Strength Of Materials Formulas Pdf

[Bertoldo Beyer]

Inmaterials science, the strength of a material is its ability to withstand an applied load without failure. A load applied to a mechanical member will induce internal forces within the member called stresses when those forces are expressed on a unit basis. The stresses acting on the material cause deformation of the material in various manner. Deformation of the material is called strain when those deformations too are placed on a unit basis. The applied loads may be axial (tensile or compressive), or shear . The stresses and strains that develop within a mechanical member must be calculated in order to assess the load capacity of that member. This requires a complete description of the geometry of the member, its constraints, the loads applied to the member and the properties of the material of which the member is composed. With a complete description of the loading and the geometry of the member, the state of stress and of state of strain at any point within the member can be calculated. Once the state of stress and strain within the member is known, the strength (load carrying capacity) of that member, its deformations (stiffness qualities), and its stability (ability to maintain its original configuration) can be calculated. The calculated stresses may then be compared to some measure of the strength of the member such as its material yield or ultimate strength. The calculated deflection of the member may be compared to a deflection criteria that is based on the member's use. The calculated buckling load of the member may be compared to the applied load. The calculated stiffness and mass distribution of the member may be used to calculate the member's dynamic response and then compared to the acoustic environment in which it will be used.

Proportional limit is the point on a stress-strain curve at which it begins to deviate from the straight-line relationship between stress and strain. See accompanying figure at (1 & 2).

Elastic limit is the maximum stress to which a specimen may be subjected and still return to its original length upon release of the load. A material is said to be stressed within the elastic region when the working stress does not exceed the elastic limit, and to be stressed in the plastic region when the working stress does exceed the elastic limit. The elastic limit for steel is for all practical purposes the same as its proportional limit. See accompanying figure at (1, 2).

Yield point is a point on the stress-strain curve at which there is a sudden increase in strain without a corresponding increase in stress. Not all materials have a yield point. See accompanying figure at (1).

Yield strength, Sy, is the maximum stress that can be applied without permanent deformation of the test specimen. This is the value of the stress at the elastic limit for materials for which there is an elastic limit. Because of the difficulty in determining the elastic limit, and because many materials do not have an elastic region, yield strength is often determined by the offset method as illustrated by the accompanying figure at (3). Yield strength in such a case is the stress value on the stress-strain curve corresponding to a definite amount of permanent set or strain, usually 0.1 or 0.2 per cent of the original dimension.

Metal deformation is proportional to the imposed loads over a range of loads. Since stress is proportional to load and strain is proportional to deformation, this implies that stress is proportional to strain. Hooke's Law is the statement of that proportionality. Stress σ = = E Strain ε The constant, E, is the modulus of elasticity, Young's modulus or the tensile modulus and is the material's stiffness. Young's modulus is in terms of 106 psi or 103 kg/mm2. If a material obeys Hooke's Law it is elastic. The modulus is insensitive to a material's temper. Normal force is directly dependent upon the elastic modulus.

The greatest stress at which a material is capable of sustaining the applied load without deviating from the proportionality of stress to strain. Expressed in psi (kg/mm2). Ultimate strength (tensile) The maximum stress a material withstands when subjected to an applied load. Dividing the load at failure by the original cross sectional area determines the value. Elastic limit The point on the stress-strain curve beyond which the material permanently deforms after removing the load .

Point at which material exceeds the elastic limit and will not return to its origin shape or length if the stress is removed. This value is determined by evaluating a stress-strain diagram produced during a tensile test.

The ratio of the lateral to longitudinal strain is Poisson's ratio for a given material. lateral strain = longitudinal strain Poisson's ratio is a dimensionless constant used for stress and deflection analysis of structures such as beams, plates, shells and rotating discs.

When bending a piece of metal, one surface of the material stretches in tension while the opposite surface compresses. It follows that there is a line or region of zero stress between the two surfaces, called the neutral axis. Make the following assumptions in simple bending theory: The beam is initially straight, unstressed and symmetric The material of the beam is linearly elastic, homogeneous and isotropic. The proportional limit is not exceeded. Young's modulus for the material is the same in tension and compression All deflections are small, so that planar cross-sections remain planar before and after bending. Using classical beam formulas and section properties, the following relationship can be derived:

Strength of materials, also know as mechanics of materials, is focused on analyzing stresses and deflections in materials under load. Knowledge of stresses and deflections allows for the safe design of structures that are capable of supporting their intended loads.

When a force is applied to a structural member, that member will develop both stress and strain as a result of the force. Stress is the force carried by the member per unit area, and typical units are lbf/in2 (psi) for US Customary units and N/m2 (Pa) for SI units:

where F is the applied force and A is the cross-sectional area over which the force acts. The applied force will cause the structural member to deform by some length, in proportion to its stiffness. Strain is the ratio of the deformation to the original length of the part:

Axial stress and bending stress are both forms of normal stress, σ, since the direction of the force is normal to the area resisting the force. Transverse shear stress and torsional stress are both forms of shear stress, τ, since the direction of the force is parallel to the area resisting the force.

In the equations for axial stress and transverse shear stress, F is the force and A is the cross-sectional area of the member. In the equation for bending stress, M is the bending moment, y is the distance between the centroidal axis and the outer surface, and Ic is the centroidal moment of inertia of the cross section about the appropriate axis. In the equation for torsional stress, T is the torsion, r is the radius, and J is the polar moment of inertia of the cross section.

In the case of axial stress over a straight section, the stress is distributed uniformly over the entire area. In the case of shear stress, the distribution is maximum at the center of the cross section; however, the average stress is given by τ = F/A, and this average shear stress is commonly used in stress calculations. More discussion can be found in the section on shear stresses in beams. In the case of bending stress and torsional stress, the maximum stress occurs at the outer surface. More discussion can be found in the section on bending stresses in beams.

Just as the primary types of stress are normal and shear stress, the primary types of strain are normal strain and shear strain. In the case of normal strain, the deformation is normal to the area carrying the force:

In the case of torsional strain, the member twists by an angle, ϕ, about its axis. The maximum shear strain occurs on the outer surface. In the case of a round bar, the maximum shear strain is given by:

Hooke's law is analogous to the spring force equation, F = k δ. Essentially, everything can be treated as a spring. Hooke's Law can be rearranged to give the deformation (elongation) in the material:

When force is applied to a structural member, that member deforms and stores potential energy, just like a spring. The strain energy (i.e. the amount of potential energy stored due to the deformation) is equal to the work expended in deforming the member. The total strain energy corresponds to the area under the load deflection curve, and has units of in-lbf in US Customary units and N-m in SI units. The elastic strain energy can be recovered, so if the deformation remains within the elastic limit, then all of the strain energy can be recovered.

Note that there are two equations for strain energy within the elastic limit. The first equation is based on the area under the load deflection curve. The second equation is based on the equation for the potential energy stored in a spring. Both equations give the same result, they are just derived somewhat differently.

Stiffness, commonly referred to as the spring constant, is the force required to deform a structural member by a unit length. All structures can be treated as collections of springs, and the forces and deformations in the structure are related by the spring equation:

If the deflection is known, then the stiffness of the member can be found by solving k = F/δmax. However, the maximum deflection is typically not known, and so the stiffness must be calculated by other means. Beam deflection tables can be used for common cases. The two most useful stiffness equations to know are those for a beam with an axially applied load, and for a cantilever beam with an end load. Note that stiffness is a function of the material's elastic modulus, E, the geometry of the part, and the loading configuration.

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