UNIT-II
ELASTICITY
INTRODUCTION
Elastic body is deformed in response to stress. There are two
types of deformation:
Change in volume and shape.
Consider the interior of a deformed body:
At point P, force dF acts on any infinitesimal area dS Stress,
with respect to direction n, is a vector: lim(dF/dS) (as dS→0).
Stress is measured in [Newton/m2], or Pascal.
For small deformations, most elastic materials, such
as springs, exhibit linear elasticity. This means that they
are characterized by a linear relationship between stress
and strain (the relative amount of deformation).
This idea was first formulated by Robert Hooke in 1675.
This law can be stated as a relationship
between force F and displacement x,
where k is a constant known as
the rate or spring constant. It can also be stated
as a relationship
between stress σ and strain :
where E is known as the elastic
modulus or Young's modulus.
Although the general proportionality constant between stress and
strain in three dimensions is a 4th order tensor, systems that
exhibit symmetry, such as a one-dimensional rod, can often be
reduced to applications of Hooke's law. However, most materials are
elastic only under relatively small deformations, and so several
conditions must be fulfilled so that Hooke's law is a good
approximation. Because Hooke's law neglects higher order terms (so
only the linear term dominates), in certain cases, such as rubbery
materials, these conditions may not hold.
POISSON'S RATIO
BENDING OF BEAMS
Bending moments are produced by transverse loads applied to
beams. The simplest case is the cantilever beam, widely
encountered in balconies, aircraft wings, diving boards etc.
The bending moment acting on a section of the beam, due
to an applied transverse force, is given by the product of the
applied force and its distance from that section. It thus has units
of N m. It is balanced by the internal
moment arising from the stresses generated. This is given by a
summation of all of the internal moments acting on individual
elements within the section. These are given by the force acting on
the element (stress times area of element) multiplied by its
distance from the neutral axis, y.
Balancing the external and internal moments during the bending
of a cantilever beam. Therefore, the bending moment, M, in a
loaded beam can be written in the form
The concept of the curvature of a beam, κ, is central
to the understanding of beam bending. The figure below, which
refers now to a solid beam, rather than the hollow pole shown in
the previous section, shows that the axial strain, ε, is given
by the ratio y / R. Equivalently, 1/R (the
"curvature", κ) is equal to the through-thickness gradient of axial
strain. It follows that the axial stress at a
distance y from the Neutral axis of the beam is
given by, σ = E κ y
Relation between the radius of curvature, R, beam curvature, κ,
and the strains within a beam subjected to a bending moment.
The bending moment can thus be expressed as
This can be presented more compactly by
defining I (the second moment of area , or
"moment of inertia") as
The units of I are m 4. The value
of I is dependent solely on the beam sectional shape.
Then I is calculated for two simple shapes. The moment can now
be written as, M = κ E I
These equations allow the curvature distribution along the
length of a beam (i.e. its shape), and the stress distribution
within it, to be calculated for any given set of applied forces.
The following simulation implements these equations for a
user-controlled beam shape and set of forces. The 3-point bending
and 4-point bending loading configurations in this simulation are
SYMMETRICAL, with the upward forces, denoted by arrows, outside of
the downward force(s), denoted by hooks.
UNIFORM AND NON-UNIFORM BENDING
CANTILEVER
A cantilever is a beam anchored at only one
end. The beam carries the load to the support where it is resisted
by moment and shear stress. Cantilever construction
allows for overhanging structures without external bracing.
Cantilevers can also be constructed
with trussesor slabs.
This is in contrast to a simply supported beam such as those
found in a post and lintel system. A simply supported
beam is supported at both ends with loads applied between the
supports.
Cantilevers are widely found in construction, notably
in cantilever bridges and balconies. In cantilever
bridges the cantilevers are usually built as pairs, with each
cantilever used to support one end of a central section.
The Forth Bridge in Scotland is an example of a
cantilever truss bridge.
Temporary cantilevers are often used in construction. The
partially constructed structure creates a cantilever, but the
completed structure does not act as a cantilever. This is very
helpful when temporary supports, or falsework, cannot be used
to support the structure while it is being built (e.g., over a busy
roadway or river, or in a deep valley). So some truss arch
bridges (see Navajo Bridge) are built from each side as
cantilevers until the spans reach each other and are then jacked
apart to stress them in compression before final joining. Nearly
all cable-stayed bridges are built using cantilevers as
this is one of their chief advantages. Many box girder bridges are
built segmentally, or in short pieces. This type of
construction lends itself well to balanced cantilever construction
where the bridge is built in both directions from a single
support.
These structures are highly based on torque and
rotational equilibrium. In an architectural application, Frank
Lloyd Wright's Falling water used cantilevers to project
large balconies. The East Stand at Elland Road Stadium in
Leeds was, when completed, the largest cantilever stand in the
world holding 17,000 spectators. The roof built over
the stands at Old Trafford Football Ground uses a
cantilever so that no supports will block views of the field. The
old, now demolished Miami Stadium had a similar roof over
the spectator area. The largest cantilever in Europe is located
at St James' Park in Newcastle-Upon-Tyne, the home
stadium of Newcastle United F.C.
Less obvious examples of cantilevers are free-standing
(vertical) radio towers without guy-wires,
and chimneys, which resist being blown over by the wind
through cantilever action at their base.
In materials science, shear
modulus or modulus of rigidity, denoted by G, or
sometimes S or μ, is defined as the ratio
of shear stress to theshear strain:[1]
where
= shear stress;
is the force which acts
is the area on which the force acts
in engineering, = shear strain. Elsewhere,
is the transverse displacement
is the initial length
Shear modulus' derived SI unit is
the pascal (Pa), although it is usually expressed
in gigapascals (GPa) or in thousands of pounds per square
inch (kpsi).
A modulus of elasticity equal to the ratio of the
tangential force per unit area to the resulting angular
deformation. Symbol G
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