Statics Outline

8/11/2019 Statics Outline

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Where is a unit vector. There are two cases in determining ; by direction cosines and by the

coordinates of any two points on the line of action of the force.

Given the direction cosines:

Given any two points P1(x1, y1) and P2(x2, y2) on the line of action of the force:

Where

i,j, and kare unit vectors in the direction of x, y and z respectively.

Note:

Also note the following:

Thus,

In simplest term

The above rectangular representation of a force is applicable in both 2D and 3D forces.

Moment of a ForceMoment is the measure of the capacity or ability of the force to produce twisting or turning effect

about an axis. This axis is perpendicular to the plane containing the line of action of the force. The

magnitude of moment is equal to the product of the force and the perpendicular distance from the


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axis to the line of action of the force. The intersection of the plane and the axis is commonly called

the moment center, and the perpendicular distance from the moment center to the line of action of

the force is called moment arm.

From the figure above, O is the moment center and d is the moment arm. The moment M of force F

about point O is equal to the product of F and d.

Couples

Couple is a system of forces whose magnitude of the resultant is zero and yet has a moment sum.

Geometrically, couple is composed of two equal forces that are parallel to each other and acting in

opposite direction. The magnitude of the couple is given by

Where are the two forces and is the moment arm, or the perpendicular distance between the

forces.
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Couple is independent of the moment center, thus, the effect is unchanged in the following

conditions.

The couple is rotated through any angle in its plane.

The couple is shifted to any other position in its plane. The couple is shifted to a parallel plane.

In a case where a system is composed entirely of couples in the same plane or parallel planes,

the resultant is a couple whose magnitude is the algebraic sum of the original couples.

Resultant of Concurrent Force SystemResultant of a force system is a force or a couple that will have the same effect to the body, both in

translation and rotation, if all the forces are removed and replaced by the resultant.

The equation involving the resultant of force system are the following

1.The x-component of the resultant is equal to the summation of forces in the x-direction.

2.The y-component of the resultant is equal to the summation of forces in the y-direction.

3.The z-component of the resultant is equal to the summation of forces in the z-direction.

Note that according to the type of force system, one or two or three of the equations above will be

used in finding the resultant.

Resultant of Coplanar Concurrent Force System

The line of action of each forces in coplanar concurrent force system are on the same plane. All ofthese forces meet at a common point, thus concurrent. In x-y plane, the resultant can be found by

the following formulas:


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Resultant of Spatial Concurrent Force System

Spatial concurrent forces (forces in 3-dimensional space) meet at a common point but do not lie in a

single plane. The resultant can be found as follows:


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Direction Cosines

Vector Notation of the Resultant

Where

Resultant of Parallel Force System

Coplanar Parallel Force System

Parallel forces can be in the same or in opposite directions. The sign of the direction can be chosen

arbitrarily, meaning, taking one direction as positive makes the opposite direction negative. Thecomplete definition of the resultant is according to its magnitude, direction, and line of action.


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Resultant of Distributed Loads

The resultant of a distributed load is equal to the area of the load diagram. It is acting at the centroid

of that area as indicated. The figure below shows the three common distributed loads namely;

rectangular load, triangular load, and trapezoidal load.

Rectangular Load

Triangular Load

Trapezoidal Load

Spatial Parallel Force SystemThe resultant of parallel forces in space will act at the point where it will create equivalent

translational and rotational (moment) effects in the system.


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In vector notation, the resultant of forces are as follows...

Note:

Two parallel forces that are equal in magnitude, opposite in direction, and not colinear will create a

rotation effect. This type of pair is called a Couple. The placement of a couple in the plane is

immaterial, meaning, its rotational effect to the body is not a function of its placement. The

magnitude of the couple is given by

Where F = the magnitude of the two equal opposing forces and d is the perpendicular distance

between these forces.


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Equilibrium of Force SystemThe body is said to be in equilibrium if the resultant of all forces acting on it is zero. There are two

major types of static equilibrium, namely, translational equilibrium and rotational equilibrium.

Formulas

Concurrent force system

Parallel Force System

Non-Concurrent Non-Parallel Force System

In static, a body is said to be in equilibrium when the force system acting upon it has a zero

resultant.

Conditions of Static Equilibrium of Concurrent Forces

The sum of all forces in the x-direction or horizontal is zero.

or

The sum of all forces in the y-direction or vertical is zero.

or

Important Points for Equilibrium Forces

Two forces are in equilibrium if they are equal and oppositely directed.

Three coplanar forces in equilibrium are concurrent. Three or more concurrent forces in equilibrium form a close polygon when connected in

head-to-tail manner.


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Equilibrium of Parallel Force System

Conditions for Equilibrium of Parallel Forces

The sum of all the forces is zero.

The sum of moment at any point O is zero.

Equilibrium of Non-Concurrent Force System

There are three equilibrium conditions that can be used for non-concurrent, non-parallel force

system.

The sum of all forces in the x-direction or horizontal is zero.

or

The sum of all forces in the y-direction or vertical is zero.

or

The sum of moment at any point O is zero.

The three equilibrium conditions can solved up to three unknowns in the system. If the system

involves more than three unknowns, it is called indeterminate. Indeterminate structures are beyond

the scope of Engineering Mechanics, it is one of the topics inStrength of Materials andTheory of

Structures.

Analysis of Structures

There are many kinds of structure. This section will limit to those that are pin-connected. Two types

of pin-connected structures will be covered here;pin-connected trussesandpin-connected frames.

In the actual structure, the joints may be welded, riveted, or bolted to a gusset plate at the joint.However as long as the center-line of the member coincide at the joint, the assumption of a pinned

joint maybe used.

Analysis of Simple Trusses

An ideal truss is a structure which is composed completely of axial members that are assumed to be

weightless. Members are connected by pinned joints, forming triangular substructures within the
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main structure and with the external loads applied only at the joints.

In real trusses, of course, the members have weight, but it is often much less than the applied load

and may be neglected with little error. Sometimes, the weight maybe included by dividing the weightin half and allowing half the weight to act at each end of the member.

Our primary interest is to know the forces acting in the bars and upon the pins of the structure. Each

member of the truss is either in tension or compression. A member in tension causes forces which

pull away from its end joints whereas a member in compression causes forces which push towards

the end joints.

Method of Joints | Analysis of Simple Trusses

Method of Joints

The free-body diagram of any joint is aconcurrent force system in which the summation of moment

will be of no help. Recall that only two equilibrium equations can be written

and

This means that to solve completely for the forces acting on a joint, we must select a joint with no

more than two unknown forces involved. This can be started by selecting a joint acted on by only two

members. We can assume any unknown member to be either tension or compression. If negative

value is obtained, this means that the force is opposite in action to that of the assumed direction.

Once the forces in one joint are determined, their effects on adjacent joints are known. We thencontinue solving on successive joints until all members have been found.

Method of Sections | Analysis of Simple Trusses

Method of Sections

In this method, we will cut the truss into two sections by passing a cutting plane through the

members whose internal forces we wish to determine. This method permits us to solve directly any
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member by analyzing the left or the right section of the cutting plane. To remain each section in

equilibrium, the cut members will be replaced by forces equivalent to the internal load transmitted to

the members. Each section may constitute ofnon-concurrent force system from which three

equilibrium equations can be written.

, , and

Because we can only solve up to three unknowns, it is important not to cut more than three

members of the truss. Depending on the type of truss and which members to solve, one may have to

repeat Method of Sections more than once to determine all the desired forces.

Method of Members | Frames Containing Three-Force

Members

A three-force member is in general a non-axial memberthat is not simply in tension or compression.A member of this kind has shear forces perpendicular to the member and subjected to bending

loads. If forces are applied to more than two positions on the member, it is three-force member. Any

beam is a three-force member according to the above definition.

Frames are pin-connected structures with some or all members are three-force members. To

analyze a frame, we can disconnect the three-force member from the structure and draw the free-

body diagram of the member. This approach is called the method of members.

In this method, three equilibrium equations can be written

, , and

Below is a figure that shows the difference between axial and non-axial (three-force) members.
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FrictionFriction is the contact resistance exerted by one body when the second body moves or tends to

move past the first body. Friction is a retarding force that always acts opposite to the motion or to the

tendency to move.

Types of Friction

Dry Friction

Dry friction, also called Coulomb friction, occurs when unlubricated surfaces of two solids are in

contact and slide or tend to slide from each other. If lubricant separates these two surfaces, the

friction created is called lubricated friction. This section will deal only with dry friction.

Fluid Friction

Fluid friction occurs when layers of two viscous fluids moves at different velocities. The relative

velocity between layers causes frictional forces between fluid elements, thus, no fluid friction occurs

when there is no relative velocity.

Skin friction

Skin friction also called friction drag is a component of the force resisting the motion of a solid body

through a fluid.

Internal Friction

Internal friction is associated with shear deformation of the solid materials subjected to cyclical

loading. As deformation undergo during loading, internal friction may accompany this deformation.

Elements of Dry Friction

= Total reaction perpendicular to the contact surface= Friction force

= Coefficient of friction

= Resultant of f and N

= angle of friction


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Formulas for dry friction

Consider the block shown to the right that weighs . It is placed

upon a plane that inclined at an angle with the horizontal.

If the maximum available friction force is less

than thus, the block will slide down the plane.

If the friction force will just equate to thus,

the block is in impending motion down the plane.

If the maximum available frictional resistance is

greater than thus, the block is stationary.

We can therefore conclude that the maximum angle that a plane may be inclined without

causing the body to slide down is equal to the angle of friction .

Centroids and Centers of Gravity

Centroids of Composite Figures

Center of gravity of a homogeneous flat plate

Centroids of areas

Centroids of lines

Center of Gravity of Bodies and Centroids of Volumes

Center of gravity of bodies


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Centroids of volumes

Centroids Determined by Integration

Centroid of area

Centroid of lines

Center of gravity of bodies

Centroids of volumes


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Centroids of Common Geometric Shapes

Rectangle Area and Centroid

Triangle Area and Centroid

Circle Area and Centroid

Semicircle Area and Centroid


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Semicircular Arc Length and Centroid

Quarter Circle Area and Centroid

Sector of a Circle Area and Centroid


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Circular Arc Length and Centroid

Ellipse Area and Centroid

Half Ellipse Area and Centroid


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Quarter Ellipse Area and Centroid

Parabolic Segment Area and Centroid

Spandrel Area and Centroid

Statics Outline

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