Chap. 4 Force System Resultants · Moment of a force about a point or axis ... force F is applied...

Chap. 4 Force System Resultants

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Chapter Outline

Moment of a Force Scalar FormationCross ProductMoment of Force Vector FormulationPrinciple of MomentsMoment of a Force about a Specified AxisMoment of a CoupleSimplification of a Force and Couple SystemFurther Simplification of a Force and Couple SystemReduction of a Simple Distributed Loading

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Moment of a Force Scalar Formation

Moment of a force about a point or axis a measure of the tendency of the force to cause a body to rotate about the point or axisTorque tendency of rotation caused by Fx or simple moment (Mo) z

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MagnitudeFor magnitude of MO,

MO = Fd (Nm)where d = perpendicular distance from O to its line of action of force

DirectionDirection using right hand rule

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Resultant Moment Resultant moment, MRo = moments of all the forces

MRo = Fd

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Cross Product

Cross Product (vector product)C = A

B

MagnitudeC = AB sin

Direction1800

BCACABC C

u)sin( =

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Cross Product

Laws of Operation

)()()(

)()()()(

DABADBA

BABABABAABBA

+=+

====

)()()( BADABDA +=+

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Cross Product

Cartesian Vector Formulation

),,(),,( zyxzyx BBBAAABA =

00

0

=========

kkijkjikikjjjkijjkikjiii

kBABAjBABAiBABA xyyxzxxzyzzy )()()( ++=

yx

yx

zyx

zyx

zyx

zyx

BBAAji

BBBAAAkji

BBBAAAkji

BA ==or

(+)

(-)

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Moment of a Force-Vector Formulation

MO = r

F (=rB

F) MO = r Fsin= F(r sin) = Fd

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Transmissibility of a Force

MO = rA

F = rB

F = rC

F

A,B,CF

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Vector Formulation

zyx

zyxO

FFFrrrkji

FrM ==

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Resultant Moment of a System of Forces

)( FrMOR =

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Example 4.4 Two forces act on the rod. Determine the resultant moment they create about the flange at O. Express the result as a Cartesian vector.

A(0, 5, 0), B(4, 5, -2)

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Position vectors are directed from point O to each force as shown.These vectors are

The resultant moment about O is

{ }{ }m 254

m 5kjir

jr

B

A

+==

( )

{ } mkN 604030 304080254

204060050

+=

+

=

+==

kji

kjikji

FrFrFrM BAOr

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Principle of Moments Varignons Theorem

MO = r

F1 + r

F2 + r

F3 + r

F4 + ...

= r

( F1 + F2 + )

= r

F

*

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p. 133, 44. Two men exert forces of F = 400 N and P = 250 N on the ropes. Determine the moment of each force about A. Which way will the pole rotate, clockwise or counterclockwise?

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424. In order to raise the lamp post from the position shown, force F is applied to the cable. If F = 1000 N, determine the moment produced by F about point A.

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p. 137, 4-36 The wheelbarrow and its contents have a center of mass at G. If F = 100 N, and the resultant moment produced by force F and the weight about the axle at A is zero, determine the mass of the wheelbarrow and its contents.

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p. 138, 4-47 The force F = {6i + 8j +10k} N creates a moment about point O of MO = {-14i + 8j + 2k} Nm. If the force passes through a point having an x coordinate of 1 m, determine the y and z coordinates of the point. Also, realizing that MO = Fd, determine the perpendicular distance d from point O to the line of action of F.

Chap. 4-2 Force System Resultants

4-2-2

MyF(y-axis)momentMoy(projection)

1. Oy Mo (=rF)

2. (j)Mo dot

productMy (= Mo

j)3. My = [j (rF)] j

Moment of a Force About a Specified Axis

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Moment of a Force About a Specified Axis

aOOa uMcosMM ==

zyx

zyx

azayax

a

FFFrrr

uuu

)Fr(u

=

=

[ ] aaaaa u)Fr(uuMM ==

Triple scalar product

()

a-a is an arbitrary axis

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p.145 Problem 4-56Determine the moment produced by force F about segment AB of the pipe assembly. Express the result as a Cartesian vector.

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p.147 Problem 4-65The A-frame is being hoisted into an upright position by the vertical force of F = 400 N. Determine the moment of this force about the y axis when the frame is in the position shown.

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Moment of a Couple

F r Fr

F )r-r( )F(r )F(- r M

AB

AB

BAO

==

=+=

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Moment of a Couple

Scalar FormulationM=Fd

Vector FormulationM=r x F

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Moment of a Couple

Equivalent Couples

Resultant Couple Moment( )

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Example 4.12

Determine the couple moment acting on the pipe. Segment AB is directed 30 below the xy plane.

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SOLUTION I (VECTOR ANALYSIS)

Take moment about point O, M = rA

(-250k) + rB

(250k)

= (0.8j)

(-250k) + (0.6cos30i + 0.8j 0.6sin30k)

(250k)

= {-130j}N.cm Take moment about point AM = rAB

(250k)

= (0.6cos30i 0.6sin30k)

(250k)= {-130j}N.cm

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SOLUTION II (SCALAR ANALYSIS)

Take moment about point A or B, M = Fd = 250N(0.5196m)

= 129.9N.cmApply right hand rule, M acts in the j direction

M = {-130j}N.cm

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p.155 Problem 76Determine the required magnitude of force F if the resultant couple moment on the frame is 200 NM, clockwise.

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p.158 Problem 4-92Determine the required magnitude of couple moments so that the resultant couple moment is MR = {-300i + 450j - 600k} Nm.

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p.159 Problem 4-102If F1 = 500 N and F2 = 1000 N, determine the magnitude and coordinate direction angles of the resultant couple moment.

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Chap. 4-3 Force System Resultants

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Simplification of a Force and Couple System

An equivalent system is when the external effects are the same as those caused by the original force and couple moment systemExternal effects of a system is the translating and rotating motion of the body Or refers to the reactive forces at the supports if the body is held fixed

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Equivalent System

Point O is On the Line of Action

Point O Is Not On the Line of Action

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Equivalent SystemResultant of a Force and Couple System

Force Summation : FR = F

Moment Summation : MRO = MC +Mo

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Ex. 4-15, replace the system by an equivalent resultant force and couple moment acting on point O.

m50,300)N(-166.4,-6 04.1666.24911.015.0kji

0)(0,-400,30

FrFrMM

N ) 800- 166.4, 250,- ( F

mN)300,400,0(MN)0,4.166,6.249(u300F

1.0)15.0()0,1.0,15.0(u

Nk800F

2B1CRO

CB2

22CB

1

=

+=

++=

=

===

+

=

=

0

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p. 169, 4-116 Replace the force system acting on the pipe assembly by a resultant force and couple moment at point O. Express the results in Cartesian vector form.

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Further Reduction of a Force and Couple System

dFM RRO =

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Further Simplification of a Force and Couple System

Concurrent Force SystemA concurrent force system is where lines of action of all the forces intersect at a common point O

= FFR

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Coplanar Force SystemLines of action of all the forces lie in the same plane Resultant force of this system also lies in this plane

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Parallel Force SystemConsists of forces that are all parallel to the z axisResultant force at point O must also be parallel to this axis

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Reduction to a Wrench ()

+=

=

MMMFMd )F(M

||RO

RR||

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p.180, 4-124. Replace the force and couple moment system acting on the overhang beam by a resultant force, and specify its location along AB measured from point A.

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p.182, 4-140. Replace the three forces acting on the plate by a wrench. Specify the magnitude of the force and couple moment for the wrench and the point P(y, z) where its line of action intersects the plate.

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Reduction of a Simple Distributed Loading

)y,x(centroid

concentrated force

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Reduction of a Simple Distributed Loading

Magnitude

Location

AdAdx)x(wFL AR

===

AxdA

dAxdA

x

FxdxxxwMM

A

A

R

ROO

=

=

=

=

)(:

first moment

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p.189, 4-144. Replace the distributed loading by an equivalent resultant force and specify its location, measured from point A.

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p.191, 4-152. Wind has blown sand over a platform such that the intensity of the load can be approximated by the function w=(0.5x3) N/m. Simplify this distributed loading to an equivalent resultant force and specify the magnitude and location of the force, measured from A.

m00.81250

10000x

m10000N

x101

dxx21dAx

1.25kNF1250N

x81

dxx21dAF

wdxdA

10

0

5

10

0

4

R

10

0

4

10

0

3R

==

=

=

=

==

=

==

=

Ans

Ans

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p.192, 4-160. The distributed load acts on the beam as shown. Determine the magnitude of the equivalent resultant force and specify its location, measured from point A.

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Chap 4-1-2012(compressed)Chap. 4 Force System ResultantsChapter OutlineMoment of a Force Scalar FormationMoment of a Force Scalar FormationMoment of a Force Scalar FormationCross ProductCross ProductCross ProductMoment of a Force-Vector FormulationMoment of a Force-Vector FormulationMoment of a Force-Vector FormulationMoment of a Force-Vector FormulationExample 4.4 Two forces act on the rod. Determine the resultant moment they create about the flange at O. Express the result as a Cartesian vector. 14Principle of Moments Varignons Theoremp. 133, 44. Two men exert forces of F = 400 N and P = 250 N on the ropes. Determine the moment of each force about A. Which way will the pole rotate, clockwise or counterclockwise?424. In order to raise the lamp post from the position shown, force F is applied to the cable. If F = 1000 N, determine the moment produced by F about point A. 18p. 137, 4-36 The wheelbarrow and its contents have a center of mass at G. If F = 100 N, and the resultant moment produced by force F and the weight about the axle at A is zero, determine the mass of the wheelbarrow and its contents. 20p. 138, 4-47 The force F = {6i + 8j +10k} N creates a moment about point O of MO = {-14i + 8j + 2k} Nm. If the force passes through a point having an x coordinate of 1 m, determine the y and z coordinates of the point. Also, realizing that MO = Fd, determine the perpendicular distance d from point O to the line of action of F.

Chap 4-2-2012(compressed)Chap. 4-2 Force System ResultantsMoment of a Force About a Specified Axis Moment of a Force About a Specified Axis 4 5 6 7 8 9Moment of a Couple Moment of a CoupleMoment of a CoupleExample 4.12SOLUTION I (VECTOR ANALYSIS)SOLUTION II (SCALAR ANALYSIS) 16 17 18 19 20 21 22

Chap 4-3-2012(compressed)Chap. 4-3 Force System ResultantsSimplification of a Force and Couple SystemEquivalent System Equivalent SystemEx. 4-15, replace the system by an equivalent resultant force and couple moment acting on point O.p. 169, 4-116 Replace the force system acting on the pipe assembly by a resultant force and couple moment at point O. Express the results in Cartesian vector form. 7Further Reduction of a Force and Couple SystemFurther Simplification of a Force and Couple SystemFurther Simplification of a Force and Couple SystemFurther Simplification of a Force and Couple SystemFurther Simplification of a Force and Couple Systemp.180, 4-124. Replace the force and couple moment system acting on the overhang beam by a resultant force, and specify its location along AB measured from point A. 14 15p.182, 4-140. Replace the three forces acting on the plate by a wrench. Specify the magnitude of the force and couple moment for the wrench and the point P(y, z) where its line of action intersects the plate. 17 18Reduction of a Simple Distributed Loading Reduction of a Simple Distributed Loadingp.189, 4-144. Replace the distributed loading by an equivalent resultant force and specify its location, measured from point A. 22p.191, 4-152. Wind has blown sand over a platform such that the intensity of the load can be approximated by the function w=(0.5x3) N/m. Simplify this distributed loading to an equivalent resultant force and specify the magnitude and location of the force, measured from A. 24 25

Chap. 4 Force System Resultants · Moment of a force about a point or axis ... force F is applied...

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Transcript of Chap. 4 Force System Resultants · Moment of a force about a point or axis ... force F is applied...