### Stress- strain relationship of steel

Besides the strength of material, the stiffness of a material is frequently of equal importance. Other mechanical properties such as hardness, toughness, and ductility also determine the selection of a material but have a lesser degree of importance.

Experimental setup

A specimen, in tension test, is gripped between the jaws of a testing machine. The values of the load and elongation in a specified length, called the gage length, are observed simultaneously. These data are then plotted on a graph with the ordinate representing the load and the abscissa representing the elongation.

Here, plotting is not done for load against extension but for unit load or stress against unit elongation or strain. This facilitates to compare the properties of one specimen with those of other specimens and the resulting diagram is called a stress-strain diagram.

Stress

Stress is defined as load per unit area. Stress is expressed symbolically as
σ = P / A . . . . . . . . . . . . . (1)
where σ = sress
A= cross-sectional area
It is noticed that maximum stress in tension or compression occurs over a section normal to the load.

Units

In SI stress in expressed as MN/m2 or Mpa.
In U.S customary unit, stress is expressed as Ksi ( Kips/in2) and in certain applications, such as soil mechanics it is also common to measure stress in units of psf ( lb/ft2).
Although the equation (1) is fairly simple, it requires care discussion. Dividing load by area does not gives the stress at all points in the cross-sectional area; it merely determines the average stress. A more precise definition of stress is obtained by dividing the differential load dp by the differential area over which it acts:
σ = dp/dA . . . . . . . . . . . . . . (2)
Strain
To obtain the unit deformation or strain, Є, we devise the elongation ∂ by the length L in which it was measured, thereby obtaining

Є =∂/L . . . . . . . . . . . . . . (3)

The strain so computed, however, measures only the average value of strain. The correct expression for strain at any position is

Є = d∂/dL . . . . . . . . . . . . . . (4)

Where, d∂ is the differential elongation of the differential length dL. Thus equation (4) determines the average strain in a length so small that the strain must be constant over the length. However, under certain conditions the strain may be assumed constant and its value is computed from equation (3). These conditions are As follows:

1) The specimen must be of constant cross section.
2) The materials must be homogeneous.
3) The load must be axial, that is, produce uniform stress.

Units

Though strain represents a change in length divided by the original length, strain is a dimensionless quantity. However, it is common to use a units of meters per meter (m/m) or inches per inch (in/in) to represent strain. In engineering work, strains of the order of 1.0 x 10-3 are frequently encountered.
Different concepts developed from the stress –strain curve are as follows :

1) Proportional limit.
2) Elastic limit.
3) Yield point.
4) Ultimate stress or ultimate strength.
5) Rupture strength.

1) Proportional limit

The initial straight line portion from origin o shows proportionality between stress and strain. The point beyond which proportionality ends is called proportional limit. It has been noticed that this proportionality does not extend throughout the diagram. Beyond this point, the stress is no longer proportional to strain. The proportional limit is very important as all subsequent theory involving the behavior of elastic bodies is based on this proportionality, providing a upper limit of usable stress of a material.

2) Elastic limit

This is the stress beyond which the material will not return to its original shape when unloaded but will retain a permanent deformation called permanent set.

3) Yield point

This is the point at which there is an appreciable elongation or yielding of material without any corresponding increase of load; indeed, the load may actually decrease while yielding occurs.

Determination of yield point

The phenomenon of yielding is peculiar to structural steel; other grades of steel and steel-alloy or other materials do not posses it, as in indicated by the typical stress-strain curves of these materials shown in figure

These curves, incidentally, are typical for a first loading of materials that contain appreciable residual stresses produced by manufacturing or aging processes. After repeated loading, these residual stresses are removed and the stress-strain curves become practically straight, as can be demonstrated in the testing laboratory. The yield strength is closely associated with the yield point. For materials that do not have a well defined yield point, yield strength is determined by the offset method.

Offset Method

This consists of drawing a line parallel to the initial tangent of the stress-strain curve, this line being started at an arbitrary offset strain, usually of 0.2% (0.002 m/m or 0.002 in/in). as shown in figure, the intersection of this line, with the stress-strain curves is called yield strength.

Ultimate strength

The ultimate stress or ultimate strength as it is more commonly called is the highest ordinates on the stress-strain curve.

Rupture strength

This is the stress at failure. For structural steel it is somewhat lower than ultimate strength because the rupture strength is computed by dividing the rupture load by the original cross-sectional area, which although convenient, is incorrect. The error is caused the phenomenon known as necking. As failure occur, the material stretches very rapidly and simultaneously narrows down, as shown in figure; so that the rupture load is actually distributed over a smaller area.
If the rupture area is measured after failure occurs and divided into the rupture load, the result is truer value of the actual failure stress. Although this is considerably higher than the ultimate strength, the ultimate is commonly taken as the maximum stress of the material.