ARC 241 – Structural Analysis I - Lecture 5, …ammar.kunet.com/notes/arc241/Lecture-5.pdfReplace the loading on the frame by a single resultant force. Specify where its line of

ARC241 – Structural Analysis I Lecture 5, Sections ST4.5 – ST4.10 ST4.5) Moment of a Force about a Specified Axis ST4.6) Moment of a Couple ST4.7) Equivalent System ST4.8) Resultant of a Force and a Couple System ST4.9) Further Reduction of a Force and Couple ST4.10) Reduction of Simple Distributed Loading

ARC241 – Structural Analysis I Dr. Ammar T. Al-Sayegh

ARC241 – Structural Analysis I Dr. Ammar T. Al-Sayegh 1

ST4.5) Moment of Force about a Specified Axis

Moment and its axis are always ⊥ to the plane containing the force and moment arm.

In some problems we need to find the component of moment along a specified axis that passes through a point (2 methods are available).


If the line of action of F is ⊥ to an axis aa, the magnitude of moment around aa is given by

Scalar Analysis:

2-Steps:

a)

But on Ob axis, so

b)

1- Step

But need ⊥ distance to y axis

mNmNM O .10)5.0)(20( ==

mNmNM y .6).10(53

==

mNmNM y .6)3.0)(20( ==


Vector Analysis:

2-Steps:

a)

But on Ob axis, so

b)

1- Step

But need ⊥ distance to y axis

mNjikjiFrM AO .}68{)20()4.03.0( +−=−×+=×=

mNjjiuMM aOy ⋅=⋅+−=⋅= 6)68(

)( FruM ay ×⋅=


Vector Analysis (General):

The magnitude of a moment around an arbitrary axis, aa`:

This is called “Triple Scalar Product”, given by:

For the moment vector zyx

zyx

aaa

a

FFFrrr

uuuM

xxx

=

)( FruM aa ×⋅=

aaaa uFruuM )]([Ma ×⋅==

The resultant moment of a series of forces:

∑∑ ×=×⋅= )()]([Ma FruFru aa


Problem ST4.58: Determine the resultant moment of the two forces about the Oa axis. Express the result as a Cartesian Vector.


Problem ST4.58: The hood of the automobile is supported by the strut AB, which exerts a force of F = 100 N on that hood. Determine the moment of this force about the hinged axis y.


Problem ST4.65: If a torque of 80 N.m is required to loosen that bolt at A, determine the force P that must be applied perpendicular to the handle of the flex-headed ratchet wrench.


ST4.6) Moment of a Couple

Couple: two parallel forces that have the same magnitude, have opposite directions, and are separated by a ⊥ distance d.

The only effect of a couple is to produce a rotation or tendency of rotation in a specified direction (resultant force = 0)

Moment produced by a couple can be determined by finding the sum of the moments of both couple forces about any arbitrary point, O.

To simplify the problem, take moments about a point lying on the line of action of one of the forces (force at chosen point produce zero moment)


If point A is chosen M = r × F -A couple moment is a free vector that can act at any point since M depends only upon the position vector r directed between the forces and not the position vectors rA and rB directed from the arbitrary point O to the forces.

- Two couples are said to be equivalent if they produce the same moment. Thus, their forces must lie either same plane or parallel planes.


Scalar Analysis:

Magnitude:

Direction:

Right-Hand Rule

Vector Analysis:

Resultant

FdM =

Fr×=M

∑ ×= FrM


Problem ST4.74: The resultant couple moment created by two couples acting on the disk is MR = {10k} kN.cm. Determine the magnitude of force T.


Problem ST4.78: Two couples act on the beam. Determine the magnitude of F so that the resultant couple moment is 450 N.m, counterclockwise. Where on the beam does the resultant couple act on?


Problem ST4.80: If the couple moment acting on the pipe has a magnitude of 400 N.m, determine the magnitude F of the vertical force applied to each wrench.

Illustration: Horizontal force on a stick Illustration: Vertical force on a stick


Illustration: Reducing multiple forces to single force with moment

Illustration: Reducing multiple force to single force without moment


Problem ST4-116: Replace the loading acting on the beam by a single resultant force. Specify where the force acts, measured from B.


Problem ST4-121: Replace the loading on the frame by a single resultant force. Specify where its line of action intersects member CD, measured from end C.


Reduction of Distributed Load: - In many situations a very large surface of a body may be subjected to distributed loadings (e.g. wind, fluids, weight of material). - Intensity of these loads at each point on the surface is defined as pressure p. - pressure = p = force per unit area = N/m2 = Pascal = Pa. -1 Pa = 1 N/m2. - In this section most common case will be considered - Most common case uniform along one axis of a flat rectangular body upon which the loading is applied.


Example of Uniform Loading: - Direction and intensity of the pressure load is indicated by arrows (Load-intensity diagram).

- The entire loading on the plate is therefore a system of parallel forces, infinite in number and each acting on a separate differential area of the plate.

- The loading function, p = p (x) Pa, is only a function of x since the pressure is uniform along the y axis.

-If we multiply p = p(x) by the width = a meters of the plate we obtain w = [p(x) N/m2 ] a m = w (x)N/m.

- It is measured as force per unit length rather than force per unit area.


2-D Representation of Uniform Load:

-Load intensity can be represented by a system of coplanar parallel forces in 2D.

-Using integration, this system of forces can be simplified to a single resultant force FR and its location can be specified.

(Area Under Curve)

(Centroid of Area Under Curve)


∫∫ ===LL

R AdAdxxwF )(

∫

∫

∫

∫==

L

L

L

L

dA

xdA

dxxw

dxxxwx

)(

)(

Problem ST4-139: The loading on the bookshelf is distributed as shown. Determine the magnitude of the equivalent resultant location, measured from point O.


Problem ST4-141: Replace the loading by an equivalent force and couple moment acting at point O.


Problem ST4-148: Replace the distributed loading by an equivalent resultant force and specify its location, measured from point A.


Problem ST4-153: Replace the distributed loading by an equivalent resultant force and specify where its line of action intersects member BC, measured from C.


Problem ST4-154: Replace the distributed loading by an equivalent resultant force and specify its line of action intersects the beam measured from point O.


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ARC 241 – Structural Analysis I - Lecture 5, …ammar.kunet.com/notes/arc241/Lecture-5.pdfReplace the loading on the frame by a single resultant force. Specify where its line of