NOTES ENGINEERING MECHANICS STATICS
2013-CIV-81 | Muhammad Adnan Ghous
Mechanics
Contents
Force system
Equilibrium
Conditions of Equilibrium
Properties of plain area(center of mass, center of area, moment of
inertia)
Friction
Columb theory
Virtual work
Truss and beams(optional)
Properties of plain area
Plane
Any two dimensional surface is called a plane. It is always defined
by the other normal. On plane 3rd dimension is zero.
Local axis
Find the position of every point with respect to their own
reference.
Global axis
We measure the position of the all points with respect to the one
axis only called global axis.
Types of forces
In order to determine the effect of applied forces in a body it is important to
know the nature and point of application (center of mass of center of
gravity) of force with respect to the form or mode of application there are
following types of forces
Concentrated force When a force is applied over a region whose
dimensions are negligible as compared to other pertinent, also known
as point load .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
D
istributed force When forces are applied over a region whose
dimensions are not negligible compared with other pertinent
dimensions. In such case we must account for the actual manner
in which force is distributed. We do this by summing the effect of
distributed force over the entire region using mathematical
integration. This requires that we know the intensity of the force
at any location.
Units Newtons (SI)
There are following three categories of distributed force
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/m 3 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 = xW
- 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:
