Guide for home task CALCULATION OF WING LOADS. National Aerospace University named after N.Ye. Zhukovsky Kharkiv Aviation Institute

Transcription

1 National Aerospace University named after N.Ye. Zhukovsky Kharkiv Aviation Institute Guide for home task by course Strength of airplanes and helicopters CALCULATION OF WING LOADS Kharkiv, 2014

2 Manual is dedicated to highly-aspect ratio wing loading. It consists of the following sections: wing geometric parameters and mass data estimation; loads calculation and diagrams plotting of shear forces, bending and reduced moments on wing span. 2

3 INTRODUCTION The given home task is continuation of your home task on department "Designing of planes and helicopters ". In this task you have defined take-off mass of the plane, its cruiser speed, mass of a wing, mass of fuel, mass of a power-plant, mass of the landing gear, mass of useful loading. From this task you have all geometrical sizes of the plane: the wing area, wing span, swept-back wing, chords of a wing, position of engines, the landing gear, etc. For this plane you should calculated limit load factors in this home task on discipline "Strength of planes and helicopters ". The given home task should be included into your course project and after your course project should be included into your baccalaureate project. For all students we give critical loading condition C - flight on cruise speed VC on cruise altitude H C with maximal maneuvering load factor n l. Students are obliged to follow these requirements according to international standard: 1. All diagrams at the figures should contain starting and ending values of the illustrated variable (not literal expression of the variable). 2. All diagrams should be built to some scale (the scale should be the same for all diagrams illustrated at one figure). The shape of the curves should correspond to the functions. 3. All calculations should be made with high accuracy. 4. The cover page is executed according to Appendix # Each table must be on one page. 6. Each table and drawing must have heading. 7. Standard rule for whole world from the beginning you should write down formula, next step you should substituted numbers and at last write down result with units. 8. You should not rewrite reference data, drawings from manual, and explanations. 9. Home task must be printed. In explanatory book you should print home task content in next execution sequence: 1. Three main views of your plane. 2. Table #1. Main data of the plane. 3. Determination of limit load factor. 4. Air loads allocation on wing span. 5. The wing structure mass load allocation. 6. Calculation of the total distributed load on a wing. 7. The shear forces, bending and reduced moments diagrams plotting. 8. Load checking for wing root cross section. 9. Calculation of shear force s position in the design cross section 10. Filling the result table. 3

4 1. PLAIN GENERAL DATA By data your plane you should fill the next table. Table #1 Main data of the plane Airplane category Take-off mass (kg)-m t Design airspeed (km/h) - V Design cruise altitude of flight (m) - H Wing mass (kg) - m w Total fuel mass (kg) - m f Mass of engine (kg)- m e Mass of main leg of landing gear (kg)- m ml Wing area (m 2 ) - S w Wing span (m) (for swept wing) - L w Wing taper Wing aspect ratio Swept wing (degree by 25% chord) Comment. a. Masses in integer kg. b. Speeds in integer km/h 4

5 1.1. WING S GENERAL DATA The airplane category (transport, normal, acrobatic, etc.), take-off mass М t are given to the final development s assignment. From previous course project on discipline Airframe the following data is made out: 1) Wing geometrical data; 2) Masses of wing-accommodated units (masses of engines, landing gears, external and internal fuel tanks, etc.), and each of theme s centers of gravity; 3) Additional data: engine type, thrust of engine, maximal flight speed, landing speed, cruise speed and the flight range. These data may be necessary in future; 4) Airfoil s average relative thickness and number. If the geometrical sizes not specified in exposition (lengths of root and tip chords, centers of gravity aggregates for units etc.) are possible to remove directly from the drawing. It is impossible to select a delta-wing airplane as in the given manual the design procedure of a highly-aspect wing is explained as the prototype WING S GEOMETRICAL DATA Geometrical data of a wing make out from exposition of the airplane. Under these data it is necessary to execute figure of a half-wing in two projections (top and front views). If a plane has swept-back wing and a sweep angle on a leading edge are more than 15 it is necessary to enter an equivalent straight wing and all further calculations to carry out for this equivalent wing. A straight wing enter by turn of a swept-back half-wing so that the stiffness axis of a straight wing was perpendicular to axes of a fuselage, thus the root b r and tip b t chords sizes decrease, and the size L w /2 of a semi span is increased. The sizes L w /2, b r and b t, are required for the further calculations and they are taken directly from the figure. At turn of a sweptback wing it is possible to mean, that the stiffness axis is located on distance 0.4 b from the leading edge where b is the wing chord. At calculation of L w /2 semi span size the plane's design is taken into account: a low-wing, a mid-wing or a high-wing monoplane. In these designs carrying ability of the fuselage part of wing owing to influence of interference is various. For a low-wing monoplane it is recommended do not take into account bearing ability of an fuselage part of wing and to accept value equal to distance from the tip of a half-wing (straightened in case of a swept-back wing) up to an onboard rib as L w /2 parameter. For a mid-wing monoplane and a high-wing monoplane in quality semi span we receive the distance from the tip of a wing up to an axis of the plane as L w /2. The wing area S w is determined under the formula: S w = 0.5 (b r + b t ) L w. (1.2.1) 5

6 The received value of the area must coincide with the area of the airplane with swept-back wing for mid-wing and high-wing airplanes. For convenience of realization of the further calculations figures of a wing (see fig , and 1.2.3) should contain a maximum quantity of the information. So, on a top view of a half-wing following characteristic lines are put by dotted, a stroke dotted or by light lines: a center-of-pressure line, a center of gravity (c.g.) line of cross sections of a wing and lines of spars. The locations of aggregates' centers of gravity (landing gears, engines, fuel tanks etc.) are indicated by the sign, and value and direction of the appropriate mass concentrated forces - by vectors. The areas are occupied by fuel tanks, on both projections are shaded. Centers of tanks weight are also indicated by the sign. In figures the geometrical sizes and numerical values of the concentrated forces are put down. The explanatory book should contain geometrical and aerodynamic characteristics of the chosen airfoil. In the final development it is supposed, that all wing cross sections have the same aerofoil. The relative coordinate of a center-of-pressure line can be found under the scheme: the given design limit loading condition - lift coefficient Су (for a cases B, C and D it is calculated) - the appropriate angle of attack α (see ap. 1) - relative coordinate of center-of-pressure line С cp. The wing's gravity center in cross section is usually located on a distance of % of a chord from the leading edge. By this value are set, considering power (or weakness) the high-lift devices of the airplane - prototype located in a tail part of a wing. Gravity centers coordinates of aggregates are made out from the description of the airplane - prototype or chosen independently, being guided by the knowledge acquired in another subject matters of aggregates' design features. For example, the center of gravity for a turbojet is placed in area of the turbine compressor, but not in area of a jet nozzle. At gravity center position s estimation of landing gear primary struts if the ones are located in a wing, it is possible to use the following statistical data: l l = ( )L w, b l = ( ) b, where the L w is a wing span, the b is a wing chord; the l l - is the landing gear base, the b l - is a distance from the leading edge wing to gravity center of the primary strut in a retracted position. 6

7 engine fuel tank front spar leading edge reduced axis b r b t L w /2 rear spar rear spar q center of gravity q t q a 0 q f Z Q Q c 0 Q t Q d Fig Diagrams of distributed loads and shear forces 7

8 M c Z M tot M d M, knm m z, kn Z M z, knm M z, c Z M z,t M z, d Fig Diagrams of bending moment M and reduced moment M z. 8

9 X equivalent straight wing Z axis of stiffnes s c k r k center of pressure e d swept wing reduced axis h Z center of wing gravity stiffness axis center of gravity for k-th aggregate center of fuel gravity Fig Plotting of equivalent straight wihg. 9

10 2. DETERMINATION OF LIMIT LOAD FACTOR. For a strength calculation it is necessary to know design airspeeds and load factors. For determination of these values for transport airplanes initial conditions are a take-off mass Mt [kg], L w - wing span, S w wing area, cruising airspeed of an airplane VC at a cruise altitude of flight Hc, the wing sweep till 0.25 of chord χ In design office (DO) calculations are carried out for all altitude range. In this HT you have the cruising airspeed V C and cruising altitude H C from technical data of plane. For your plane you should choose wing airfoil. The more speed of the plane, the thinner wing airfoil is. At speeds km/h airfoils are recommended with relative thickness c = %. For planes which flies with speeds km/h airfoils are recommended with relative thickness c = 8-12 %. According to ARU-25, AR-25, FAR-25, JAR-25 maneuvering maximum limit load factor does not depend on an altitude and en-route weight and are determined by the formula: l nymaxman 2.1 ; (2.1) Mt 4536 where M t is the design maximum takeoff mass in kilograms; except that n l y max man may not be less than 2.5 and need not be greater than WING S MASS DATA Units' masses (if these data are absent in the description of the prototype) are set with the help of statistical data for the transport airplanes adduced in tab Thus the mass of one of primary struts makes usually 45 % from mass of the whole landing gear. From home task on department "Designing of planes and helicopters" you have total fuel mass in wing. The half-wings each have three fuel tanks from safety conditions as minimal. In this project you can suppose that fuel load is concentrated for simplicity. From statistic we know that in the first tank is placed 45% from total fuel mass in half-wing, in the second tank 35%, in the third tank 20% (see fig. 3.1). Approximately by axes X relative coordinates of centers of mass for tanks are equal x f =0.4=40% from leading edge. Approximately under axes Z relative coordinates of centers of mass for the first tanks is equal z =0.2, for the second tank - z =0.5, for the third tank - z =0.8. From the point of view of strength fuel is expedient to place in a wing. Therefore in the final development it is necessary to place the greatest possible fuel content in a wing and the rest of fuel to place in a fuselage. If in a wing there are freights dropped in flight (external fuel tanks) or fuel from wing fuel tanks is consumed non-uniformly in this case strength of the given cross 10

11 section of a wing is calculated from the loadings appropriate not to take-off mass M t, but to the flight M fl one. Table 3.1. Assemblages and payloads relative mass in the percent share from transport airplane take-off mass Take-off mass, tons М t Assemblages relative mass % М w The wing М l The landing gear The power plant М pp Jet planes М pp Turboprops planes Total load М tl Note: the total load M tl is equal to the sum of the fuel and the payload. Let in a wing there is a freight dropped in flight with weight of G* (the tanksection containing fuel by weight G*), which gravity centre is located in the А-А cross section with coordinate z (fig. 3.2). Bending moment in the designing cross section 1-1 depends from a relative А-А section's positioning and Ру force coordinate which is the resultant of an air loading, operating on a segment covered with S cut area, located on the right of the 1-1 cross section. Considering approximately, that air loading is constant on all wing area, we can write down: Scut Scut P y Mtg Gt S w S (3.1) w where Gt Mtg - is take-off weight of plane, S w wing area. 0 If the G* load is present, the М bending moment in the 1-1 cross section is defined by the formula: 0 Scut * М = Gt z0 G (z z 1 ). (3.2) Sw At the G* load s dropping the Р у force is decreased by the value * Scut ΔP y (Gt 2G ). (3.3) S w * That s why the М load s post dropping bending moment in 1-1 section is equal to 11

12 X * М = S 2S G z G z. cut * cut t 0 0 Sw Sw x f Front spar c.g. 1-st fuel tank c.g. 2-nd fuel tank c.g. 3-rd fuel tank Z Rear spar Fig Disposition of fuel tanks. Fig Disposition of dropping cargo G*and force P y relatively designing cross section I-I. The 0 М and * М, moments are equal to each other if the given identity is right * z z z1 z 0(2S cut / S w ). If the load has the z z * coordinate, than at its dropping the bending moment is increased in the 1-1 section. * М > 0 М, therefore, 12

13 Thus, to the 1-1 designing cross section a case when freights dropped in flight are not taken into account, and fuel from tanks sections is consumed which gravity centers coordinates exceed the z* is more dangerous. At this stage the calculations are necessary to perform for the G fl flight mass which can be received, subtracting from the G t take-off mass the dropped freights and burnt out fuel. Mass of the dropped freights and burnt out fuel in the further calculations is not taken into account. The z 0 parameter is defined from the geometrical construction (fig.3.3) or under the formula 0 l b 2a z0. 3 b a For all student designing cross section is assigned under z =0.2. In this case designing flight mass M fl is equal: where M f is total fuel mass. M fl Mt 0.2M f, (3.4) Fig The scheme of calculation for coordinate z WING S LOADS CALCULATION The wing is influenced by the air forces allocated on a surface and mass forces caused by a wing structure and by the wing-arranged fuel, the concentrated forces from the wing - arranged units' masses. Mass forces are parallel to air forces, but are directed to the opposite side. The fuel tank is expedient to divide on tanksections and mass of everyone tank-section to concentrate in its gravity center. Then the fuel-distributed load is possible to replace by a set from the concentrated forces. In speed coordinate system the air loading resultant Р a has two components: the Y - lift directed perpendicularly to vector of flight speed, and the X - drag force directed on flight (fig. 4.1). The components are bounded with the Р a load by the reliance: 13

14 Y P a cos ; X Pa sin ; (4.1) C x arctg, C y where the Cx and Cy are the drag and the lift wing s coefficients that are estimated on the wing s data from table for angle of attack, corresponding the given design critical loading condition. Analogically mass forces are divided. The lift coefficient is calculated from equation of equilibrium: 2 l l HV nm fl g ngfl Cy Sw. 2 In the SI we have from this formula: l 2n M fl g C y 2 HV S, w where ρ H is air density on H C in SI, V cruise airspeed in m/s, M fl designing flight mass of plane in kg mass. M n Q n Y α P a θ α X V Q t M t Fig Distribution of the load on axes. By the value of C y you can estimate the angle of attack α with accurate within 1 o, drag coefficient C x and the relative coordinate of pressure center C cp. On the basis of stated it is enough to plot diagrams of the shear force and bending moment on a wing from effect of the efforts parallel plane YOZ in speed coordinate system. The shear force and bending moment in the given cross section 14

15 from the loads parallel to a plane XOZ, we receive by multiplication of force and the moment, taken in this section from available diagrams, on value tg α. The wing strength is determined in ultimate, instead of a limit loading condition. Then also diagrams of shear forces and bending moments it is convenient to plot from ultimate, instead from limit loadings. At calculation of ultimate loads in the beginning we find the ultimate load factor under the formula: u l n n f, (4.2) where the n is the limit load factor for the given design critical loading condition; the f - is the safety factor. According to AR-25, FAR-25, JAR-25 unless otherwise specified, a factor of safety of f=1.5 must be applied to the prescribed limit load are considered external load on the structure. Under value n u it is possible to find the ultimate loads. So, lift and a component along an axis Y from resultant mass load of a wing structure are found by the formulas: u u Y n M g [N]; n G fl fl u u P nmg yw ng w w [N]. (4.3) Load components acting along the Y axis from effect of a concentrated mass of the aggregate is calculated under the formula: P u u y.ag n G nm аg ag g [N], (4.4) where the G аg - is the unit s weight [N] AIR LOADS ALLOCATION ON THE WING SPAN. The Y air load is allocated according to the relative circulation low, i.e. nl f gm а fl z q y ( z ) 1.05 L Г(z)[N/m], z, (4.1.1) w 0,5L w where Г ( z ) - is relative circulation, M fl - is the designing flight mass of the plane, n l, f limit load factor and factor of safety, L w wingspan. For distributed load we have next sign convention - if distributed load is directed upward it has positive sign, if distributed load is directed downward it has negative sign. The function Г ( z ) depends from many factors, from which in the given work you should take into account only the dependence from wing taper and sweepback. Wing taper is designated through and is equal to: b / 0 b t 15

16 The Г ( z ) = Г f ( z ) function values for plane straight trapezoidal center-section-less wing are reduced in table # Essential influence on distribution of air loading renders a wing sweep. Relative circulation in this case is determined by the formula: Г ( z ) = Г f ( z ) + ( z ), (4.1.2) where s ( z ) is amendment on the wing sweep. This amendment is calculated by the formulas: о ( ) ( 45 ), (4.1.3) о 45 where the is the designing wing sweep on the chord s fourth, angle in degree. Table # Relative circulation on wingspan straight trapezoidal center-section-less flat wing Г f (5 10) z =2z/l = 1 = 2 = 3 = 4 = ,2721 1,3435 1,3859 1, ,2624 1,3298 1,3701 1, ,1196 1,2363 1,2908 1,3245 1, ,1096 1,1890 1,2228 1,2524 1, ,0961 1,1299 1,1484 1,1601 1, ,0765 1,0590 1,0570 1,0543 1, ,0457 0, ,9419 0, ,9954 0,8988 0,8538 0,8271 0, ,9138 0,8032 0,7430 0,7051 0, ,7597 0,6513 0,6090 0,5434 0, , ,4092 0, Comment. 1. Wing has not center-section (2 l c = 0). 2. Wing is flat. 3. Wing aspect ratio is equal to Lw 2 Sw. 4. Wing taper is equal to b 0 / bt. 5. For low-wing monoplane Г f is given from board rib, for mid-wing and high-wing Г f is given from axial rib. 6. If wing taper differentiates from table data, valises Г f are calculated by linear interpolation. 16

17 Table # The Г (45 ) correction s allocation for the wing with the following parameters = 5, = 2, = 45. 2z/l Г s (45 ) 2z/l Г s (45 ) i Z Table # a The q y, q w, q f and q distributed loads calculation s scheme b( z ), Г f Г Г kn q a kn, q w, kn y y m m m qt, m The ( z )(45 ) wing sweep correction value (having the following parameters: aspect ratio = 5; wing taper = 2; sweep angle on the chord s fourth = 45 ) is reduced in table It is possible to use the Г allocation (see tab.2.1.2) with another and parameter values THE WING STRUCTURE MASS LOAD ALLOCATION. In approximate calculations it is possible to consider, that load per unit of wing span mass forces is proportional to chords. Then the next formula is used: l l w n fg f g w w q y ( z ) b( z ) n M b( z ), (4.2.1) S S w w 17

18 where the b(z) is the wing chord, M w the wing mass. The length of wing chord (see column #3 in table 4.1.3) is computed by formulas: b( z ) b r ( br b t )z, (4.2.2) where b r is root chord of wing, b t tip chord, z - relative coordinate of cross section (column #2). After the component calculations it is possible to compute the total distributed wing load, acting in the direction of the axis Y in the speed coordinate system. Calculations are put into the tab.# 4. At this action the coordinates origin is put into the wing root and cross sections are enumerated from the wing root in the wing tip direction, beginning from the i = 0. The letter Z accentuates relative coordinate Z 2 z / Lw. Since on the site Z = cross sections the q a diagram are moved away from straight line, it is necessary to introduce the cross section with the Z = 0.95 coordinate (see tab. # # 4.1.1, 4.1.3) CALCULATION OF THE TOTAL DISTRIBUTED LOAD ON A WING The total distributed wing load is calculated under the formula: a w qt qy qy, (4.3.1) a w It is also necessary to plot the q, q and q functions in the same y y coordinate system and in the same scale (see fig ). In this formula you should summarize in algebraic sense with account of sign. The concentrated mass forces from aggregates also put on figure of a wing (see fig ). Thus it is convenient to show forces by vectors and to put down a value of these forces. Instead of ultimate mass force's value to indicate value of aggregate weights it is not recommended, as it is an additional source of errors THE CHEAR FORCES, BENDING AND REDUCED MOMENTS DIAGRAMS PLOTTING In the beginning functions shear force Q d ( z ) and bending moment M d ( z ) from the distributed load q (z) are found on the wing span. For this purpose integrals are calculated by a tabulated way with trapezoids method. M z Q q( z ) dz, L z w 2 L w 2 Q(z)dz (4.4.1) You must yourself to determine signs for q, Q, M according to sign convention from strength of materials, see fig. #

19 The calculation scheme is given in the tab. # 4.4.1, which includes the following values: Δzi 0.5(zi 1 z i)lw; z 11 =0, (i =10, 9,..., 1,0), ΔQi 0.5(qi 1 q i )Δzi, Q 11 = 0, (i = 10, 9,..., 1, 0), Q Q Q, Q 11 = 0; (i = 10, 9,..., 1, 0) M i i i 1 Q Q i Δ 0.5 ΔZ, M 11 =0, (i =10, 9,..., 1, 0) i i 1 i M i M i M i 1, M 11 = 0; (i = 10, 9,..., 1,0) (4.4.2) where z 10 is distance between cross-section number 10 and cross-section number 11 and so on; accordingly Q 11 =0 is increment of shear force in crosssection number 11 from distributed loads out tip wing, Q 10 - is increment of shear force in cross-section number 10 from distributed loads on site between 10 and 11 cross-sections and so on; Q 11 =0 is shear force in cross-section number 11 from distributed loads out tip wing, Q 10 - is shear force in cross-section number 10 from distributed loads on site between 10 and 11 cross-sections and so on; M 11 =0 - is increment of bending moment in cross-section number 11 from distributed loads out tip wing, M 10 - is increment of bending moment in cross-section number 10 from distributed loads on site between 10 and 11 cross-sections and so on; M 11 =0 is bending moment in cross-section number 11 from distributed loads out tip wing, M 10 - is bending moment in cross-section number 10 from distributed loads on site between 10 and 11 cross-sections and so on. Q>0 M>0 Fig. # Sign convention for a shear force Q and bending moment M. The table # is constructed in the assumption, that integration implements by a trapezoids method. The origin is placed in the wing root section, sections are numbered from a root to a wing tip, since i=0. 19

20 After filling of tab. # by the calculated shear forces Q and bending moments M (on fig are shown by dashed lines) diagrams are plotted. Diagrams of bending moments are plotted on tension fibers of wing. Also it is necessary to result the shear forces and bending moments affected by the P agr y concentrated mass forces (in the same coordinate systems, that Q and M, and in the same scale) diagrams. However the sign of these diagrams is opposite to one of diagrams Q and M. On fig light lines show these diagrams from concentrated forces (table # 4.4.2). In concentrated mass forces you must include all aggregates of wing engines, landing gears, fuel tanks and so on. The calculation scheme is given in the tab. # 4.4.2, which includes the following values: Q ic = P i agr from (4.4) where i - is number of cross section in which this unit is placed; in any cross sections Q ic = 0. In table #4.4.2 for example concentrated force is given only in cross section i= 9. You can rewrite columns 2 and 3 from previous table. Δz 0.5(z z )L ; z 11 =0, (i =10, 9,..., 1,0), M i i 1 Q ΔQ Q Q Q i w, (i = 10, 9,..., 1, 0), ic ic i 1c i Δ 0.5 ΔZ, M 11c =0, (i =10, 9,..., 1, 0) M ic i 1c ic M M, M 11с = 0; (i = 10, , 0) (4.4.3) iс i 1с i 1с where Q 11c =0 is shear force in cross-section number 11 from concentrated loads in the tip wing, Q 10c - is shear force in cross-section number 10 from concentrated loads, Q 9 - is shear force in cross-section number 9 from concentrated loads which has jump in this cross-section and two values one previous value - 0 and new value Q 9c and so on; M 11c =0 - is increment of bending moment in cross-section number 11 from concentrated loads out tip wing, M 10c - is increment of bending moment in cross-section number 10 from concentrated loads on site between 10 and 11 cross-sections and so on; M 11c =0 is bending moment in cross-section number 11 from concentrated loads out tip wing, M 10c - is bending moment in crosssection number 10 from concentrated loads on site between 10 and 11 crosssections and so on. You must know that increment of bending moment from concentrated force and bending moment from concentrated force you can calculate for next cross-section with number i-1=8 in our example see fig and table # Folding appropriate diagrams algebraically (table # 4.4.3), you should plot total diagrams Q and tot M tot (on fig are shown by continuous lines). The calculation scheme is given in the tab. # 4.4.3, which includes the following values: Q id - is shear force from distributed loads from table # 4.4.1; Q ic - is shear force from concentrated loads from table # 4.4.2; Q tot = Q id + Q ic with account signs; 20

21 M id - is bending moment from distributed loads from table # 4.4.1; M ic - is bending moment from concentrated loads from table # 4.4.2; M tot = M id + M ic with account signs. As a wing is calculated on strength in connected coordinate system for design cross section determination of shear forces and bending moments are carried out in this coordinate system. In connected coordinate system the t axis is directed on a chord of a wing, an axis n - is perpendicular to it. Table # The Q d (z) shear forces and the М d (z) bending moment are affected by the q(z) distributed load. M id, kn m M id, kn m i z Δ Zi, q, i t Q id, Q id, m kn kn kn m q Q 0 Q 0 M 0 M 0 t Δ Z Δ Z q Q t 9 9 Q 9 M 9 M 9 q Q 10 Q 10 =Q 10 M 10 M 10 = M 10 t10 q t11 21

22 Table # The Q(z) shear forces and the М(z) bending moment are affected by the concentrated load. Z i M iс, kn m M iс, kn m i Δ Z i, Q iс, Q iс, m kn kn Q 0 M 0 M Q 7 =Q Q 8 =Q 9 M 8 M 8 =M Z 9 Q 9 Q 9 =Q 9 / Z Table # The total Q tot (z) shear forces and the total М tot (z) bending moment are affected by all forces. i Q id, kn Q ic, kn Q itot, kn M id, kn*m M ic, KN*m M itot, kn m An origin is placed in the gravity centre of cross section. According to fig. 4.1 it is possible to write down: 22

23 cos( ) Qn Qtot, cos sin( ) Q t Q, (4.4.4) tot cos sin( ) M n M tot, cos cos( ) M t M tot, cos where the is the angle of attack; - is the angle between total aerodynamic force P a and lift force Y (see 4.1), the Q tot and M tot are shear force and bending moment values in the design cross section in speed coordinate system, taken from the diagrams (fig ) and table # 4.4.3; the Q n and Q t are shear forces; the M n and M t are bending moments vectors in the design cross section in the connected coordinate system. If <, which is usual for the A critical loading condition, than - <0 and, hence, the Q t and M n vectors change their direction to the opposite one. At plotting of the diagram of the reduced moments in the beginning we set a position of an axis of reduction (fig ). This axis is parallel to an axis Z (strictly speaking, the axis of reduction should be a parallel axis of stiffness centers of a wing). Further you should plot a diagram of the distributed reduced moments m z affected by the distributed loads q a and q w. For the moment m z the formula is received, writing п п down the moment from the specified loads concerning an axis of reduction. At calculation m z it is necessary to mean, that the adduced moments are calculated in connected coordinate system. Thus assume, that frontal components lay in a plane XOZ and, hence, moment about axis of reduction does not give (fig ). reduced axis q a e q w Fig Positions of loads in cross section. 23

24 For distributed loads q a п formulas: q a п q and q w п (see fig ) it is possible to record a cos( ) q, y cos w cos( ) q, (4.4.5) y cos w п q w d qae m z п n (4.4.6) where the e, and the d are distances from load points q a and q w to the reduction п п axis. The moment is considered like positive if it acts on pitching relative to the reduction axis. The е, and d values are taken from the fig You can compute their by formulas: d z tgγ xc.gb 0.5L z xc.gb, i i i w i i ei zitgγ xc.pbi 0.5Lwzi xc.p [ bi, 0.8( br b t ) tgγ, L where z i - is the relative coordinate z for i-th cross section (column 2 from table 4.4.2), x c.g - is relative coordinate of wing center of gravity Integrating the diagram m z, we receive the reduced moments M zd affected by the distributed loads. The scheme of calculation is shown in tab in which designations is entered: M ( m m ) z /, M z M 0 ; zid zi 1 z,i i 2 z,i,d z,i 1,d z,i,d w, 11 z, 11 M M ΔM, (i = 10, 9,..., 0). In an explanatory book it is necessary to plot diagrams m z and M z (a diagram M z is shown on fig by a dashed line). In a coordinate system for the moments M z also it is necessary to result a diagram of the reduced moments affected by concentrated masses (on fig it is shown by a light line). Affected by a concentrated mass of the i-th aggregate the increment of the moment M z,c,i is found out by the formula: cos( θ α) ΔMz,c,i Pyag,i ri Py,ag,iri, (4.4.7) cosθ where the r i is the distance from the i-th concentrated mass gravity center to reduction axis (it is measured on the drawing), P yag is design weight from formula (2.4). The moment M z,,c,i is positive if it acts on pitching. This increment you have only in point where you have aggregates. In any points this increment is equal zero. Reduced moment M z,c,i is calculated by the formula: 24

25 ΔMz,c,11 Mz,c,11 0; M M ΔM, (i = 10, 9,..., 0). (4.4.8) z,c,i z,c,i 1 z,c,i In the point with aggregates we have jumps of reduced moment (see fig ). For this table we take M z i d from table # and total reduced moment we compute with account of signs upon formula: M z tot = M z d + M z c (4.4.9) i Z i, m Table # Reduced moments calculation scheme from distributed loads q a, n i kn /m e i, m q w, п i kn /m d i, m m z i, kn M zid, kn m M z id, kn m e q a п 0 Z q a п 10 q a п Z 11 0 e 10 e 11 q w п 0 q w п 10 q w п 11 d 0 m z 0 M z 0d d 10 m 10 M z 10d z M z 10d d 11 m z It is also necessary to plot the it is shown by the solid line). M z,tot total reduced moment diagram (on fig. 25

26 Table # Calculation scheme of reduced moment from concentrated loads and from all loads. I P y,ag,i kn r i m M z,c,i kn*m M z,c,i, kn*m M zd, kn*m M ztot, kn*m LOAD CHECKING FOR WING ROOT CROSS SECTION Shear forces, bending and reduced moment s values are checked in the root cross section under the formulas: Q n fg[0.5(m M ) M ], [kn], l agr r tot fl w k k M n fg[0.5(m M )C M c ], [knm], (4.5.1) l agr r tot fl w k k k l agr Mz r tot n fg(0.5m fle 0.5Mwd Mk r k ), [knm] k. Here M fl is designing flight mass of plane from (3.4), M w the wing mass, С is the distance from root section to the air resultant load point; c k - is the distance from root section to the k-th aggregate's gravity center and fuel tanks; e and d are distances from an axis of reduction to points of interception of a plane z=c with a center-of-pressure line and with a c.g. line; r k - is the distance from an axis of reduction to the k-th aggregate centre of gravity and fuel tanks. In list of aggregates you should include all aggregates of wing engines, landing gears, fuel tanks and so on. Value C is found with the help of geometrical construction or under the formula: 26

27 С Lw 2 η 6 1 η, (4.5.2) where the η is the wing taper. Values c k, and r k are taken from fig.1.1.3, and parameters e and d values from drawing (see fig.1.1.3) in the z = c cross section. Summation in the right parts of adduced formulas is distributed to all concentrated masses located in one half-wing. An error of calculation of valuesq Σ r, M Σ r and M Σ zr in relation to the appropriate values taken from tables in root cross section should not exceed value 1, 10 and 15 % accordingly CALCULATION OF SHEAR FORCE S POSITION IN THE DESIGN CROSS SECTION Upon values of shear force and the reduced moment in design cross section, it is possible to find out a shear force load point on a chord of a wing of design cross section: Σ z Σ xr M Q. (4.6.1) The x r coordinate is count off from the reduction axis. The resultant position is necessary to be shown by an asterisk on the wing s top view (see fig ). 27

28 Table # Results of calculations Design loading condition C Design flight mass (kg) - M fl Limit load factor - n l Safety factor - f Ultimate load factor - n u Fuel mass in 1-st fuel tank (kg) - m f1 Fuel mass in 2-nd fuel tank (kg) ) - m f2 Fuel mass in 3-rd fuel tank (kg) - m f3 Wing span (m) (for equivalent wing)- L we Wing taper Wing aspect ratio Root wing chord (m) (for equivalent wing) - b r Tip wing chord (m) (for equivalent wing) - b t Relative thickness of airfoil (%) - c Number of airfoil Position of front spar (in % from chord) Position of rear spar (in % from chord) Designing cross section z 0.2 The normal bending moment for designing cross section (kn*m, form ) The tangential bending moment for designing cross section (kn*m, form ) The normal shear force for designing cross section (kn, form ) The tangential shear force for designing cross section (kn, form ) The distance from reduced axis up to application point of resultant shear force (m) The angle of attack (degree) The angle between resultant air force and lift force (degree) Comment. a. Masses in integer kg. 28

29 APPENDIXIES 29

30 Characteristic of airfoil Appendix #1 Geometric characteristic of airfoil (in % from chord) The airfoil NACA 0009 Aerodynamic characteristic of airfoil X Yt Yb h Cy Сх Ccp и

31 Geometric characteristic of airfoil (in % from chord) The airfoil NACA 0012 Aerodynamic characteristic of airfoil X Yt Yb h Cy Сх Ccp

32 Geometric characteristic of airfoil (in % from chord) The airfoil NACA 0015 Aerodynamic characteristic of airfoil X Yt Yb h Cy Сх Ccp

33 The airfoil NACA Geometric characteristic of airfoil (in % from chord) Aerodynamic characteristic of airfoil X Y t Y b h С у C x C m C cp З '

34 The airfoil NACA Geometrical characteristic of airfoil (in % from chord) Aerodynamic characteristic of airfoil X Yt Yb Ym h Cy Cx Cm Ccp

35 The airfoil NACA Geometric characteristic of airfoil Aerodynamic characteristic of airfoil (in % from chord) X Yt Yb Ym h Cy Сх Ccp о

36 The airfoil NACA Geometric characteristic of airfoil Aerodynamic characteristic of airfoil (in % from chord) X Yt Yb h Cy Сх C cp ,17 0,

37 The airfoil NACA Geometric characteristic of airfoil Aerodynamic characteristic of airfoil (in % from chord) X Yt Yb h Cy Сх C cp

38 The airfoil NACA Geometric characteristic of airfoil (in % from chord) Aerodynamic characteristic of airfoil X Yt Yb Ym h Cy Сх Cm Ccp

39 The airfoil NACA Geometric characteristic of airfoil Aerodynamic characteristic of airfoil (in % from chord) X Yt Yb Ym h Cy Сх Ccp

40 The airfoil NACA 2312 Geometric characteristic of airfoil (in % from chord) Aerodynamic characteristic of airfoil X Yt Yb Ym h Cy Сх Ccp

41 The airfoil NACA Geometric characteristic of airfoil Aerodynamic characteristic of airfoil (in % from chord) X Yt Yb h Cy Сх C cp

42 The airfoil NACA-2412 Geometric characteristic of airfoil (in % from chord) Aerodynamic characteristic of airfoil X Yt Yb Ym h Cy Cх Cm Ccp

43 The airfoil NACA-2415 Geometric characteristic of airfoil (in % from chord) Aerodynamic characteristic of airfoil X Yt Yb Ym h Cy Cx Cm Ccp ,18 0,013-0,050 1,25 2,71-2,06 0,33 4, ,02 0,010 0,035 2,5 3,71-2,86 0, ,13 0,012 0,0735 0, ,07-3,84 0,62 8,91 2 0,28 0,016 0,110 0,392 7,5 6,06-4,47 0,80 10,53 4 0,42 0,020 0,145 0, ,83-4,90 0,87 11,73 6 0,57 0,030 0,182 0, ,97-5, ,39 8 0,71 0,042 0,218 0, ,70-5, , ,86 0,056 0,255 0, ,17-5,70 1,74 14, ,00 0,071 0,288 0, ,38-5, , ,15 0,090 0,326 0, ,25-5,25 2,00 14, ,28 0,112 0,360 0, ,57-4, , ,40 0,136 0,390 0, ,50-3, , ,50 0,160 0,415 0, ,10-3,05 1,53 9, ,54 0,192 0,425 0, ,41-2,15 1,13 6, ,41 0,280 0,441 0, ,45-1,17 0,64 3, ,31 0,332 0,439 0, ,34-0,68 0,33 2, ,20 0,383 0,425 0, ,10 0,415 0,378 43

44 The airfoil NACA-2409 Geometric characteristic of airfoil (in % from chord) Aerodynamic characteristic of airfoil X Yt Yb Ym h Cy Cx Cm Ccp ,25 1,62-1,23 0,195 2, ,192 0,012-0,004 2,5 2,27-1,66 0,305 3,93-2 0,00 0,008 0, ,2-2,15 0,525 5,35 0 0,13 0,008 0,076 0,588 7,5 3,87-2,44 0,715 6,31 2 0,29 0,0128 0,118 0, ,43-2,60 0,915 7,03 4 0,43 0,020 0,150 0, ,25-2,77 1,24 8,02 6 0,58 0,028 0,188 0, ,81-2,79 1,51 8,60 8 0,72 0,040 0,224 0, ,18-2,74 1,72 8, ,88 0,054 0,264 0, ,38-2,62 1,88 9, ,02 0,070 0,298 0, ,35-2,35 2,00 8, ,18 0,090 0,336 0, ,92-2,02 1,95 7, ,30 0,112 0,370 0, ,22-1,63 1,795 6, ,43 0,140 0,402 0, ,27-1,24 1,515 5, ,50 0,180 0,416 0, ,10-0,85 1,125 3, ,30 0,270 0,444 0, ,72-0,47 0,625 2, ,16 0,370 0,430 0, ,94-0,28 0,33 1, ,08 0,420 0, ,00 0,410 0,410 44

45 The airfoil NACA Geometric characteristic of airfoil (in % from chord) Aerodynamic characteristic of airfoil X Yt Yb Ym h Cy Cx Cm Ccp ,21 0,014-0,042 1,25 3,34-1,54 0,90 4, ,06 0,011-0,006 2,5 4,44-2,25 1,095 6,69 0 0,09 0,0082 0,029 0, ,89-3,04 1,425 8,93 2 0,23 0,014 0,063 0,274 7,5 6,91-3,61 1,65 10,52 4 0,39 0,018 0,101 0, ,64-4,09 1,78 11,73 6 0,53 0,027 0,135 0, ,52-4,84 1,84 13,36 8 0,69 0,038 0,173 0, ,92-5,41 1,76 14, ,83 0,051 0,206 0, ,08-5,78 1,65 14, ,98 0,068 0,242 0, ,05-5,96 1,55 15, ,13 0,088 0,278 0, ,59-5,92 1,34 14, ,27 0,108 0,312 0, ,74-5,50 1,12 13, ,40 0,132 0,343 0, ,61-4,81 0,90 11, ,52 0,158 0,372 0, ,25-3,91 0,67 9,16 22,2 1,61 0,190 0,393 0, ,73-2,83 0,45 6,56 22,2 1,36 0,245 0,375 0, ,04-1,59 0,23 3, ,27 0,288 0,379 0, ,12-0,90 0,12 2, ,18 0,338 0,382 0, ,01 0,372 0,368 45

46 The airfoil NACA Geometric characteristic of airfoil (in % from chord) Aerodynamic characteristic of airfoil X Yt Yb Ym h Cy Cx Cm Ccp ,22 0,012-0,0415 1,25 2,04-0,91 0,07 2, ,09 0,009-0,013 2,5 2,83-1,19 0,82 4,02 0 0,09 0,0066 0,031 0, ,93-1,44 1,25 5,37 2 0,225 0,011 0,063 0,280 7,5 4,70-1,63 1,54 6,33 4 0,39 0,0165 0,103 0, ,26-1,79 1,74 7,05 6 0,53 0,023 0,137 0, ,85-2,17 1,84 9,02 8 0,69 0,035 0,175 0, ,06-2,55 2,26 8, ,83 0,050 0,209 0, ,11-2,80 1,66 8, ,975 0,066 0,244 0, ,05-2,96 1,55 9, ,12 0,088 0,279 0, ,69-3,03 1,33 8, ,29 0,110 0,320 0, ,09-2,86 1,12 7, ,40 0,133 0,347 0, ,32-2,53 0,89 6,85 20,3 1,55 0,170 0,383 0, ,42-2,08 0,72 5,50 20,3 1,30 0,232 0,383 0, ,41-1,51 0,45 3, ,25 0,290 0,401 0, ,31-0,86 0,23 2, ,16 0,360 0,420 0, ,72-0,50 0,11 1, ,08 0,410 0, ,95 0,389 0,409 46