Optimality Criteria

Abstract

A first optimality criterion, that of the fully stressed design, was already introduced in the previous chapter. The buckling of a circular tube in compression is used to illustrate a second criterion, that of simultaneous buckling modes. In fact, when the tube forms part of a truss structure, this might be seen as a logical extension of the principle of the fully stressed design. This second optimality criterion leads directly to an efficiency formula, expressing the maximum stress that can be achieved in a thin tube or other component in terms of a suitable structural index and the elastic modulus of the material. The concept of the design space, widely used in subsequent chapters, is introduced with the circular tube. A third criterion is developed for the maximum stiffness of a structure, on the basis of a simple truss but taken in principle to apply more widely. It is shown that under certain conditions, a fully stressed design, with maximum strength-to-weight ratio, also has maximum stiffness. A spreadsheet program is presented for the optimization of circular and square tubes in compression, subject to dimensional restrictions and specified maximum allowable stress.

Exercises

• 2.1 Verify the efficiency formula for a square section tube in Table 2.1. For a typical aluminium alloy, with an allowable compressive stress of 300 N/mm² and elastic modulus 72,000 N/mm², at what value of structural index is the maximum stress limited by the allowable stress of the material?

  To derive the efficiency formula, assume that the thickness of the tube is small compared with its width, i.e. use simplified formulae for A and I similar to those for the circular tube in Sect. 2.1.

• 2.2 Derive the efficiency of a thin tube of hexagonal cross section (6 equal sides) and simply supported length L under a compressive load P. To compare the hexagonal tube with circular and square tubes, plot the maximum stress of the three sections for a chosen material over a realistic range of structural index.

  Verify that the second moment of area of a thin hexagonal section about any axis through its middle point is $5b^{3} t/2$ , where t is the thickness of the tube and b is the mean width of each side. For the local buckling stress, use the formula: $\upsigma_{L} = 3.62E\;{\kern 1pt} \left( {t/b} \right)^{2}$.

• 2.3 Verify the relation $W\;{\kern 1pt} \propto \;{\kern 1pt} \left( {P/L^{2} } \right)^{ - 1/3} PL$ in Sect. 2.1.1 for the minimum weight of a circular tube in compression. Derive a similar relation for a square tube.

  Use the efficiency formulae in Table 2.1.

• 2.4 Draw the design space for the three-bar truss made of different materials in Sect. 1.1.1.

  Draw the design space with variables $A_{1}$ and $A_{2}$ . Use the formulae in Eqs. (1.1) and (1.2) for the stress in the bars, with allowable stresses as in Table 1.2 and other data in Fig. 1.7 . Plot the maximum stress constraints for both the single material and for the two different materials, Notice that in the second case, the stress in the outer bars will be the critical design condition unless these two bars are removed altogether.

• 2.5 Draw the design space for the three-bar truss under alternative loads in Sect. 1.1.2. Verify the minimum volume of the truss given in that section.

  Draw the design space with variables $A_{{{\kern 1pt} 1}}$ and $A_{2}$ . Use the formulae in Eqs. (1.3)–(1.5) for the stress in the bars. ‘Goal Seek’ can be used in Excel to solve for $A_{2}$ for a series of values of $A_{{{\kern 1pt} 1}}$ in Eq. (1.3). Take $P =$ 100 kN, $\upsigma_{0} =$ 300 N/mm² and $L =$ 1000 mm, as in Sect. 1.1.2.

• 2.6 Use the spreadsheet ‘Circular Tube in Compression’ with different values of minimum thickness to explore the effect of this on the achieved efficiency of the tube.

  Use the values of P, L and E already on the spreadsheet (or any other convenient values). Set $d_{ \hbox{min} } = 0$ , and $d_{ \hbox{max} }$ and $\upsigma_{c}$ large enough to ensure they have no effect. Make a plot of η and d over a range of minimum thickness.

• 2.7 Use the spreadsheet ‘Circular Tube in Compression’ to make a plot of maximum stress against structural index for circular tubes made of various different materials.

  The allowable compressive stress and elastic modulus of different grades of aluminium alloy, steel, titanium and other materials are widely available. Choose P and L for values of structural index that give realistic stress levels for the materials.

• 2.8 Modify the spreadsheet ‘Square Tube in Compression’ for the optimization of a hexagonal tube in compression.

  Use the formulae in Exercise 2.2. Compare the efficiency with the previously calculated value.

• 2.9 Use the spreadsheet ‘Truss with Tubular Members’ to find the optimum layout of the seven-bar truss under buckling constraints.

  Use the material and other data already on the spreadsheet. Try different starting points for the optimization to verify that the same result is obtained. Note the deflection of the truss and the stress in each member of the optimized truss.

• 2.10 Verify the deflection $\updelta$ given in the spreadsheet for the optimized truss in the previous exercise.

  Deflection δ can be calculated from Eqs. (2.7) to (2.9).

• 2.11 Use the spreadsheet ‘Truss with Tubular Members’ to verify that the optimum design for stiffness, in the absence of constraints, has equal stress in all members and has the same shape as the optimum design for minimum weight.

  Use the material and other data already on the spreadsheet (or any other convenient values). First remove all constraints currently in the Solver dialog box. Add a new constraint to set the volume V of the truss to any reasonable value. Deflection δ at the point of loading is given on the spreadsheet. Change the objective to deflection δ in the dialog box, and use Solver to optimize the truss for minimum δ.

• 2.12 Modify the spreadsheet ‘Truss with Tubular Members’ for a load applied vertically upwards at mid-span.

  Members 2 and 4 are now in compression, members 1 and 3 in tension. Compare the strength-to-weight ratio $P/W$ with that for a downwards applied load.