Cylinders find many applications, two of the most common categories being :
- fluid containers such as pipes and pressure vessels - interference-fitted bearing bushes, sleeves and the like.
Cylinders can act as beams or shafts eg. ( load building blocks which have already been examined ) but in the present context cylinders are loaded primarily by internal and external ( gauge ) pressures pi and po due to adjacent fluids or to contacting cylindrical surfaces.
The following notes examine cylinder safety and derive appropriate design equations. Since the analysis presupposes an elastic material, superposition may be applied if other loading mechanisms such as bending or torsion occur. The treatment ignores end effects and the axial variation of pressures and stresses - so all cross- sections are loaded and stressed identically, and cylinder length is of no consequence. Secondary stresses arising from incompatibility between cylinder and end closure are examined in the context of pressure vessels.
The stress state in the wall is essentially triaxial and initial analysis gives the principal components without need for resolution :
[LEFT][SIZE=4]Thin Thick Limiting proportions (approx) Di/t > 20 Di/t < 25 Analytic treatment simple approximation accurate Statically determinate yes no Stress state membrane - ie. biaxial triaxial Stress - radial, σr zero varies with radius - tangential, σt uniform varies with radius - axial, σa uniform uniform
The limiting proportions - the dividing line between thin and thick - is not hard and fast. Whether we choose to treat a particular cylinder as thin or as thick is a matter only of the accuracy we seek from the mathematical model together with the effort we’re prepared to expend in solving the model. For example, if in a particular design the thin model results in a cylinder with Di/t = 7 then we’d either [/size][/left]
[SIZE=4]Physically, thin cylinders are not suitable when the external fluid pressure is much greater than the internal, unless the cylinder is supported or stiffened against local buckling. We shall not consider thin cylinders with net external pressures sufficient to cause buckling, unless the cylinders are obviously adequately supported as in figure (a) below. Axial stress
Cylinders are classed as being either :
A fluid container sealed by a piston is open © - in this case an external axial force Fa is mandatory for equilibrium of the overall piston-and-cylinder assembly (a bicycle pump is a common example). If Ai is the internal circular area then a free body of either piston or cylinder end © must have Fa = pi Ai and there is no need for axial wall stresses to equilibrate the fluid pressure.
If an axial stress does exist then it is uniform across the cylinder wall, no matter whether the cylinder is thin or thick. The stress may be found easily from equilibrium in the axial direction, aided by a free body viewed in side sectional elevation. Considering the free body of one end of an internally pressurised closed cylinder (d) the fluid pressure resultant pi Ai is equilibrated by the wall stress resultant σa Aa - where the annular wall area is Aa = Ao - Ai in which Ao is the outside circular area.
In the more general case where an external fluid pressure also exists :
[SIZE=4]COLOR=blue σa = ( pi Ai - po Ao ) / Aa [/color]
A free body (e) of part of a pipe which connects two vessels (not shown) might be thought open, since no ends are evident and the fluid pressure is self-equilibrating across the free body. However each connected vessel acts as a pipe closure, so the pipe is in fact closed and axial stresses must occur. Thin cylinders
The tangential stress is uniform across the wall; so, from equilibrium of the free body : σt = ( pi Di - po Do ) / 2 t
Since Di / t > > 1, this equation with ( i), yields :
COLOR=blue σt = Δp Di / 2 t where Δp = pi - po
σa = Δp Di / 4 t (CLOSED) or σa = 0 (OPEN) [/color]
[SIZE=4][COLOR=blue]The radial stress is zero, the tangential stress is always the principal of greatest magnitude, and the axial stress is either zero in the case of open thin cylinders or half the tangential stress in closed thin cylinders. The Mohr’s circles thus appear as shown. Derivation of the design equations corresponding to the various failure theories is left as an exercise for the reader :-