BUCKLING OF WING SPARS UNDER COMBINED LOADING

Transcription

1 5 TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES BUCLING OF WING SPARS UNDER COMBINED LOADING David ennedy*, Dharesh C. Patel*, Carol A. Featherston* *Cardiff School of Engineering, Cardiff University, Queen s Buildings, The Parade, Cardiff CF4 AA, U.. eywords: spars, critical uckling, copression, ending, shear Astract Design rules for the prediction of critical uckling stresses in aircraft wing spars are extended to cover loading cases which include unifor transverse copression. Curve fitting for pure loading cases enales nuerical predictions to e ade of the stiffener diensions required to provide effective siple support to the skin, and of the critical uckling stress for panels of aritrary diensions. A proposed extension of the ethod to coined loading cases is outlined. Introduction Spars in aircraft wing oxes are long integrally achined channel sections consisting of panels separated y vertical stiffeners, as shown in Fig. (a. The purpose of the spar is to provide one of the ain load paths for vertical shear loading. Therefore critical uckling is a ajor consideration in the design of spar panels, either as a priary design criterion or to indicate the onset of postuckling. For the siplified infinitely long spar odels of Fig. (, theory has long een availale [] to predict the critical shear uckling stress for a range of skin and stiffener diensions. Corresponding results for longitudinal copressive loading and in-plane ending have recently een presented []. Coined loading cases are often handled using epirical design rules, or interaction equations, representing (usually quadratic interaction surfaces relating two (or ore of the coponent stresses at critical uckling [-6]. A previous study [] verified the interaction equation SS a I I I I I I I I a (a x SS ( y Copression σ Bending σ x Shear σ s (c Copression σ Bending σ y Fig. (a Diagraatic representation of a wing spar. ( Siplified odel showing siply supported and claped edge conditions on the longitudinal edges. (c Loading cases.

2 DAVID ENNEDY, DHARMESH C. PATEL, CAROL A. FEATHERSTON R R + R = ( + x s for a siply supported isotropic plate with a wide range of aspect ratios, and hence for spar panels, such as those of Fig.(a, whose stiffeners effectively provide siple support to the skin. Here R = σ σ ( Rx = σ x σx R = s σ s σ s where σ, σ x and σ s are the longitudinal copressive, ending and shear stresses at critical uckling in a coined loading case, as shown in Fig. (c, while σ, σ x and σ s are the critical uckling stresses for the respective single loading cases. This paper extends the analysis to spar panels loaded in unifor transverse copression σ. After studying the effect of stiffener size on the critical uckling stress σ, critical uckling stresses are deterined for siply supported plates under coined loading cases which include transverse copression, providing a generalisation of Eq. (. Finally, for the single loading cases, an attept is ade to quantify the stiffener sizes needed to provide effective siple support and the reduction in critical uckling stress when saller stiffeners are used. This enales reliale critical uckling predictions to e ade using siple calculations ased on the spar geoetry. The results in this paper were otained using the software VICONOPT [7], which covers prisatic asselies of rectangular plates, each of which can carry any coination of the in-plane stresses σ, σ s and σ of Fig. (c, which are assued to e invariant in the longitudinal (x direction. A longitudinal ending stress σ is odelled y dividing the x plate into n longitudinal strips carrying longitudinal copressive stresses σ x[ (( i n ], i =,, n. Accura was guaranteed y using n = 7, although in ost cases adequate results are otained with n =. Transverse ending stresses σ y were not considered. The analysis is ased on the exact solution of the governing differential equations of the plates, yielding exact stiffness atrices whose eleents are transcendental functions of the load factor. In the siplest for of the analysis [8], the uckling ode is assued to vary sinusoidally in the x direction with halfwavelength λ. Shear loaded panels are ore accurately odelled y coining such responses [9] for an infinitely long panel whose end supports repeat at intervals of the panel length l. This approach, which has een adopted in the present paper, also perits the stiffeners to e odelled as transverse ea supports with no torsional stiffness. Pure Loading Cases. Review of Previous Results Critical uckling results for spar panels have previously een otained [] for the three load cases of unifor longitudinal copression σ, longitudinal ending x σ and in-plane shear σ s, for the two longitudinal edge support conditions shown in Fig. (. For each case, the VICONOPT results were verified y coparison with pulished results [] or y finite eleent analysis. Design curves for isotropic spar panels are otained as non-diensional plots of the critical uckling paraeter against the stiffener paraeter µ, defined y = σ Et ( tss µ = a t where σ ( =, x, s is the critical uckling stress, E is Young s odulus, the panel has width a and stiffener spacing, while the stiffeners are odelled as lades with depth s

3 BUCLING OF WING SPARS UNDER COMBINED LOADING /a=.9 /a=.8 /a=.67 /a=.58 /a=.4 /a=. Siple support Fig.. Critical uckling paraeter against stiffener paraeter µ for a spar panel with siply supported longitudinal edges, loaded in longitudinal copression. The faint line connects the transition points µ=µ s, eyond which the stiffeners effectively provide siple support. and thickness t s. The torsional stiffness of the stiffeners is neglected. Typical plots for longitudinal copression and shear loading are shown in Figs. and, respectively, for a range of aspect ratios a. Each plot shows a transition etween the liiting cases of an infinitely long unstiffened plate (at µ = and a panel with transverse siple supports repeating at intervals of. During this transition there are changes in the uckling ode shape. In the case of longitudinal copression, the points µ = µ s at which the stiffeners effectively provide siple support are well defined (and plotted in Fig., denoting an arupt change in ode shape. If µ > µ s, atches analytical predictions [] and shows no further increase. In contrast, there is a gradual transition of the shear uckling odes associated with Fig. so that µ s is less well defined and the liiting value of ust e found asyptotically /a=.9 /a=.8 /a=.67 /a=.58 /a=.4 /a=. Fig. Critical uckling paraeter against stiffener paraeter µ for a spar panel with siply supported longitudinal edges, loaded in shear.

4 DAVID ENNEDY, DHARMESH C. PATEL, CAROL A. FEATHERSTON /a=.9 /a=.8 /a=.67 /a=.58 /a=.4 /a=. Fig. 4. Critical uckling paraeter against stiffener paraeter µ for a spar panel with siply supported longitudinal edges, loaded in transverse copression.. Unifor Transverse Copression Using VICONOPT analysis, critical uckling stresses have een found for spar panels loaded in unifor transverse copression σ, for a range of aspect ratios a and stiffener sizes. The results, represented y the paraeters and µ of Eq. ( with σ = σ, are shown in Fig. 4. When µ =, the critical uckling stress atches that of an infinitely long unstiffened plate of width a under unifor transverse copression, which is equivalent to colun uckling for a plate of length a with free longitudinal edges under unifor longitudinal copression. For large values of µ, the stiffeners effectively provide siple support and atches analytical predictions for a siply supported plate []. This condition is reached at a well defined point µ = µ s, which (unlike the loading cases of Figs. and is relatively insensitive to the panel aspect ratio. The draatic change in slope at µ = µ s for each of the plots of Fig. 4 indicates a change in the shape of the critical uckling ode. Non-unifor transverse loading cannot e odelled easily y VICONOPT and so the transverse ending case σ y has not een considered in the present work. However for µ > µ s, soe analogies with the longitudinal ending case σ x can e drawn y considering a siply supported plate with the x and y axes interchanged. Coined Loading Cases. Review of Previous Results Figures -4 show that a spar panel with vertical stiffeners of stiffness µ µ s can e adequately odelled y a siply supported plate of width a and length, representing the portion of skin etween adjacent stiffeners. Such plates of varying aspect ratio a have previously een analysed [] for coined loading cases including unifor longitudinal copression σ, longitudinal ending σ x and in-plane shear σ s. Typical interaction curves are shown in Fig. 5. The results show that, except for soe extree aspect ratios, the predictions given y Eq. ( are accurate to within a few percent and are alost always conservative, i.e. underestiating the stresses at critical uckling. 4

5 BUCLING OF WING SPARS UNDER COMBINED LOADING R Rs= Rs=. Rs=.4 Rs=.6 Rs=.8 Rs= (Eq. Rs=. (Eq. Rs=.4 (Eq. Rs=.6 (Eq. Rs=.8 (Eq Fig. 5. Critical uckling criteria for a siply supported plate with /a=, loaded in longitudinal copression σ, longitudinal ending σ x and in-plane shear σ s. Each solid curve shows VICONOPT results for the copressionending interaction at a particular value of the shear stress paraeter R s, while the corresponding dashed curve shows the criteria predicted y Eq. (.. Inclusion of Transverse Copression The critical uckling criterion for a siply supported plate of width a and length, loaded in unifor longitudinal and transverse copression, has een derived [] as σ π = + n Et ( ν a σ + n a (4 where the uckling ode has and n halfwaves in the longitudinal and transverse directions, respectively, and ν is Poisson s ratio. The linear relationship of Eq. (4 eans that the interactio n diagra etween σ and σ is a series of intersecting straight lines. For exaple, if a = and σ and σ are oth copressive (i.e. positive, then critical uckling always occurs with = n =. However the critical uckling ode can have a higher value of or n if one of the stresses is Rx tensile (i.e. negative. Results otained using VICONOPT atch this analytical prediction. It is therefore suggested that, for coined loading cases including transverse copression, the ratio R = σ σ (5 plays a siilar role to that of R in the interaction equation. This has een confired y studies of the interaction etween transverse copression and shear loading for aspect ratios in the range.5 a, giving the revised, usually conservative, critical uckling criterion R R + R + R = (6 + x s More evidence is needed to confir that the critical uckling criterion can e further extended to give where R R + R + R + R = (7 + x s x Ry = σ σ (8 y y 5

6 DAVID ENNEDY, DHARMESH C. PATEL, CAROL A. FEATHERSTON and σ y is the critical uckling stress under transverse ending. 4. Prediction of Critical Buckling Stress 4. Paraeter Estiation The effects of stiffener size on the critical uckling stress of a wing spar have een shown in Figs. -4 for pure loading cases. This section illustrates, for a panel of aritrary aspect ratio, the estiation of the stiffness paraeter µ s eyond which the stiffeners effectively provide siple support. Estiates are also given of the critical uckling paraeter when µ < µ s. The analysis is presented for panels loaded in unifor longitudinal copression, using the results of Fig., and can readily e applied to other pure loading cases. The faint line in Fig. connects the points where µ = µ s for different aspect ratios a. Least squares fitting of a cuic polynoial to this data gives the relationship ( a µ = (9 s ( a. 984( a with correlation coefficient r = The asyptote µ s =. is assued when a >. Attepts were ade to fit polynoial curves to the against µ curves of Fig., ut this proved unreliale due to the changes in gradient at µ = µ s. In general a etter fit was otained y using the Boltzann curve = ( µ α ( µ α e + e ( µ α ( µ α e + e ( illustrated in Fig. 6. Here varies onotonically etween a lower liit and an upper liit, taking the value.5( + at µ = µ. The paraeter α controls the rate of increase fro to. Least squares curve fitting gave the paraeters µ =.5 =.56 =.498 α =.89 ( with correlation coefficient r = for the aspect ratio a =. 5 (which is not shown in Fig.. As expected, the fitted values of and are close to the values of at µ = and µ = µ s, respectively, and µ. 5µ s. However considering only the points in the range µ < µ gave an even etter fit µ Fig. 6. Boltzann curve of Eq. (, with =, =, µ =.5 and α=. s =.59 =.6 =.8 α =.8 ( with correlation coefficient r = Although this refineent tends to overestiate the upper liit of, it has een adopted hereafter in a hyrid for which predicts fro the Boltzann curve when µ < µ s and fro analytical results for siply supported plates [] when µ µ s. The hyrid approach correctly predicts the changes in gradient in the against µ curves at µ = µ s. Repeating the Boltzann curve fitting for different aspect ratios a gives failies of values for the paraeters,, µ and α. Next, these paraeters are each fitted y cuic polynoials in a, with correlation coefficient r >.999 in each case, to give 6

7 BUCLING OF WING SPARS UNDER COMBINED LOADING µ = = = α = ( a ( a +.745( a +.56( a ( a +.7( a.9( a ( a.89( a.997( a ( a.5956( a 4. Nuerical Predictions ( Using the paraeters estalished in section 4., it is now possile to predict the critical uckling paraeter for wing spars of aritrary skin and stiffener diensions. A siple Visual Basic coputer progra has een written to the specification of Tale. Fig. 7 gives soe illustrative results for a spar panel loaded in unifor longitudinal copression, for three different aspect ratios a. The results show that the values of predicted y the progra are conservative and accurate to within a few percent. Tale. Prediction of critical uckling paraeter for a pure loading case. Step Task Input skin diensions a,, t and stiffener diensions s, t s. Calculate µ using Eq. (. Calculate µ s using Eq. (9. 4(a If µ < µ s, calculate using Boltzann curve of Eq. ( with paraeters,, µ, α given y Eq. (. 4( If µ µ s, calculate using analytical results for a siply supported plate []. In order to extend the predictions to coined loading cases, it is convenient to define a stress vector { σ, σ, σ, σ, σ } s = (4 x s It is required to find the critical uckling factor F, such that uckling occurs when y { σ σ, σ, σ σ } s = F (5, x s, y where σ ( =, x, s,, y represents a set of ase stress values. Steps and 4 of Tale ust e carried out for each coponent of stress σ, giving a faily of paraeters, fro which the critical uckling stresses σ for each of the pure loading cases can e calculated /a=.67 (actual /a=.67 (predicted /a=.5 (actual /a=.5 (predicted /a=. (actual /a=. (predicted Fig. 7. Actual and predicted critical uckling paraeters for a spar panel with siply supported longitudinal edges, loaded in longitudinal copression. 7

8 DAVID ENNEDY, DHARMESH C. PATEL, CAROL A. FEATHERSTON using Eq. (. Then, using ase values of the stress ratios R (6 = σ σ in Eqs. (7 and (5, the critical uckling criterion F is otained fro the equation F ( Rx + Rs + Rx + F ( R + R = which always has one positive root. 5 Concluding Rearks (7 Paraetric studies have previously een carried out, using an exact strip ethod, on the effect of stiffener diensions on the critical uckling of aircraft wing spars sujected separately to longitudinal copression, in-plane shear and ending stresses. In each case, increasing the stiffener second oent of area increases the critical uckling stress fro its theoretical value for an unstiffened plate of infinite length to a higher value atching that of an unstiffened plate with siply supported ends and length equal to the stiffener spacing. Results for coined loading cases can e predicted using a siple interaction equation. The present work has extended the analysis to wing spars loaded in pure transverse copression, and has also estalished interaction relationships for coined load cases which include transverse copression. Nuerical predictions have een ade of the stiffener diensions required to provide effective siple support to the skin, for a range of panel aspect ratios. For pure loading cases, the relationship etween critical uckling stress and stiffener diensions has een fitted to a non-diensional Boltzann curve, so that the critical uckling stress can e estiated siply and accurately for panels of aritrary diensions. A proposed extension of the nuerical predictions to coined loading cases has een outlined. The proposed ethods for predicting critical uckling stress are expected to e of value in the aerospace industry, where design curves and other siple heuristic ethods continue to e used in the initial design stage ecause they provide quick solutions without the need for extensive data preparation. Acknowledgeents The coputer progra used to otain the results of section 4. was written y Mr Jonathon Bell. References [] Engineering Sciences Data Unit. Flat panels in shear. Buckling of Long Panels with Transverse Stiffeners. ESDU.., 97. [] Featherston CA, ennedy D, Weston DJ and offi. Design rules for panel uckling under coined shear, copression and ending. Proc 4th Int. Conf. on Thin-Walled Structures, Loughorough, U, pp 5-6, 4. [] Johnson AE Jr and Buchert P. Critical coinations of ending, shear and transverse copressive stresses for uckling of infinitely long flat plates. NACA Technical Note 56, 95. [4] Gerard G and Becker H, Handook of structural staility. Part uckling of flat plates. NACA Technical Note 78, 957. [5] Plank RJ and Willias FW. Critical uckling of soe stiffened panels in copression, shear and ending. Aeronautical Quarterly, Vol. 5, Part, pp 65-79, 974. [6] Ueda Y, Rashed SMH and Paik J. Buckling and ultiate strength interaction in plates and stiffened panels under coined inplane iaxial and shearing forces. Marine Structures, Vol. 8, No., pp -6, 995. [7] ennedy D, Willias FW and Anderson MS. Buckling and viration analysis of lainated panels using VICONOPT. ASCE Journal of Aerospace Engineering, Vol. 7, No., pp 45-6, 994. [8] Wittrick WH and Willias FW. Buckling and viration of anisotropic or isotropic plate asselies under coined loadings. International Journal of Mechanical Sciences, Vol. 6, No. 4, pp 9-9, 974. [9] Anderson MS, Willias FW and Wright CJ. Buckling and viration of any prisatic assely of shear and copression loaded anisotropic plates with an aritrary supporting structure. International Journal of Mechanical Sciences, Vol. 5, No. 8, pp , 98. [] Tioshenko SP and Gere JM. Theory of elastic staility. nd edition, McGraw Hill, 96. 8