OPTIMUM DESIGN OF STIFFENED PANEL USING THE METHOD OF MATHEMATICAL PROGRAMMING

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1 ICA 000 COGRE OPTIMUM EIG OF TIFFEE PAEL UIG THE METHO OF MATHEMATICAL PROGRAMMIG o. Ing. Anonín Píš!k C. Ing. P"esl Hobza Insiue of Aerospae Engineering Faul of Mehanial Engineering rno Universi of Tehnolog Tehniká Czeh Republi Kewords: opiizaion panel sruure Absra The opiu design proble is one of he inverse asks ha eans he sruure paraeers are alulaed aording o a known rieria of he seleion aheaiall desribed as a os funion. The faor deerining he developen of opiu design besides he hardware equipen is he developing of he new ehods and new aheaial prograing algorihs. The opiu design of a skin sringer panel using he ehod of non-linear prograing of Hielblaua [] is presened in his paper. The os funion is he ass of he panel. The onsrains has o be hosen so ha our panel should saisfied he requireens for loal and global panel bukling loal and global sringer bukling and i should saisf as well as he sai resisane deands. esign paraeers basi diensions and he loading of he panel Opiu design of our panel will be defined b following design paraeers: - skin hikness n v - nuber of sringers () v - referene sringer hikness h v - referene sringer heigh Panel skin is loaded b he noral sresses σ σ and shear sress τ. The following noaion will be inrodued for he oninuous noral edge loading and he shear loading : σ σ τ () upposing ha he sringers are loaded b he fore orresponding wih he σ sress hen V F σ () v v L F h V Figure. asi diensions and he posiive orienaion of he panel loading. A.

2 Píš!k A. Hobza P. Coninuous loading a are onsan hrough he panel edge and he noral sresses generaed in sringers are he sae. he sringer hikness we will undersand he haraerisi diension of a general seion. The real seion area and he oens of ineria will be deerined using he diensionless oeffiiens: Fv KFV hv v I V K I v hv () Coeffiiens K FV a K I are for hosen seion onsan. For opiu panel design we will suppose he loading whih is desribed on figure and we will onsider hese panel failures:. loal skin bukling beween wo sringers. sringer bukling. global bukling of he panel. reahing he UT or YT of he used aerial Loal skin bukling beween wo sringers The skin beween wo sringer as well as whole panel an be loaded b he linear fores a. The bukling rieria of he loal bukling of he skin is desribed b following equaion: σ kr σ kr τ kr (5) where σkr σkr a τ kr are he riial sresses of he loal bukling for eah single loading The riial sress will be of he following for σ ikr k σ ( i ) (6) i E where k k k are he oeffiiens of sabili a σ E is so alled Euler s riial sress whih will be of he following for π. E σe v v µ ( ) ( n ) A ( n ). (7) The sabili oeffiiens k k a k depend on he edge boundar ondiions and diensions. If he oeffiiens k i alos does no hange and i is possible o onsider he as independen on he diensions ( Ab v ) hen edge raio is A/ b v for he onsidered wing panels. ringer bukling he sringer design paraeers v a h v is undersood he haraerisi sringer diensions The sringer an be a general arbirar seion. For diension opuing following assupion will be onsidered:. he noral sringer sress has he sae value wihσ in he skin.. he noral sringer sress has o be equal o he riial bukling sringer sress. σv σ σvkr. (8) ine for he skin and he sringer he ondiion of he sae deforaion ( ε εv) is valid we an obain he sringer noral sress using jus he Hook s law σv E ε µ. (9) Criial sringer bukling sress will be desribed using he idenial for wih he riial skin sress σ k σ (0) where σ ve vkr v ve π E ( µ ) v h v. () ringer oeffiien of sabili k v is a onsan for hosen sringer..

3 OPTIMUM EIG OF TIFFEE PAEL UIG THE METHO OF MATHEMATICAL PROGRAMIG Global bukling of he panel The global bukling of he panel is suh a shape when he big deforaion in several half waves of our panel suppored b wo spars and wo ribs arrives. upposing ha he invesigaed panel is an orhorophi plae sipl suppored on eah edge. The loading is desribed in figure. The panel is loaded b he oninuous noral load and he shear flow on eah edge. The referene hikness will be Fv nv KFV nv v hv. () iilarl like in ase of loal skin bukling we an use for global bukling he sae equaion σkr σ kr τ kr () where σkr σkr a τ kr are he riial global bukling sresses for eah siple loading. For he applied loads in he sringer direion he differenial sabili equaion beoes o w w w w 0 () Finall we an obain he following equaion for he global bukling riial sress π A A σkr A A. (5) Where: E (6) ( µ ) skin lindrial siffness of an uni widh E I (7) global longiudinal panel lindrial siffness and I I (8) I is he oen of ineria of he enire seion in he enre of he gravi of he panel seion. The differenial equaion for he riial sress of global bukling of he panel loaded perpendiularl o he sringers beoes w w w.. w 0 (9). and hen σ kr π A π E A ( µ ) E I (0) The riial global bukling sress of he panel loaded b he shear flow is desribed b he following equaion w w w w 0 () For he plaes wih he onsiderable anisorop when he siffness in one direion is uh higher hen he one in oher direion he following equaion is valid for he panel loaded wih he single shear τ kr π nv κ γ nv κγ A nv κγ () 5 Cos funion and he funion of he panel onsrains fro he poin of view of he aheaial prograing. For opiu design of a siffened panel one varian of he Hielblau and Paviani nuerial opiisaion ehod was developed whih is published in lieraure [] fro where we have also used he odified ode developed in FORTRA. The ode is able o find he.

4 Píš!k A. Hobza P. eree of he funion W( ) in ase of he equali onsrains h i ( ) 0 i... and he inequali onsrains g i ( ) 0 i...p T where (... n ) is he veor of design variables. The following funions define he proble of he opiu design of a siffened panel: Equali onsrain is no presribed 0. Inequali onsrains:. Loal skin bukling beween wo sringers is given b he equaion (5) whih beoes: g ( ) where A k A k A. k ( ) A A. and π E A ( µ ). Modifing he equaions (8) a (9) we an obain he following equaion for he loal sringer bukling g ( ) A µ v π E kv Av ( µ ).. The global panel bukling is given b inequali () whih we odif o obain: g ( ) when [( α ) G] A α ( ~ ) A. ( G) α A G G ( G) α G K ( µ ). κ ~ K Fv. The srengh is inluded in he onsrain g ( ) R. 5. The liiaion of he nuber of sringers is one of he design-ehnologial liiaions: g5( ) nv a where n v a engaged aiu nuber of sringers 6. Liiaion of he iniu skin hikness is given b he funion: g ( ). 6 in 7. The aiu skin hikness is liied siilarl g ( ). 7 a The hikness in and a are he inpu daa..

5 OPTIMUM EIG OF TIFFEE PAEL UIG THE METHO OF MATHEMATICAL PROGRAMIG 8. The sringer heigh is defined b he funion g8( ) hv a. 9. Liiaion in equaion F V I V W kg 6 5 g ( ) 9 assures he regulari of he aspoi values of oeffiiens of sabili k. W Than he os funion is obained as follows W( ) ρ ( A K A). Fv F V 6 Eaples As an eaple we ake a wing panel for a sall ranspor airraf. The inpu daa are: A 00 [] panel lengh ( rib spaing ) 5 [] panel widh 65 [/] load in he sringer direion 0 [/] raverse load 75 [/] shear load K Fv 5 [] sringer area oeffiien K 96 [] sringer oen of ineria oeffiien K V 6 [] sringer sabili oeffiien κ [] influene of sringer aser R 80 [MPa] allowed ension sress I V uber of sringers n v Figure. Relaion beween he panel opiu diensions and he nuber of he sringers. The haraerisis of he Z sringer orrespond wih he profile PR 05 and he opuaions fro lieraure []. The ehod presened in his paper was used o design opiu nuber of sringers for a wing skin-sringer panel using he sae inpu daa. Figure shows he resuls for he differen nubers of sringers varing fro o 0. For he ehnologiall favorable hikness.5 opiu nuber of sringers is 6. oe ha all he designs saisf he redisribuion load requireens however ass differene beween he opiu panel and one sringer panel is alos double. 7 Conlusion Appliaion of Maheaial - Prograing Mehod o he opiu sruural design of a siffened panel is aep o lass he opiisaion alulaion o general sruure.5

6 Píš!k A. Hobza P. design using he nuerial ehods. espie of soe shorage he resuls are valuable for he designers beause he provide he in advane iporan inforaion abou basi design paraeers. ased on alulaion ino whih is neessar o add eperiene and ehnologial speial requireens i is possible o hoose he wing onepion so ha he global ass effeive sruure of he plane ould be reahed. The alulaion enables o follow up design paraeers sensiivi of he ass effeive skin-sringer sruures. 8 Referenes [] LipinE.K.: Opialnoje projekorovanije podkreplenh panelej p!i kobinirovanno nagru"enii. U#onnje zapisk CAGI I 978. [] Hielblau.M.: Prikladnoje nelinejnoje projekirovanije Mir Moskva 975. [] Tioshenko.: Theor of Elasi abili. MGraw- Hill ook Copan ew York 96. [] Píš$kA.: Kandidáská disera#ní práe rno 980. [5] irjuk.i. LipinE.K. FrolovV.M.: Meod projekirovanija konsrukij saoleov. Mašinosrojenije 977. [6] GallagherR.H. ZienkiewizO.C.: Opiu ruural esign John Wile and ons London 97..6