Reinforced Shell Structures

Abstract

Formulae are given for the bending and shear stress in a long shell structure such as an aircraft wing, together with formulae for the different modes of buckling. These are used in the optimization of some reinforced shell structures. Alternative methods for modelling the reinforcement are discussed. At higher stresses, yielding of the material causes a redistribution of stress in the cross section and reduction in buckling stress. The Ramberg–Osgood formula is used to define the stress–strain curve of the material and to derive formulae for the tangent and secant moduli. Efficiency formulae are developed for stiffened panels in compression and shear. A spreadsheet program for a panel with integral, unflanged stiffeners under compression and shear optimizes the cross section, subject to minimum stiffener spacing and plate thickness, with reduced moduli for buckling. The same program is incorporated in a spreadsheet for the optimization of a rectangular box beam under bending and shear, with variable rib spacing, subject to material stress limits and buckling in different modes. A third spreadsheet program optimizes the cross section of a circular fuselage section, with variable skin thickness and stringer dimensions around the cross section.

Exercises

Unless otherwise indicated, in exercises 7.3–7.10 use the loading and other data already entered in the original spreadsheet.

• 7.1 Make a hand calculation of the flexural and local buckling stresses of the panel with integral, unflanged stiffeners in Fig. 7.14, loaded in axial compression. The stiffener spacing is 120 mm, the stiffener height is 30 mm, and the thickness of both plate and stiffeners is 3.0 mm. The effective length of the panel is 800 mm, and the elastic modulus of the material is 72,800 N/mm². Calculate the efficiency of the panel, based on the lesser of the two buckling stresses.

  Note that the stiffener height is measured to the mid-plane of the plate. Take the local buckling coefficient $K = 3.95$ in Fig. 7.15 . Assume no reduction in modulus due to yielding. Calculate the efficiency from Eq. (7.27).

• 7.2 A stiffened shear web of height $h = 500$ mm and stiffener spacing $d = 250$ mm has stiffeners designed to satisfy the criterion $\upmu = \upmu_{c}$ (taken to be the optimum). The stiffeners have a coefficient $C = 1.0$. The elastic modulus $E = 72{,}800$ N/mm² (neglect any effect of yielding). Calculate the equivalent shear stress $\uptau^{\prime}$ in the web under a shear force $Q = 60{ , 120 , 240}$ kN. Use the three results to deduce an efficiency formula for a stiffened shear web of this type in the form:

  $$ \uptau^{\prime} = \upalpha \,E^{1 - n} \left( {q/h} \right)^{n} $$

  Take $\upmu_{c}$ and $K$ from Table 7.3 , with the long edges of the web simply-supported. Use Eqs. (7.31) and (7.30) to calculate t and $t^{\prime}$ , and calculate the equivalent shear stress at each load. Make a ${ \log }\left( {\uptau^{\prime}} \right)$-$\log \left( {q/h} \right)$ plot to verify the above relation. Deduce values of α and n from the log-log plot, or numerically.

• 7.3 Use the spreadsheet ‘Stiffened Panel’ to optimize a panel with minimum stiffener spacing $b{\kern 1pt}_{\hbox{min} } = 100$ mm and minimum plate thickness $t_{{{\kern 1pt} \hbox{min} }} = 1.0$ mm, and with these dimensional constraints removed. Determine also the maximum efficiency of the panel.

  Set $b{\kern 1pt}_{\hbox{min} }$ and $t_{{{\kern 1pt} \hbox{min} }}$ to the required values and optimize the panel. Compare the dimensions of the panel and its efficiency with those of a panel with these limits removed. For maximum efficiency, we should avoid any reduction in modulus. Set $b_{\,\hbox{min} }$ and $t{\kern 1pt}_{\hbox{min} }$ to zero, and reduce the loading intensity p until $E_{t} = E_{s} = E$ after optimization. Try different initial design variables to ensure that a true optimum has been found, and not a local minimum.

• 7.4 Use the spreadsheet ‘Stiffened Panel’ to verify the hand calculation in Exercise 7.1.

  Enter the dimensions and effective length from Exercise 7.1 (no optimization required). Compare the second moment of area and buckling stresses with the calculated values. For the correct efficiency, the compressive stress in the panel has to match the lesser of the flexural and local buckling stresses. Try different values of loading intensity p until it is close enough (alternatively use Goal Seek in Excel). Verify that there is negligible reduction in modulus due to yielding.

• 7.5 Use the spreadsheet ‘Stiffened Panel’ to show the sensitivity of the flexural and local buckling stresses to stiffener spacing, stiffener height, plate thickness and stiffener thickness for the panel in Exercise 7.1.

  Enter the dimensions and effective length from Exercise 7.1. Make a one per cent increase in each dimension in turn, and note the increase or decrease in flexural and local buckling stress. The data can be used to select the best dimension to change to improve one buckling stress with the least effect on the other, or to obtain constraint gradient data.

• 7.6 Use the spreadsheet ‘Rectangular Box Beam’ to optimize the beam for a range of rib spacing from 100 to 1000 mm, under the bending moment, shear force and other data given in the spreadsheet.

  Remove rib spacing from the list of variables in Solver before optimizing the beam for different rib spacing. Try different initial design variables to verify convergence. Make a plot of minimum mass per metre against rib spacing.

• 7.7 Optimize the beam in the spreadsheet ‘Rectangular Box Beam’ under a range of bending moment M from 100 to 1000 kNm with corresponding shear force $Q = \frac{M}{5}$ (in kN), to show the relation between mass per metre of the beam and applied loading.

  Set the minimum cross-sectional dimensions to zero to have no effect on the optimized beam. Ensure that rib spacing is included in the list of variables before optimizing the beam under different loading. Make a plot of mass per metre against bending moment. Note the relatively small increase in mass per metre with increase in loading.

• 7.8 Use the spreadsheet ‘Circular Fuselage Section’ to show the effect of different frame spacing on the minimum mass per metre of the fuselage structure, both with minimum skin thickness $t{\kern 1pt}_{\hbox{min} } = 1.6$ mm and minimum stringer spacing $b_{{{\kern 1pt} \hbox{min} }} = 120$ mm and with no minimum skin thickness and stringer spacing.

  Repeat the optimization for different frame spacing over a range from 200 to 1000 mm. Note the change in the mass of the frames and the mass of the fuselage shell with different frame spacing. Add frame spacing to the list of variables in Solver for the optimum frame spacing.

• 7.9 Calculate the efficiency of the stringer-skin panel in the spreadsheet ‘Circular Fuselage Section’, at the point of maximum compressive stress, after optimization with minimum skin thickness $t{\kern 1pt}_{\hbox{min} } = 1.6$ mm, minimum stringer spacing $b_{{{\kern 1pt} \hbox{min} }} = 120$ mm and frame spacing $L = 500$ mm. Compare the efficiency with that of the same panel with no constraint on skin thickness and stringer spacing.

  Deduce the loading intensity p from the compressive stress and optimized dimensions of the panel, and substitute in Eq. (7.27) for the efficiency. For the maximum efficiency, set the constraints on minimum skin thickness and stringer spacing to zero. Repeat the optimization and recalculate the efficiency. This is the maximum efficiency for this type of panel with $d/h = 0.4$ . Observe the difference in dimensions of the panel with and without dimensional constraints.

• 7.10 Add a constraint on the ratio of flexural to local buckling stress: $\upsigma_{F} /\upsigma_{L} \le \;\,0.85$ to the spreadsheet ‘Circular Fuselage Section’, to reduce the imperfection sensitivity of the panels. Take $b{\kern 1pt}_{\hbox{min} } = 120$ mm and $t_{{{\kern 1pt} \hbox{min} }} = 1.6$ mm. Compare the optimized mass per metre with $\upsigma_{F} /\upsigma_{L} \le \;\,0.85$ with the optimized mass per metre with no added constraint.

  There are substantial margins of safety on allowable stress in both load cases. To avoid the limit on the number of constraints in Solver, these constraints may be removed and replaced by constraints on $\upsigma_{F} /\upsigma_{L}$ at each location. Note the small increase in mass as a result of the constraint on $\upsigma_{F} /\upsigma_{L}$.