Stress, Strain and Deflection

Stress and Strain


Stresses are either tensile or compressive. Structural materials are chosen by their ability to resist tensile or compressive forces, depending upon the application.  Most materials are better at resisting one or the other.  For instance, concrete is strong in compression and relatively weak in tension. Steel is equally strong in both tension and compression. There are two types of forces in structural engineering: tension and compression.

TENSION:Pulling force
Imagine the force felt in your arms while you hang from a bar.
A structural element subjected to tension is elongated.
 
COMPRESSION:Squeezing force
Imagine the force felt in your arms while you stand on your hands.
A structural element subjected to compression is shortened.

Stress is defined as force per unit area that the force acts upon.  Thus,
Stresses are either tensile or compressive. Structural materials are chosen by their ability to resist tensile or compressive forces, depending upon the application.  Most materials are better at resisting one or the other.  For instance, concrete is strong in compression and relatively weak in tension. Steel is equally strong in both tension and compression.


An example of a tensile structural element would be a cable.

An example of a compressive structural element would be a column.

Strain is defined as the change in length of a stressed structural element divided by the original length of the unstressed element. Thus,

\[ \epsilon =\frac{\Delta L}{L_0}\]

where, \[=\Delta L = L’-L_0,\]

A material’s tensile strength is determined in the laboratory by pulling on a specimen until it breaks. While the test is conducted, both the stress and strain are recorded. The maximum stress that the specimen can withstand is called the ultimate strength of that particular material. From a design stand-point, we are mainly interested in the stress where the material stops behaving elastically.

A material behaves elastically when it returns to its original shape when an applied load is no longer applied. This point is found by plotting stress versus strain during the test and determining the stress at which the plot becomes non-linear. This stress is called the yield stress, sy.

The slope of the stress-strain curve in the elastic region is defined as the elastic modulus, E. Structures should be designed so that any applied load would not cause the stress in the structure to be greater than sy.

Beams are structural elements that are subjected to bending forces. When bending occurs, the beam is subjected to tension and compression simultaneously.

Imagine a sponge beam. Say we draw a grid on the side of the beam, so that the sponge is divided into two rows of rectangles of equal length, Lo, and height, h/2.

When a force is applied to the beam, the rectangles deform.

The tops of the upper row of rectangles are shortened, and the bottoms of the lower row of rectangles are elongated. Thus, we see that the top of the beam is in compression and the bottom of the beam is in tension.

Notice that the middle of the beam is in neither tension or compression. This is called the neutral axis. The bending stress at the neutral axis is zero.

The key to designing a beam is to locate the point of maximum stress. For a simply-supported beam under a uniform load, the maximum stress occurs at the center point. The maximum compressive stress at the top of the beam, scmax, and the maximum tensile stress at the bottom of the beam, stmax, are given by the following equations:

where h is the height of the beam, b is the width of the beam, and Mmax is the maximum moment at the midspan of the beam.

Bending and deflection and its equations

Figure ii-1) stresses and moments acting over a cross-section of a beam; (a) a crosssection, (b) normal and shear stresses acting over the cross-section, (c) the moment
and shear force resultant of the normal and shear stresses
Figure ii-2) Sign Convention for Moments and Shear Forces

The beams of fig ii-1 and ii-2 show the normal stress and deflection one would expect when a beam bends downward. There are situations when parts of a beam bend upwards, and in these cases the signs of the normal stresses will be opposite to those shown in first figure. However, the moments (and shear forces) shown in Fig ii-1 will be regarded as positive. This sign convention to be used is shown in Fig. ii-2

Relationship between moment, shear force and loading

Relationships between the applied loads and the internal shear force and bending moment in a beam can be established by considering a small beam element, of width dx,

Plane sections remain plane in the elementary beam theory
Deformation of material fibres in an element of beam

A beam is a three-dimensional object, and so will in general experience a fairly complex three-dimensional stress state. We will show in what follows that a simple one-dimensional approximation,\[\sigma=\epsilon E\], whilst disregarding all other stresses and strains, will be sufficiently accurate for our purposes.

Flexural Stress of a beam


Consider a beam AB which is initially straight and horizontal when unloaded. If under the action of loads the beam deflect to a position A’B’ under load or infact we say that the axis of the beam bends to a shape A’B’. It is customary to call A’B’ the curved axis of the beam as the elastic line or deflection curve. In the case of a beam bent by transverse loads acting in a plane of symmetry, the bending moment M varies along the length of the beam and we represent the variation of bending moment in B.M diagram. Futher, it is assumed that the simple bending theory equation holds good.

Deflection equation of a beam
Integration Method of the deflection of a beam
Example of a simply supported beam by Direct Integration method
Couples and Forces


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