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:
∫ xdW = x W
- 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
Choose dV = πr2dy
not dA = dxdy
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
Trapezoidal load
Broken into a rectangular
and a triangular load.
General load distribution:
Linear distributed load