Trusses

• Consist entirely of straight two-force members, connected at joints
• Example on board: decompose truss into two-force members and joints, and show how forces meet at joints

Trusses: Method of Joints:-

• Typically used to find forces in all or several of the members
• Each joint is a particle
• Particle equilibrium in 2D:

• For each joint, we have 2 equations, therefore, we can solve for 2 unknowns
• Must start process at a joint with only 2 unknown forces
• Find a joint with only two unknown forces

-First, may need to draw an FBD of the entire truss and find support reactions

• Draw FBD of selected joint

-Draw each force along the direction of the member
-Draw in tension (away from joint)
-Find angle of force from truss geometry
-Resolve angled forces into x, y components

• Apply equilibrium, to solve for 2 unknown forces

A + sign: the force is tension (T)

A – sign: the force is compression (C)

• Find the next joint that has only 2 unknown forces and repeat the process

–Typically this is adjacent to the prior joint

• Repeat with additional joints until all member forces are known.
• Remember to specify (T) or (C) for each force!

Analysis of Structures

•Trusses

–Designed to support loads

–Consist entirely of two-force members

•Frames

–Designed to support loads

–Include one or more multi-force members

•Machines

–Designed to transmit and/or modify forces

–Include one or more multi-force members

Two-Force Members:-

• Pinned at both ends (both joints)
• No applied forces between joints
• No applied moments
• Line of action of forces is directed along a line drawn between the two joints 

Vectors

The vectors can be solved by

1. Law of sine and law of cosines (two forces)
2. Graphically
3. Equilibrium

• Table
• Sum of values

Types of Forces(Loads)

1.Point loads - concentrated forces exerted at point or location

2.Distributed loads - a force applied along a length or over an area.  The distribution can be uniform or non-uniform.

Resultant Forces-

If two forces P and Q acting on a particle A may be replaced by a single force R, which has the same effect on the particle.

Resultant Forces-

• This force is called the resultant of the forces P and Q and may be obtained by constructing a parallelogram, using P and Q as two sides of the parallelogram.  The diagonal that pass through A represents the resultant.
• This is known as the parallelogram law for the addition of two forces.  This law is based on experimental evidence,; it can not be proved or derived mathematically. 
• For multiple forces action on a point, the forces can be broken into the components of x and y.
  

Principle of Transmissibility

The principle of transmissibility states that the condition of equilibrium or of motion of a rigid body will remain unchanged if a force F action at a given point of the rigid body is replace by a force F’ of the same magnitude and the same direction, but acting at a different point, provided that the two forces have the same line of action.

Line of action

Scalar Quantity,Vector Quantities

Scalar Quantity has magnitude only (not direction) and can be indicated by a point on a scale. Examples are temperature, mass, time and dollars.

Vector Quantities have magnitude and direction.  Examples are wind velocity, distance between to points on a map and forces.

Collinear : If several forces lie along the same line-of –action, they are said to be collinear.

Coplanar When all forces acting on a body are in the same plane, the forces are coplanar.

Type of Vectors

Free Vector - is vector which may be freely moved creating couples in space.

Sliding Vector - forces action on a rigid body  are represented by vectors which may move or slid along their line of action.

Bound  Vector or Fixed Vector - can not be moved without modifying the conditions of the problem.

Addition of Couples

• Consider two intersecting planes P1 andP2 with each containing a couple
  

•  Resultants of the vectors also form a couple
  

•  By Varigon’s theorem
  

• Sum of two couples is also a couple that is equalto the vector sum of the two couples

Moment of a Couple

• Two forces F and -F having the same magnitude, parallel lines of action, and opposite sense are said to form a couple.
•  Moment of the couple,
  

•  The moment vector of the couple is independent of the choice of the origin of the coordinate axes, i.e., it is a free vector that can be applied at any point with the same effect.
• Two couples will have equal moments if
  

•  the two couples lie in parallel planes, and
• the two couples have the same sense or the tendency to cause rotation in the same direction.

Concurrent Force Systems

A concurrent force system contains forces whose lines-of action meet at some one point.

Forces may be tensile (pulling).

Forces may be compressive (pushing)

Force exerted on a body has two effects:

•    The external effect, which is tendency to change the motion of the body or to develop resisting forces in the body
•    The internal effect, which is the tendency to deform the body.
• If the force system acting on a body produces no external effect, the forces are said to be in balance and the body experience no change in motion is said to be in equilibrium. The process of reducing a force system to a simpler equivalent stem is called a reduction. The process of expanding a force or a force system into a less simple equivalent system is called a resolution.
• A force is a vector quantity that, when applied to some rigid body, has a tendency to produce translation (movement in a straight line) or translation and rotation of body.  When problems are given, a force may also be referred to as a load or weight.
  Characteristics of force are the: magnitude,direction(orientation) and point of application.

The Laws of Dry Friction. Coefficients of Friction

• Block of weight W placed on horizontal surface. Forces acting on block are its weight and reaction of surface N.
•  Small horizontal force P applied to block. For block to remain stationary, in equilibrium, a horizontal component F of the surface reaction is required. F is a static-friction force.
•  As P increases, the static-friction force F increases as well until it reaches a maximum value Fm.
  

•  Further increase in P causes the block to begin to move as F drops to a smaller kinetic-friction
  

Friction

• In preceding lectures, it was assumed that surfaces in contact were either frictionless (surfaces could move freely with respect to each other) or rough (tangential forces prevent relative motion between surfaces).
• Actually, no perfectly frictionless surface exists. For two surfaces in contact, tangential forces, called friction forces, will develop if one attempts to move one relative to the other.
•  However, the friction forces are limited in magnitude and will not prevent motion if sufficiently large forces are applied.
• The distinction between frictionless and rough is, therefore, a matter of degree.
• There are two types of friction: dry or Coulomb friction and fluid friction. Fluid friction applies to lubricated mechanisms. The present discussion is limited to dry friction between nonlubricated surfaces.

Addition of Forces by Summing Components 2

• Wish to find the resultant of 3 or more concurrent forces
  
• Resolve each force into rectangular components
  
• The scalar components of the resultant are equal to the sum of the corresponding scalar components of the given forces

• To find the resultant magnitude and direction
  

MECHANICS

Introduction

The objective for the current chapter is to investigate the effects of forces on particles:
- replacing multiple forces acting on a particle with a single equivalent or resultant force
 - relations between forces acting on a particle that is in a state of equilibrium

The focus on particles does not imply a restriction to miniscule bodies. Rather, the study is restricted to analyses in which the size and shape of the bodies is not significant so that all forces may be assumed to be applied at a single point.

Resultant of Two Forces

• force: action of one body on another; characterized by its point of application, magnitude, line of action, and sense.
• Experimental evidence shows that the combined effect of two forces may be represented by a single resultant force.
• The resultant is equivalent to the diagonal of a parallelogram which contains the two forces in adjacent legs.
• Force is a vector quantity.

Vectors

• Vector: parameters possessing magnitude and direction which add according to the parallelogram law. Examples: displacements, velocities, accelerations.
• Scalar: parameters possessing magnitude but not direction. Examples: mass, volume, temperature
• Vector classifications:

1. Fixed or bound vectors have well defined points of application that cannot be changed without affecting an analysis.
2. Free vectors may be freely moved in space without changing their effect on an analysis.
3. Sliding vectors may be applied anywhere along their line of action without affecting an analysis.

• Equal vectors have the same magnitude and direction.
• Negative vector of a given vector has the same magnitude and the opposite direction.

Rectangular Components of a Force: Unit Vectors

• May resolve a force vector into perpendicular components so that the resulting parallelogram is a rectangle.Fx & Fy are referred to as rectangular vector components                                                

• Vector components may be expressed as products of the unit vectors with the scalar magnitudes of the

vector components