Statics Notes 2013 [Force Systems]Dr. Sameer Shadeed 1 Statics-61110 Dr. Sameer Shadeed An-Najah National University College of Engineering Chapter [2]

Statics Notes 2013 [Force Systems] Dr. Sameer Shadeed1

Statics-61110

Dr. Sameer Shadeed

An-Najah National UniversityCollege of Engineering

Chapter [2]

Force Systems


Chapter Objectives

Students will be able to:

1. Resolve a 2D and 3D vector into components

2. Work with 2D and 3D vectors using Cartesian vector notations

3. Estimate the resultant of forces and couples in 2D and 3D


There are four concurrent cable forces acting on the bracket

How do you determine the resultant force acting on the bracket ?

Applications


Given the forces in the cables, how will you determine the resultant force acting at D, the top of the tower?

Applications


Scalars and Vectors

Scalars: A mathematical quantity

possessing magnitude only (e.g. area,

volume, mass, energy)

Vectors: A mathematical quantity

possessing magnitude and direction (e.g.

forces, velocity, displacement)


Types of Vectors

A Free vector: is a vector whose action is not confined to or associated with a unique line in space.

A Sliding vector: is a vector which has a unique line of action in space but not a unique point of application.

A Fixed vector: is a vector for which a unique point of application is specified and thus cannot be moved without modifying the conditions of the problem.


Working with Vectors

For two vectors to be equal they must have the same:

1. Magnitude

2. Direction

V V

But they do not need to have the same point of application

A negative vector of a given vector has same magnitude but opposite direction

V and -V are equal and opposite V + (-V) = 0

V -V


Vector Operations

Product of a scalar and a vector

V + V + V = 3V (the number 3 is a scalar)

This is a vector in the same direction as V but 3 times as long

(+n)V = vector same direction as V, n times as long

(-n)V = vector opposite direction as V, n times as long


Vector Addition

The sum of two vectors can be obtained by attaching the two vectors to the same point and constructing a parallelogram (Parallelogram law)

As vector: V = V1+ V2

Addition of vectors is commutative: V1 + V2 = V2 + V1

The sum of three vector (Parallelogram law) V1

V2

V3

RR1

R = R1 + V3 = V1 + V2 + V3

R1 = V1 + V2

Vector addition is associative: V1 + V2 + V3 = (V1 + V2 ) + V3

= V1 + (V2 + V3 )

As scalar: V ≠ V1+ V2


Vector Subtraction

Vector Subtraction: is the addition of the corresponding negative vector


Two-Dimensional Force System

Rectangular Components: is the most common two-dimensional resolution of a force vector

where Fx and Fy are the vector components of F in the x- and y-directions

jFiFF yx

yx FFF

In terms of the unit vectors i and j, Fx = Fxi and Fy = Fyj

where the scalers Fx and Fy are the x and y scaler components of the vector F


The scalar components can be positive or negative, depending on the quadrant into which F points

The magnitude and direction of F is expressed by:

Two-Dimensional Force System


Determining the Components of a Force

Dimensions are not always given in horizontal and vertical directions

Angles need not be measured counterclockwise from the x-axis

The origin of coordinates need not be on the line of action of a force

Therefore, it is essential that we be able to determine the correct components of a force no matter how the axes are originated or how angles are measured


The term means “the algebric sum of x-scalar components”

Resultant of Two Concurrent Forces

From which we can conclude that:

or


Example 1


Example 1 (Solution)




Example 2






Example 3




Example 4








Moment (Moment about a Point)

Moment: is the measure of the tendency of the force to make a rigid body rotate about a point or fixed axis

The magnitude of the moment M is proportional both to the magnitude of the force F and the moment arm d

The basic units of the moment in SI units are newton-meters (N.m), and in the U.S. Custamary system are pound-feet (Ib.ft)


Moment direction: a plus sign (+) for counterclockwise moments and a minus sign (−) for clockwise moments, or vice versa

The moment is a vector M perpendicular to the plane of the body

The sense of M depends on the direction in which F tends to rotate the body

The right hand rule is used to identify this sense

Moment (Moment about a Point)


where r is a posision vector which runs from the moment reference point A to any point on the line of action of F

The moment of F about point A may be represented by the cross-product expression

Moment (The Cross Product)

The order of r × F of the vectors must be maintained because F × r would produce a vector with a sense opposite to that of M

F × r = -M


The magnitude of the moment M is gevin by:

M = F (r sin α)

From the shown diagram:

M = Fd

r sin α = d

Moment (The Cross Product)


Moment (Varignon’s Theorem)

The moment of a force about any point is equal to the sum of the moments of the components of the force about the same point

R = P + QMo = r × R

Mo = r × P + r × Q

Mo = Rd = Pp − Qq


Example 5








Example 6

Find:

1. Moment of the 100 N force about O

2. Magnitude of a horizontal force applied at A which create the same moment about O

3. The smallest force applied at A which creates the same moment about O

Given:



1. Moment of the 100 N force about O

Mo = Fd = 100(5 cos60o) = 250 N.m

2. Magnitude of a horizontal force applied at A which create the same moment about O

Mo = 250 = PL L = 5 sin 60o = 4.33 m

P = 250/4.33 = 57.7 N



3. The smallest force applied at A which creates the same moment about O

Mo = Pd = P × 5 = 250

P is smallest when d in M = Fd is a maximum

This occurs when P is perpendicular to the lever

P = 250/5 = 50 N


Couple

Couple: is the moment produced by two equal, opposite, and non-collinear forces

The combined moment of the two forces about axis normal to their plane and passing through any point such as O in their plane is the couple M

M = Fd

M = F(a + d) - Fa


Couple (Vector Algebra Method)

With the cross-product notation, the combined moment about point O of the forces forming the couple is

where rA and rB are position vectors which run from point O to arbitrary points A and B on the lines of action of F and –F

M = r × F

M = rA × F + rB × (–F) = (rA – rB) × F

Because rA – rB = r, M can be express as:


The Sense of a Couple Vector

The sense of a couple vector M can represent as clockwise or counterclockwise by one of the shown conventions


A couple is not affected if the forces act in a different but parallel plane

Equivalent Couple

Changing the values of F and d does not change a given couple as long the product Fd remains the same


The combination of the force and couple in the

shown figure is refered to as a force – couple

system

Force – Couple System


Example 7




Example 8




Resultants

The Resultant of a system of forces is the simplest force combination which can replace the original forces without altering the external effect on the rigid body to which the forces are applied


Resultants

Algebraic Method


Resultants

Algebraic Method


Resultants

Principle of Moments: This process is summarized in equation form by

If the resultant force R for a given force system is zero, the resultant of the systam need not be zero because the resultant may be a couple

The resultant force in the shown figure have a zero resultant force but have a resultant clockwise couple M = F3d


Example 9










Three Dimensional Force System

The Force F acting at point O in the shown figure has the rectangular components Fx, Fy and Fz where

Rectangular Components



The above equation may write as

nF is a unit vector which characterizes the direction of F


If the coordinates of points A and B of the shown figure are known, the force F may be written as

(a) Specification by two points on the line of action of the force

Thus the x, y, and z scalar components of F are the scalar coefficients of the unit vectors i, j, and k, respectively



Consider the geometry of the shown figure and assuming that the angles θ and Φ are known. First Resolve F into horizantal and vertical components

(b) Specification by two angles which orient the line of action of the force

Then resolve the horizontal components Fxy into x- and y-components

Fx, Fy, and Fz are the desired scalar components of F



The dot product of two vectors P and Q is defined as the product of their magnitudes times the cosine of the angle α between them

Dot Product

If n is a unit vector in a specified direction, the projection of F in the n-direction has the magnitude Fn = F.n

The projection of F in the n-direction can be expressed as a vector quantity by multiplying its scalar component (F.n) with the unit vector n to give Fn = (F.n)n = F.nn




Dot Product


Angle between two Vectors



Example 10











Moment and Couple

Moments in 3D can be calculated using scalar (2D) approach but it can be difficult and time consuming

Thus, it is often easier to use a mathematical approach called the vector cross product

Using the vector cross product, MO = r F

Here r is the position vector from point O to any point on the line of action of F


Cross Product

In general, the cross product of two vectors A and B results in another vector C , i.e., C = A B

The magnitude and direction of the resulting vector can be written as: C = A B = A B sin

The right hand rule is a useful tool for determining the direction of the vector resulting from a cross product


Evaluating the Cross Product

Each component can be determined using 22 determinants

Of even more utility, the cross product can be written as


Cartesian Vector Formulation

jki

kji

ii

ˆˆˆ

ˆˆˆ

0ˆˆ

ikj

jj

kij

ˆˆˆ

0ˆˆ

ˆˆˆ

0ˆˆ

ˆˆˆ

ˆˆˆ

kk

ijk

jik

Here is a simple way of remembering this

What is ?

ji ˆˆ

ji ˆˆ

k̂Magnitude: = (1)(1)sin90o = 1

Direction:


Moment In 3D

By expanding the above equation using 22 determinants, we get

So, the cross-product expression for the moment MO may be written in the determinant form as:

MO = (ryFZ − rZ Fy)i − (rxFz − rzFx )j + (rxFy − ryFx )k


In the shown figure, the three components of a force F acting at a point located relative to O by the vector r is illustrated

The scalar magnitude of the moments of these forces about the positive x-, y-, and z-axes through O can be obtained from the moment-arm rule, and are

Which agree with the relative terms in the determinant expresion for the cross product r × F

Moment In 3D


Moment about an Arbitrary Axis

Our goal is to find the moment of F about any axis λ through O

First compute the moment of F about any arbitrary point O that lies on the λ axis using the cross product

MO = r F

Now, find the component of MO along the axis λ using the dot product

Mλ = MO •n

Here n is a unit vector in the λ-direction


To obtain the vector expression for the moment Mλ of F about λ, multiply the magnitude by the direction unit vector n to obtain

The expression r × F•n is known as a triple scalar product

The triple scalar product may be represented by the determinant

where α, β, and γ are the direction cosines of the unit vector n

Moment about an Arbitrary Axis


Varignon’s Theorem in 3D In the shown figure, a system of concurrent forces F1, F2, F3, .....

The sum of the moments about O of these forces is

Mo = ∑ (r × F) = r × R

Using the symbol Mo to represent the sum of the

moments on the left side of the a bove equation, we have


Couples in 3D

A couple is defined as two parallel forces with the same magnitude but opposite in direction separated by a perpendicular distance d

where rA and rB are position vectors which run from point O to to arbitrary points A and B on the lines of action of F and –F

M = r × F

M = rA × F + rB × (–F) = (rA – rB) × F

Because rA – rB = r, M can be express as

With the cross-product notation, the combined moment about point O of the forces forming the couple is


Couples in 3D

The moment of a couple is free vector (The moment of a couple is the same about all points)

whereas the moment of a force about a point (which is also the same about a defined axis through the point) is a sliding vector

Couple vectors obey all of the rules which govern vector quantities


Example 11






Example 12




Example 13






Example 14







Resultants

The Resultant of a system of forces is the

simplest force combination which can replace

the original forces without altering the external

effect on the rigid body to which the forces are

applied


Resultants of A force System

When several forces act on a body, you can

move each force and its associated couple

moment to a common point O


Resultants of A force System

In general, the force system can be expressed as

Now you can add all the forces and couple moments together and find one resultant force-couple moment pair


Resultants for Several Special Force Systems

Concurrent Forces: When forces are concurrent at a point, there are no moments about the point of concurrency

Parallel Forces: For a system of parallel forces not all in the same plane, the magnitude of the parallel resultant force R is simply the magnitude of algebraic sum of the given forces. The position of its line of action is obtained from the principle of moments by requiring that r × F = Mo

Coplanar Forces: Are discussed earlier


Wrench Resultants: When the resultant couple vector M is parallel to the resultant force R

A wrench is positive if the couple and force vectors points in the same direction and negative if they points in opposite directions

Resultants for Several Special Force Systems


Example 15




Example 16






Example 17







Statics Notes 2013 [Force Systems]Dr. Sameer Shadeed 1 Statics-61110 Dr. Sameer Shadeed An-Najah National University College of Engineering Chapter [2]

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