PROVIDING HELICOPTER FATIGUE STRENGTH: UNIT LOADS [Fundamental Mechanical and Strength Sciences]

by

© Ph. D. & Dr. Sc. Lev Gelimson

RUAG Aerospace Services GmbH, Germany

Key words: Configuration, condition, general strength theory, stress concentration, fatigue.

Abstract. By critical conditions, there are normal and shear stresses in typical helicopter configurations. Common strength criteria hold for the simplest cases of a material and static loading, and safety factors for uniaxial stress under simple loading. General strength theory includes critical state theory and general reserve theory based on introduced pure stresses as their own reversed uniaxial safety factors with using equidangerous cycles and Haigh's diagram. Each principal stress is a certain function of the initial data in a problem. A general reserve is based on worst-case combining the values of the individual reserves for the problem input and output parameters in their ranges given by these reserves and is significantly less than safety factors optimistic and dangerous. As compared to the stress concentration factor, an introduced equivalent stress concentration factor has many advantages. At a round hole in a plate loaded by stresses far from the hole, it is between 2 and 3 or a little greater. The different radii of a rivet and a hole substantially increase the maximum contact pressure between them. All results apply to predicting fatigue strength, e.g., in aircraft building.

1. INTRODUCTION

Helicopter structural elements need sufficient static and fatigue strengths to provide reliability. Common strength criteria usually hold for isotropic materials with equal strengths in tension and compression under static loading. For an arbitrarily anisotropic material with unequal strengths in tensions and compressions under any nonstationary loading with turning the main directions of the stress state at a point to be considered, there are no applicable and, all the more, adequate propositions at all. The usual safety factor holds for a uniaxial stress state under simple loading whose all parameters are directly proportional to a certain common one. The strength concentration factor is based on one positive component of a 3-dimentional stress state. For a typical plate having a round hole and loaded by stresses far from it, the maximum stress concentration factor is not known. The same holds for substantially increasing the maximum contact pressure between a rivet and a hole having different radii. This work is dedicated to creating new concepts, theories, and methods to improve predicting the fatigue strengths of responsible structural elements, for example, in aircraft building.

2. SOME HELICOPTER DESIGN FUNDAMENTALS

2.1. Configurations

A helicopter has some typical configurations [1], e.g., weight empty, basic, litter evacuation, critical forward center of gravity, critical aft center of gravity, design alternate forward center of gravity, design alternate aft center of gravity, 7 troop carrier, and 11 troop carrier.

2.2. Conditions

A helicopter works by some typical conditions1 subdivided into four groups: quasistatic, flight (or flying), landing, and handling. Each condition is either critical (with decisive influence on the reliability) for a certain structure element under consideration or noncritical for it.

2.3. Approaches

Discretization: To locate particular forces, moments, stresses, etc., a whole helicopter is mentally subdivided into relatively small parts with one inch dimensions and numbering in the longitudinal, vertical, and transversal directions begins at a certain point of a helicopter1.

Nearness: Configurations are considered near if their decisive parameters such as dimensions, weight, the position of the center of gravity, etc., are near to one another.

Proportionality: Configurations are considered proportional if their decisive parameters are near to be naturally proportional to one another, e.g., their weights are near to be proportional to the third powers of their corresponding dimensions. If such a proportionality factor is not near to 1, then the scale factor has to be also considered.

Similarity: To effectively use available experience, known data and design solutions for similar structures previously enough tested can be taken as a fundament for new ones.

Artificiality: If needed for comparing, not available data on a configuration can be artificially created using available data on similar (near and/or proportional) configurations.

Distribution: To obtain unavailable new configuration data using known configuration data, their corresponding parameters differences can be reasonably distributed in the configuration volume with conserving known data on a new configuration, e.g., the gravity center position.

Envelope: For a new helicopter configuration, there can be that under all critical conditions the most dangerous parameters (shears, moments, stresses, etc.) are between the corresponding ones for two similar configurations previously enough tested. In such a case it can be expected that this new configuration has enough reliability, too.

2.4. Equilibrium criteria

Vanishing sums of longitudinal, vertical, and transversal forces, pitch, roll, and yaw moments.

3. CLASSICAL STRENGTH SCIENCE

3.1. Critical (ultimate) states

The limiting (also called critical or ultimate) stress states [2, 3] are determined by a limiting criterion [4, 5] whose usual form is one of the following four:

σ_e = F(σ_j[o] | j = 1, 2, 3) = σ_L (1)

(σ_e = F(σ₁ , σ₂ , σ₃) = σ_L), (2)

σ_e = F(σ_jk | j, k = 1, 2, 3) = σ_L ,

F(σ_j[o] | j = 1, 2, 3) = 0,

F(σ_jk | j, k = 1, 2, 3) = 0.

The two first of them explicitly define, by σ_e = F, an equidangerous (so-called equivalent, or effective) uniaxial stress, σ_e , also in non-limiting stress states. The equality F = σ_L holds if and only if a stress state is limiting correspondingly to σ_L . In all these formulae, F is a function of the triple of the principal stresses σ_j ordered (σ_1o ≥ σ_2o ≥ σ_3o), which is indicated by the optional index "o", in the case of their nonsymmetrical occurrence in F , or of the normal, σ_jj , and shear, σ_jk , or τ_jk , with j ≠ k , j , k = 1, 2, 3, stresses forming a stress tensor and possibly of some material constants. A limiting stress, σ_l , can be a yield stress, σ_y , or an ultimate strength, σ_u , equal in tension and compression, or different in tension, σ_t , σ_ty , σ_tu , and in compression, σ_c , σ_cy , σ_cu > 0, possibly in directions j, k, which is indicated by the additional indices j, k.

Each limiting criterion means a concrete definition of the function F, e.g., for isotropic ductile materials (σ_L = σ_t = σ_c), the Tresca criterion of the maximum shear stress [4, 5]

σ_e = σ_1o - σ_3o = σ_L , (3)

the Huber-von-Mises-Hencky criterion of the octahedral shear stress (or distortion energy) [4, 5]

σ_e = σ_i ={[(σ₁₁ - σ₂₂)² + (σ₂₂ - σ₃₃)² + (σ₃₃ - σ₁₁)² + 6(σ₁₂)² + 6(σ₂₃)² + 6(σ₃₁)²]/2}^1/2 = σ_L (4)

with the stress intensity, σ_i , the both being yield criteria. For isotropic brittle materials with σ_t ≠ σ_c , the failure criteria used are the Galilei criterion of the maximum normal stress [4, 5]

-σ_c ≤ σ_j ≤ σ_t , j = 1, 2, 3, (5)

with at least one equality, the Coulomb criterion of internal friction [4, 5]

σ_e = σ_1o - χσ_3o = σ_t (χ = σ_t/σ_c) (6)

and the Pisarenko-Lebedev criterion [4, 5]

σ_e = (1 - χ)σ_1o + χσ_i = σ_t (7)

that should give [4, 5] criterion (4) by χ = 1 and criterion (5) by χ = 0. For orthotropic ductile materials with equal strengths in each tension and compression, σ_tj = σ_cj (= σ_Lj), j = 1, 2, 3, and only if the principal directions of a stress state coincide with the basic orthotropy directions 1, 2, and 3, one may use the Hu-Marin criterion [4, 5]

σ₁²/(σ_L1)² + σ₂²/(σ_L2)² + σ₃²/(σ_L3)² - σ₁σ₂/(σ_L1σ_L2) - σ₂σ₃/(σ_L2σ_L3) - σ₃σ₁/(σ_L3σ_L1) = 1; (8)

for anisotropic materials with σ_tj = σ_cj in each direction j , the von Mises-Hill criterion [4, 5] with material constants F, G, H, L, M, N:

F(σ₁₁ - σ₂₂)² + G(σ₂₂ - σ₃₃)² + H(σ₃₃ - σ₁₁)² + 2L(σ₁₂)² + 2M(σ₂₃)² + 2N(σ₃₁)² = 1; (9)

for anisotropic materials possibly with σ_tj ≠ σ_cj in some direction j , the Tsai criterion [4, 5] with material constants of the forms F_α and F_αβ by the Einstein tensor notation:

F_ασ_α + F_αβσ_ασ_β = 1 (α, β = 1, 2, ... , 6; σ_j = σ_jj , j = 1, 2, 3; σ₄ = σ₁₂ , σ₅ = σ₂₃ , σ₆ = σ₃₁). (10)

For uniaxial cyclic loading (fatigue) of an isotropic ductile material with σ_t = σ_c, one can use the Goodman linear approximation of the Haigh diagram [4, 5]

σ_m/σ_L + σ_a/σ_-1 = 1

(σ_m = (σ_max + σ_min)/2, σ_a = (σ_max - σ_min)/2)

with the mean stress σ_m of a cycle with maximum σ_max and minimum σ_min stresses, its amplitude stress σ_a, and the ultimate amplitude stress σ_-1 of a symmetric cycle (σ_min = -σ_max). For a symmetric cycle of a uniaxial normal stress combined with a symmetric cycle of a shear stress (usually bending combined with torsion), one may use the elliptic relation [4, 5]

(σ_a/σ_-1)² + (τ_a/τ_-1)² = 1

where τ_a is the amplitude stress of the shear cycle and τ_-1 is its ultimate amplitude stress. By nonstationarily loading an arbitrarily anisitropic solid with turning directions 1, 2, 3 of the principal stresses σ₁ ≥ σ₂ ≥ σ₃ , there are no known propositions to formulate such criteria [4, 5].

3.2. Noncritical states

It is not sufficient to only determine a usual safety factor, n_L , as the limiting stress, σ_L , divided by the equivalent stress, σ_e . Experiments can define in this aspect a limiting surface only:

F(σ₁ , σ₂ , σ₃) = σ_L ,

F^γ(σ₁ , σ₂ , σ₃)/σ_L^γ-1 = σ_L

both equivalent by any nonzero number γ . So, instead of

σ_e = F(σ₁ , σ₂ , σ₃),

we can consider

σ_eγ = F^γ(σ₁ , σ₂ , σ₃)/σ_L^γ-1

as the equivalent stress also for nonlimiting states when n_L ≠ 1. The usual safety factor [6],

n_Lγ = σ_L/σ_eγ = σ_L^γ/F^γ(σ₁ , σ₂ , σ₃) = n_L^γ,

can then take on any positive values when choosing suitable values of γ .

4. GENERAL STRENGTH THEORY

4.1. Generalization methods for critical state criteria

The main idea of the present methods is that, in principle, limiting criteria (distinct in their usual forms for different materials and loading conditions) must be sufficiently universal fundamental laws of nature. But, for example, the third strength theory (3) in its usual form

σ_e = σ₁ - σ₃ = σ_L (11)

can model the limiting surface for a certain ductile material only and cannot be applied to any brittle one with two different limiting stresses σ_t in tension and σ_c > 0 in compression. Assume criterion (11) be an expression of a certain temporarily unknown sufficiently general criterion now applied to a certain ductile material with σ_L . Then try to determine, for such a desired criterion, its form that may not include this specific constant of a certain material. By dimensionality and similarity theories, divide principal stresses by σ_L > 0 with designations

σ_j/σ_L = σ_j⁰ (j = 1, 2, 3),

σ_e/σ_L = σ_e⁰ = 1/n_L (12)

where n_L is the reserve of σ_e with respect to σ_L . Criterion (11) becomes pure (dimensionless)

σ_e⁰ = σ₁⁰ - σ₃⁰ = 1 (13)

without evident constants of a material and holds for any ductile material independently of the specific value of σ_L . Such a circumstance makes it possible, to further generalize criterion (13) by giving, as compared with transformation (12), a more general meaning to the reduced relative stresses σ_j⁰ introduced in this way. For a brittle solid and each index j ∈ {1, 2, 3, e},

σ_j⁰ = σ_j/σ_t (σ_j ≥ 0), σ_j⁰ = σ_j/σ_c (σ_j ≤ 0). (14)

Analyzing [7 - 9] experimental data [4, 5] for different ductile and brittle solids convincingly shows the following. Transformation (14) independent of limiting criteria themselves, indeed, unifies them and hence is an immediate expression of a limiting criteria generalization method itself. Moreover, in the space of the relative principal stresses, σ₁⁰, σ₂⁰, σ₃⁰, and especially in the corresponding plane, those unified data evidently cluster near the limiting surfaces and curves by criterion (13) and the fourth strength theory (4) also generalized

σ_e⁰ = σ_i⁰ = {[(σ₁⁰ - σ₂⁰)² + (σ₂⁰ - σ₃⁰)² + (σ₃⁰ - σ₁⁰)²]/2}^1/2 = 1. (15)

Generalizing an arbitrary limiting criterion of the form (1) gives

σ_e⁰ = F(σ₁⁰, σ₂⁰, σ₃⁰) = 1. (16)

Criterion (16) and its particular cases (13) and (15) are invariant and universal in the space of the relative (reduced) principal stresses, σ₁⁰, σ₂⁰, σ₃⁰, but have forms depending on the signs of the usual principal stresses, σ₁ , σ₂ , σ₃ , in their own space, which is shown [7 - 9] in detail. For orthotropic materials, the principal directions, 1, 2, 3, of a stress state coinciding with the basic orthotropy directions at the same solid's point, a generalization of (12) and (14) gives

σ_j⁰ = σ_j/σ_tj (σ_j ≥ 0), σ_j⁰ = σ_j/σ_cj (σ_j ≤ 0) (17)

with possible reindexation of the relative (reduced) principal stresses to obtain inequalities

σ₁⁰ ≥ σ₂⁰ ≥ σ₃⁰.

For any stationarily loading an arbitrary anisotropic solid, such a generalization is

σ_j⁰ = σ_j/|σ_Lj|. (18)

Here, for any principal stress σ_j , σ_Lj is its limiting value which has the direction and sign of σ_j and acts at the same point of a solid. The both other principal stresses vanish, and all the other loading conditions at the same point are the same as in the reality. Such recomprehending σ_tj and σ_cj (18) generalizes transformation (17) and preserves form (16) unlike criteria (9, 10). They are too complicated and have nothing in common with universal laws of nature always simple enough in their initial forms. Such a natural transformation (18) is not the only for a brittle solid with different strengths in tension and compression. In this case, by uniaxial cyclic loading in any principal direction, j , the limiting amplitude stress, σ_aj , can reach its peak, σ_ajmax , by a possibly nonzero mean stress, σ_m0j , of a cycle that is asymmetric in this nonzero case. Then also the stress state with σ_j = σ_m0j , j = 1, 2, 3, as opposed to the stress state with σ_j = 0 in (18), can be considered initial instead of the zero stress state. If

σ_m0j = (σ_tj - σ_cj)/2,

then a material with unequal strengths in tension and compression can be considered as one having these initial stresses -σ_m0j . This is a summary effect of microstresses and submicrostresses causing unequiresistibility as a phenomenological macroresult. The corresponding generalization of (18), for nonstationary loading at an arbitrary instant of time, t , from its interval, T = [t₀ , t₁], leads to transformation

σ_j⁰(t) = [σ_j(t) - σ_m0j(t)]/|σ_Lj(t) - σ_m0j(t)|. (19)

For each uniaxial stress process, σ_j(t), its reserve, n_j , is determined by the similar limiting process, n_jσ_j(t). For it, damage accumulation [4, 5] can also be taken into account. The equidangerous cycle of the reduced stresses with mean stress σ_mj⁰ and amplitude one σ_aj⁰ is determined by formula (19). Then stationary vectorial reduced stress

σ_j⁰ = (σ_mj⁰, σ_aj⁰)

can be found via a limiting amplitude diagram [7-9]. Ultimately, the postulate on the universality of function (16) gives (as a basis for hierarchies of strength laws of nature) criterion [7-9]

σ_e⁰ = max{sup_t∈Tmax_ju(t)F(σ_1u⁰(t), σ_2u⁰(t), σ_3u⁰(t)), max_ju|F(σ_1u⁰, σ_2u⁰, σ_3u⁰)|} = 1. (20)

Here choose the most dangerous, possibly depending on t , permutations of the stationary indexes, ju , of the unordered reduced principal stresses without inequalities σ_1u⁰ ≥ σ_2u⁰ ≥ σ_3u⁰.

4.2. Correction methods for critical state criteria

The third (3) and fourth (4) strength theories are not sensitive at all to hydrostatic tensions or compressions, the third one to the intermediate principal stress, σ₂ , too, also important [9]. The idea and physical sense of linearly correcting [7-9] a critical state criterion are essentially the hypothesis on a linear influence of the principal stresses on reaching a limiting state. Namely, in limiting states, suppose the equivalent stress be no constant as in (1) but a linear function of the principal stresses in accordance with equation

λ₀F(σ₁ , σ₂ , σ₃) + λ₁σ₁ + λ₂σ₂ + λ₃σ₃ = λ₄ (21)

where λ₀ , λ₁ , λ₂ , λ₃ , λ₄ are constants. The essence of a proposed approach to linearly correcting critical (limiting) criteria may be shown, e.g., in connection with the third strength theory (3). Substituting (11) into (21) and taking the data on uniaxial tension and compression into account give the criterion with an additional constant, x , of the material:

σ_e = σ₁ + xσ₂ - σ₃ = σ_L . (22)

Criterion (22) can also be obtained by taking the more specific hypothesis on the linear influence of the intermediate principal stress, σ₂ , on reaching a limiting state. The introduction of the pure (dimensionless) constant of a material, x , additional to the unique constant of a material σ_L , is justified as follows. The limiting stress, σ_L , is not sufficient to take into account the influence of the intermediate principal stress, σ₂ , and of hydrostatic tensions and compressions on reaching a limiting state. The physical sense of this constant, x , is that it is the uniaxial limiting stress, σ_L , divided by the limiting stress in hydrostatic tension, σ_ttt (σ₁ = σ₂ = σ₃ = σ_ttt). The last can be hardly determined directly but may be obtained by using the data on a third experiment with σ₂ ≠ 0, hence pure shear is not suitable. In biaxial compression and triaxial tension and compression we have, respectively:

x = 1 - σ_L/σ_cc (σ₁ = 0, σ₂ = σ₃ = -σ_cc),

x = 2 - σ_L/σ_tcc (σ₁ = σ_tcc , σ₂ = σ₃ = -σ_tcc).

Analogously correcting the fourth strength theory (4) and general criterion (1) gives criteria

σ_e = σ_i = {[(σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₃ - σ₁)²]/2}^1/2 + xσ₂ = σ_L ,

σ_e = F(σ₁ , σ₂ , σ₃) + xσ₂ = σ_L .

Further generalizing method (21) to extend [10, 11] strength laws hierarchies (20) gives equations

G(σ₁⁰, σ₂⁰, σ₃⁰) = λ₀ + λ₁σ₁⁰ + λ₂σ₂⁰ + λ₃σ₃⁰, (23)

G(σ₁⁰, σ₂⁰, σ₃⁰) = H(σ₁⁰, σ₂⁰, σ₃⁰) (24)

where G, H are certain (maybe unknown unlike F) different functions of the reduced (relative) principal normal stresses and possibly of some pure constants of a solid's material [10, 11]. It is desirable to choose functions G and H having obvious physical sense and possibly least numbers of material's parameters. Their values may be selected from experimental data on uniaxial critical states and other simple ones. Another idea to correct critical state criteria is as follows. In many cases practically important, not only the ultimate normal stress σ_u , but also the ultimate shear stress τ_u and, due to them both, their ratio σ_u/τ_u are available [2, 3]. For estimating the danger of complex stress states in a ductile isotropic material with equal strength in tension and compression, the yield normal stress σ_y and the yield shear stress τ_y (both to be substituted for σ_u and τ_u , respectively), as well as strength criteria (3) and (4) are commonly used. But they both, as well as other known criteria with pre-defined values of the σ_u/τ_u ratio, too, give no possibility to consider such additional data and, therefore, the distinctive features of many materials. That is why it is necessary to develop methods of correcting strength criteria in order to ensure taking the values of τ_u and σ_u/τ_u , if available, into account. To begin with, consider the classical problem of combined bending and torsion with a normal stress σ and a shear stress τ at a point of a solid. The principal stresses σ₁ , σ₂ , σ₃ commonly ordered (σ₁ ≥ σ₂ ≥ σ₃) are [6]

σ₁ = (σ + (σ² + 4τ²)^1/2)/2, σ₂ = 0, σ₃ = (σ - (σ² + 4τ²)^1/2)/2. (26)

According to criterion (3) with the pre-defined value 2 of σ_u/τ_u , we obtain for (26) a criterion [6]

σ_e = (σ² + 4τ²)^1/2 ≤ σ_u . (27)

According to criterion (4) with pre-defined value 3^1/2 of σ_u/τ_u , we obtain for (26) a criterion [6]

σ_e = (σ² + 3τ²)^1/2 ≤ σ_u . (28)

In the both cases, it is simply impossible to take any other value of the σ_u/τ_u ratio into account even if that is available. This lack can be very important when that value substantially deviates from the pre-defined one. Additionally, it would be a good practice to unify the both particular criteria (27) and (28). And namely this unification problem is relatively simple and gives further ideas to solve much more complex ones. It is easy to see that the factors 4 and 3 in the both particular criteria (27) and (28) are the corresponding pre-defined values 2 and 3^1/2 of the σ_u/τ_u ratio which are raised to the 2nd power, i.e., the unified particular criterion is

σ_e = (σ² + (σ_u/τ_u)²τ²)^1/2 ≤ σ_u . (29)

To understand the naturalness of this simple formula, divide all its parts by σ_u :

(σ_e⁰ = ) σ_e/σ_u = ((σ/σ_u)² + (τ/τ_u)²)^1/2 ≤ σ_u/σ_u ( = 1). (30)

Hence each normal and shear stress is divided by its limiting value, which apparently develops the idea of the strength transformation methods. Note that the last two formulas not only unify the results of using criteria (3) and (4), but also for the first time provide taking any value of σ_u/τ_u into account and thus completely solving problems in the simple case of combined bending and torsion [6], as well as of similar biaxial stress states [6]. Let us extend the present method for the general case of a triaxial stress state. It is suitable to first consider criterion (4). To take into account the unique additional material constant σ_u/τ_u , it is natural for the first attempt to include one unknown factor k into the corresponding formula (keeping it symmetric relatively to the principal normal stresses) in form

σ_e = (σ₁² + σ₂² + σ₃² + k(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u . (31)

Note that the factor at the sum σ₁² + σ₂² + σ₃² should be namely unit only to provide that the criterion holds for a uniaxial tension. Thus such a suitable introduction of an unknown factor seems to be unique. To determine its value by using combined bending and tension once more, substitute formulae (26) to (31). In this particular case we obtain

σ_e = (σ² + (2 - k)τ²))^1/2 ≤ σ_u .

Then comparing that with formula (29) gives

k = 2 - (σ_u/τ_u)² (32)

and finally the corrected criterion

σ_e = (σ₁² + σ₂² + σ₃² + (2 - (σ_u/τ_u)²)(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u . (33)

Now we see that we could initially choose, instead of (31), the formula

σ_e = (((σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₁ - σ₃)²)/2 + k(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u (34)

leading to

k = 3 - (σ_u/τ_u)² (35)

instead of (32) and finally, instead of (33), to the equivalent corrected criterion

σ_e = (((σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₁ - σ₃)²)/2 + (3 - (σ_u/τ_u)²)(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u . (36)

Now let us consider the strength criterion of the maximum shear stress (3), whose correction is much more complicated. First of all, it is impossible to keep the criterion’s linear form by correction because its factors 1 and -1 are unique to provide its applicability to a uniaxial tension and compression and introducing σ₂ brings nothing at least in the particular case of combined bending and torsion (σ₂ vanishes) and thus cannot, all the more, solve the general problem. That is why some complication is now necessary. Using the obtained experience with criterion (4), it is natural to choose a similar quadratic form of a formula like (34)

σ_e = ((σ₁ - σ₃)² + kσ₁σ₃)^1/2 ≤ σ_u (37)

taking into account the specific distinctive features of the initial formulae (3) and (4) of the both criteria and using one additional constant k of a material. The same combined bending and torsion leads to

σ_e = (σ² + (4 - k)τ²))^1/2 ≤ σ_u (38)

and, comparing this formula with (29), to

k = 4 - (σ_u/τ_u)² (39)

and finally to

σ_e = ((σ₁ - σ₃)² + (4 - (σ_u/τ_u)²)σ₁σ₃)^1/2 ≤ σ_u (40)

or, equivalently,

σ_e = (σ₁² + σ₃² + (2 - (σ_u/τ_u)²)σ₁σ₃)^1/2 ≤ σ_u . (41)

Formula (41) is much more similar to (33) than (40) and shows that we could initially choose

σ_e = (σ₁² + σ₃² + kσ₁σ₃)^1/2 ≤ σ_u (42)

instead of formula (37), which would lead us to formula (41) again. It is well known [4, 5] that for many ductile materials, experimental data on strength is often placed between the curves given by the both criteria (3) and (4). It is very natural to expect: The smaller the deviation of the σ_u/τ_u ratio from its value pre-defined by a strength criterion, the more precision of it by modeling experimental data. That is why one can use the unified criterion as the linear combination of the both criteria both using their initial forms (3) and (4)

σ_e = ((σ_u/τ_u)² - 3)(σ₁ - σ₃) + (4 - (σ_u/τ_u)²)(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u (43)

or already corrected by the present methods

σ_e = ((σ_u/τ_u)² - 3)(σ₁² + σ₃² + (2 - (σ_u/τ_u)²)σ₁σ₃)^1/2 +

(4 - (σ_u/τ_u)²)(σ₁² + σ₂² + σ₃² + (2 - (σ_u/τ_u)²)(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u . (44)

4.3. General reserve theory

The usual safety factor can give too optimistic and therefore very dangerous values. We have in example 1

(σ₁ = 250 MPa, σ₂ = 240 MPa, σ₃ = 210 MPa, σ_L = 235 MPa)

n_L = 5.9, n_Lσ₁ - σ₃/n_L = 1439 MPa >> σ_L .

In example 2, a bar with strengths in tension and in compression

σ_t = 100 MPa, σ_c = 800 MPa

is contracted and stretched by two pairs of forces independently causing the stresses

σ = σ^- + σ⁺ = -500 MPa + 400 MPa = -100 MPa,

n_L = σ_c/|σ| = 8, n_Lσ^- + σ⁺/n_L = -3950 MPa << -σ_c , σ^-/n_L + n_Lσ⁺ = 3137.5 MPa >> σ_L .

The main idea [7-9, 12] to realistically determine the reserve of a system under consideration is separately taking the reserves of its original parameters into account, each of these reserves being expressed via a common additional number. It is obtained from the condition that, by the worst realizable combination of the values of these parameters arbitrarily modified within the bounds determined by the corresponding reserves, the state of at least one element of the system becomes limiting, no element of it being in an overlimiting state. This is a further generalization of the universalization methods for critical state criteria. In a general problem, for any function of an arbitrary set of variables, where (α) means that index α ∈ Α is optional,

z = f[_α∈Α z_α], Z = f[_α∈Α Z_α], z_(α) ∈ Z_(α).

The genuine values of the independent variables, z_α , and of the dependent one, z , usually deviate from their values calculated. Those should belong to their admissible sets (domains), [Z_(α)], if the problem has certain limitations like strength criteria in strength problems. If

f[_α∈Α [Z_α]] ⊆ [Z],

the problem has been already solved. Otherwise, it is necessary to determine such a combination of the restrictions, Z_α , of the admissible sets, [Z_α], that the inclusion

f[_α∈Α Z_α] ⊆ [Z]

is true. For the existence of the numeric measures of those restrictions, or the reserves of the independent variables, it is sufficient that, for any α∈Α, [Z_(α)] is included into a certain Hilbert space, L_(α). It has the norm, ||z_(α)||_(α), of each element, z_(α), and the scalar product, (z_(α), z’_(α))_(α), of each pair of elements, z_(α) and z’_(α). The additive approach to obtaining reserves develops, generalizes, and extends the relative error in a certain sense. That naturally determines the neighborhood, Z_(α)(δ_(α), z_0(α)), of set Z_(α) with respect to element z_0(α) ∈ L_(α) with error δ_(α) ≥ 0 as the set of all z’_(α) ∈ L_(α) with

||z’_(α) - z_(α)||_(α) ≤ δ_(α)||z_(α) - z_0(α)||_(α).

The additive reserve of set Z_(α) by set [Z_(α)] with respect to element z_0(α) is defined as

n_a(α) = 1 + sup{δ_(α) ≥ 0: Z_(α)(δ_(α), z_0(α)) ⊆ [Z_(α)]}.

The multiplicative approach to obtaining reserves develops, generalizes, and extends the reserve factor in a certain sense. That gives the neighborhood,

Z_(α)(n_(α)exp(iφ_(α)), z_0(α)),

of set Z_(α) with respect to element z_0(α) ∈ L_(α) as the set of all z’_(α) ∈ L_(α). Here

0 ≤ φ_(α) ≤ π, i² = -1, n_(α)^-1||z_(α) - z_0(α)||_(α) ≤ ||z’_(α) - z_0(α)||_(α) ≤ n_(α)||z_(α) - z_0(α)||_(α)

where n_(α) ≥ 1 is a multiplicative reserve,

arccos[(z’_(α) - z_0(α), z_(α) - z_0(α))_(α)/(||z’_(α) - z_0(α)||_(α) ||z_(α) - z_0(α)||_(α))] ≤ φ_(α).

If the dimensionality of space L_(α) is at least two, then two independent parameters n(α) and φ_(α) can be used, but possibly with relation φ(α)(n(α)). If L_(α) is one-dimensional, then φ_(α) = 0. The multiplicative reserve of set Z_(α) by set [Z_(α)] with respect to element z_0(α) is defined as

n_m(α) = sup{n_(α) ≥ 1: Z(_α)(n_(α)exp(iφ_(α)), z_0(α)) ⊆ [Z_(α)]}.

By any of the both approaches, reserves n_α can be expressed via different nondecreasing functions of an additive reserve, n_fa , or a multiplicative one, n_fm , respectively, the both being common for reserves n_α and determined by the condition that there is an element z ∈ Z in a limiting state by the worst realizable combination of all z_α :

n_fa = sup{n ≥ 1: f[_α∈Α Z_α(n_α(n), z_0α)] ⊆ [Z]},

n_fm = sup{n ≥ 1: f[_α∈Α Z_α(n_α(n)exp(iφ_α(n_α(n))), z_0α) ⊆ [Z]}.

For simply (proportionally) loading, the multiplicative reserve is obtained from the condition

F(n_fmσ₁ , n_fmσ₂ , n_fmσ₃) = σ_L .

In the simplest case of the equal reserves of all z_α , in example 1 we obtain n_fa = 1.423, n_fm = 1.5 and in example 2 n_fmt = 1.25, n_fmc = 2 in tension and compression, respectively. These realistic reserves are significantly less than the usual ones too optimistic.

5. FATIGUE RESISTANCE BY STRESS CONCENTRATIONS

5.1. Equivalent stress concentration factor

The stress concentration factor [6] determine maximum stress σ_max via the nominal one σ_nom :

C = σ_max/σ_nom (45)

implicitly considering the positive values of individual stresses only. But they can be zero or negative and bring ambiguity to defining σ_max (either by algebraic values or by moduli). There are different choices of individual stresses in a non-uniaxial stress state. The roles of the chosen individual stresses in complex stress states at different places with maximal and conditionally nominal stresses can be very distinct leading to great errors in determining the stress concentration factor. No individual stress alone evaluates the danger of a complex stress state. Correcting usually leads to many different cases caused by the number of the individual stresses. Let us use the equivalent stresses [6] instead of the individual ones by defining both the maximum and the nominal stresses. To determine the maximum equivalent stress σ_emax via the nominal equivalent stress σ_enom , the equivalent stress concentration factor

C_e = lim σ_emax/σ_enom (46)

as σ_enom approaches its real value is proposed instead of (45). Due to this limit, the definition also holds if σ_enom vanishes. Otherwise, i.e. by nonzero σ_enom , we simply have

C_e = σ_emax/σ_enom . (47)

This definition holds in the general case of a complex stress state with different individual components having any signs independently from the relations between the algebraic values of these components. The roles of individual stresses at different places of a machine part are completely taken into account. The equivalent stresses much more adequately evaluate the danger of a complex stress state than any individual stress alone. It is not necessary to consider many different cases caused by the number of the individual stresses, their signs, and the relations between these stresses.

5.2. Maximum stress concentrations at round holes

A design scheme is a plate infinite in the both directions and having a finite (i.e., infinitely small in comparison with the both plate dimensions besides the thickness) circular hole under two-directional uniform loading with generally unequal σ_x (along the x-axis) and σ_y (along the y-axis, no yield stress) in the conditional infinity [13]:

σ_emax = max{|3σ_x - σ_y|, |3σ_y - σ_x|},

σ_enom = max{|σ_x|, |σ_y|, |σ_y - σ_x|},

C_e = lim σ_emax/σ_enom = lim max{|3σ_x - σ_y|, |3σ_y - σ_x|}/max{|σ_x|, |σ_y|, |σ_y - σ_x|} (48)

as max{|σ_x|, |σ_y|, |σ_y - σ_x|} approaches its real value which can also vanish by criterion (3) and

σ_emax = max{|3σ_x - σ_y|, |3σ_y - σ_x|},

σ_enom = {[σ_x² + σ_y² + (σ_y - σ_x)²]/2}^1/2 = (σ_x² - σ_xσ_y + σ_y²)^1/2 ,

C_e = lim σ_emax/σ_enom = lim max{|3σ_x - σ_y|, |3σ_y - σ_x|}/(σ_x² - σ_xσ_y + σ_y²)^1/2 (49)

as (σ_x² - σ_xσ_y + σ_y²)^1/2 approaches its real value which can also vanish by criterion (4). In the particular case of pure shear by criteria (3) and (4), respectively,

σ_emax = 4|σ_x|, σ_enom = 2|σ_x|, C_e = lim σ_emax/σ_enom = 2, (50)

σ_emax = 4|σ_x|, σ_enom = 3^1/2|σ_x|, C_e = lim σ_emax/σ_enom = 4/3^1/2 (51)

as |σ_x| approaches its real value which can also vanish, both instead of C = 4 due to [13] (45). Analysis of (48) and (49) by all possible relations between σ_x and σ_y shows the bounds

C_emin = 2 (σ_x = σ_y), C_emax = 3 (σ_x/σ_y = 0 or ∞),

C_emin = 2 (σ_x = σ_y), C_emax = 14/21^1/2 ≈ 3.055 (σ_x/σ_y = 1/5 or 5)

by (3) and (4), respectively.

5.3. Maximum contact pressure between a rivet and a hole edge with distinct diameters

The classical Hertz solution for a contact of two cylinders [14] giving maximum contact pressure

q_max = (1/(2π(1-ν²))pE(R - r)/(rR))^1/2

vanishing in the theoretical limiting case R = r (the hole and rivet diameters, respectively) cannot be applied to this problem. And the Hertz assumption [14] that the contact width is much less than r does not hold. The simplest method of improving the classical solution for the internal contact of two cylindrical surfaces whose radii r and R (or curvatures 1/r and 1/R) can be the same seems to be adding to q_max a constant corresponding to the homogeneous distribution of p (force per unit length) on the width 2R:

q_max = (1/(2π(1-ν²))pE(R - r)/(rR))^1/2 + p/(2R).

Such an approach is apparently conservative and returns the correct result by R = r.

6. BASIC RESULTS FOR A HELICOPTER UNDER UNIT LOADS

Comparing all the corresponding distributions (first separately for the design alternate forward and aft configurations of the helicopter and then unifying all the data to use the envelopes1 created by the both comparison bases) have convincingly shown the following:

1. From the nose of the helicopter to its gravity center zone, the stresses holding in the longitudinal vertical plane in the helicopter configurations to be investigated are some more dangerous than the same stress envelope created by the both comparison bases.

2. In the helicopter gravity center zone, the stresses acting in the longitudinal vertical plane in the helicopter configurations to be investigated are about 20 % greater than the maximum of the same stress envelope created by the both comparison bases.

3. From the gravity center zone of the helicopter up to its tail, the stresses holding in the longitudinal vertical plane in the helicopter configurations to be investigated are contained in the same stress envelope created by the both comparison bases.

The obtained results have provided optimizing the helicopter by locally improving the fatigue strength15 of the relevant elements of the design alternate helicopter configurations.

7. CONCLUSIONS

1. Helicopter design needs optimizing its configurations under unit loads.

2. The common strength criteria, safety factor, and strength concentration factor have restricted and vague applicability and adequacy ranges to optimize strength.

3. The proposed general strength theory including general reserves along with the introduced equivalent stress concentration factor and the corrected formula for the rivet contact pressure is adequate and very suitable for solving typical strength problems.

4. Applying the concepts and methods of general strength theory provides optimizing helicopter configurations under unit loads and improving its fatigue strength.

8. BIBLIOGRAPHY

1. Bell Reports 205-099-004, 205-099-006, 205-099-007, and 205-099-011.

2. Military Handbook. Metallic Materials and Elements for Aerospace Vehicle Structures. MIL-HDBK-5H, 1998

3. Handbuch Struktur-Berechnung. Prof. Dr.-Ing. L. Schwarmann. Industrie-Ausschuss-Struktur-Berechnungsunterlagen, Bremen, 1998

4. Pisarenko, G. S., and Lebedev, A. A.: Deformation and Strength of Materials in Complex Stress State, Izd. Naukova Dumka, Kiev, 1976 (Russ.)

5. Lebedev A. A., Koval'chuk B. I., Giginjak F. F., and Lamashevsky V. P.: Handbook of Mechanical Properties of Structural Materials at a Complex Stress State. Edited by Prof. A. A. Lebedev, Academician of the NAS of Ukraine. Begell House, Inc., N. Y., Wallingford, U. K., 2001

6. Pisarenko, G. S. y otros: Manual de Resistencia de Materiales. Editorial MIR, Moscú, 1989

7. Lev Gelimson: General Strength Theory. Drukar Publishers, Sumy, 1993

8. Lev Gelimson: The generalized structure for critical state criteria. Transactions of the Ukraine Glass Institute, 1 (1994), 204-209

9. Lev Gelimson: Elastic Mathematics. General Strength Theory. The ”Collegium” International Academy of Sciences Publishers, Munich (Germany), 2004

10. P. W. Bridgman, Collected Experimental Papers, Vols. 1 to 7, Harvard University Press Publ., Cambridge (Massachusetts), 1964

11. Lev Gelimson: Basic New Mathematics, Drukar Publishers, Sumy, 1995

12. Lev Gelimson: General strength theory. Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin, Publisher Prof. Dr. habil. V. Mairanowski, 3 (2003), Berlin

13. Timoshenko, S. P.: Theory of Elasticity, 3rd ed. McGraw-Hill, New York, 1970

14. Hertz, H.: Über die Berührung fester elastischer Körper. J. reine und angewandte Matematik, 92 (1882), 156-171

15. Bruhn, E. F.: Analysis and Design of Flight Vehicle Structures. Jacobs Publishing, Inc., Indianapolis (IN), 1973