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Stress and Strain Elasticity
Normal Strain In an axially loaded member, normal strain, is the change in the length, with respect to the original length, L. = L It is UNITLESS, but may be called strain or microstrain ().
s
Shearing Strain
L
In a member loaded with shear forces, shear strain, is the change in the sheared side, s with respect to the original height, L. For small angles: tan . = s = tan L In a member subjected to twisting, the shearing strain is a measure of the angle of twist with respect to the length and distance from the center, : = L Testing of Load vs. Strain
Behavior of materials can be measured by recording deformation with respect to the size of the load. For members with constant cross section area, we can plot stress vs. strain. BRITTLE MATERIALS - ceramics, glass, stone, cast iron; show abrupt fracture at small strains. DUCTILE MATERIALS plastics, steel; show a yield point and large strains (considered plastic) and "necking" (give warning of failure) SEMI-BRITTLE MATERIALS concrete; show no real yield point, small strains, but have some "strain-hardening".
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Linear-Elastic Behavior In the straight portion of the stress-strain diagram, the materials are elastic, which means if they are loaded and unloaded no permanent deformation occurs.
True Stress & Engineering Stress
f E 1
True stress takes into account that the area of the cross section changes with loading. Engineering stress uses the original area of the cross section.
Hooke's Law Modulus of Elasticity
In the linear-elastic range, the slope of the stress-strain diagram is constant, and has a value of E, called Modulus of Elasticity or Young's Modulus. f = E Isotropic Materials have the same E with any direction of loading. Anisotropic Materials have different E's with the direction of loading. Orthotropic Materials have directionally based E's
Plastic Behavior & Fatigue
Permanent deformations happen outside the linear-elastic range and are called plastic deformations. Fatigue is damage caused by reversal of loading.
The proportional limit (at the end of the elastic range) is the greatest stress valid using Hooke's law. The elastic limit is the maximum stress that can be applied before permanent deformation would appear upon unloading. The yield point (at the yield stress) is where a ductile material continues to elongate without an increase of load. (May not be well defined on the stress-strain plot.) The ultimate strength is the largest stress a material will see before rupturing, also called the tensile strength. The rupture strength is the stress at the point of rupture or failure. It may not coincide with the ultimate strength in ductile materials. In brittle materials, it will be the same as the ultimate strength.
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The fatigue is strength the stress at failure when a member is subjected to reverse cycles of stress (up & down or compression & tension). This can happen at much lower values than the ultimate strength of a material. Toughness of a material is how much work (a combination of stress and strain) us used for fracture. It is the area under the stress-strain curve.
Concrete does not respond well to tension and is tested in compression. The strength at crushing is called the compression strength. Materials that have time dependent elongations when loaded are said to have creep. Concrete and wood creep. Concrete also has the property of shrinking over time.
y
Poisson's Ratio
For an isometric material that is homogeneous, the properties are the same for the cross section: y = z z There exists a linear relationship while in the linear-elastic range of the material between longitudinal strain and lateral strain:
x
=-
y lateral strain =- =- z axial strain x x
y = z = -
f x
E
Positive strain results from an increase in length with respect to overall length. Negative strain results from a decrease in length with respect to overall length.
is the Poisson's ratio and has a value between 0 and , depending on the material
Relation of Stress to Strain
f =
P P f ; = and E = so E = A which rearranges to: L A L
=
PL AE
Orthotropic Materials
One class of non-isotropic materials is orthotropic materials that have directionally based values of modulus of elasticity and Poisson's ratio (E, ). Ex: plywood, laminates, fiber reinforced polymers with direction fibers
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Stress Concentrations
In some sudden changes of cross section, the stress concentration changes (and is why we used average normal stress)....