5 Distributed Forces

5.1 Introduction - Concentrated forces are models. These forces do not exist

in the exact sense.

- Every external force applied to a body is distributed over a

finite contact area.

Example:

Force exerted by the pavement on an automobile tire.

Two cases to be considered:

- Force is applied to the tire over its entire area of contact →

distributed force

- Force acts on the car as a whole → concentrated force

Internal loads are always distributed forces → stresses

Types of distributed forces

1. Line distributed force

- distributed along a line

- the intensity of this force is expressed as force per

unit length of line (N/m), lb/ft

2. Area distributed force

- These forces are distributed over an area.

- The intensity of these forces is expressed as force

per unit area

- This intensity is called pressure for the action of

fluid forces and stress for the internal distribution of

forces in solids.

- The basic unit for pressure or stress in SI is the

newton per square meter (N/m2), which is called

pascal (Pa).

- In the U.S. customary system of units, the unit for

pressure or stress is pound per square inch (lb/in.2).

3. Volume distributed force

- These forces are distributed over the volume of a

body and are called body forces.

- Examples of body forces are gravitational attraction

and the weight.

- The intensity of gravitational force is the specific

weight γ = ρg, where ρ is the density of the body

and g is the acceleration due to gravity.

- The units for the intensity of body forces are N/m3

in SI units and lb/in3 in the U.S. customary system.

Section A: Center of Mass and Centroids

5.2 Center of Mass

Center of gravity

It exists no unique center of gravity in the exact sense.

Determining the center of gravity

Assume: A uniform and parallel force field due to the

gravitational attraction of the earth.

- To determine mathematically the location of the

center of gravity of any body, we apply the

principle of moments to the gravitational forces.

- Principle of moments: The moment of the resultant

gravitational force W about any axis equals the sum

of the moments about the same axis of the

gravitational force dW acting on all particles treated

as infinitesimal elements of the body (The sum of

moments equal the moment of the sum).

- Principle of moments about the y-axis:

∫ = WxxdW

- For all three coordinates of the center of gravity G

we get:

- With W = mg and dW = gdm, we get:

or in vector form:

- With dm = ρ dV, we obtain

Center of Mass versus Center of Gravity

- The equations in which g not appears define the

center of mass

- The center of mass coincides with the center of

gravity as long as the gravity field is treated as

uniform and parallel.

- The center of mass is unique.

- It is meaningless to speak of the center of gravity of

a body which is removed from the gravitational

field of the earth, since no gravitational forces

would act on it.

- The calculation of the position of the center of mass

may be simplified by:

- The intelligent choice of the position of reference

axis.

- The type of the coordinates (rectangular, polar)

- Consideration of symmetry.

- Whenever there exists a line or plane of symmetry

in a homogeneous body, a coordinate axis or plane

should be chosen to coincide with this line or plane.

The center of mass will always lie on such a line or

plane.

5.3 Centroids of Lines, Areas, and

Volumes

- Assume ρ = const. → ρ will cancel from the previous equations.

- The remaining expressions of the equations a purely

geometrical property of the body →the centroid.

- The term centroid is used when the calculation

concerns a geometrical shape only.

- If the density is uniform throughout the body, the

positions of the centroid and center of mass are

identical.

Calculation of centroids:

1. Lines:

- Consider a wire or rod of length L, with constant

cross-sectional area A and constant density ρ. - The element has a mass dm = ρAdL. - The coordinates of the centroid are given by:

- In general, the centroid C will not lie on the line.

2. Areas:

- When a body of density has a small but constant

thickness t, we can model it as a surface area A.

- The mass of an element becomes dm = ρ tdA. - If ρ and t are constant over the entire area, then the coordinates of the centroid may be given as:

- The centroid C for the curved surface will in general

not lie on the surface.

3. Volumes

- For a general body of volume V and density ρ, the element has a mass dm = ρdV.

- If ρ is constant over the entire volume then the coordinates of the centroid may be given as:

Choice of Element for Integration

The principal difficulty with a theory often lies not in its

concepts but in the procedure for applying it.

The following five guidelines will be useful for the

choice of the differential element and setting up the

integrals.

1. Order of Element

Whenever possible, a first-order differential element

should be selected.

Choose dA = ldy not dA = dxdy

Choose dV = πr2dy not dV = dxdydz

2. Continuity

Whenever possible, we choose an element which can be

integrated in one continuous operation to cover the

figure.

Horizontal strip dA = ydx requires

only one integral.

Vertical strip dA = xdy requires two

separate integrals because of

discontinuity at x = x1.

3. Discarding Higher-Order Terms

Higher-order terms may always be dropped compared

with lower-order terms.

Select dA=ydx not dA=

ydx + 0.5dxdy

In the limit, of course, there

is no error.

4. Choice of Coordinates

We choose the coordinate system which best matches the

boundaries of the figure.

Choose rectangular coordinate system for this figure

Choose rectangular coordinate system for this figure

5. Centroidal Coordinate of Element

When a first- or second-order differential element is

chosen, it is essential to use coordinate of the centroid of

the element for the moment arm in expressing the

moment of the differential element.

It is essential to recognize that the subscript c serves as a

reminder that the moment arms appearing in the

numerators of the integral expressions for moments are

always the coordinates of the centroids of the particular

element chosen.

5.4 Composite Bodies and Figures;

Approximations

When a body or figure can be conveniently divided into

several parts whose mass centers are easily determined,

we use the principle of moments and treat each part as a

finite element of the whole.

For the x-coordinate of the center of mass of the body

shown in the figure we get:

In general, the coordinate of the mass center are given as:

An approximation method

Centroidal coordinates:

5.6 Beams External Effects

- Beams are structural members which offer

resistance to bending.

- Most beams are long prismatic bars.

- The loads are usually applied normal to the axes of

the beams.

- To analyze the load-carrying capacities of beams we

must:

- Determine the external loading and reactions

acting on a beam as a whole.

- Calculate the distribution along the beam of the

internal force and moment.

Types of Beams

Statically determinate beams:

Equilibrium Equations

Statics only

Statically indeterminate beams

Equilibrium Equations + Elastic Equations

Statics + Mechanics of Materials

In the following only statically determinate beams will

be considered.

Distributed Loads

Constant distributed load Linear distributed load

Trapezoidal load

Broken into a rectangular

and a triangular load.

General load distribution: