Embedded Files
Warm it up huh? what changes does it make? area? volume? length maybe? Any of these may happen if you warm up an object, but how much? That depends on the nature of the object, shown by coefficient of linear thermal expansion, α:
δthermal = αL(t - t0)
Hooke's law for thermal stress: σth = Eϵth
The lower, the value α, the lower the sensitivity of the material to extreme temperature. When mixing two elements together, the difference between their α, can be useful such as in bimetallic elements.
Other types of stress are longitudinal, circumferential, and radial stresses that is for pressure vessels, which are grouped into thin-wall and thick-wall pressure vessels. If the ration of the pressure tank's internal radius, Ri to its thickness t, is more than 10 (or diameter Di/t > 20) then we may say that the tank is a thin-wall pressure vessel. In this case, the radial stress is close to zero and can be neglected. Otherwise, it is a thick-wall class and the radial stress varies throughout the thickness of the tank.
Thin-walled tanks
Hoop stress, σh (aka, tangential or circumferential stress) for the thin-wall vessel under internal pressure is in the direction of wall's cross-section:
σh = pD/2t.
Axial stress, σl (aka, longitudinal stress), is the stress on the wall's longitudinal axis produced by the pressure inside:
σa = pD/4t = σh/2
For a thin-walled spherical tank any surface stress is longitudinal stress, σa = pD/4t.
Hoops and axial stresses are the principal stress in pressure vessels with internal pressure.
Tanks under external pressure fail due to buckling, not yielding and therefore, no thin-wall tank equation is valid for external pressure cases.
Chapter 14 of the mechanics of material class deals with this subject thoroughly. Enjoy!
Thick-walled tanks
Radial and tangential stress vary through the wall thickness but the axial does not.
At any point, all the longitudinal, hoop, and radial stresses are principal stresses and their max value is always at the inner surface of the wall for both internal and external pressure.
Here is a good video that explains thick-wall tank stresses and how to calculate them. The thing is that in this case, since the thickness is large, we will have two diameters to work with, the inner and outer diameters.
Next section is see these stresses in action through structural beams and columns.
If we have a closed box (not round thin-walled shell), the shear stress works its way out through the perimeter of the shell's wall. Still, if we know the thickness and the area enclosed inside the shell, we can determine the shear stress resulted by the applied torque. In this case we have a shear flow which is the product of shear stress and thickness. This shear flow is constant throughout the shell, even if the thickness changes. That is higher stress for thinner sections and vice versa.
Torsional stiffness (aka, torsional spring constant or twisting moment per radian of twist) is shown by k, or c equals torque divided by angle of twist. Basically it means how much torque is required to twist the shaft per unit of angle.
Next type of stress is the shear stress, τ, under torsion (torque), T. This stress is different if the rod under torsion is solid or hollow or if it is round or not. Here is a good page online (accessed 9/15/15) for details and equations of shear stress under different case.
Page updated
Google Sites
Report abuse