Fig. 1. Loading of load-carrying rib

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1 Plan: 1. Calculation o load-carrying ribs. Standard (normal) ribs calculation 3. Frames calculation. Lecture # 5(13). Frames and ribs strength analysis. 1. Calculation o load-carrying ribs On load-carrying ribs except distributed air and mass loads the additional concentrated loads act rom a landing gear, rom the engine, uel, and so on. At calculation o loadcarrying ribs we use ollowing assumptions. The wing skin and the spar webs work only on shear. The load-carrying rib is rigidly connected to a skin o a wing and o spar walls. Let's consider a load-carrying rib calculation technique. Let us have a load-carrying rib o a thin-walled design (ig. 1), loaded by a concentrated orce P, or example, rom engine mounting. Fig. 1. Loading o load-carrying rib The shear low rom the shear orce Q u in ront q Q and rear q rq spar webs are determined by the ollowing ormulas, as they are distributed by proportionally to bending rigidity: QuI q Q (1) I I QuIr q rq =, I Ir where I, I r are moments o inertia accordingly or ront and rear spars, Q u is shear orce in this cross section rom engine weight. r 1

2 For calculation o constant SF on whole contour let us write the equation o equilibrium or moments relative point 1 on ront spar: Pa q 0 ( b sp h + b sp h r ) + q rq b sp h r = 0 () Р Fig.. Analytical model o load-carrying rib. Whence: Pa qrqbsphr q0, bsp( h hr ) where b sp distance between spars, h, h r accordingly heights o ront and rear spars. For calculation o total reactions in spar webs we'll have the ollowing ormulas: Q = ( q Q + q 0 ) h, (4) Q r = ( q Qз - q 0 ) h з. Further we build shear orces diagrams and the diagrams or bending moments on length o a rib (see ig. 1). max max M hf п, Q h, where F the rib lange cross-sectional area; the thickness o a rib web; h the altitude o a rib web.. Standard (normal) ribs calculation The standard rib calculation is basically by nothing diers rom a load-carring rib calculation. At the precise standard rib calculation its also accepted, that a wing skin and spars serve as the ribs legs. At the approximated calculation its considered, that the rib has only spar legs and the wing skin reactions are not taken into account. At the calculation on external loads operating on a rib, in approximated statement we need to determine the legs reactions and to plot the shear orces and bending moments on length o rib diagrams (ig. ).

3 Fig. 3. Diagrams o shear orces and bending moment or rib. The normal stresses in langes and the shear stresses in a rib web can be determined by the ormulas M hf, Q h. It is possible to determine the shear stresses in a web with acilitation holes by the approximated ormula Q ( h d ), where d the hole diameter. I a standard rib web is cut in a middle part on a diagonal, then in the rib calculation it should be considered as a rame, and it is necessary to actuate in the rib area langes and part o a web. 3. Frames calculation. As against ribs the air load come on a rame contour is inappreciable and it is usually not taken into account, taking into consideration only the orces rom aggregates or reights with mounting points located on a rame. There, where the air load is great, it should also be taken into account. Thus, the rame loading is determined completely. On ig. 4, 5 the analytical models o loaded rames are shown. In a ig. 4 there is the load rom the aggregate, in a ig. 5 the load rom vertical unit sectionals. Now it is necessary to calculate these rames by known rom mechanic o structure methods. 3

4 Fig. 4 Fig. 5 Fig. 6 Sometimes or calculations simpliication it is possible to use the shel solutions available in the reerence literature. With the reerence to ring rames o constant crosssection the ormulas and plots o the bending moments or three kinds o concentrated loading by radial orce P (ig. 6), tangent T (ig. 67 and moment m (ig. 8) are resulted. Using a superposition principle with the help o these solutions it is possible receiving the bending moments or rings o constant cross-section at any loading easily. With some approximation these solutions can be used and or calculation o ring rames with eebly varied cross-section by a contour. In plane's uselage there is a usually located the great quantity o rames not carrying the concentrated load (normal rames). For this rames (especially in a large plane's uselages) their can is essential a load being a corollary o uselage axis bending deormation. Let's allocate a ring o length dx rom a uselage with the help o cross sections and then igure it in a deormed condition under some bending moment М z operation (ig. 9). In a ig. 10 its cross-section is shown schematically. Let's allocate rom a design one o stringers with a skin aixed to it (ig. 11). Fig. 7 Fig. 8 4

5 Fig. 9 Fig. 10 It is easy to see, that the P i orces give vertical component P dx i dn P sin Pi. As M z 1 Pi i Fy i i I z and M z EI z, than we'll get M z dn i ifydx i i EI z (1) These orces together with у i coordinate change the sign and thereore the ring appears loaded as it is igured in a ig. 1. Toting orce (1) by all ring, it is easy to convince, that they are mutually counterbalanced. Thin skins are not capable to counteract the cross-section lattening eect and this role is laid to normal rames. I the arrangement pitch o the last is equal to a than each o normal rames loading will be recorded as: a Mz i Ni Fydx i i EI a z, at the stationary M z values and it gives Mza ifiyi Ni EI z. () 5

6 Fig. 11 Fig. 1 The normal rame should counteract lattening made by this load, which application scheme is similar igured on a ig. 1. The calculation can be conducted by known mechanics methods. However it is necessary to mean, that the suicient normal rame strength detected by such calculation, does not guarantee the uselage will not latten. This means that as a result o uselage rame lattening under an system o N i orces operation the inertia moment o area I z will decrease and that will cause the urther lattening loads growth [see ormula ()]. At some normal rame bending stiness value such process can appear continuous and the uselage will be destroyed. This phenomenon has a character o stability loss and it appears more probable in case o large plane uselages. With some reserve or the normal ring rame minimum inertia moment o area can be determined by the empirical ormula 4 M z R a m I r 0.7 (3) E I z Here Мz the bending moment o a uselage cross-section by a rame; R the circle radius o a uselage cross-section; a a pitch o normal rames statement; the "thickness" o a uselage skin with the stringers, "spread" on a contour. It is equal to the area o a stringer with an aixed skin, divided on a stringer pitch. It is supposed here, that m is constant over the contour: st t sk m, t EI z - bending uselage stiness in this cross-section. In summary let's mark, that rames calculation methods explained here [except or the ormula (3)] are rested in essence on a hypothesis about a nondeormability o a rame contour in its plane. The similar calculations with reerence to ribs were justiied enough by that the last usually have the web. The rib deormations in its plane appear minor in this case, and they can be neglected. 6

7 Other picture is observed in a rame. The last, as a rule, one has essentially changes its shape under load. The rame contour points displacements, observed at it, call additional shits in a skin and by that change the bending moments on a rame. Thus the system o additional stress resultants and moments appears, essentially varying a character o a stress state as o a rame, and o skin. The rames calculation including its deormations in its planes is represented by considerably more composite problem. This problem is not illuminated here. Let's mark only, that the rectiication to the rame elasticity will reduce the bending moments o this rame and will give an additional shit load to the applied skin. 7