Engineering Mechanics: Statics Chapter 2: Force Systems

Engineering Mechanics: Engineering Mechanics: Statics Statics

Chapter 2: Force Systems

ForceForce SystemsSystems

Part A: Two Dimensional Force Systems

ForceForce An action of one body on another Vector quantity

External and Internal forces

Mechanics of Rigid bodies: Principle of Transmissibility• Specify magnitude, direction, line of action• No need to specify point of application

Concurrent forces• Lines of action intersect at a point

Vector Components Vector Components A vector can be resolved into several vector components

Vector sum of the components must equal the original vector

Do not confused vector components with perpendicular projections

2D force systems•Most common 2D resolution of a force vector

•Express in terms of unit vectors ,

Rectangular ComponentsRectangular Components

F

x

y

i

xF

yF

j i j

ˆ ˆ

cos , sin x y x y

x y

F F F F i F j

F F F F

2 2x yF F F F

1tan y

x

F

F

Scalar components – can be positive and negative

2D Force Systems2D Force Systems Rectangular components are convenient for finding

the sum or resultant of two (or more) forces which are concurrent

R

1 2 1 1 2 2

1 2 1 2

ˆ ˆ ˆ ˆ ( ) ( )

ˆ ˆ = ( ) ( )

x y x y

x x y y

R F F F i F j F i F j

F F i F F j

Actual problems do not come with reference axes. Choose the most convenient

one!

Example 2.1Example 2.1

The link is subjected to two forces F1 and F2. Determine the magnitude and direction of the resultant force.

2 2236.8 582.8

629 N

RF N N

1 582.8tan236.8

67.9

NN

Solution

Example 2/1 (p. 29) Example 2/1 (p. 29)

Determine the x and y scalar components of each of the three forces

Unit vectors

• = Unit vector in direction of

cos direction cosinex

x

V

V

Rectangular componentsRectangular components

V

n

x

y

i

xV

yV

j

ˆ ˆˆ ˆ

ˆ ˆ cos cos

x y yx

x y

V i V j VVVn i j

V V V V

i j

n

V

x

y

2 2cos cos 1x y

The line of action of the 34-kN force runs through the points A and B as shown in the figure.

(a) Determine the x and y scalar component of F.

(b) Write F in vector form.

Problem 2/4Problem 2/4

MomentMoment In addition to tendency to move a body

in the direction of its application, a force tends to rotate a body about an axis.

The axis is any line which neither intersects nor is parallel to the line of action

This rotational tendency is known as the moment M of the force Proportional to force F and the

perpendicular distance from the axis to the line of action of the force d

The magnitude of M is M = Fd

MomentMoment The moment is a vector M perpendicular

to the plane of the body. Sense of M is determined by the right-

hand rule Direction of the thumb = arrowhead Fingers curled in the direction of the

rotational tendency

In a given plane (2D),we may speak of moment about a point which means moment with respect to an axis normal to the plane and passing through the point.

+, - signs are used for moment directions – must be consistent throughout the problem!

MomentMoment A vector approach for moment

calculations is proper for 3D problems. Moment of F about point A maybe

represented by the cross-product

where r = a position vector from point A to any point on the line of action of F

M = r x F

M = Fr sin = Fd

Example 2/5 (p. 40)Example 2/5 (p. 40)

Calculate the magnitude of the moment about the base point O of the 600-N force by using both scalar and vector approaches.

Problem 2/43 Problem 2/43

(a) Calculate the moment of the 90-N force about point O for the condition = 15º. (b) Determine the value of for which the moment about O is (b.1) zero (b.2) a maximum

CoupleCouple Moment produced by two equal, opposite,

and noncollinear forces = couple

Moment of a couple has the same value for all moment center

Vector approach

Couple M is a free vector

M = F(a+d) – Fa = Fd

M = rA x F + rB x (-F) = (rA - rB) x F = r x F

CoupleCouple Equivalent couples

Change of values F and d Force in different directions but parallel plane Product Fd remains the same

Force-Couple SystemsForce-Couple Systems Replacement of a force by a force and a couple Force F is replaced by a parallel force F and a

counterclockwise couple Fd

Example Replace the force by an equivalent system at point O

Also, reverse the problem by the replacement of a force and a couple by a single force

Problem 2/67Problem 2/67

The wrench is subjected to the 200-N force and the force P as shown. If the equivalent of the two forces is a for R at O and a couple expressed as the vector M = 20 kN.m, determine the vector expressions for P and R

ResultantsResultants The simplest force combination which can

replace the original forces without changing the external effect on the rigid body

Resultant = a force-couple system

1 2 3

2 2

-1

, , ( ) ( )

= tan

x x y y x y

y

x

R F F F F

R F R F R F F

R

R

ResultantsResultants Choose a reference point (point O) and

move all forces to that point Add all forces at O to form the resultant

force R and add all moment to form the resultant couple MO

Find the line of action of R by requiring R to have a moment of MO

( )

= O

O

R F

M M Fd

Rd M

Problem 2/79Problem 2/79

Replace the three forces acting on the bent pipe by a single equivalent force R. Specify the distance x from point O to the point on the x-axis through which the line of action of R passes.

ForceForce SystemsSystems

Part B: Three Dimensional Force Systems

Rectangular components in 3D

•Express in terms of unit vectors , ,

• cosx, cosy , cosz are the direction cosines

• cosx = l, cosy = m, cos z= n

Three-Dimensional Force Three-Dimensional Force SystemSystem

ˆ ˆ ˆ x y zF F i F j Fk

2 2 2x y zF F F F

i j k

cos , cos , cosx x y y z zF F F F F F

ˆ ˆ ˆ ( )F F li mj nk

Rectangular components in 3D

• If the coordinates of points A and B on the line of action are known,

• If two angles and which orient the line of action of the force are known,

Three-Dimensional Force Three-Dimensional Force SystemSystem

2 1 2 1 2 1

2 2 22 1 2 1 2 1

ˆ ˆ ˆ( ) ( ) ( )

( ) ( ) ( )F

x x i y y j z z kABF Fn F F

AB x x y y z z

cos , sin

cos cos , cos sinxy z

x y

F F F F

F F F F

Problem 2/98Problem 2/98 The cable exerts a tension of 2 kN on the fixed bracket at

A. Write the vector expression for the tension T.

Dot product

Orthogonal projection of Fcos of F in the direction of Q Orthogonal projection of Qcos of Q in the direction of F

We can express Fx = Fcosx of the force F as Fx =

If the projection of F in the n-direction is

Three-Dimensional Force Three-Dimensional Force SystemSystem

cosP Q PQ

F i

F n

ExampleExample Find the projection of T along the line OA

Moment of force F about the axis through point O is

r runs from O to any point on the line of action of F Point O and force F establish a plane A The vector Mo is normal to the plane in the direction

established by the right-hand rule

Evaluating the cross product

Moment and CoupleMoment and Couple

MO = r x F

ˆ ˆ ˆ

O x y z

x y z

i j k

M r r r

F F F

Moment about an arbitrary axis

known as triple scalar product (see appendix C/7)

The triple scalar product may be represented by the determinant

where l, m, n are the direction cosines of the unit vector n

Moment and CoupleMoment and Couple

( )M r F n n

x y z

x y z

r r r

M M F F F

l m n

A tension T of magniture 10 kN is applied to the cable attached to the top A of the rigid mast and secured to the ground at B. Determine the moment Mz of T about the z-axis passing through the base O.

Sample Problem 2/10 Sample Problem 2/10

A force system can be reduced to a resultant force and a resultant couple

ResultantsResultants

1 2 3

1 2 3 ( )

R F F F F

M M M M r F

Any general force systems can be represented by a wrench

Wrench ResultantsWrench Resultants

Replace the two forces and single couple by an equivalent force-couple system at point A

Determine the wrench resultant and the coordinate in the xy plane through which the resultant force of the wrench acts

Problem 2/143Problem 2/143

Special cases• Concurrent forces – no moments about point of

concurrency• Coplanar forces – 2D• Parallel forces (not in the same plane) – magnitude of

resultant = algebraic sum of the forces• Wrench resultant – resultant couple M is parallel to the

resultant force R• Example of positive wrench = screw driver

ResultantsResultants

Replace the resultant of the force system acting on the pipe assembly by a single force R at A and a couple M

Determine the wrench resultant and the coordinate in the xy plane through which the resultant force of the wrench acts

Problem 2/142Problem 2/142