03-Strain - Strain Civ E 270 Khattak N Strain Deformation A...

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Strain Civ E 270 Khattak, N 1 F ° s L o ± ² / 2 Strain Deformation A body subjected to external loads would go through a change in its size and shape. These size and shape changes are called deformations which may be either large enough to be easily seen or so small that precise instruments have to be used to measure them. Loads that produce deformations can be either external forces or temperature changes. A body undergoing a temperature increase would expand (and hence deform) and a temperature decrease would contract. Strain The deformation per unit of length is defined as strain. Strain can be either: Normal Strain – Longitudinal deformation per unit of length In the case of the shown cantilever beam, subjected to an axial load P at its end, average normal strain will be expressed as: o avg L δ ε = where L o is the original length of the member The strain symbol ° is read as epsilon If the length of the member was to approach a small length ³ L and the deformation in that small length ³ L is ³° , then the equation for normal strain for this very short length ³ L can be given as: L L Δ Δ = Δ δ ε 0 lim Strain is a ratio of two lengths; hence it is a dimensionless quantity. However, it is sometimes expressed as a length ratio unit, i.e. mm/mm. Strains measured for most engineering applications are very small and are sometimes reported as microstrains ( ±° ) , where 1 ±° = 10 -6 strain. Shear Strain – Angle change per unit length Consider the distortion of the plate under the force F as shown in the figure. If the angle between the sides of the plate before the deformation was ² / 2 and it was ± after the distortion took place, the change in angle that occurs is defined as shear strain and is denoted by the symbol ´ (gamma), measured in radians. In this figure: ´ = ² /2 ± Also, 0 L Tan s δ γ = Since, we are usually dealing with very small deformations, therefore Tan ´ = ´ (in radians) ° 0 L s δ γ = If the length L o approaches a very small value ³ L, we can write the limiting equation of shear strain, L s L Δ Δ = Δ δ γ 0 lim P L o °
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Strain Civ E 270 Khattak, N 2 Example 3.1 A prismatic cantilever beam undergoes an increase in temperature, which elongates the beam along its axis. What is the normal strain in
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