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5. DISTRIBUTED FORCES. CENTRES OF GRAVITY

5.1 Distributed forces. In engineering problems we often have to deal with

loads distributed along a line, over an area or within a certain volume. Let us

examine some simple cases of distributed forces.

A system of distributed forces along a line can be characterized by the load

per unit length of the line of application, which is called the intensity q. The

dimension of intensity is newton per meter: N/m.

1) Forces uniformly distributed along a straight line (Fig.5.1). The intensity q of

such system is a constant quantity.

2) Forces distributed along a straight line according to a linear law (Fig.5.2). An

example of such a load is the pressure of water against a dam, which drops from

a maximum at the bottom to zero at the surface. For such forces the intensity

varies from zero to qm.

qmq

Fig. 5.1 Fig. 5.2

3) Forces uniformly distributed along the arc of a circle (Fig.5.3). An example of

such forces is the hydrostatic pressure on the sides of a cylindrical vessel.

4) Forces uniformly distributed along a curved line. An example of such forces

are gravity forces of a wire lying in thexy plane (Fig.5.4).

z y

x0

R

Fig. 5.3 Fig. 5.4

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5.2 Centre of gravity of a two-dimensional body. Centroids of areas and

lines. We have assumed so far that the attraction exerted by the earth on rigid

body could be represented by a single force W. This force, called the weight of

the body, was to be applied at the centre of gravity of the body. Actually, the

earth exerts a force on each of the particles forming the body. The action of the

earth on a rigid body should thus be represented by a large number of small

forces distributed over the entire body. We shall see in this chapter, however,

that all these small forces may be replaced by a single equivalent force W. We

shall also learn to determine the centre of gravity, i.e., the point of application of

the resultant W, for various shapes of bodies.

Let us consider a flat horizontal plane (Fig.5.5). We may divide the plate into

n small elements. The forces exerted by the earth on elements of plate will be

denoted by W1, W2,, Wn, respectively. These forces are directed towards

the centre of the earth; however, for all practical purposes they may be assumed

parallel. Their resultant is therefore a single force in the same direction. The

magnitude W of this force is obtained by the adding the magnitudes of the

elementary weights,

:Fz W = W1 + W2 + Wn (5.1)

Fig. 5.5 Fig. 5.6

G

yG x

y

z

W0

xG

x

y

W

x

yz

0

To obtain the coordinatesxG andyG of the point G where the resultant W should

be applied, we write that the moments ofW about they and x axes are equal to

the sum of the corresponding moments of the elementary weights,

:My xGW = x1W1 + x2W2 + + xnWn

:Mx yGW = y1W1 + y2W2 + + ynWn (5.2)

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If we now increase the number of elements into which the plate is divided and

simultaneously decrease the size of each element, we obtain at the limit

following expressions:

(5.3)= dWW = xdWWxG = ydWWyG

These equations define the weight W and the coordinatesxG andyG of the centre

of gravity G of a flat plate. The same equations may be derived for a wire lying

in thexy plane (Fig.5.6). We shall observe, in the latter case, that the centre of

gravity G will generally not be located on the wire.

In the case of homogeneous plates of uniform thickness (Fig.5.7a) the centre

of gravity G coincides with the centroid Cof the area A of the plate, the latter

being define as follows:

= xdAAxC (5.4)= ydAAyC

xC

yC

y

x

xC

yCC

A

L

y

0 0

C

a) b)

Fig. 5.7

The integral is known as the first moment of the area A with respect to

the y axis.Similarly the integral defines the first moment of the area A

with respect to the x axis. It is seen from eqs. (5.4) that, if the centroid of an area

is located on a coordinate axis, the first moment of the area with respect to this

axis is zero.

xdA

ydA

Similarly the centre of gravity of the homogeneous wire of uniform cross

section coincides with the centroid Cof the line L defining the shape of the line

L (Fig.5.7b). The coordinatesxC andyC of the centroid of the lineL are obtained

from the equations

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(5.5)= xdLLxC = ydLLyC

Centroids of common shapes of areas and lines are shown in Fig.5.8

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5.3 Centre of gravity of a three-dimensional body. Centroid of a volume.

The method of determination of the centre of gravity G of a three-dimensional

body is, in principle, similar to that applied to plates in previous section. The

body, however, is divided into small volume elements of weighs W. Increasing

the number of elements and simultaneously decreasing the size of each element,

we obtain at the limit

(5.6)= xdWWxG = ydWWyG = zdWWzG

If the body is made of a homogeneous material, the equations

(5.7)= xdVVxC = ydVVyC = zdVVzC

defining the centroid C of the volume V of the body may be used to a

determination of its centre of gravity.

The integral is known as thefirst moment of the volume with respect to

the yz plane. Similarly, the integrals and define the first moments

of the volume with respect to the zx plane and the xy plane, respectively. It is

seen from eqs. (5.7) that, if the centroid of a volume is located in a coordinate

plane, the first moment of the volume with respect to that plane is zero.

xdV

ydV zdV

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A volume is said to be symmetrical with respect to a given plane if to every

point P of thevolume corresponds a point P of the same volume, such that the

line PP is perpendicular to the given plane and divided into two equal parts by

that plane. The plane is said to be aplane of symmetry for a given volume. When

a volume Vpossesses two planes of symmetry, the centroid of the volume must

be located on the line of intersection of the two planes. Finally, when a volume

V possesses three planes of symmetry which intersect in a well-defined point

(i.e., not along a common line), the point of intersection of the three planes must

coincide with the centroid of the volume. This property enables us to determine

immediately the centroid of the volume of spheres, ellipsoides, cubes,

rectangular parallelepipeds, etc.

Centroids of unsymmetrical volumes or of volumes possessing only one or

two planes of symmetry should be determined by integration. Centroids of

common shapes of volumes are shown in Fig.5.9. It should be observed that the

centroid of a volume of revolution in general does not coincide of its cross

section. Thus, the centroid of hemisphere is different from that of a semi circulararea, and the centroid of a cone is different from that of a triangle.

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Fig. 5.9

5.4 Determination of gravity centres of composite plates, wires and bodies.

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In many instances a flat plate may be divided into rectangles, triangles, or other

common shapes shown in Fig.5.8. The abscissaxGof its centre of gravity G may

be determined from the abscissasx1,x2, of the centres of gravity of the various

parts by expressing that the moment of the weight of the whole plate about y

axis is equal to the sum of the moments of the weights of the various parts about

the same axis (Fig.5.10). The ordinateyGof the centre of gravity of the plate is

found in a similar way by equating moments about the x axis. Thus, denoting by

W the magnitude of the weight of a plate and by W1, W2,, Wn the magnitudes

of the weights of its parts, we obtain:

:My iiG WxWx = (5.8a)

:Mx iiG WyWy = (5.8b)

zy

x

G

W

y

x

z

0 0

G2G1

G3W1 W2

W3

xG

yG

Fig. 5.10

If the plate is homogeneous and of uniform thickness, the centre of gravity

coincides with the centroid Cof its area. The abscissaxC and ordinateyC of the

centroid of the area may be then determined by expressing that the first moment

of the composite area with respect to either y orx axis, respectively is equal to

the sum of the first moments of the elementary areas with respect to the same

axis. Thus, coordinates of the area centroid can be determined from the

following equations

yM : = iiC AxAx (5.9a)

xM : iiC AyAy = (5.9b)

Care should be taken to record the moment of each area with the appropriate

sign. First moments of areas, just like moments of forces, may be positive and

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negative. For example, an area whose centroid is located to the left of the y axis

will have a negative first moment with respect to that axis. Also, the area of a

hole should be recorded with a negative sign (Fig.5.11)

y

xA1 A2 A3

x1x2

x3

xi A xiA

A1 Semicircle - + -

A2 Full rectangle + + +

A3 Circular hole + - -

Fig. 5.11

Similarly, it is possible in many cases to determine the centre of gravity of a

composite wire/body or centroid of a composite line/volume by dividing the

wire/body or line/volume into simpler elements. Proceeding analogously as in

the case of plates, we obtain the following equations defining the coordinates xG,

yG andzG of the composite body gravity centre G :

iiG WxWx = ,iiG WyWy = ,

iiG WzWz = (5.10)5.5 Determination of centroids by integration. The centroid of an area

bounded by analytical curves (i.e., curves defined by algebraic equations) is

usually determined by computing the integrals in Eqs.5.4

= xdAAxC (5.4)= ydAAyCIf the element of area dA is chosen equal to a small square of sides dx and dy,

the determination of each of these integrals requires a double integration in xandy. A double integration is also necessary if polar coordinates are used.

In most cases, however, it is possible to determine the coordinates of the

centroid of an area by performing a single integration. This is achieved by

choosing for dA a thin rectangle or strip, or a thin sector of pie-shaped element

(Fig.5.12); the centroid of the thin rectangle is located at its centre, and the

centroid of the thin sector at a distance3

2 rfrom its vertex (as for triangle). The

coordinates of the centroid of the area under consideration are then obtained by

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expressing that the first moment of the entire area with respect to each of the

coordinate axes is equal to the sum (or integral) of the corresponding moments

of the elements area. Denoting by xeland yel the coordinates of the centroid of

the element dA, we write

yM : (5.11a)dAxAx elG =

xM : (5.11b)dAyAy elG =

y

x

xel

yel

y

x

xel

yel

a

x

y

Q(x,y)

P(x,y)

dx

dy

0 0

x

xel

yelx

y

0

r

2/3r

R( ,r)

y

x

a) b) c)

xel = x

2

xax el

+= = cos

3

r2x el

yel = y/2 yel = y = sin3

r2yel

dA = ydx dA = (a x)dy = dr2

1dA 2

Fig.5. 12

If the area itself is not already known, it may also be computed from these

elements.

The coordinates xel and yel of the centroid of the element of area should be

expressed in terms of the coordinates of a point located on the curve bounding

the area under consideration. Also, the element of the area dA should be

expressed in terms of the coordinates of the point and their differentials. This

has been done in Fig.5.12 for three common types of elements; the pie-shaped

element of part c should be used when the equation of the curve bounding the

area is given in polar coordinates. The appropriate expressions should be

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substituted in formulae (5.11), and the equation of the curve should be used to

express one of the coordinates in terms of the other. The integration is thus

reduced to a single integration which may be performed according to the usual

rules of calculus.

A similar procedure may be applied to the determination of centroids of a

volume bounded by analytical surfaces.

Examples of this method of problem solution will be given in the course of

tutorials.

5.6 Theorems of Pappus Guldinus. These theorems deal with surfaces and

bodies of revolution.

A surface of revolution is a surface, which may be generated by rotating a

plane curve about a fixed axis lying in the same plane. For example (Fig.5.13),

the surface of a sphere may be obtained by rotating a semicircular arc ABC

about the diameterAC; the surface of a cone by rotating a straight lineAB about

an axis AC; the surface of a torus or ring by rotating the circumference of a

circle about nonintersecting axis.

Sphere Cone Torus

A

B

C A

BB

C CA

Fig. 5.13

A body of revolution is a body which may be generated by rotating a plane

area about a fixed axis. A solid sphere may be obtained by rotating a semi-

circular area, a cone by rotating triangular area, and a solid torus by rotating a

full circular area (Fig.5.14).

Sphere Cone Torus

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Fig. 5.14

Theorem I. The area of a surface of revolution is equal to the length of the

generating curve times the distance traveled by the centroid of the curve while

the surface is being generated, i.e.

A = 2yCL (5.12)

Theorem II.The volume of a body of revolution is equal to the generating

area times the distance traveled by the centroid of the area while the body is

being generated, i.e.

V = 2yCA (5.13)

Fig.5.15 shows most important relation between curve element dL and a surface

of revolution, while Fig.5.16 shows relation between area element dA and a

volume generated by this element. For proofs of both theorems please consult

Beer & Johnston p. 184.

The theorems of Pappus-Guldin offer a simple way for computing the area of

surfaces of revolution. They may also be used conversely to determine the

centroid of a plane curve when the area of the surface generated by the curve is

known or to determine the centroid of a plane area when the volume of the body

generated by the area is known.

5.7 Distributed loads on beams. The concept of centroid of an area may be

used to solve other problems besides those dealing with the weight of flat plates.

Consider, for example, a beam supporting a distributed load; this load may

consist of the weight of materials supported directly or indirectly by the beam,

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or it may be caused by wind or hydrostatic pressure. The distributed load may be

represented by plotting the load w supported per unit length (Fig.5.17); this load

will be expressed in N/m. The magnitude of the force exerted on an element of

beam of length dx is dW= wdx, and the total load supported by the beam is

x

L

dx

w

dw = dA

0

wdw

x

L

0

w

x

C

W = A

P

xP

W

xC

a) b)

(5.14)=L

0

wdxW

Fig. 5.17

But the product wdx is equal in magnitude to the element of area dA shown in

Fig.5.17a, and W is thus equal in magnitude to the total area A under the load

curve,

(5.15) == AdAW

We shall now determine where a single concentrated load W, of the same

magnitude Was the total distributed load, should be applied on the beam if it is

to produce the same reactions at supports (Fig.5.17b). This concentrated load W,

which represents the resultant of the given distributed loading, should be

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equivalent to this loading as far as the free-body diagram of the entire beam is

concerned. The point of application P of the equivalent concentrated load W

will therefore be obtained by expressing that the moment ofW about point O is

equal to the sum of the moments of the elementary loads dW about O :

(5.16)= xdWWxP

or, since dW= wdx = dA and W = A

(5.17)=L

0

P xdAAx

Since the integral represents the first moment with respect to the w axis of the

area under the load curve, it may be replaced by the product xCA. We have

thereforexP = xC , wherexC is the distance from the w axis to the centroid Cof

the areaA (i.e. area representing the distributed load).

A distributed load on beam may be replaced by a concentrated load; the

magnitude of this single load is equal to the area under the load curve, and its

line of action passes through the centroid of that area. It should be noted,

however, that the concentrated load is equivalent to the given loading only as far

as external forces are concerned. It may be used to determine reactions but

should not be used to compute internal forces and deflections.

Bibliography

Beer F.P, Johnston E.R., Jr., Vector Mechanics for Engineers, McGraw-Hill

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