Simple stresses are produced by constant conditions of loading on elements
that can be represented as beams, rods, or bars. The table summarizes
information pertaining to the calculation of simple stresses. Following is an explanation of
the symbols used in simple stress formulae: σ = simple normal (tensile or compressive)
stress in pounds per square inch; τ = simple shear stress in pounds per square inch; F =
external force in pounds; V = shearing force in pounds; M = bending moment in inch-pounds;
T = torsional moment in inch-pounds; A = cross-sectional area in square inches; Z
= section modulus in inches; Zp = polar section modulus in inches; I = moment of inertia
in inches; J = polar moment of inertia in inches; a = area of the web of wide flange and I
beams in square inches; y = perpendicular distance from axis through center of gravity of
cross-sectional area to stressed fiber in inches; c = radial distance from center of gravity to
stressed fiber in inches.
Showing posts with label Mechanics and Strength of Materials. Show all posts
Showing posts with label Mechanics and Strength of Materials. Show all posts
Velocity and Acceleration
VELOCITY AND ACCELERATION
Motion is a progressive change of position of a body. Velocity is the rate of motion, that
is, the rate of change of position. When the velocity of a body is the same at every moment
during which the motion takes place, the latter is called uniform motion. When the velocity
is variable and constantly increasing, the rate at which it changes is called acceleration;
that is, acceleration is the rate at which the velocity of a body changes in a unit of time, as
the change in feet per second, in one second. When the motion is decreasing instead of
increasing, it is called retarded motion, and the rate at which the motion is retarded is frequently
called the deceleration. If the acceleration is uniform, the motion is called uniformly
accelerated motion. An example of such motion is found in that of falling bodies.
Newton's Laws of Motion.—The first clear statement of the fundamental relations existing
between force and motion was made in the seventeenth century by Sir Isaac Newton,
the English mathematician and physicist. It was put in the form of three laws, which are
given as originally stated by Newton:
1) Every body continues in its state of rest, or uniform motion in a straight line, except in
so far as it may be compelled by force to change that state.
2) Change of motion is proportional to the force applied and takes place in the direction
in which that force acts.
3) To every action there is always an equal reaction; or, the mutual actions of two bodies
are always equal and oppositely directed.
Motion with Constant Velocity.—In the formulas that follow, S = distance moved; V =
velocity; t = time of motion, θ = angle of rotation, and ω = angular velocity; the usual units
for these quantities are, respectively, feet, feet per second, seconds, radians, and radians
per second. Any other consistent set of units may be employed.
Constant Linear Velocity:
S = V × t V = S ÷ t t = S ÷ V
Constant Angular Velocity:
θ = ωt ω = θ ÷ t t = θ ÷ ω
Relation between Angular Motion and Linear Motion: The relation between the angular velocity of a rotating body and the linear velocity of a point at a distance r feet from the center of rotation is:
V(ft per sec) = r(ft) × ω(radians per sec)
Similarly, the distance moved by the point during rotation through angle θ is:
S(ft) = r(ft) × θ(radians)
Linear Motion with Constant Acceleration.—The relations between distance, velocity, and time for linear motion with constant or uniform acceleration are given by the formulas in the accompanying Table 1. In these formulas, the acceleration is assumed to be in the same direction as the initial velocity; hence, if the acceleration in a particular problem should happen to be in a direction opposite that of the initial velocity, then a should be replaced by − a. Thus, for example, the formula Vf = Vo + at becomes Vf = Vo − at when a
and Vo are opposite in direction.
Definition of Mechanics
MECHANICS
The science of mechanics deals with the effects of forces in causing or preventing
motion. Statics is the branch of mechanics that deals with bodies in equilibrium,
i.e., the forces acting on them cause them to remain at rest or to move with uniform velocity.
Dynamics is the branch of mechanics that deals with bodies not in equilibrium, i.e., the
forces acting on them cause them to move with non-uniform velocity. Kinetics is the
branch of dynamics that deals with both the forces acting on bodies and the motions that
they cause. Kinematics is the branch of dynamics that deals only with the motions of bodies
without reference to the forces that cause them.
Definitions of certain terms and quantities as used in mechanics follow:
Force may be defined simply as a push or a pull; the push or pull may result from the
force of contact between bodies or from a force, such as magnetism or gravitation, in which
no direct contact takes place.
Matter is any substance that occupies space; gases, liquids, solids, electrons, atoms,
molecules, etc., all fit this definition.
Inertia is the property of matter that causes it to resist any change in its motion or state of
rest.
Mass is a measure of the inertia of a body.
Work, in mechanics, is the product of force times distance and is expressed by a combination
of units of force and distance, as foot-pounds, inch-pounds, meter-kilograms, etc.
Power, in mechanics, is the product of force times distance divided by time; it measures
the performance of a given amount of work in a given time. It is the rate of doing work and
as such is expressed in foot-pounds per minute, foot-pounds per second, kilogram-meters
per second, etc.
Horsepower is the unit of power that has been adopted for engineering work. One horsepower
is equal to 33,000 foot-pounds per minute or 550 foot-pounds per second. The kilowatt,
used in electrical work, equals 1.34 horsepower; or 1 horsepower equals 0.746
kilowatt.
Torque or moment of a force is a measure of the tendency of the force to rotate the body
upon which it acts about an axis. The magnitude of the moment due to a force acting in a
plane perpendicular to some axis is obtained by multiplying the force by the perpendicular
distance from the axis to the line of action of the force. (If the axis of rotation is not perpendicular
to the plane of the force, then the components of the force in a plane perpendicular
to the axis of rotation are used to find the resultant moment of the force by finding the
moment of each component and adding these component moments algebraically.)
Moment or torque is commonly expressed in pound-feet, pound-inches, kilogram-meters,
etc.
Velocity is the time-rate of change of distance and is expressed as distance divided by
time, that is, feet per second, miles per hour, centimeters per second, meters per second,
etc.
Acceleration is defined as the time-rate of change of velocity and is expressed as velocity
divided by time or as distance divided by time squared, that is, in feet per second, per
second or feet per second squared; inches per second, per second or inches per second
squared; centimeters per second, per second or centimeters per second squared; etc.
Flywheel Calculations
Flywheels may be classified as balance wheels or as flywheel pulleys. The object of all
flywheels is to equalize the energy exerted and the work done and thereby prevent excessive
or sudden changes of speed. The permissible speed variation is an important factor in
all flywheel designs. The allowable speed change varies considerably for different classes
of machinery; for instance, it is about 1 or 2 per cent in steam engines, while in punching
and shearing machinery a speed variation of 20 per cent may be allowed.
The function of a balance wheel is to absorb and equalize energy in case the resistance to
motion, or driving power, varies throughout the cycle. Therefore, the rim section is generally
quite heavy and is designed with reference to the energy that must be stored in it to prevent
excessive speed variations and, with reference to the strength necessary to withstand
safely the stresses resulting from the required speed. The rims of most balance wheels are
either square or nearly square in section, but flywheel pulleys are commonly made wide to
accommodate a belt and relatively thin in a radial direction, although this is not an invariable
rule.
Flywheels, in general, may either be formed of a solid or one-piece section, or they may
be of sectional construction. Flywheels in diameters up to about eight feet are usually cast
solid, the hubs sometimes being divided to relieve cooling stresses. Flywheels ranging
from, say, eight feet to fifteen feet in diameter, are commonly cast in half sections, and the
larger sizes in several sections, the number of which may equal the number of arms in the
wheel. Sectional flywheels may be divided into two general classes. One class includes
cast wheels which are formed of sections principally because a solid casting would be too
large to transport readily. The second class includes wheels of sectional construction
which, by reason of the materials used and the special arrangement of the sections, enables
much higher peripheral speeds to be obtained safely than would be possible with ordinary
sectional wheels of the type not designed especially for high speeds. Various designs have
been built to withstand the extreme stresses encountered in some classes of service. The
rims in some designs are laminated, being partly or entirely formed of numerous segment shaped
steel plates. Another type of flywheel, which is superior to an ordinary sectional
wheel, has a solid cast-iron rim connected to the hub by disk-shaped steel plates instead of
cast spokes.
Steel wheels may be divided into three distinct types, including 1) those having the center
and rim built up entirely of steel plates; 2) those having a cast-iron center and steel
rim; and 3) those having a cast-steel center and rim formed of steel plates.
Wheels having wire-wound rims have been used to a limited extent when extremely high
speeds have been necessary.
When the rim is formed of sections held together by joints it is very important to design
these joints properly. The ordinary bolted and flanged rim joints located between the arms
average about 20 per cent of the strength of a solid rim and about 25 per cent is the maximum
strength obtainable for a joint of this kind. However, by placing the joints at the ends
of the arms instead of between them, an efficiency of 50 per cent of the strength of the rim
may be obtained, because the joint is not subjected to the outward bending stresses
between the arms but is directly supported by the arm, the end of which is secured to the rim
just beneath the joint. When the rim sections of heavy balance wheels are held together by
steel links shrunk into place, an efficiency of 60 per cent may be obtained; and by using a
rim of box or I-section, a link type of joint connection may have an efficiency of 100 percent.
STRENGTH OF MATERIALS
Introduction
Strength of materials deals with the relations between the external forces applied to elastic bodies and the resulting deformations and stresses. In the design of structures and machines, the application of the principles of strength of materials is necessary if satisfactory materials are to be utilized and adequate proportions obtained to resist functional forces.
Forces are produced by the action of gravity, by accelerations and impacts of moving
parts, by gasses and fluids under pressure, by the transmission of mechanical power, etc. In order to analyze the stresses and deflections of a body, the magnitudes, directions and points of application of forces acting on the body must be known. Information given in the Mechanics section provides the basis for evaluating force systems.
The time element in the application of a force on a body is an important consideration.
Thus a force may be static or change so slowly that its maximum value can be treated as if it were static; it may be suddenly applied, as with an impact; or it may have a repetitive or cyclic behavior.