1 Introduction

Different and often contradictory requirements are imposed on aircrafts, for example, it must have good flight and performance characteristics, but it must also be strong enough and provide the required design life with a minimum weight. The minimum weight of the aircraft airframe structure while providing a given level of reliability is one of the main criteria determining the perfection of the aircraft design.

One of the most critical aircraft structures is the wing, which is designed to create aerodynamic lift and ensure the transverse stability of the passenger aircraft [1, 2]. The weight of the wing can reach 35–40% of the airframe weight that is why the wings of modern aircraft are mainly made of polymer composite materials [3]. Continuous growth of speed and height of aircraft flight has a decisive influence on the changes in their aerodynamic layout and structural-power diagrams, which leads to significant changes in the shape and thickness of the wing profiles. All this requires further development and improvement of methods for calculating the strength of aircraft structures, in particular the wing.

The strength of wing structures is one of the main factors of flight safety of any aircraft. Its design is a complex task involving the development and analysis of a large number of options and schemes, which leads to significant time and financial costs. There are a large number of scientific works in the field of aerodynamics and loads on the wing, in which the strength calculation is carried out, experimental studies are taken into account and simulation is carried out to obtain the most accurate data, for example, [4, 5]. In this case, when calculating the loads acting on the wing, surface forces (distributed aerodynamic loads directly acting on the wing fabric covering) and mass forces (primarily the weight of the wing structure) are considered [6].

The purpose of this work, which is a continuation of earlier research by the authors [7, 8], is to develop an engineering methodology for the strength calculation of a caisson aircraft wing based on an analytical model and statistical modeling to assess the impact of the safety factor on the probability of its failure-free operation. Hereinafter, the safety factor will be understood as the value by which the design load on the wing is multiplied to take into account the influence of such factors as technological defects, randomly occurring loads, etc.

2 Theoretical basis

Theoretical analysis of the technical systems functioning processes, to which the aircraft wing belongs, is based on the choice of certain models or calculation schemes. In this case, two approaches to the analysis are possible: deterministic (cause–effect) and probabilistic-statistical one. In the first case, it is assumed that all factors influencing the model behavior are known and unchangeable. Therefore, with the help of deterministic methods it is possible to analyze the functioning of the system only with the initial data that are available to the researcher. However, since the behavior of real systems has, as a rule, a random character, the results obtained on the basis of deterministic models are not always correct.

The probability-statistical methods are based on taking into account the action on the system of many factors, each of which has a stochastic (random) nature. In this case on the basis of the totality of these random factors, a decisive rule is formed, with the help of which a decision is made whether the system belongs to one of two possible states (dichotomy): "serviceable" or "defective" one [9]. When using the probabilistic-statistical approach, the effectiveness of decisions depends on factors that are random quantities with known statistical characteristics. In practice, probabilistic-statistical methods are often used when the conclusions made on the basis of sample data are transferred to the entire population, for example, in relation to the study of the reliability of aircraft wings, the previously obtained statistical material on the loads and the strength of the wing structure is used.

One of the best known and most used probability-statistical methods is Monte Carlo method: numerical method of solving various problems by simulating random events, based on obtaining a large number of realizations of random variables, which are formed so that their probability characteristics coincide with similar values of the problem to be solved [10, 11]. Monte Carlo method organizes a procedure for drawing random phenomena, the result of which is also random. As a rule, this method is used when modeling complex systems where there are many interacting random factors, as well as when checking the applicability of simpler analytical methods and finding out the conditions for their applicability [12].

As applied to the considered problem, there are two methods of determining the probability of failure-free operation of an aircraft wing structure: spatial–temporal and quasi-static one. The spatial–temporal method allows determining the probability of operability of a structure at all stages of flight, i.e., to estimate the probability of failure at a certain time interval. To implement the spatial–temporal methodology, it is necessary to know in advance the correlation functions of the parameters by which loads, load capacity and state parameters of the structure are calculated. During the design period, the correlation functions cannot be found using statistical processing of test results, because the structure does not yet exist in the metal. Borrowing initial data from test results of previously created structures requires proving the validity of their transfer.

The quasi-static methodology considers the probability of structural failure only at the most critical points, and the calculation is performed at a fixed point in time, at which the loading of the wing structure is most dangerous. Loads and load capacity of the wing in this approach are random values, which necessitates the use of statistical modeling in the calculations.

3 Methodology

Structurally, the caisson wing of an aircraft consists of a power set, the basis of which is formed by the fabric covering, the caisson and the ribcage located across the wing and forming its profile. Fig. 1 shows schematically a caisson wing pinched along one edge under the action of uniform transverse pressure P and having a span l and an end chord b. For further analysis, we introduce the following designations of the wing profile parameters (Fig. 2): δ is the fabric covering, δn is the ribcage thickness, bn is the ribcage width, δk is the caisson thickness, b0 is the side chord value, bk is the size of the root chord of the caisson, z(x) is the variable building height, c = z(x)/b(x) is the relative profile thickness, η = b0/b is the wing contracture and χ is the angle of sweep along the neutral axis of the caisson. We will also assume that the origin of coordinates (x = 0) is in the termination in the center of the wing. It should be noted that in the loading scheme of the caisson wing (Fig. 1) it is assumed that the pressure on it is constant and does not change either along the chord or along the span of the wing. This approach, which is often used for engineering calculations and does not affect the very idea of the proposed method, can significantly simplify mathematical calculations.

Fig. 1
figure 1

Loading scheme of the caisson wing

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Fig. 2
figure 2

Profile of the caisson wing and its parameters

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To obtain the analytical solution of the problem, we assume that the section of the wing can be replaced by the reduced section in the form shown in the Fig. 3 and in the Fig. 4, where the ratio of the aircraft fabric covering to the flat form kz = z0(x)/z(x) is selected on the basis of equality of moments of inertia of the original and reduced sections.

Fig. 3
figure 3

Reduced cross section of the wing profile

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Fig. 4
figure 4

Cross section of ribcage

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We assume that the form of vibration (deflection w) for the wing is modeled as for a cantilever beam and has the following form:

$$ w = w_{0} \left( {1 - \cos \left( {\frac{\pi x}{{2l}}} \right)} \right), $$
(1)

where w0 is the deflection amplitude (maximum deflection).

The choice of a suitable form of vibration serves as the main point in the considered problem. For elastic cantilever, the permissible shape of the deformed axis must satisfy the conditions of zero deflection and angle of rotation in the embedment as well as maximum deflection and zero second derivative (absence of moment of forces) at the free end. Expression (1) satisfies all the above boundary conditions.

Assuming that the deformations are small (Bernoulli–Euler elastic bending [13]), we use the expression for the potential energy of the wing deformation:

$$ U_{kp} = U_{ob} + U_{k} + U_{n} , $$
(2)

where Uob. is the epy potential energy of deformation of the wing fabric covering, Uk is the potential energy of the caisson deformation and Un is the potential energy of deformation of the ribcage.

Hereinafter, we will mainly use the results of [7, 8] without deriving the corresponding expressions. Taking it into account, we write the components of the potential energy in formula (2) in the following form:

$$ \begin{aligned} U_{ob} & = \frac{E}{2}\mathop \int \limits_{0}^{1} I_{ob} \frac{{{\text{d}}^{2} w}}{{{\text{d}}x^{2} }}{\text{d}}x, \\ U_{k} & = \frac{E}{2}\mathop \int \limits_{0}^{{\frac{l}{\cos \left( \chi \right)}}} I_{k} \frac{{{\text{d}}^{2} w_{k} }}{{{\text{d}}x^{2} }}{\text{d}}x, \\ U_{n} & = U_{nst} + U_{npol} = \frac{E}{2}\mathop \int \limits_{0}^{{\delta_{n} }} I_{nst} \frac{{{\text{d}}^{2} w}}{{{\text{d}}x^{2} }}{\text{d}}x \\ & \quad+ \frac{E}{2}\mathop \int \limits_{0}^{{b_{n} }} 2I_{npol} \frac{{{\text{d}}^{2} w}}{{{\text{d}}x^{2} }}{\text{d}}x, \\ \end{aligned} $$
(3)

where \({I}_{ob},{I}_{k},{I}_{nst},{I}_{npol}\) are the moments of inertia of fabric covering, caisson, ribcage wall and ribcage flange, respectively, E is the modulus of elasticity of the fabric covering, ribcage and caisson material, χ is the angle of sweep along the neutral axis of the caisson (Fig. 2).

Caisson deflection wk by analogy with formula (1) is determined by the following expression:

$$ w_{k} = w_{0} \left( {1 - \cos \left( {\frac{\pi x\cos \left( \chi \right)}{{2l}}} \right)} \right), $$
(4)

and the moment of inertia of the fabric covering with the following formula:

$$ I_{ob} = 2\left( {\frac{{\delta^{3} b\left( x \right)}}{12} + b\left( x \right)\delta z_{0}^{2} \left( x \right)} \right), $$
(5)

where b(x) is the wing chord in the section with coordinate x, z0(x) is the height of the reduced section with coordinate x (Fig. 2).

$$ b\left( x \right) = b_{0} \left( {1 - \frac{x}{2l}} \right),z_{0} \left( x \right) = k_{z} z\left( x \right) = k_{z} \left( {\frac{b\left( x \right)c - \delta }{2}} \right). $$

The value of the moment of inertia of the caisson is the following:

$$ I_{k} = \frac{{b_{k} \left( x \right)h_{k1}^{3} \left( x \right) - \left( {b_{k} \left( x \right) - 2\delta_{k} } \right)h_{k2}^{3} \left( x \right)}}{12}, $$
(6)

where bk(x) is the chord of the caisson in the section with the coordinate x, hk1(x) is the outer height of the caisson with coordinate x, hk2(x) is internal height of the caisson with coordinate x (Fig. 5).

$$ b_{k} \left( x \right) = b\left( x \right)k, h_{k1} \left( x \right) = b\left( x \right)c - 2\delta ,h_{k2} \left( x \right) = h_{k1} \left( x \right) - 2\delta_{k} , $$

where k is the ratio of the caisson chord to the wing chord in the section with coordinate x.

Fig. 5
figure 5

Reduced section with coordinate x

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The value of the moment of inertia of the ribcage tube is written in the following form:

$${I}_{n}={I}_{nst}+{I}_{np}=\frac{b(x){\left({h}_{n}\left(x\right)-{\delta }_{n}\right)}^{3}}{12}+2\left(b\left(x\right){\delta }_{n}{\left(\frac{{h}_{n}\left(x\right)}{2}\right)}^{2}+\frac{b(x){\delta }_{n}^{3}}{12}\right),$$
(7)

where hn(x) is the construction height of ribcage in the section x.

$$ h_{n} \left( x \right) = 2k_{z} \left( {z\left( x \right) - \frac{{\delta - \delta_{n} }}{2}} \right). $$

Substituting expressions (1, 4, 5, 6, 7) into (3) and integrating by coordinate x, we obtain the relations for calculating the potential deformation energies of fabric covering, caisson and ribcage.

The potential deformation energy of the fabric covering is the following:

$$ U_{ob} = Ew_{0}^{2} Q_{ob} . $$

Expression for Qob is determined by the following ratio:

$$ \begin{aligned} Q_{ob} & = \frac{1}{{48B_{1}^{4} }}\left( \begin{gathered} 3B_{1} \sin \left( {2B_{1} l} \right)\left( {G_{5} \left( {2B_{1}^{2} l\left( {G_{1} l^{2} + G_{2} l + G_{5} } \right) - 3G_{1} l - G_{2} } \right) + 2G_{4} B_{1}^{2} } \right) \hfill \\ + 3G_{5} \cos \left( {B_{1} l} \right)^{2} \left( {2B_{1}^{2} \left( {6G_{1} l^{2} + 2G_{2} l + G_{3} } \right) - 3G_{1} } \right) \hfill \\ + B_{1}^{2} \left( {B_{1}^{2} l\left( {G_{5} l\left( {3G_{1} l^{2} + 4G_{2} l + 6G_{3} } \right) + 12G_{4} } \right) - 3G_{5} \left( {3G_{1} l^{2} + 2G_{2} + 2G_{3} } \right)} \right) + 9G_{1} G_{5} \hfill \\ \end{gathered} \right), \\ B_{1} & = \frac{\pi }{2l},G_{1} = 2b_{0}^{3} \delta k_{z}^{2} B_{3}^{3} c^{2} ,G_{2} = 2b_{0}^{2} \delta k_{z}^{2} B_{3}^{2} c\left( {2\delta - b_{0} c} \right),B_{3} = \eta - \frac{1}{\eta l}, \\ G_{3} & = - 2b_{0} \delta B_{3} \left( {k_{z}^{2} \left( {\delta - b_{0} c} \right)^{2} + \frac{{\delta^{2} }}{3}} \right), G_{4} = \frac{{b_{0} \delta^{3} \pi^{4} }}{{96l^{4} }},G_{5} = \frac{{\pi^{4} }}{{64l^{4} }}. \\ \end{aligned} $$

The potential energy of deformation of the caisson is the following:

$${U}_{k}=E{w}_{0}^{2}{Q}_{k},$$

expression for Qk is determined by the following ratio:

$$ Q_{k} = \frac{{L_{5} }}{{24B_{2}^{4} }}\left( \begin{gathered} 2B_{2} \sin \left( {2B_{2} r} \right)\left( {2B_{2}^{2} \left( {L_{1} r^{3} + L_{2} r^{2} + L_{3} r + L_{4} } \right) - 3L_{1} r - L_{2} } \right) \hfill \\ + 3\cos \left( {B_{2} r} \right)^{2} \left( {2B_{2}^{2} \left( {L_{1} r^{2} + 2L_{2} r + L_{3} } \right) - 3L_{1} } \right) + 9L_{1} \hfill \\ + B_{2}^{2} \left( {B_{2}^{2} \left( {3L_{1} r^{4} + 4L_{2} r^{3} + 2L_{3} r^{2} + 12L_{4} } \right) - 9L_{1} r^{2} - 6L_{2} r - 6L_{3} } \right) \hfill \\ \end{gathered} \right), $$

where

\(B_{2} = \frac{\pi \cos ( \chi)}{{2l}},L_{1} = 2\delta_{k} b_{0}^{3} B_{3}^{3} c^{2} ( {1 + 3k} ),\) \(L_{2} = 3b_{0}^{3} B_{3}^{2} c( k( b_{0} c - 2\delta)^{2} - ( b_{0} c - 2\delta - 2\delta_{k}) ( k( b_{0} c - 2\delta - 2\delta_{k}) + b_{0} k - 2\delta_{k})),\) \( L_{3} = b_{0} B_{3} ( 2( b_{0} c - 2\delta - 2\delta_{k})( k( 2b_{0} c - \delta - \delta_{k}) - 3c\delta_{k}) - k( b_{0} c - 2\delta)^{2} (3c - b_{0} c + 2\delta)), \) \(L_{4} = b_{0} k\left( {b_{0} c - 2\delta } \right)^{3} - \left( {b_{0} c - 2\delta - 2\delta_{k} } \right)^{2} \left( {b_{0} k - 2\delta_{k} } \right),\) \(L_{5} = \frac{{\pi^{4} \cos \left( \chi \right)^{4} }}{{192l^{4} }}.\)

The potential energy of deformation of the ribcage is the following:

$$ U_{nst} = Ew_{0}^{2} \left( {Q_{nst} + Q_{npol} } \right). $$

Expression for Qnst is determined by the following ratio:

$$ Q_{nst} = \frac{{N_{5} }}{{240B_{1}^{5} }}\left( \begin{gathered} 15\sin \left( {2B_{1} \delta_{n} } \right)\left( {B_{1} \left( {\begin{array}{*{20}c} {2B_{1}^{4} \left( {N_{1} \delta_{n}^{4} + N_{2} \delta_{n}^{3} + N_{3} \delta_{n}^{2} + N_{4} \delta_{n} + N_{6} } \right) + } \\ { + 3N_{1} \left( {1 - 2\delta_{n}^{2} } \right) - 3N_{2} \delta_{n} - N_{3} } \\ \end{array} } \right)} \right) \hfill \\ + 15\cos \left( {B_{1} \delta_{n} } \right)^{2} \left( {2B_{1}^{2} \left( {4N_{1} \delta_{n}^{3} + 3N_{2} \delta_{n}^{2} + 2N_{3} \delta_{n} + N_{4} } \right) - 3N_{2} - 12N_{1} \delta_{n} } \right) \hfill \\ + B_{1}^{2} \left( {\begin{array}{*{20}c} {B_{1}^{2} \delta_{n} \left( {15N_{2} \delta_{n}^{3} + 20N_{3} \delta_{n}^{2} + \delta_{n} \left( {60N_{4} + 12N_{1} } \right) + 120N_{6} } \right) - } \\ { - 15\left( {4N_{1} \delta_{n}^{3} + 3N_{2} \delta_{n}^{2} + 2N_{3} \delta_{n} + 2N_{4} } \right) + 45\left( {N_{2} + 2N_{1} \delta_{n} } \right)} \\ \end{array} } \right) \hfill \\ \end{gathered} \right), $$

where \(N_{1} = - b_{0}^{3} k_{z}^{3} B_{3}^{4} c^{3} ,N_{2} = b_{0}^{2} k_{z}^{2} B_{3}^{2} c^{2} \left( {b_{0} c - 3\left( {k_{z} \left( {b_{0} c - 2\delta - \delta_{n} } \right) - \delta_{n} } \right)} \right)\), \(N_{3} = 3b_{0} ck_{z} B_{3}^{2} ( k_{z} ( b_{0} c - 2\delta - \delta_{n} ) - \delta_{n} ) ( b_{0} ck_{z} + ( k_{z} ( b_{0} c - 2\delta - \delta_{n}) - \delta_{n}))\), \(N_{4} = - B_{3} ( k_{z} ( b_{0} c - 2\delta - \delta_{n} ) - \delta_{n})^{2} ( 3b_{0} ck_{z} + ( k_{z} ( b_{0} c - 2\delta - \delta_{n} ) - \delta_{n} ))\), \(.\)

Expression for Qnpol is determined by the following ratio:

$$ Q_{npol} = \frac{{M_{5} }}{{48B_{1}^{4} }}\left( \begin{gathered} 3B_{1} \sin \left( {2B_{1} b_{n} } \right)\left( {2B_{1}^{2} \left( {M_{1} b_{n}^{3} + M_{2} b_{n}^{2} - M_{3} b_{n} + M_{4} } \right) - 3M_{1} b_{n} - M_{2} } \right) \hfill \\ + 3\cos \left( {B_{1} b_{n} } \right)^{2} \left( {2B_{1}^{2} \left( {3M_{1} b_{n}^{2} + 2M_{2} b_{n} - M_{3} } \right) - 3M_{1} } \right) \hfill \\ + B_{1}^{2} \left( {B_{1}^{2} b_{n} \left( {3M_{1} b_{n}^{3} + 4M_{2} b_{n}^{2} - 6M_{3} b_{n} + 12M_{4} } \right) + 9M_{1} \left( {1 - b_{n}^{2} } \right) + 6M_{3} } \right) \hfill \\ \end{gathered} \right), $$

where \(M_{1} = - 2b_{0}^{3} \delta_{n} k_{z} B_{3}^{3} c^{2} ,M_{2} = 2b_{0}^{2} \delta_{n} k_{z}^{2} B_{3}^{2} c\left( {2b_{0} c - 2\delta - \delta_{n} } \right),\) \(M_{3} = b_{0} \delta_{n} B_{3} \left( {\frac{{\delta_{n}^{2} }}{6} + 2k_{z}^{2} \left( {\frac{{b_{0} c}}{2} - \delta - \frac{{\delta_{n} }}{2}} \right)\left( {\frac{{5b_{0} c}}{2} - \delta - \frac{{\delta_{n} }}{2}} \right)} \right)\), \(M_{4} = b_{0} \delta_{n} \left( {2k_{z}^{2} \left( {\frac{{b_{0} c}}{2} - \delta - \frac{{\delta_{n} }}{2}} \right)^{2} + \frac{{\delta_{n}^{2} }}{6}} \right),M_{5} = \frac{{\pi^{4} }}{{16l^{4} }}.\)

Maximum possible load operation with pressure P, made over the wing is equal to the following:

$$ A = P\mathop \int \limits_{0}^{l} b\left( x \right)w{\text{d}}x = Pw_{0} Y. $$
(8)

Here Y is defined by the following expression:

$$ Y = \frac{{\begin{aligned}b_{0} \left( {2B_{3} \left( {1 - \cos \left( {B_{1} l} \right)} \right) + B_{1} \left( {B_{1} l\left( {B_{3} l - 2} \right)}\right.}\right. \\ \left.{\left.{+ 2\sin \left( {B_{1} l} \right)\left( {1 - B_{3} l} \right)} \right)} \right)\end{aligned}}}{{2B_{1}^{2} }}. $$

By equating expressions (2) and (8) for potential energy of deformation and work of external forces and transforming the obtained results, we get an expression for calculating the maximum deflection amplitude of the caisson wing:

$$ w_{0} = \frac{PY}{{E\left( {Q_{ob} + Q_{k} + Q_{nst} + Q_{npol} } \right)}}. $$
(9)

The expression for the bending moment is the following:

$$ M_{u} = EI_{kp} \frac{{{\text{d}}^{2} w}}{{{\text{d}}x^{2} }} = \frac{{2I_{kp} \sigma_{kp} }}{z\left( x \right)}, $$

where σmkp is the maximum effective stresses in the wing, Ikp is the total moment of inertia of the fabric covering, caisson and ribcage, the formulas for calculating which are given above.

Taking into account (1), (4) and (9), we obtain the following relations for calculating the maximum effective stresses (at x = 0) in the wing fabric covering and caisson:

$$ \sigma_{{ob{\text{max}}}} = \frac{{fw_{0} \pi^{2} E\left( {b_{0} c - \delta } \right)}}{{16l^{2} }}, $$
(10)
$$ \sigma_{{k{\text{max}}}} = \frac{{fw_{0} \pi^{2} E{\text{cos}}^{2} \left( \chi \right)\left( {b_{0} c - 2\delta } \right)}}{{16l^{2} }}, $$
(11)

where f is the safety factor, w0 is defined by the expression (9).

Further calculation will be based on the following condition:

$$\left\{{\sigma }_{ob\mathrm{max}},{\sigma }_{k\mathrm{max}}\right\}\le {\sigma }_{0.2},$$

where σ0,2 is the conditional yield strength of the material of the wing structure.

4 Results

As an example of applying the above methodology to calculate the strength characteristics of the wing, the authors considered a wing with the following characteristics: b0 = 390 mm; l = 878 mm; E = 0.72 1011 Pa, μ = 0.3; η = 2, χ = 6 deg., δ = 2 mm; δn = 2 mm, δk = 4 mm; kz = 0.595; c = 0.1, k = 0.36, σ0,2 = 300 × 106 Pa, at the acting external pressure P = 49,000 Pa.

Preliminarily, the specified wing was calculated by the finite element method and designed in the program for the calculation and optimization of the wing according to the method outlined in [14]. The results of calculation by the finite element method agree satisfactorily with the results obtained by dependences (10) and (11). The difference in the calculation results for the maximum effective stresses was about 6.5%, which indicates the validity of the adopted mathematical model for assessing the load-carrying capacity of the caisson wing.

The constructed mathematical model of the load capacity allows conducting a numerical experiment to calculate the probability of failure-free operation of the considered wing, using Monte Carlo method. As it was noted earlier, there are two methods for determining the probability of failure-free operation of a structure: spatial–temporal and quasi-static one. While the first of them makes it possible to determine the probability of design operability at all stages of flight, the second evaluates the probability of design failure only at the most critical points. At the design stage, the quasi-static methodology can be used, for the implementation of which it is sufficient to know the distribution laws of bearing capacity and loads acting on the structure.

We will calculate the random bearing capacity of a structure by statistical method using the analytical deterministic model presented in [15, 16] and the probabilistic characteristics of disturbing parameters known in advance. The perturbing parameters of the carrying capacity of the structure in the model include: physical and mechanical characteristics of structural materials (conventional yield strength) and dimensions of structural elements (thickness of the fabric covering, flanges and walls of the caisson). The random nature of these parameters is mainly due to manufacturing errors. The numerical values of these parameters are chosen according to [17].

The load acting in the process of statistical tests is also a random value, and at the stage of designing the aircraft there is no absolutely reliable data on the loads it has to withstand. With the increase of number of parameters, taken into account in calculation of random value of acting load, the law of distribution of this load will tend to normal. The greatest influence on the probabilistic characteristics of the acting load in subsonic flight mode is caused by flight speed and wind disturbances. When flying at low altitudes over hilly terrain, wind disturbances can lead to significant increase of aircraft loads.

We represent the wind load as a sum of the static component, i.e., steady-state flow velocity and the dynamic component caused by pulsation of the wind velocity head. The wind impact is taken into account by changing the angle of attack by the value of

$${\alpha }_{ber}=\frac{W}{V},$$
(12)

where W is the random wind gust speed at a given altitude, and V is the aircraft flight speed.

To approximate the wind speed distribution, we will use the two-parameter Weibull distribution law [18, 19], which gives a good agreement with the measurement data. Then the distribution density function can be written in the following form:

$$v\left(W\right)=\frac{\gamma }{\psi }{\left(\frac{W}{\psi }\right)}^{\gamma -1}\mathrm{exp}{\left(\frac{W}{\psi }\right)}^{\gamma },$$
(13)

where γ is the scale parameter close to the value of the average wind speed, and ψ is the parameter of the shape of the distribution curve. Graph of the probability density function for the average wind speed at ψ = 12 m/s and γ = 1.2 \(\upgamma =1.5\) \(\upgamma =1.5\) (which corresponds to the flight mode at an altitude of 50 m for the area of northern Russia [15]) is shown in the Fig. 6.

Fig. 6
figure 6

Graph of the probability density function of wind gust speed

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Considering that the load acting on the wing is the sum of the loads from the steady angle of attack and the random angle of attack caused by the wind gust, we get the following ratio:

$$P={C}_{y}^{\alpha }\left({\alpha }_{\mathrm{prog}}+{\alpha }_{\mathrm{ber}}\right)\frac{\rho {V}^{2}}{2}.$$
(14)

Here V is the aircraft flight speed, assumed in the calculations to be 240 m/s, Cαy is the partial derivative of the lift coefficient on the angle of attack, αnpog is the program (steady-state) angle of attack, and ρ is the air density at the altitude of the aircraft.

The block diagram of the algorithm for calculating the probability of no-failure operation of the wing in question is shown in the Fig. 7. The result of the program's work is the coefficient value, which is necessary to obtain the specified level of no-failure operation.

Fig. 7
figure 7

Block diagram of the statistical testing algorithm

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In the block diagram the following designations are taken: β is the given probability of no-failure operation; βraz is the no-failure rate; f is the safety factor; N is the number of experiments; P is the random destructive load for the wing; Praz is the load with safety factor; δ is the structural thicknesses; E is the flexural modulus; Pber is the random value of the wing load capacity; σber, δ, E are the random values of element thicknesses, yield strength and elastic modulus, respectively.

During modeling, the most loaded point on the aircraft flight path was taken as the design point and for it, in accordance with the deterministic approach, the value of the carrying capacity was calculated, which means the load that leads to local or total destruction of the structure. The load-carrying capacity was calculated by the method of successive approximations with the step Δ by the safety factor and a given accuracy ε of calculating the probability of no-failure operation. At each calculation step, random values of the thicknesses of structural elements (fabric covering, flange and spar wall), yield strength and elastic modulus were calculated, with the distribution of these quantities considered normal with the corresponding values of mathematical expectation and standard deviation [17]. Then, on the basis of comparison of the random value of the wing load-carrying capacity and the random acting load, a conclusion was made about the serviceability of the structure in the given iteration. Then, the number of m experiments, in which the design retained its operability. Relation m to the total number of experiments N is the frequency of no-failure operation for a given point of the trajectory, which, with a large number of trials, is reduced to the probability of no-failure operation. According to the results of the comparison of the no-failure rate β with a given initial probability of no-failure operation βraz the decision was made to increase or decrease the safety factor, after which the experiments were performed again until the difference \(\left| {\beta_{raz} - \beta } \right|\) does not become lower than the specified calculation accuracy. As a result of the calculations, a safety factor f was obtained, providing a given probability of failure-free operation.

The Fig. 8 shows the dependence of the probability of no-failure operation on the safety factor for the most interesting, from the point of view of engineering practice, range of no-failure operation β = [0.99…0.999].

Fig. 8
figure 8

Dependence of the coefficient of safety f from the probability of β of fault-free operation of the design

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As an example, the Fig. 9 shows a histogram of the wing load-carrying capacity and load for β = 0.999 and f = 1.31. The safety factor on this histogram can also be defined as the ratio of the value of the expectation of the carrying capacity to the value of the expectation of the acting load.

Fig. 9
figure 9

Histograms of load distribution and wing load-carrying capacity for f = 1.31, β = 0.999

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Table 1 shows the results of calculations for three alternative versions of the consoles with different probabilities of failure-free operation, pre-designed in [8], with the mass of the wing console determined by the following expression:

$$ m_{kp} = m_{ob} + m_{kess} + m_{ner} , $$
(15)

where \(m_{ob} = 2\rho \delta \mathop \int \limits_{0}^{l} b\left( x \right){\text{d}}x\) is the fabric covering weight, \(m_{kess} = 2\rho \delta_{k} \left[ {\mathop \int \limits_{0}^{{\frac{l}{\cos \left( \chi \right)}}} b_{k} \left( x \right){\text{d}}x + \mathop \int \limits_{0}^{{\frac{l}{\cos \left( \chi \right)}}} h_{k1} \left( x \right){\text{d}}x} \right]\) is the caisson weight, \(m_{ner} = 2\rho \delta_{n} \left[ {\mathop \int \limits_{0}^{{b_{n} }} b\left( x \right){\text{d}}x + \mathop \int \limits_{0}^{{d_{n} }} h_{n} \left( x \right){\text{d}}x} \right]\) is the ribcage weight.

Table 1 Calculation results for the wing console
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5 Discussion

The authors of the article believe that the probabilistic-statistical approach to the choice of the safety factor, based on the notion of maximum loads and load-carrying capacity of structures as random quantities, better reflects the peculiar nature of randomly significant quantities than the deterministic approach. Besides, the probabilistic approach makes it possible to establish a qualitative connection between the safety factor and the probability of no-failure operation of an aircraft structure, which is not always possible with the deterministic approach. The presence of such a relation makes it possible in the process of designing, using the statistics of destructions, to set an acceptable value of the probability of no-failure operation, which would provide a sufficiently high level of reliability of the aircraft design and would not lead to an excessive increase in its mass.

The authors can also agree that the probabilistic approach to the choice of the safety factor is rather complicated and assumes a large and reliable statistical material on loads and on the strength of the wing structure. However, this drawback can be eliminated by the accumulation and systematization of such material, which will allow the use of probabilistic-statistical approach to the selection of the safety factor in practical activities in the design of aircraft.

6 Conclusion

The article proposes a probabilistic-time approach to solving the actual problem of aircraft airframe design: assessment of the strength of the caisson wing structure. On the basis of earlier studies of the authors, an engineering methodology for the strength calculation of the aircraft caisson wing was developed, involving an analytical model and statistical modeling in order to assess the impact of the safety factor on the probability of its failure-free operation. This methodology can be widely used in the design of aircraft as statistical material on the wing loads and its strength characteristics is accumulated. Numerical experiments based on Monte Carlo method for calculating the probability of no-failure operation of the caisson wing have been conducted. The dependences of the probability of no-failure operation on the safety factor for the most interesting, from the viewpoint of engineering practice, the non-failure range from 0.99 to 0.999 were obtained. The dependences between the safety factor, the probability of no-failure operation and the mass of the wing are given.