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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

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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.

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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 HC with maximal maneuvering load factor nl.

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 # 3. 5. 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.

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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)-Mt

Design airspeed (km/h) - V Design cruise altitude of flight (m) - H Wing mass (kg) - mw

Total fuel mass (kg) - mf Mass of engine (kg)- me

Mass of main leg of landing gear (kg)- mml Wing area (m2) - Sw Wing span (m) (for swept wing) - Lw

Wing taper Wing aspect ratio Swept wing (degree by 25% chord) - 0.25

Comment.

a. Masses in integer kg. b. Speeds in integer km/h

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1.1. WING’S GENERAL DATA

The airplane category (transport, normal, acrobatic, etc.), take-off mass Мt 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.

1.2. 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 br and tip bt chords sizes decrease, and the size Lw/2 of a semi span is increased. The sizes Lw/2, br and bt, are required for the further calculations and they are taken directly from the figure. At turn of a swept-back 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 Lw/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 Lw/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 Lw/2. The wing area Sw is determined under the formula:

Sw = 0.5 (br + bt) Lw . (1.2.1)

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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. 1.2.1, 1.2.2 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 Сcp.

The wing's gravity center in cross section is usually located on a distance of 40 - 45 % 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: ll = (0.2 … 0.25)Lw, bl = (0.3… 0.7) b, where the Lw is a wing span, the b is a wing chord; the ll - is the landing gear base, the bl - is a distance from the leading edge wing to gravity center of the primary strut in a retracted position.

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Fig.1.2.1. Diagrams of distributed loads and shear forces

front spar leading edge

center of gravity

fuel tankengine

rear spar

q

qt qa

qf Z

Q

Qt

Qd

Qc

0

Lw/2

0

rear spar

br bt

reduced axis

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Fig.1.2.2. Diagrams of bending moment M and reduced moment Mz.

M, kNm

Mtot

Md

Mc

Z

mz, kN Z

Mz, kNm Z

Mz,t

Mz, d

Mz, c

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Fig. 1.2.3. Plotting of equivalent straight wihg.

equivalent straight wing

swept wing

axis of stiffness

reduced axis

center of pressure

center of wing gravity

stiffness axis

Z

center of fuel gravity

e

Z

d

h

X

rkck

center of gravity for k-th aggregate

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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], Lw - wing span, Sw – wing area, cruising airspeed of an airplane VC at a cruise altitude of flight Hc, the wing sweep till 0.25 of chord – χ0.25 .

In design office (DO) calculations are carried out for all altitude range. In this HT you have the cruising airspeed VC and cruising altitude HC 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 400-700 km/h airfoils are recommended with relative thickness c = 12-15 %. For planes which flies with speeds 700-950 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:

ly max man

t

108862.1

4536n

M

; (2.1)

where Mt is the design maximum takeoff mass in kilograms; except that nl

y max man may not be less than 2.5 and need not be greater than 3.8.

3. 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. 3.1.. 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 fx =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

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section of a wing is calculated from the loadings appropriate not to take-off mass Mt, but to the flight Mfl 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 10 50 100 150 200

Assemblages relative mass %

М w The wing

12.2 10.2 9.5 9.1 8.8

М l The landing gear

4.5 4.0 3.8 3.7 3.6 The power plant

М pp Jet planes

12.3 11.0 10.5 10.2 10.0

М pp Turboprops planes

16.4 15.6 15.3 15.1 15.0 Total load

Мtl 43.3 45.8 53.7 61.4 67.6

Note: the total load Mtl 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 tank-section 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 Scut 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:

cut cuty t t

w w

S SP M g G

S S (3.1)

where t tG M g - is take-off weight of plane, Sw – wing area.

If the G* load is present, the 0М bending moment in the 1-1 cross section is defined by the formula:

0М = *cutt 0 1

w

SG z G ( z z )

S . (3.2)

At the G* load’s dropping the Ру force is decreased by the value

* cuty t

w

SΔP (G 2G )

S . (3.3)

That’s why the *М load’s post dropping bending moment in 1-1 section is equal to

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*М = *cut cutt 0 0

w w

S 2SG z G z

S S .

Fig. 3.1. Disposition of fuel tanks.

Fig. 3.2. Disposition of dropping cargo G*and force Py relatively designing cross section I-I.

The 0М and *М , moments are equal to each other if the given identity is right *

1 0 cut wz z z z ( 2S / S ) .

If the load has the z z * coordinate, than at its dropping *М > 0М , therefore, the bending moment is increased in the 1-1 section.

1-st fuel tank 2-nd fuel

tank

3-rd fuel tank

c.g. c.g. c.g.

xf

Front spar

X

Z

Rear spar

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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 Gfl flight mass which can be received, subtracting from the Gt 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 z0 parameter is defined from the geometrical construction (fig.3.3) or under the formula

zl b a

b a0

0

3

2

.

For all student designing cross section is assigned under z =0.2. In this case

designing flight mass Mfl is equal:

fl t fM M 0.2M , (3.4)

where Mf is total fuel mass.

Fig. 3.3. The scheme of calculation for coordinate z0.

4. 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 tank-sections 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 Рa 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 Рa load by the reliance:

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cos P aY ; sinPX a ; (4.1)

C

C

y

xarctg ,

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:

2l l H

fl fl y w

Vn M g n G C S

2

.

In the SI we have from this formula: l

fly 2

H w

2n M gC

V S ,

where ρH – is air density on HC in SI, V – cruise airspeed in m/s, Mfl – designing flight mass of plane in kg mass.

Fig. 4.1. Distribution of the load on axes.

By the value of Cy you can estimate the angle of attack α with accurate within 1o, drag coefficient Cx and the relative coordinate of pressure center Ccp.

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

V

α

α

θ

Pa

Y

X

Qn

Qt

Mt

Mn

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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 lfn n , (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 nu 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 uflfl

Y 1.05 1.05n M gn G [N];

u uwyw w

n M gn GP [N]. (4.3)

Load components acting along the Y axis from effect of a concentrated mass of the aggregate is calculated under the formula:

u uagy .ag аg

n M gn GP [N], (4.4)

where the Gаg - is the unit’s weight [N].

4.1. AIR LOADS ALLOCATION ON THE WING SPAN.

The Y air load is allocated according to the relative circulation low, i.e.

а

y

ln f g M flLw

q ( z ) 1.05 Г ( z )[N/m] , wL5,0

zz , (4.1.1)

where )z(Г - is relative circulation, Mfl - is the designing flight mass of the plane, nl, f – limit load factor and factor of safety, Lw – 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 )z(Г 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:

tbb /0

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The )z(Г = )z(Г f function values for plane straight trapezoidal center-section-less

wing are reduced in table # 4.1.1. Essential influence on distribution of air loading renders a wing sweep.

Relative circulation in this case is determined by the formula: )z(Г = )z(Г f + )z( , (4.1.2)

where )z(s is amendment on the wing sweep. This amendment is calculated by the formulas:

оо )()(

4545

, (4.1.3)

where the is the designing wing sweep on the chord’s fourth, angle in degree.

Table # 4.1.1 Relative circulation on wingspan straight trapezoidal center-section-less flat wing

Гf (5 10)

z =2z/l = 1 = 2 = 3 = 4 = 5

0.0 1.1225 1,2721 1,3435 1,3859 1,4157 0.1 1.1261 1,2624 1,3298 1,3701 1,39870.2 1,1196 1,2363 1,2908 1,3245 1,3490 0.3 1,1096 1,1890 1,2228 1,2524 1,2711 0.4 1,0961 1,1299 1,1484 1,1601 1,1703 0.5 1,0765 1,0590 1,0570 1,0543 1,0561 0.6 1,0457 0,9814 0.9571 0,9419 0,93430.7 0,9954 0,8988 0,8538 0,8271 0,8098 0.8 0,9138 0,8032 0,7430 0,7051 0,6784 0.9 0,7597 0,6513 0,6090 0,5434 0,5115

0.95 0.6599 0,5151 0.4593 0,4092 0,3798 1 0 0 0 0 0

Comment. 1. Wing has not center-section (2 lc = 0). 2. Wing is flat.

3. Wing aspect ratio is equal to wSwL 2 .

4. Wing taper is equal to tb/b0 . 5. For low-wing monoplane Гf is given from board rib, for mid-wing and high-wing Гf is given from axial rib. 6. If wing taper differentiates from table data, valises Гf are calculated by linear interpolation.

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Table # 4.1.2 The Г(45°) correction’s allocation for the wing with the following parameters

= 5, = 2, = 45°. 2z/l Гs (45) 2z/l Гs (45) 0 -0.235 0.6 0.073

0.1 -0.175 0.7 0.111 0.2 -0.123 0.8 0.135 0.3 -0.072 0.9 0.140 0.4 -0.025 0.95 0.125 0.5 0.025 1.00 0

Table # 4.1.3

The ,, qq way qf and q distributed loads calculation’s scheme

i Z

b( z ),m

Гf Г Г m

kN,q

a

y

m

kN,q

w

y ,t

kNq

m

1 2 3 4 5 6 7 8 9 0 0 1 0.1 2 0.2 3 0.3 4 0.4 5 0.5 6 0.6 7 0.7 8 0.8 9 0.9 10 0.95 11 1.0 0 0 0 0

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 4.1.2. It is possible to use the Г allocation (see tab.2.1.2) with another and parameter values.

4.2. 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: ll

w w wy

w w

f gn fGq ( z ) b( z ) b( z )

S Sn M

, (4.2.1)

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where the b(z) is the wing chord, Mw – the wing mass. The length of wing chord (see column #3 in table 4.1.3) is computed by

formulas:

r r tb( z ) b ( b b )z , (4.2.2)

where br – is root chord of wing, bt – 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 wL/z2Z

. Since on the site Z

= 1 … 0.9 cross sections the qa 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).

4.3. CALCULATION OF THE TOTAL DISTRIBUTED LOAD ON A WING

The total distributed wing load is calculated under the formula: a w

t y yq q q , (4.3.1)

It is also necessary to plot the a w

y y,q q and q functions in the same

coordinate system and in the same scale (see fig. 1.1.1). 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. 1.1.1). 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.

4.4. THE CHEAR FORCES, BENDING AND REDUCED MOMENTS’ DIAGRAMS

PLOTTING

In the beginning functions shear force )z(Qd and bending moment )z(M d 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.

z

2Lw

dz)z(qQ ,

w

z

L2

M 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. # 4.4.1.

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The calculation scheme is given in the tab. # 4.4.1, which includes the following values:

i 1 ii wΔz 0.5( z z )L ; ∆z11=0, (i =10, 9, ... , 1,0),

i i 1 i iΔQ 0.5( q q )Δz , Q11 = 0, (i = 10, 9, ... , 1, 0), QQQ iii 1 , Q11 = 0; (i = 10, 9, ... , 1, 0)

ii i 1 iΔ 0.5 ΔZQ QM

, M11 =0, (i =10, 9, ... , 1, 0)

MMM iii 1 , M11 = 0; (i = 10, 9, ... , 1,0) (4.4.2) where ∆z10 – is distance between cross-section number 10 and cross-section number 11 and so on; accordingly ∆Q11=0 – is increment of shear force in cross-section number 11 from distributed loads out tip wing, ∆Q10 - is increment of shear force in cross-section number 10 from distributed loads on site between 10 and 11 cross-sections and so on; Q11=0 – is shear force in cross-section number 11 from distributed loads out tip wing, Q10 - is shear force in cross-section number 10 from distributed loads on site between 10 and 11 cross-sections and so on; ∆M11=0 - is increment of bending moment in cross-section number 11 from distributed loads out tip wing, ∆M10 - is increment of bending moment in cross-section number 10 from distributed loads on site between 10 and 11 cross-sections and so on; M11=0 – is bending moment in cross-section number 11 from distributed loads out tip wing, M10 - is bending moment in cross-section number 10 from distributed loads on site between 10 and 11 cross-sections and so on.

Fig. # 4.4.1. Sign convention for a shear force Q and bending moment M. The table # 4.4.1 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.

Q>0

M>0

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After filling of tab. # 4.4.1 by the calculated shear forces Q and bending moments M (on fig.1.1.2 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 Pagr

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.1.1.2 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: Qic= Pi agr from (4.4) where i - is number of cross section in which this unit is placed; in any cross sections Qic = 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.

i 1 ii wΔz 0.5( z z )L ; ∆z11=0, (i =10, 9, ... , 1,0),

ic ic i 1cQ ΔQ Q

, (i = 10, 9, ... , 1, 0),

iic i 1c icΔ 0.5 ΔZQ QM

, M11c =0, (i =10, 9, ... , 1, 0)

MMM сiсiiс 11 , M11с = 0; (i = 10, 9... 1, 0) (4.4.3) where Q11c=0 – is shear force in cross-section number 11 from concentrated loads in the tip wing, Q10c - is shear force in cross-section number 10 from concentrated loads, Q9 - 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 ∆Q9c and so on; ∆M11c=0 - is increment of bending moment in cross-section number 11 from concentrated loads out tip wing, ∆M10c - is increment of bending moment in cross-section number 10 from concentrated loads on site between 10 and 11 cross-sections and so on; M11c=0 – is bending moment in cross-section number 11 from concentrated loads out tip wing, M10c - is bending moment in cross-section number 10 from concentrated loads on site between 10 and 11 cross-sections 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. 1.1.2 and table # 4.4.2.

Folding appropriate diagrams algebraically (table # 4.4.3), you should plot total

diagrams totQ and totM (on fig. 1.1.2 are shown by continuous lines). The

calculation scheme is given in the tab. # 4.4.3, which includes the following values: Qid - is shear force from distributed loads from table # 4.4.1; Qic - is shear force from concentrated loads from table # 4.4.2; Qtot = Qid+ Qic with account signs;

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Mid - is bending moment from distributed loads from table # 4.4.1; Mic - is bending moment from concentrated loads from table # 4.4.2; Mtot = Mid + Mic 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 # 4.4.1 The Qd(z) shear forces and the Мd(z) bending moment are affected by the

q(z) distributed load. i

iz ZiΔ , m

,tq

kN

m

Q id ,

kN

Qid,

kN

M id ,

kNm M id ,

kN m

1 2 3 4 5 6 7 8 0 0

0tq ∆Q0 Q0

∆M0 M 0

1 2 3 4 5 6 7 8 ... … … … … … … 9 0.9 9ZΔ

9tq Q9 Q9

M 9 M 9

10 0.95 10ZΔ 10t

q Q10 Q10 =Q10 M 10 M 10 = M 10

11 1.0 0 11t

q 0 0 0 0

22

Table # 4.4.2 The Q(z) shear forces and the М(z) bending moment are affected by the

concentrated load. i iZ

iZΔ ,

m Qiс ,

kN

Qiс,

kN

M iс ,

kNm M iс ,

kN m

1 2 3 4 5 6 70 0 0 Q0 ∆M0 M 0

1 2 3 4 5 6 ... 7 Q7=Q9 … … 8 ... … … Q8=Q9 ∆M8 ∆M8 =M8

9 0.9 ∆Z9 Q9 Q9 =Q9 /0 0 0

10 0.95 ∆Z10 0 0 0 0 11 1.0 0 0 0 0 0

Table # 4.4.3

The total Qtot(z) shear forces and the total Мtot(z) bending moment are affected by all forces.

i Qid, kN

Qic, kN

Qitot, kN

Mid, kN*m

Mic, KN*m

Mitot, kNm

1 2 3 4 5 6 70 1 2 3 4 5 6 7 8 9 10 11

An origin is placed in the gravity centre of cross section. According to fig. 4.1 it

is possible to write down:

23

cos

)cos(QQ totn

,

cos

)sin(tot

QQt , (4.4.4)

cos

)sin(totn MM

,

cos

)cos(tott MM

,

where the − is the angle of attack; - is the angle between total aerodynamic force Pa and lift force Y (see 4.1), the Qtot and Mtot are shear force and bending moment values in the design cross section in speed coordinate system, taken from the diagrams (fig. 1.1.2) and table # 4.4.3; the Qn and Qt are shear forces; the Mn and Mt 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 Qt and Mn 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. 1.2.1). 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 mz affected by the

distributed loads qa

п and q

w

п. For the moment mz the formula is received, writing

down the moment from the specified loads concerning an axis of reduction. At calculation mz 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. 4.4.2).

Fig. 4.4.2. Positions of loads in cross section.

qw

qa

reduced axis

e

24

For distributed loads qa

п and q

w

п (see fig. 4.4.2) it is possible to record

formulas:

cos

)cos(qq a

yaп

,

cos

)cos(qq w

ywп

, (4.4.5)

z

w ad q eqm п n (4.4.6)

where the e, and the d are distances from load points qa

пand q

w

п to the reduction

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. 1.2.3.

You can compute their by formulas:

c .g c .gii i i w id z tgγ x b 0.5L z x b ,

c . p c . p [ii i i w ie z tgγ x b 0.5L z x b ,

r t

w

0.8( b b )tgγ

L

,

where iz - is the relative coordinate z for i-th cross section (column 2 from table

4.4.2), c .gx - is relative coordinate of wing center of gravity Integrating the diagram mz, we receive the reduced moments Mzd affected by

the distributed loads. The scheme of calculation is shown in tab. 4.4.4 in which designations is entered:

,/z)mm(M ii,zzizid 21 01111 ,z,z MM ;

z ,i ,d z ,i 1 ,d z ,i ,dM M ΔM , (i = 10, 9,..., 0).

In an explanatory book it is necessary to plot diagrams mz and Mz (a diagram

Mz is shown on fig.1.2.2 by a dashed line). In a coordinate system for the moments Mz also it is necessary to result a diagram of the reduced moments affected by concentrated masses (on fig. 1.2.2 it is shown by a light line).

Affected by a concentrated mass of the i-th aggregate the increment of the moment ∆ z ,c ,iM is found out by the formula:

z ,c ,i yag ,i i y ,ag ,i i

cos(θ α )ΔM P r P r

cosθ

, (4.4.7)

where the ri is the distance from the i-th concentrated mass gravity center to reduction axis (it is measured on the drawing), Pyag – is design weight from formula (2.4). The moment z ,,c ,iM 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 Mz,c,i is calculated by the formula:

25

z ,c ,11 z ,c ,11ΔM M 0 ;

z ,c ,i z ,c ,i 1 z ,c ,iM M ΔM , (i = 10, 9,..., 0). (4.4.8)

In the point with aggregates we have jumps of reduced moment (see fig. 1.1.2). For this table we take Mz i d from table # 4.4.4 and total reduced moment we

compute with account of signs upon formula: Mz tot = Mz d + Mz c (4.4.9)

Table # 4.4.4

Reduced moments calculation scheme from distributed loads

i iZ , m

qa

in,

kN /m

ie , m

qw

iп,

kN /m

id , m

m iz ,

kN zidM ,

kN m

M idz ,

kN m

1 2 3 4 5 6 7 8 9 0 q

a

0п 0e q

w

0п d0 m 0z M dz 0

1 2 3 4 5 6 7 8 9

10 10Z qa

10п e10 q

w

10п d10 m 10z M dz 10 M dz 10

11 11Z qa

11п e11 q

w

11п d11 m 11z 0 0

It is also necessary to plot the z ,totM total reduced moment diagram (on fig.

1.2.2 it is shown by the solid line).

26

Table # 4.4.5 Calculation scheme of reduced moment from concentrated loads and from all loads.

I Py,ag,i

kN ri

m ∆Mz,c ,i

kN*m Mz,c,i, kN*m

Mzd, kN*m

Mztot, kN*m

1 2 3 4 5 6 70 1 2 3 4 5 6 7 8 9 10 11

4.5. 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:

l agrr tot fl w k

k

Q n fg[0.5( M M ) M ], [kN] ,

l agrr tot fl w k k

k

M n fg[0.5( M M )C M c ], [kNm] , (4.5.1)

l agrz r tot fl w k k

k

M n fg( 0.5M e 0.5M d M r ), [kNm] .

Here Mfl is designing flight mass of plane from (3.4), Mw – the wing mass, С is the distance from root section to the air resultant load point; ck - 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; rk - 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:

27

wL 2 ηС

6 1 η

, (4.5.2)

where the η – is the wing taper. Values ck, and rk 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 valuesQΣr ,

M Σr 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.

4.6. 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:

QMΣΣ

zrx . (4.6.1)

The xr 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. 1.1.1).

28

Table # 4.6.1 Results of calculations Design loading condition C Design flight mass (kg) - Mfl Limit load factor - nl Safety factor - f Ultimate load factor - nu Fuel mass in 1-st fuel tank (kg) - mf1 Fuel mass in 2-nd fuel tank (kg) ) - mf2 Fuel mass in 3-rd fuel tank (kg) - mf3 Wing span (m) (for equivalent wing)- Lwe Wing taper Wing aspect ratio Root wing chord (m) (for equivalent wing) - br Tip wing chord (m) (for equivalent wing) - bt 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. 4.4.4)

The tangential bending moment for designing cross section (kN*m, form. 4.4.4)

The normal shear force for designing cross section (kN, form. 4.4.4)

The tangential shear force for designing cross section (kN, form. 4.4.4)

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.

29

APPENDIXIES

30

Appendix #1 Characteristic of airfoil

The airfoil NACA – 0009

Geometric characteristic of airfoil

(in % from chord) Aerodynamic characteristic of airfoil

X Yt Yb h ° Cy Сх Ccp

0 0 0 0 -4 -0.30 0.014 -

2.5 1.96 -1.96 3.92 -2 -0.16 0.008 -

5 2.67 -2.67 5.34 0 0.00 0.0064 -

7.5 3.15 -3.15 6.30 2 0.16 0.008 0.240

10 3.51 -3.51 7.02 4 0.30 0.014 0.240

15 4.01 -4.01 8.02 6 0.45 0.020 0.240

20 4.30 -4.30 8.60 8 0.60 0.032 0.240

25 4.46 -4.46 8.92 10 0.74 0.042 0.240

30 4.50 -4.50 9.00 12 0.90 0.059 0.240

40 4.35 -4.35 8.70 14 1.05 0.077 0.240

50 3.97 -3.97 7.94 16 1.19 0.098 0.240

60 3.42 -3.42 6.84 18 1.30 0.120 0.24и

70 2.75 -2.75 5.50 20 1.17 0.165 0.266

80 1.97 -1.97 3.94 21 1.06 0.280 0.324

90 1.09 -1.09 2.18 22 0.96 0.340 0.362

100 0 0 0 24 0.91 0.392 0.383

31



(in % from chord) Aerodynamic characteristic of

airfoil


0 0 0 0 -4 -0.30 0.015 -

2.5 2.62 -2.62 5.24 -2 -0.15 0.009 -

5 3.56 -3.56 0.00 0 0.00 0.007 -

7.5 4.20 -4.20 8.40 2 0.15 0.009 0.244

10 4.68 -4.68 9.36 4 0.30 0.015 0.244

15 5.34 -5.34 10.68 6 0.445 0.020 0.244

20 5.74 -5.74 11.48 8 0.60 0.033 0.244

25 5.94 -5.94 11.88 10 0.745 0.041 0.244

30 6.00 -6.00 12.00 12 0.90 0.059 0.244

40 5.80 -5.80 11.60 14 1.045 0.075 0.244

50 5.29 -5.29 10.58 16 1.20 0.096 0.244

60 4.56 -4.56 9.12 18 1.32 0.119 0.244

70 3.66 -3.66 7.32 20 1.46 0.142 0.244

80 2.62 -2.62 5.24 21 1.55 0.173 0.244

90 1.45 -1.45 2.90 22 1.20 0.262 0.301

100 0 0 0 24 1.09 0.322 0.335

32


Geometric characteristic of

airfoil (in % from chord)

Aerodynamic characteristic of airfoil


0 0 0 0 -4 -0.30 0.014 -

2.5 3.27 -3.27 6.54 -2 -0.15 0.009 -

5 4.44 -4.44 8.88 0 0.00 0.007 0.238

7.5 5.25 -5.25 10.50 2 0.15 0.009 0.238

10 5.85 -5.85 11.70 4 0.30 0.014 0.238

15 6.68 -6.68 13.36 6 0.45 0.020 0.238

20 7.17 -7.17 14.34 8 0.60 0.031 0.238

25 7.43 -7.43 14.86 10 0.74 0.042 0.238

30 7.50 -7.50 15.00 12 0.89 0.060 0.238

40 7.25 -7.25 14.50 14 1.02 0.075 0.233

50 6.62 -6.62 13.24 16 1.17 0.095 0.238

60 5.70 -5.70 11.40 18 1.30 0.119 0.238

70 4.58 -4.58 9.16 20 1.42 0.140 0.238

80 3.28 -3.28 6.56 21 1.55 0.178 0.238

90 1.81 -1.81 3.62 22 1.29 0.210 0.284

100 0 0 0 24 1.21 0.269 0.300

33

The airfoil NACA-21012


(in % from chord)


X Yt Yb h Су Cx Cm Ccp

0 0 0 0 -4 -0.26 0.014 -0.062 —

1.25 2.95 -0.90 3.85 -2 -0.20 0.0095 -0.024 --- 2.5 3.72 -1.45 5.17 0 0.035 0.0071 0.0072 0.206 5 4.67 -2.44 8.11 2 0.20 0.011 0.046 0.230

7.5 5.28 -З.12 8.40 4 0.36 0.017 0.0814 0.232 10 5.72 -3.64 9.36 6 0.50 0.0225 0.1165 0.23315 6.33 -4.36 10.69 8 0.65 0.034 0.152 0.234 20 6.67 -4.80 11.47 10 0.80 0.047 0.187 0.23425 6.82 -5.07 11.89 12 0.95 0.065 0.222 0.234 30 6.82 -5.18 12.00 14 1.09 0.083 0.255 0.233 40 6.52 -5.10 11.622 16 1.23 0.114 0.288 0.23450 5.89 -4.71 10.60 18 1.36 0.128 0.319 0.234 60 5.04 -4.09 9.13 20.8 1.50 0.160 0.352 0.234 70 4.03 -3.30 7.33 21 1.52 0.182 0.354 0.234 80 2.86 -2.38 5.24 21 1.20 0.252 0.352 '0.293 90 1.5757 -1.32 2.89 22 1.12 0.281 0.353 0.315 95 0.87 -0.75 1.62 24 1.02 0.341 0.360 0.353

100 0 0 0 26 0.96 0.392 0.346 0.360 30 0.88 0.464 0.347 0.394

34


Geometrical characteristic of airfoil


X Yt Yb Ym h Cy Cx Cm Ccp

0 0 0 0 0 -4 -0.25 0.0092 -0.054 ---

1.25 2.84 -1.10 0.87 3.94 -2 -0.10 0.008 -0.019 ---

2.5 3.76 -1.60 1.08 5.36 0 0.05 0.0073 0.017 0.336

5 4.97 -2.17 1.40 7.14 2 0.20 0.009 0.052 0.260

7.5 5.71 -2.68 1.52 8.39 4 0.37 0.016 0.092 0.249

10 6.22 -3.15 1.54 9.37 6 0.50 0.022 0.123 0.246

15 6.80 -3.89 1.46 10.69 8 0.66 0.034 0.161 0.244

20 7.11 -4.38 1.37 11.49 10 0.80 0.048 0.195 0.244

25 7.23 -4.66 1.29 11.89 12 0.97 0.063 0.237 0.244

30 7.22 -4.80 1.21 12.02 14 1.10 0.082 0.268 0.244

40 6.85 -4.76 1.05 11.61 16 1.24 0.105 0.300 0.244

50 6.17 -4.42 0.88 10.59 18 1.38 0.130 0.337 0.244

60 5.27 -3.85 0.71 9.12 20 1.50 0.156 0.366 0.244

70 4.19 -3.14 0.53 7.33 22 1.60 0.180 0.389 0.245

80 2.99 -2.26 0.37 5.25 22 1.26 0.252 0.368 0.292

90 1.63 -1.26 0.19 2.89 24 1.13 0.320 0.378 0.334

95 0.89 -0.71 0.09 1.60 26 1.04 0.372 0.377 0.363

100 0 0 0 0 30 0.94 0.454 0.372 0.395

35

The airfoil NACA - 2210



X Yt Yb Ym h ° Cy Сх Ccp

0 0 0 о 0 0 0.120 0.010 0.467

2.5 2.92 -1.52 0.70 4.44 2 0.262 0.013 0.339

5 4.02 -1.96 1.03 5.98 4 0.403 0.020 0.304

7.5 4.83 -2.17 1.33 7.00 6 0.545 0.029 0.291

10 5.51 -2.47 1.59 7.98 8 0.688 0.043 0.279

15 6.40 -2.50 1.96 9.00 10 0.827 0.058 0.273

20 6.78 -2.78 2.00 9.56 12 0.960 0.074 0.267

25 6.94 -2.96 1.99 9.90 14 1.080 0.094 0.264

30 6.97 -3.03 1.97 10.00 16 1.195 0.114 0.260

40 6.75 -2.95 1.90 9.70 18 1.250 0.130 0.257

50 6.16 -2.72 1.72 8.88 20 1.162 0.163 0.283

60 5.34 -2.30 1.52 7.64 21 1.158 0.207 0.299

70 4.29 -1.81 1.24 6.10 22 1.130 0.278 0.317

80 3.19 -1.41 0.89 4.60

90 1.60 -0.74 0.43 2.34

100 0 0 0 0

36

The airfoil NACA -2212 Geometric characteristic of airfoil



0 0 0 0 -4 -0,17 0,0110 −

2.5 3.35 -1.96 5.31 -2 -0.01 0.0088 −

5 4.62 -2.55 7.17 0 0.13 0.0088 0.476

7.5 5.55 -2.89 8.44 2 0.29 0.0135 0.348

10 6.27 -3.11 9.38 4 0.43 0.0195 0.316

15 7.25 -3.44 10.69 6 0.59 0.028 0.300

20 7.74 -3.74 11.48 8 0.73 0.040 0.289

25 7.93 -3.94 11.87 10 0.88 0.055 0.283

30 7.97 -4.03 12.00 12 1.02 0.072 0.278

40 7.68 -3.92 11.60 14 1.16 0.092 0.275

50 7.02 -3.56 10.58 16 1.30 0.113 0.272

60 6.07 -3.05 9.12 18 1.42 0.139 0.270

70 4.90 -2.43 7.33 20 1.54 0.162 0.269

80 3.52 -1.74 5.26 21 1.60 0.203 0.268

90 1.93 -0.97 2.90 22 1.40 0.240 0.300

100 0 0 0 24 1.31 0.310 0.327

37

The airfoil NACA -2214

Geometric characteristic of airfoil (in % from chord)



0 0 0 0 -5.12 -0.229 0.0162 0.104

2.5 3.8 -2.41 6.21 -3.27 -0.106 0.0131 -

5 5.21 -3.15 8.36 -1.51 0.017 0.0116 -

7.5 6.23 -3.58 9.81 0.3 0.139 0.0127 0.418

10 7.06 -3.90 10.96 2.14 0.264 0.0165 0.327

15 8.20 -4.28 12.48 4.01 0.396 0.0235 0.299

20 8.69 -4.69 13.38 5.79 0.535 0.0325 0.285

25 8.92 -4.94 13.86 7.65 0.678 0.0446 0.279

30 8.97 -5.03 14.00 9.5 0.825 0.0596 0.275

40 8.68 -4.89 13.57 11.39 0.943 0.0764 0.275

50 7.88 -4.44 12.32 13.15 1.057 0.0923 0.261

60 6.05 -3.71 10.66 14.99 1.154 0.110 0.261

70 5.5 -3.02 8.52 16.94 1.226 0.1302 0.260

80 3.96 -2.18 6.44 18.65 1.257 0.1672 0.263

90 2.07 -1.21 3.28 20.43 1.214 0.2041 0.285

100 0 0 0 22.22 1.190 0.2359 0.302

38




X Yt Yb Ym h ° Cy Сх Cm Ccp

0 0 0 0 0 -4 -0.22 0.013 0.046 - 1.25 2.67 -1.23 0.77 3.90 -2 -0.08 0.00955 -0.011 - 2.5 3.61 -1.71 0.95 5.32 0 0.085 0.0071 0.028 0.3305 4.91 -2.26 1.33 7.17 2 0.24 0.012 0.065 0.270

7.5 5.80 -2.61 1.60 8.41 4 0.385 0.018 0.099 0.25710 6.43 -2.92 1.76 9.35 6 0.53 0.025 0.134 0.25315 7.19 -3.50 1.85 10.69 8 0.68 0.035 0.169 0.24820 7.50 -3.97 1.77 11.47 10 0.835 0.050 0.206 0.24725 7.60 -4.28 1.66 11.88 12 0.98 0.067 0.242 0.24730 7.55 -4.46 1.54 12.01 14 1.12 0.088 0.275 0.24540 7.14 -4.48 1.33 11.62 16 1.28 0.108 0.313 0.24450 6.41 -4.17 1.12 10.58 18 1.40 0.130 0.342 0.24560 5.47 -3.67 0.90 9.14 20 1.53 0.159 0.372 0.24370 4.36 -3.00 0.68 7.36 22 1.63 0.186 0.396 0.24380 3.08 -2.16 0.46 5.24 22 1.31 0.255 0.382 0.29290 1.68 -1.23 0.23 2.71 24 1.19 0.317 0.394 0.33195 0.92 -0.70 0.11 1.62 26 1.045 — 0.390 0.375

100 0 0 0 0 30 0.98 — 0.393 0.400

39

The airfoil NACA - 2309




0 0 0 0 -2 0.00 0.009 -

2.5 2.39 -1.58 0.405 3.97 0 0.15 0.008 0.490

5 3.36 -2.01 0.675 5.37 2 0.30 0.012 0.370

7.5 4.09 -2.24 0.925 6.33 4 0.45 0.020 0.331

10 4.67 -2.38 1.145 7.05 6 0.60 0.028 0.310

15 5.54 -2.50 1.52 8.04 8 0.75 0.040 0.299

20 6.08 -2.52 1.78 8.60 10 0.90 0.054 0.290

25 6.37 -2.51 1.93 8.88 12 1.06 0.074 0.285

30 6.50 -2.50 2.00 9.00 14 1.20 0.094 0.282

40 6.32 -2.39 1.965 8.71 16 1.34 0.120 0.279

50 5.82 -2.13 1.845 7.95 18 1.44 0.142 0.278

60 5.07 -1.78 1.645 6.85 20 1.51 0.188 0.277

70 4.11 -1.38 1.365 5.49 21 1.40 0.238 0.307

80 2.96 -0.97 0.995 3.93 22 1.30 0.310 0.342

90 1.64 -0.54 0.55 2.18 24 1.20 0.380 0.375

100 0 0 0 0

40



(in % from chord)



0 0 0 0 0 -2 0.00 0.003 -

2.5 3.11 -2.16 0.475 5.27 0 0.13 0.011 0.527

5 4.31 -2.85 0.73 7.16 2 0.30 0.014 0.377

7.5 5.18 -3.26 0.96 8.14 4 0.44 0.020 0.338

10 5.86 -3.52 1.17 9.38 6 0.58 0.028 0.310

15 6.89 -3.82 1.535 10.71 8 0.74 0.040 0.297

20 7.54 -3.94 1.80 11.48 10 0.90 0.056 0.289

25 7.88 -3.99 1.945 11.87 12 1.04 0.064 0.284

30 8.00 -4.10 2.00 12.00 14 1.18 0.090 0.273

40 7.77 -3.84 1.965 11.61 16 1.30 0.114 0.279

50 7.14 -3.45 1.845 10.59 18 1.42 0.140 0.276

60 6.21 -2.92 1.645 9.13 20 1.54 0.164 0.276

70 5.02 -2.31 1.355 7.33 21 1.61 0.200 0.276

80 3.62 -1.63 0.995 5.25 22 1.47 0.247 0.302

90 2.00 -1.91 0.545 2 91 24 1.36 0.300 0.316

100 0 0 0 0 26 1.24 0.360 0.351

41

The airfoil NACA -2315




0 0 0 0 -4 -0.19 0.013 −

2.5 3.85 -2.74 6.59 -2 -0.01 0.010 −

5 5.26 -3.66 8.92 0 0.13 0.011 0.510

7.5 6.28 -4.25 10.74 2 0.30 0.014 0.357

10 7.08 -4.66 11.74 4 0.42 0.020 0.324

15 8.25 -5.13 13.38 6 0.53 0.030 0.302

20 8.97 -5.38 14.35 8 0.72 0.040 0.292

25 9.36 -5.48 14.84 10 0.86 0.054 0.285

30 9.50 -5.50 15.00 12 1.01 0.072 0.279

40 9.22 -5.29 14.51 14 1.10 0.090 0.277

50 8.47 -4.77 13.24 16 1.30 0.110 0.273

60 7.66 -4.06 11.42 18 1.40 0.140 0.274

70 5.95 -3.22 9.17 20 1.53 0.162 0.274

80 4.29 -2.28 6.57 21 1.54 0.172 0.275

90 2.39 -1.26 3.62 22 1.44 0.230 0.297

95 1.30 -0.72 2.02 24 1.40 0.280 0.314

100 0 0 0 26 1.34 0.340 0.324

42




X Yt Yb Ym h ° Cy Cх Cm Ccp

0 0 0 0 0 -4 -0.18 0.012 0.001 --

1.25 2.15 -1.65 0.25 3.80 -2 0.00 0.0088 0.044 __

2.5 2.99 -2.27 0.36 5.26 0 0.13 0.010 0.076 0.588

5 4.13 -3.01 0.56 7.14 2 0.29 0.0128 0.119 0.397

7.5 4.96 -3.46 0.75 8.42 4 0.42 0.020 0.150 0.355

10 5.63 -3.75 0.94 9.38 6 0.58 0.030 0.189 0.326

15 6.61 -4.10 1.255 10.71 8 0.72 0.040 0.224 0.311

20 7.26 -4.23 1.515 11.49 10 0.88 0.052 0.264 0.300

25 7.67 -4.22 1.725 11.89 12 1.00 0.074 0.294 0.294

30 7.88 -4.12 1.88 12.00 14 1.16 0.090 0.334 0.288

40 7.80 -3.80 2.00 11.60 16 1.30 0.112 0.370 0.281

50 7.24 -3.34 1.95 10.58 18 1.40 0.140 0.392 0.281

60 6.36 -2.76 1.80 9.12 20 1.52 0.160 0.424 0.279

70 5.18 -2.14 1.52 7.32 22 1.60 0.192 0.444 0.278

80 3.75 -1.50 1.125 5.25 24 1.34 0.300 0.436 0.325

90 2.08 -0.82 0.63 2.90 26 1.20 0.360 0.428 0.355

95 1.14 -0.48 0.33 1.62 28 1.10 — 0.414 0.377

100 0 0 0 0

43




X Yt Yb Ym h ° Cy Cx Cm Ccp

0 0 0 0 0 -4 -0,18 0,013 -0,050 —

1,25 2,71 -2,06 0,33 4,77 -2 -0,02 0,010 0,035 —

2,5 3,71 -2,86 0,43 6.57 0 0,13 0,012 0,0735 0,557

5 5,07 -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 0,345

10 6,83 -4,90 0,87 11,73 6 0,57 0,030 0,182 0,320

15 7,97 -5,42 1.28 13,39 8 0,71 0,042 0,218 0,307

20 8,70 -5,66 1.52 14,36 10 0,86 0,056 0,255 0,297

25 9,17 -5,70 1,74 14,87 12 1,00 0,071 0,288 0,288

30 9,38 -5,62 1.88 15,00 14 1,15 0,090 0,326 0,283

40 9,25 -5,25 2,00 14,50 16 1,28 0,112 0,360 0,281

50 8,57 -4,67 1.95 13,24 18 1,40 0,136 0,390 0,278

60 7,50 -3,90 1.80 11,40 20 1,50 0,160 0,415 0,276

70 6,10 -3,05 1,53 9,15 22 1,54 0,192 0,425 0,276

80 4,41 -2,15 1,13 6,56 24 1,41 0,280 0,441 0,313

90 2,45 -1,17 0,64 3,62 26 1,31 0,332 0,439 0,335

95 1,34 -0,68 0,33 2,02 28 1,20 0,383 0,425 0,354

100 0 0 0 0 30 1,10 — 0,415 0,378

44




airfoil


0 0 0 0 0

1,25 1,62 -1,23 0,195 2,85 -4 -0,192 0,012 -0,004 —

2,5 2,27 -1,66 0,305 3,93 -2 0,00 0,008 0,044 —

5 3,2 -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 0,397

10 4,43 -2,60 0,915 7,03 4 0,43 0,020 0,150 0,352

15 5,25 -2,77 1,24 8,02 6 0,58 0,028 0,188 0,326

20 5,81 -2,79 1,51 8,60 8 0,72 0,040 0,224 0,311

25 6,18 -2,74 1,72 8,92 10 0,88 0,054 0,264 0,300

30 6,38 -2,62 1,88 9,00 12 1,02 0,070 0,298 0,293

40 6,35 -2,35 2,00 8,70 14 1,18 0,090 0,336 0,287

50 5,92 -2,02 1,95 7,94 16 1,30 0,112 0,370 0,284

60 5,22 -1,63 1,795 6,85 18 1,43 0,140 0,402 0,281

70 4,27 -1,24 1,515 5,51 20 1,50 0,180 0,416 0,277

80 3,10 -0,85 1,125 3,95 22 1,30 0,270 0,444 0,342

90 1,72 -0,47 0,625 2,19 24 1,16 0,370 0,430 0,371

95 0,94 -0,28 0,33 1,22 26 1,08 — 0,420 0,389

100 0 0 0 0 28 1,00 — 0,410 0,410

45




airfoil


0 — 0 0 0 -4 -0,21 0,014 -0,042 —

1,25 3,34 -1,54 0,90 4,90 -2 -0,06 0,011 -0,006 —

2,5 4,44 -2,25 1,095 6,69 0 0,09 0,0082 0,029 0,332

5 5,89 -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 0,259

10 7,64 -4,09 1,78 11,73 6 0,53 0,027 0,135 0,255

15 8,52 -4,84 1,84 13,36 8 0,69 0,038 0,173 0,251

20 8,92 -5,41 1,76 14,33 10 0,83 0,051 0,206 0,248

25 9,08 -5,78 1,65 14,86 12 0,98 0,068 0,242 0,247

30 9,05 -5,96 1,55 15,01 14 1,13 0,088 0,278 0,246

40 8,59 -5,92 1,34 14,51 16 1,27 0,108 0,312 0,246

50 7,74 -5,50 1,12 13,24 18 1,40 0,132 0,343 0,245

60 6,61 -4,81 0,90 11,42 20 1,52 0,158 0,372 0,244

70 5,25 -3,91 0,67 9,16 22,2 1,61 0,190 0,393 0,244

80 3,73 -2,83 0,45 6,56 22,2 1,36 0,245 0,375 0,275

90 2,04 -1,59 0,23 3,63 24 1,27 0,288 0,379 0,298

95 1,12 -0,90 0,12 2,02 26 1,18 0,338 0,382 0,324

100 0 0 0 0 30 1,01 — 0,372 0,368

46





0 — 0 0 0 -4 -0,22 0,012 -0,0415 —

1,25 2,04 -0,91 0,07 2,95 -2 -0,09 0,009 -0,013 —

2,5 2,83 -1,19 0,82 4,02 0 0,09 0,0066 0,031 0,344

5 3,93 -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 0,264

10 5,26 -1,79 1,74 7,05 6 0,53 0,023 0,137 0,258

15 5,85 -2,17 1,84 9,02 8 0,69 0,035 0,175 0,254

20 6,06 -2,55 2,26 8,61 10 0,83 0,050 0,209 0,252

25 6,11 -2,80 1,66 8,91 12 0,975 0,066 0,244 0,250

30 6,05 -2,96 1,55 9,01 14 1,12 0,088 0,279 0,249

40 5,69 -3,03 1,33 8,72 16 1,29 0,110 0,320 0,248

50 5,09 -2,86 1,12 7,95 18 1,40 0,133 0,347 0,247

60 4,32 -2,53 0,89 6,85 20,3 1,55 0,170 0,383 0,247

70 3,42 -2,08 0,72 5,50 20,3 1,30 0,232 0,383 0,295

80 2,41 -1,51 0,45 3,92 22 1,25 0,290 0,401 0,320

90 1,31 -0,86 0,23 2,17 24 1,16 0,360 0,420 0,362

95 0,72 -0,50 0,11 1,22 26 1,08 — 0,410 0,380

100 0 0 0 0 30 0,95 — 0,389 0,409

47


Geometric characteristic of airfoil (in %

from chord) Aerodynamic characteristic of

airfoil


0 0 0 0 0 -4 -0,20 0,012 -0,043 —

1,25 3,32 -0,86 1,23 4,18 -2 -0,05 0,0078 -0,007 —

2,5 4,36 -1,11 1,625 5,47 0 0,10 0,0085 0,030 0,300

5 5,69 -1,50 2,095 7,19 2 0,26 0,0128 0,067 0,257

7,5 6,48 -1,91 2,29 8,39 4 0,40 0,018 0,100 0,250

10 6,99 -2,38 2,31 9,37 6 0,55 0,027 0,137 0,249

15 7,53 -3,18 2,18 10,71 8 0,70 0,038 0,173 0,247

20 7,80 -3,68 2,06 11,48 10 0,85 0,052 0,208 0,245

25 7,87 -4,00 1,94 11,87 12 1,00 0,070 0,249 0,244

30 7,81 -4,20 1,81 12,01 14 1,16 0,090 0,283 0,244

40 7,35 -4,26 1,55 11,61 16 1,30 0,112 0,318 0,244

50 6,59 -4,00 1,30 10,59 18 1,41 0,136 0,346 0,245

60 5,60 -3,51 1,05 9,11 20 1,54 0,161 0,378 0,245

70 4,46 -2,88 0,79 7,34 21,8 1,62 0,185 0,397 0,245

80 3,15 -2,10 0,53 5,25 21,8 1,26 0,266 0,370 0,302

90 1,71 -1,19 0,26 2,90 24 1,11 0,334 0,386 0,348

95 0,93 -0,69 0,12 1,62 28 1,00 — 0,379 0,379

100 0 0 0 0 30 1,97 — 0,392 0,404

48





0 0 0 0 0 -4 -0,20 0,012 -0,035 —

1,25 2,58 -1,34 0,62 3,92 -2 -0,04 0,0075 -0,0035 —

2,5 3,50 -1,85 0,83 5,35 0 0,11 0,008 0,0391 0,356

5 4,80 -2,37 1,22 7,17 2 0,28 0,013 0,079 0,282

7,5 5,74 -2,70 1,52 8,44 4 0,42 0,019 0,125 0,298

10 6,44 -2,95 1,75 9,39 6 0,57 0,027 0,148 0,259

15 7,37 -3,34 2,015 10,71 8 0,71 0,040 0,1815 0,255

20 7,82 -3,66 2,08 11,48 10 0,86 0,054 0,217 0,252

25 7,96 -3,92 2,02 11,88 12 1,01 0,072 0,252 0,250

30 7,89 -4,11 1,89 12,00 14 1,16 0,092 0,287 0,247

40 7,44 -4,17 1,64 11,61 16 1,30 0,113 0,321 0,246

50 6,66 -3,93 1,40 10,59 18 1,43 0,140 0,352 0,247

60 5,67 -3,47 1,10 9,14 20,8 1,59 0,175 0,390 0,245

70 4,48 -2,84 0,82 7,32 21,5 1,60 0,190 0,392 0,245

80 3,18 -2,07 0,56 5,25 21,5 1,38 0,235 0,387 0,280

90 1,73 -1,18 0,28 2,91 24 1,30 0,315 0,388 0,299

95 0,94 -0,67 0,14 1,61 26 1,18 0,368 0,400 0,339

100 0 0 0 0 30 1,00 0,461 0,387 0,387

49

The airfoil CLARK-YH




0 0 0 0 --16 -0,596 0,203 0.356

2.5 3.10 -2.03 5.13 -12 -0.562 0.095 0.264

5 4.59 -2.54 7.13 -8 -0.388 0.025 0.196

7.5 5.62 -2.81 8.43 -4 -0.130 0.013 -

10 6.42 -3.03 9.45 -2 0.000 0.012 -

15 7.57 -3.24 10.81 0 0.130 0.013 0.493

20 8.33 -3.25 11.58 2 0.266 0.023 0.330

30 8.85 -3.14 11.99 4 0.400 0.072 0.278

40 8.66 -3.00 11.66 8 0.656 0.043 0.308

50 7.91 -2.84 10.75 10 0.792 0.059 0.300

60 6.71 -2.69 9.40 12 0.924 0.077 0.294

70 5.07 -2.43 7.50 16 1.166 0.118 0.286

80 3.39 -1.98 5.37 18 1.258 0.146 0.286

90 1.73 -1.21 2.94 20 1.28 0.180 0.297

95 0.90 -0.69 1.59 22 1.24 0.239 0.316

100 0.08 -0.08 0.16 24 1.148 0.289 0.344

50

The airfoil CAGI –6-8.3%




0 0 0 0 -2 0,034 0,0110 −

2.5 1.80 -0.98 2.78 0 0.168 0.012 0.619

5 2.78 -1.23 4.01 2 0.294 0.016 0.470

7.5 3.62 -1.32 4.94 4 0.428 0.022 0.298

10 4.29 -1.34 5.63 6 0.562 0.032 0.359

15 5.26 -1.34 6.60 8 0.684 0.045 0.342

20 6.05 -1.28 7.33 10 0.808 0.061 0.322

30 7.20 -1.09 8.29 12 0.922 0.067 0.303

40 7.04 -0.90 7.94 14 1.004 0.122 0.298

50 6.63 -0.60 7.23 16 1.038 0.168 0.308

60 5.82 -0.35 6.17 18 1.024 0.231 0.346

70 4.52 -0.28 4.80

80 3.04 -0.16 3.20

90 1.51 -0.07 1.58

100 0 0 0

51

The airfoil MUNK- 1




0 0 0 0 --3 -0.208 0.009 ----

2.5 1.36 -1.36 2.72 1.5 -0.104 0.008 ---

5 1.8 -1.8 3.6 0 -0.006 0.007 ---

7.5 2.1 -2.1 4.2 1.5 0.120 0.008 0.158

10 2.34 -2.34 4.68 3 0.231 0.011 0.198

15 2.67 -2.67 5.34 4.5 0.341 0.014 0.237

20 2.88 -2.88 5.76 6 0.458 0.020 0.240

30 3.05 -3.05 6.1 9 0.667 0.034 0.264

40 2.85 -2.85 5.7 12 0.782 0.101 0.275

50 2.53 -2.53 5.06 15 0.805 0.196 0.2286

60 2.08 -2.08 4.16 18 0.788 0.257 0.312

70 1.54 -1.54 3.08 21 0.742 0.297 ---

80 0.91 -0.91 1.82

90 0.20 -0.20 0.40

100 0 0 0

52

Appendix # 2 THE STANDARD ATMOSPHERE IN SYSTEM SI.

Height. Н. m

Temperature tH. 0C

Pressure.PH. Pa

Density. Н. kg/m3

Relative density =Н/0

Acoustic speed

m/s km/h

-1000 21.5 113920 1.347 1.099 344.1 1238

0 15 101325 1.225 1.000 340.2 1225

1000 8.5 89860 1.11 0.907 336.4 1211

2000 2.0 79500 1.006 0.821 332.5 1197

3000 -4.5 70130 0.909 0.742 328.5 1183

4000 -11.0 61595 0.819 0.668 324.5 1168

5000 -17.5 54000 0.736 0.601 320.5 1154

6000 -24.0 47200 0.660 0.539 316.4 1139

7000 -30.5 41060 0.590 0.482 312.2 1124

8000 -37.0 35600 0.526 0.420 308 1109

9000 -43.5 30800 0.467 0.381 303.8 1093

10000 -50.0 26400 0.413 0.337 299.4 1078

11000 -56.5 22665 0.365 0.298 295 1062

12000 -56.5 19385 0.312 0.254 295 1062

13000 -56.5 16570 0.266 0.217 295 1062

14000 -56.5 14160 0.228 0.186 295 1062

16000 -56.5 10280 0.166 0.137 295 1062

18000 -56.5 7560 0.120 0.099 295 1062

20000 -56.5 5520 0.088 0.072 295 1062

53

Appendix # 3 MINISTRY OF EDUCATION AND SCIENCE OF UKRAINE

National Aerospace University «Kharkiv Aviation Institute»

Strength Department

CALCULATION OF WING LOADS

Explanatory book

(ALL THE WAY-0000-0000LEB)

Fulfilled by:

Checked up by:

Kharkiv, 2014

54

REFERENCES

1. Евсеев Л. А. Расчет на прочность крыла большого удлинения. Харьков. ХАИ. 1985.

2. Кирпикин А. А. Расчет нагрузок на фюзеляжи самолетов и вертолетов. Харьков. ХАИ. 1992.

3. Зайцев В. Н., Рудаков В. Л. Конструкция и прочность самолетов. К. 1978.

55

CONTENTS

INTRODUCTION…………………………………………………………………..…..3 1. PLAIN GENERAL DATA……………………………………………………………...4 1. 1. WING’S GENERAL DATA……………………………………………………………5 1.2. WING’S GEOMETRICAL DATA……………………………………………………..5 2. DETERMINATION OF LIMIT LOAD FACTOR……………………………………...10 3. WING’S MASS DATA …………………………………………………………………10 4. WING’S LOADS CALCULATION …………………………………………………....13 4.1. AIR LOADS ALLOCATION ON THE WING’S SPAN…………………………….15 4.2. THE WING STRUCTURE MASS LOAD ALLOCATION………………………....17 4.3. CALCULATION OF THE TOTAL DISTRIBUTED LOAD ON A WING………….18 4.4. THE CHEAR FORCES, BENDING AND REDUCED MOMENTS’ DIAGRAMS PLOTTING …………………………………………………………………………………18

4.5. LOAD CHECKING FOR WING ROOT CROSS SECTION………………………26 4.6. CALCULATION OF SHEAR FORCE’S POSITION IN THE DESIGN CROSS SECTION…………………………………………………………………………………...27 APPENDIXIES…………………………………………………………………….……….29 APPENDIX #1.CHARACTERISTIC OF AIRFOIL………………………………………30 APPENDIX # 2. THE STANDARD ATMOSPHERE IN SYSTEM SI ………………...52 APPENDIX # 3. THE COVER PAGE …………………………………………………..53 REFERENCES…………………………………………………………………………….54

CALCULATION OF WING LOADS - k102.khai.edu · calculated limit load factors in this home task on discipline ... All calculations should be made with high ... prototype located in a

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