Machine Design

Caution: Mathematical Shortcuts Ahead

L. L. Lawson
Consulting Structural Analyst
Bothell, Wash.

When engineers set out to solve lengthy design problems they often shoot first for ball-park numbers so they can meet their deadlines. One way they do this is by using time-tested approximations for complex equations. Although these shortcuts often save time, they don’t always produce the needed results.

For example, one common mistake involves using a shortcut intended for relatively wide thin sheet to find torsional stress on a cross section that is nearly square. When this happens the error is significant and unconservative (meaning the calculated stresses are lower than the actual stresses).

Many analytical shortcuts were developed years ago for design and analysis in the aircraft industry where relatively thin elements, such as airplane skins, are amenable to quick approximation methods. But because of computers, shortcuts are unnecessary in many contemporary designs.

Computer programs let engineers perform rigorous calculations with such ease and speed that there is no reason to use approximations in final design and analysis. However shortcuts are still useful in certain preliminary design stages. One shortcut, for example, yields the hoop stress in a thin-walled cylindrical pressure vessel with more than enough accuracy for initial sizing. And the calculation is so simple that a reasonable estimate can be obtained without a pencil and paper.

Another handy use for shortcuts is in checking the results of sophisticated calculations. Minor human errors or glitches in CAD or FEA code can produce serious flaws in computer-aided design and analysis. But a quick look using a classical shortcut may catch an error and prevent problems later in the design cycle.

On the other hand, one disadvantage of many shortcuts is that they have a limited range. The pressure-vessel approximation discussed below, for example, yields results within 1% of the more rigorous Lamé solution but only when the wall thickness of the vessel is less than one-fifth of the mean radius. The difference grows to 10% when wall thickness reaches about five-eighths of the mean radius. Once these errors are known, a simple correction factor fixes them.

Ambiguous design classifications can also mislead engineers. Sometimes shortcuts are given for elements that are “thin,” “deep,” “wide,” or “very thin” without defining the terms. In such cases it is important to know the dimensional limits within which the method is acceptably accurate.

The opportunity to use shortcuts occasionally arises in special design situations. For instance, engineers often calculate the maximum longitudinal stress due to the bending moment on a hollow cylinder. This is required when designing structures such as pipes, storage tanks on supports, rocket-motor cases, and airplane fuselages.

A convenient shortcut to the approximate value comes from:

Scyl, s = M/At

= M/(πRm2)t

where Scyl, s = longitudinal stress due to bending moment on a hollow cylinder using the shortcut, psi; M = applied moment, lb-in.; A = area encompassed by mean circumference of the cylinder, in.2; t = cylinder wall thickness, in.; and Rm = mean radius of the cylinder wall, in.

By comparison, a more “rigorous” engineering method uses the equation:

Scyl, l = Mc/I

= (0.25π(R4 – r4))c/I

where Scyl, l= longitudinal stress due to bending moment on a hollow cylinder using the long method, psi; c = outside radius of cylinder, in.; I = moment of inertia of cross section, in.4; R = outside radius, in.; and r = inner radius, in.

The table, Longitudinal stress on hollow cylinders, shows the error between the rigorous method and the approximation by comparing normalized shortcut values with associated values using rigorous calculations.

The mean radius always yields more accurate results than does the inner or outer radius. Also the results are conservative because the predicted stress is always higher than that given by the more rigorous approach. The inner radius is nearly as accurate and does not require calculating the mean radius. But predicted stresses are lower than actual stresses so the inner-radius method is unconservative. Although the outer radius produces conservative results, it also leads to substantial error, even when R/t = 20 or more.

Engineers can also use shortcuts to calculate maximum shear stress due to torsion of a rectangular section. The shortcut to this solution is:

Srect, s = 3T/(bt2)

where srect, s = maximum shear stress due to torsion of a rectangular section using the shortcut, psi; T = applied torque, lb-in.; b = the longer, or base, dimension of the cross section, in.; and t = the shorter, or thickness, dimension of the cross section, in.

This approach was developed for convenience when dealing with torsion on relatively wide thin sections, which are often encountered in aircraft analysis. It works equally well on cross sections of any open shape formed from thin sheet, such as channels, angles, flat strips, and longitudinally slit cylinders. This equation is entirely satisfactory when the aspect ratio of the cross section is relatively large.

The method is unconservative because calculated stresses are lower than those predicted by more rigorous methods. The shortcut also introduces considerable error when applied to sections with relatively low b/t ratios, predicting only about five-eighths of the actual stress in the limit case of a square section with b/t equal to unity.

The classical, more rigorous approach is:

Srect, l = 3 T(b + 0.6t)/(b2 t2)

where Srect, l= maximum shear stress due to torsion of a rectangular section using the longer method, psi.

This, too, is a simplified approximation, but it is accurate within 4% for all rectangular sections. The difference between the methods can be quantified by the relation:

Srect, l(b/(b + 0.6t)) = Srect, s

The table, Torsional shear stress on rectangles, shows relative stress values at several b/t ratios.

Another acceptable shortcut helps find solutions for circumferential, or “hoop,” stress in a pressurized cylinder. This method determines the average hoop stress using:

Shoop, s = Prs/t

where shoop, s = hoop stress in a pressurized cylinder using the shortcut, psi; P = internal pressure, psi; rs = mean radius of shell, in.; and t = shell thickness, in.

This gives the average hoop stress in the wall, which is very close to the maximum stress at the inner surface of the shell when the ratio rs/t is sufficiently large.

The more rigorous solution by Lamé equation is:

Shoop, l = P(R2 + r2)/(R2 ­– r2)

where Shoop, l= hoop stress in a pressurized cylinder using the long method.

This equation is equally valid for all ratios of rs/t and yields the maximum hoop stress, which occurs at the inner surface of the shell. The approximation is reasonably accurate over a fairly large range of rs/t ratios, as shown in the table, Hoop stress in pressurized cylinders.

© 2010 Penton Media, Inc.

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