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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