplanar Force SystemThe lines of action of all the forces in the system pass through a common point and lie in the same plane.

Nonconcurrent Coplanar Force SystemThe lines of action of all the forces in the system lie in the same plane, but do not pass thorough a common point.

A B

F2F1

RA RB

A

Line of Action

All Lines of Action are Concurrent at Point A

F1F2

F3

7

Concurrent Spatial Force System Nonconcurrent Spatial Force System

1-6Newton's Laws

A particle remains at rest or continues to move along a straight line with constant velocity if the force acting on it is zero.

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, the magnitude of 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 = mawhere 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

The forces of action and reaction between two interactive bodies always have the same magnitudes and opposite directions.

1-7

v

Equilibrium

F2F1

F3

Faccelerated motion

a

F FA BForce of A on B Force of B on A

First Law

Second Law

Third Law

action - reaction

8

The Principle of Transmissibilty

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.

1-8System of Units

U.S. Customary UnitsThree Base Units:Length: foot (ft)Force: pound (lb)Time: second (s)

The base unit pound is dependent on the gravitational attraction of the earth.

The U.S. Units are a Gravitational System

S.I. UnitsThree Base Units:Length: foot (ft)Mass: kilogram (kg)Time: second (s)

The base units are all independent.

The S.I. Units are an Absolute System

The unit of mass, called the slug, is a derived unit:F = m x am = F / aslug = lb/ft/s² = lb x s²/ft

What is the weight of a mass of 1 slug? (on earth)W = m x gW = 1 slug x 32.2 ft/s²W = (1 lb x s²/ft) x 32.2 ft/s²W = 32.2 lb

The unit of force, called the newton (N) is a derived unit expressed in terms of the three base units:F = m x aF = kg x m/s² = 1 Newton (N)

What is the weight of a mass of 1 kg? (on earth)W = m x gW = 1 kg x 9.81 m/s²W = 9.81 kg x m/s² = 9.81 N

F F

9

1-9Unit Conversion

Changing units within a system is called unit reduction.Reduce U.S. Customary unit to U.S. Customary UnitReduce S.I. unit to S.I. Unit

Changing units from one system to another system is called unit conversion.Convert U.S. Customary Unit to S.I. UnitConvert S.I. Unit to U.S. Customary Unit

Examples:Convert a velocity of 30 mph into its equivalent value in m/s.

Reduce v = 55 mi/h (mph) to ft/s (fps)

Unit Conversion WILL NOT BE NEEDED to solve problems in CMGT 350.

10

1-10Consistency of Units in an Equation

Dimensional Analysis � When doing statics and strengths 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 asπr (which is a length) × r (another length).

Examples of DimensionsQuantity Dimension U.S. Units S.I. UnitsVolume L x L X L = L³ ft³ m³Acceleration (velocity / time) L / T² ft / s² m / s²Density (mass / volume) M / L³ slug / ft³ kg / m³Forces (mass x acceleration) M x L / T² slug x ft / s² = lb kg x m /s² = NWork (in 1-D, force x distance) M x L² / T² lb x ft N x mPower (work / time) M x L² / T lb x ft / s N x m / s

Important!When solving statics and strengths 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.

ExampleA 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.

11

1-11Rules for Numerical Computations

Approximate Numbers. Although some numbers that we encounter in engineering computations are exact numbers, most numbers are approximate.

Exact Numbers. Either derived from definition or obtained by counting.

Examples:One hour has exactly 60 minutes by definitionOne inch is defined to equal 25.4 mm exactlyAn automobile has four wheels by counting

Approximate numbers - usually obtained through some kind of measurement.

Examples:The distance between two points on the ground is measured to be 237.7 ftThe voltage of a house current is measured to be 115 volts.

Approximate numbers are usually written with a decimal and often include zeros that serve as placeholders.

Examples7400 20050.0057 0.708

Significant Digits. Except for the zeros used as placeholders, all the other digits in an approximate number are considered significant digits.

Examples:1760.587135030500.00408

Accuracy and PrecisionAccuracy - The accuracy of a number refers to its number of significant digits.Examples:157060.90.0805

Precision - The precision of a number is the decimal position of the last significant digit.Examples:1.35 precise to the nearest hundredths (two decimal places)0.745 precise to the nearest thousandths (three decimal places)

These zeros are placeholders.

These zeros indicate the values of those digits are zero.

Accurate to three significant digits.

Three significant digits each.

12

Rules for Numerical Computations. When calculations are preformed on approximate numbers, the results must be expressed with the proper number of digits.

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.

ExampleThe diameter of a circle measures 2.683 in. Calculate the area of the circle.

A = πr² = π(d/2)² = π d² / 4A = πr² = π(d/2)² = 3.141592654 x (2.683)² / 4

4 Exact number, does not limit the accuracy of the result

π What is the accuracy of your calculator set to?

Calculators Accuracy A = π d² / 4 Rule 13.141592654 (10 significant digits) 5.653680040 in² 5.654 in²3.1415927 (8 significant digits) 5.6536800 in² 5.654 in²3.14 (3 significant digits) 5.65 in² 5.65 in²3.1 (2 significant digits) 5.7 in² 5.7 in²

ExampleA steel plate 1.25 in. thick is coated with a thin layer of paint 0.014 in. thick. Of these two values of thickness, which one has a greater accuracy and which one has a greater precision?

Solution.The number 1.25 has three significant digits, while the number 0.014 has only two significant digits. Therefore, the thickness of the plate, 1.25 in., has a greater accuracy.

On the other hand, the number 1.25 is precise to the nearest hundredths, and the number 0.014 is precise to the nearest thousandths; therefore, the thickness of the paint has a greater precision.

d = 2.683 in

r

A

13

1-12A Brief Review of Mathematics

Mathematics Required for Statics and Strengths:ArithmeticAlgebraGeometryTrigonometry

Right Triangle -Three-sided closed figure that has a right angle (angle equals 90°)

Angles - Capital LettersSides - lowercase letters

Sum of the interior angles of a triangle is 180°

A + B + C = 180°

C = 90°

A + B = 90°

Pythagorean Theorem

c² = a² + b²

Trigonometry of Right Triangles

Sin A = =

Cos A = =

Tan A = =

A

B

C

ac

b

OppositeHypotenuse

ac

AdjacentHypotenuse

bc

OppositeAdjacent

ab

14

Oblique Triangle - None of the interior angles are equal to 90°.

Sum of the interior angles of a triangle is 180°

A + B + C = 180°

Trigonometry of Oblique Triangles

Law of Sines. The ratio of any side of a triangle to the sine function of its opposite angle is a constant:

conversely,

Law of Cosines. The square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the two sides multiplied by the cosine function of the angle between them:

a² = b² + c² - 2bc Cos A

b² = a² + c² - 2ac Cos B

c² = a² + b² - 2ab Cos C

Acute Oblique Triangle Obtuse Oblique Triangle

A

B

C

ac

bA

B

C

ac

b

aSin A

bSin B

cSin C

= =

Sin Aa

Sin Bb

Sin Cc

= =

15

How do you determine which oblique triangle trig identity to use?

Example. For the oblique triangle shown, solve for the lengths of side a and side b and for angle C.

Solution.

A

B

C

a

b

c = 20

ft50°

25°

16

Example. For the oblique triangle shown, solve for the lengths of side a and for angles B & C.

Solution.

A

B

C

a

b = 8.75 ftc =

20 ft

50°

17

Simultaneous Equations

Solving two simultaneous linear equations containing two unknowns sometimes occurs in two-dimensional equilibrium problems.

Method 1. Elimination by Substitution

Method 2: Elimination by Addition and Subtraction

Method 3: Cramer's Rule

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

4x + 1y = 10 (1) 3x - 5y = 19 (2)

Solution.

Methods for Solving Simultaneous Linear Equations

18

Example. Solve the following equations for x and y by using Elimination by Addition & Subtraction.

4x + 1y = 10 (1) 3x - 5y = 19 (2)

Solution.

19

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.

4x + 1y = 10 (1) 3x - 5y = 19 (2)

4 1 x = 103 -5 y 19

Matrix Form

Determinant of a MatrixThe determinant of a matrix is a number.

Determinant of a matrix of order 2:

Det A =a1 b1

= a1b2 - a2b1a2 b2

Cramer's Rule for Two Simultaneous Linear Equations with Two Unknowns

a1x + b1y = k1a2x + b2y = k2

x =D�

y =Dy

D D

where,

D = a1 b1 Dx = k1 b1 Dy = a1 k1a2 b2 k2 b2 a2 k2

20

Example. Solve the following equations for x and y by using Cramer's Rule.

4x + 1y = 10 (1) 3x - 5y = 19 (2)

Solution.

For Cramer's Rule for solving 3 equations and 3 unknowns see example in textbook, Pg. 26-29.

21

1-13General Procedures for Problem Solution

Extensive applications of statics and strength of materials are based on a few simple principles. The most effective way of learning this subject is to solve problems of different levels of complexity. The following general pro cedure is helpful:

1. Read the problem carefully. Identify the given data and the unknown quantities to be determined.2. Make a neat sketch showing all the quantities involved. For some problems, it may be helpful to tabulate the given data and the com puted results.3. Apply the relevant principles and express the physical conditions in mathematical form. The solution must be based on the principles and theorems presented in the text and must be executed in a logi cal manner.4. The equations obtained must be dimensionally homogeneous. Values in consistent units must be used for substitution. The answer obtained must be rounded off to the proper degree of accuracy or precision.5. Use your common sense and judgment to determine if the answer obtained is reasonable. In some problems, there are conditions in which answers can be checked. If such conditions are available, always use them to check the answers.6. The engineering profession requires work that meets high standards. Students preparing to enter an engineering career must present their work in a neat and organized fashion.

1-14Summary

Forces. Mechanics is a physical science that studies the effects of forces. Forces are vector quantities. Vector quantities are characterized by a mag nitude, a point of application, and a direction.

Types of Forces. Forces can be applied on a body by direct contact or through remote action. Forces can be concentrated at a point or distributed along a length, over an area, or throughout the entire body. External forces are exerted on the body by another body. Internal forces are the resisting forces within a body.

Types of Force Systems. Force systems can be classified into the fol lowing three types, depending on whether they are coplanar or spatial, con current or nonconcurrent.1. Concurrent-coplanar force system2. Nonconcurrent-coplanar force system3. Spatial force system

Newton's Three Laws. These three laws form the foundation for the study of Newtonian mechanics. The first law deals with conditions for equi librium of a particle and thereby lays the foundation for the study of stati cs. The second law provides the basic formulation for the study of dynamics. The third law provides the basic understanding for the nature of action and reaction forces.

The Principle of Transmissibility. The point of application of a force may be placed anywhere along the line of action of the force without chang ing the external effects of the force. However, the line of action and the direction of a force must be well defined. For the internal effect or the defor mation of a body, a force acting on the body must have a fixed point of application, and therefore the principle of superposition does not apply.

22

System of Units. Two systems of units are used in this book: the U.S. customary units and the SI units. The base units in the U.S. system are the foot, second, and pound. The base unit for force ( or weight), the pound, is dependent on gravitational attraction; it is therefore a gravitational system. The base units in the SI system are the meter, second, and kilogram. The base unit for mass, the kilogram, is independent of gravitational attraction; it is therefore an absolute system.

Rules for Numerical Computations. Calculated results should always be rounded off according to the following rules:

Rule 1 When approximate numbers are multiplied or divided, the result is expressed to the same accuracy as the least accurate number.

Rule 2 When approximate numbers are added or subtracted, the result is expressed to the same precision as the least precise number.

Mathematics Used in Mechanics. Some fundamental mathematical skills are required of the student. For example, students must be able to perform elementary algebraic manipulations, solve a right triangle using the Pythagorean theorem and trigonometric functions, solve an oblique tri angle using the law of sines and/or the law of cosines, and solve two or three simultaneous linear equations.

General Procedure for Problem Solution. Problems must be solved in a logical and orderly manner. Students must learn to analyze the problem carefully. Make necessary sketches and apply the relevant principles. Equations must be solved by using proper mathematical operations. Results should be checked against certain required conditions or judged to be reasonable using common sense. Work must be presented in a neat and organized fashion.