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7
UDC 539.3
Doi: 10.31772/2587-6066-2020-21-4-483-491
For citation: Matveev А. D. Method of equivalent strength conditions in calculations of bodies with inhomogéneos regular structure. Siberian Journal of Science and Technology. 2020, Vol. 21, No. 4, P. 483-491. Doi: 10.31772/25876066-2020-21-4-483-491
Для цитирования: Матвеев А. Д. Метод эквивалентных условий прочности в расчетах тел с неоднородной регулярной структурой // Сибирский журнал науки и технологий. 2020. Т. 21, № 4. С. 483-491. Doi: 10.31772/2587-6066-2020-21-4-483-491
METHOD OF EQUIVALENT STRENGTH CONDITIONS IN CALCULATIONS OF BODIES WITH INHOMOGENEOS REGULAR STRUCTURE
А. D. Matveev
Institute of Computational Modeling 50/44, Akademgorodok, Krasnoyarsk, 660036, Russian Federation E-mail: mtv241@mail.ru
Plates, beams and shells with a non-uniform and micro-uniform regular structure are widely used in aviation and rocket and space technology. In calculating the strength of elastic composite structures using the finite element method (FEM) it is important to know the error of the approximate solution for finding where you need to build a sequence of approximate solutions that is connected with the procedure of crushing discrete models. Implementation of the procedure for grinding (within the micro-pass) discrete models of composite structures (bodies) requires large computer resources, especially for discrete models with a microinhomogeneous structure. In this paper, we propose a method of equivalent strength conditions (MESC) for calculating elastic bodies static strength with inhomogeneous and microin-homogeneous regular structures, which is implemented via FEM using multigrid finite elements. The calculation of composite bodies' strength according to MESC is limited to the calculation of elastic isotropic homogeneous bodies strength using equivalent strength conditions, which are determined based on the strength conditions set for composite bodies. The MESC is based on the following statement. For all composite bodies V0, which are such a homogeneous
isotropic body Vb and the number of p , if the safety factor nb of the body Vb satisfies the equivalent conditions of strength pn1(1 + 5a ) < nb (1 -5^ ) < pn2(1 -5a ), the safety factor n0 of the body V0 meets the defined criteria for strength n1 < n0 < n2, where n1, n2 specified, the safety factor n0 (nb) complies with the accurate (approximate) solution of elasticity theory problem is built for body V0 (body Vb); 5a < (n2 - nj)/(n2 + n1) ; 5a is the upper 5b error
estimation of the maximum equivalent body stress Vb, corresponding to approximate solution. When constructing equivalent strength conditions, i. e when finding the equivalence p coefficient, a system of discrete models is used, dimensions of which are smaller than the dimensions of the basic composite bodies models. The implementation of MESC requires small computer resources and does not use procedures for grinding composite discrete models. Strength calculations for bodies with a microinhomogeneous structure using MESC show its high efficiency. The main procedures for implementing the MESC are briefly described.
Keywords: elasticity, composites, equivalent strength conditions, multigrid finite elements, plates, beams, shells.
МЕТОД ЭКВИВАЛЕНТНЫХ УСЛОВИЙ ПРОЧНОСТИ В РАСЧЕТАХ ТЕЛ С НЕОДНОРОДНОЙ РЕГУЛЯРНОЙ СТРУКТУРОЙ
А. Д. Матвеев
Институт вычислительного моделирования СО РАН Российская Федерация, 630036, г. Красноярск, Академгородок, стр. 50/44 E-mail: mtv241@mail.ru
Пластины, балки и оболочки с неоднородной, микронеоднородной регулярной структурой широко применяются в авиационной и ракетно-космической технике. В расчетах на прочность упругих композитных конструкций с помощью метода конечных элементов (МКЭ) важно знать погрешность приближенного решения, для нахождения которой необходимо построить последовательность приближенных решений, что связано с применением процедуры измельчения дискретных моделей. Реализация процедуры измельчения (в рамках микроподхода) дискретных моделей композитных конструкций (тел) требует больших ресурсов ЭВМ, особенно для дискретных моделей с микронеоднородной структурой. В данной работе предложен метод эквивалентных
условий прочности (МЭУП) для расчета на статическую прочность упругих тел с неоднородной и микронеоднородной регулярной структурой, который реализуется с помощью МКЭ с применением многосеточных конечных элементов. Расчет на прочность композитных тел по МЭУП сводится к расчету на прочность упругих изотропных однородных тел с применением эквивалентных условий прочности, которые определяются на основе условий прочности заданных для композитных тел. В основе МЭУП лежит следующее утверждение. Для всякого композитного тела ¥0 существуют такое изотропное однородное тело ¥ь и число р, что если коэффициент запаса пь тела ¥ь удовлетворяет эквивалентным условиям прочности вида рп1(1 + 8а) < пь (1 -Ъ2а) < рп2(1 -8а), то коэффициент запаса п0 тела ¥0 удовлетворяет заданным условиям прочности п1 < п0 < п2, где п1, п2 заданы, коэффициент запаса п0 (пь) отвечает точному (приближенному) решению задачи теории упругости, построенному для тела ¥0 (тела ¥ь), 5а < (п2 - п1)/(п2 + п1), 5а - верхняя оценка погрешности Ъь максимального эквивалентного напряжения тела ¥ь, отвечающего приближенному решению. При построении эквивалентных условий прочности, т. е. при нахождении коэффициента эквивалентности р , используется система дискретных моделей, размерности которых меньше размерностей базовых моделей композитных тел. Реализация МЭУП требует малых ресурсов ЭВМ и не использует процедуры измельчения композитных дискретных моделей. С помощью расчетов показано, что эквивалентные условия прочности, построенные для конкретного нагружения композитного тела, можно использовать для определенного вида его нагружений. Расчеты на прочность тел с микронеоднородной структурой с помощью МЭУП показывают высокую его эффективность. Кратко изложены основные процедуры реализации МЭУП.
Ключевые слова: упругость, композиты, эквивалентные условия прочности, многосеточные конечные элементы, пластины, балки, оболочки.
Introduction. Structure strength calculation is one of the most important stages in the outline design of a structure based on a structure project feasibility study. As a rule, calculations for static strength, elastic structure (body) of a certain class (for example, elements or aircraft and rocket-space structures) are carried out according to safety requirements [1-3], and limited to the equivalent structure stress determination. In this case for the body V0 the given strength conditions are n1 < n0 < n2, where n1, n2 are given, n0 is the body safety factor, V0, n0 = ctt / ct0 , ctt is the yield stress [1], ct0 is the maximum equivalent stress corresponding to the exact solution of the elasticity problem (constructed for the body V0 ). If the safety factor n0 satisfies the given strength conditions, then it is suggested that the body V0 does not collapse during operation. It should be noted that construction of analytical solutions of the three-dimensional problem of elasticity theory for composite bodies is associated with great difficulties. If the maximum equivalent stresses of the bodies is approximate, then in this case the corrected strength conditions are used [4], which pass the stress error. In the analysis of the stress-strain state (SSS), the finite element method (FEM) is widely used [5; 6]. Basic discrete models of bodies, accounting for their in-homogeneous and micro-inhomogeneous structures within the micro-approach [7], have a very high dimension. Implementation of FEM for such discrete models is very difficult, since it requires large computer resources. In addition, to determine the error in the solution, a sequence of approximate solutions constructed using refinement (within the micro approach) of discrete models is used. The grinding procedure is difficult to implement; it leads to a sharp increase in the discrete models size, making implementation of FEM challenging. To determine the SSS of composite bodies, the method of multi-
grid finite elements (MFEM) [8-14] is effectively applied, which generates discrete models, dimensions of which are 103 ^106 times less than the base models dimensions. It should be noted that FEM is a special case of MFEM. If when solving boundary value problems by FEM, multigrid finite elements (MgFE) are used [8-22], then MFEM is implemented in this case.
In this work, for calculating the strength of solid composite bodies using equivalent strengths, the method of equivalent strength (MESC) is proposed, which means calculating the strength of isotropic homogeneous bodies using equivalent strengths [23]. In this paper in contrast to [24], a theorem is formulated and proved, which underlies the MESC. In addition, the following should be noted: equivalent strength conditions are based on specified strength conditions using the equivalence coefficient p. In fact, the construction of equivalent strength islimited to determining the coefficient p, which is determined for a given composite body loading. However, it is important to note that the equivalent strength conditions constructed using the coefficient p can be used in composite body strength calculations for a certain type of its loading.
To find the coefficient p, a system of homogeneous and composite discrete models is used, dimensions of which are less than the dimensions of composite bodies models. The analysis of SSS in discrete models is carried out using the MFEM, which generates discrete models of small dimension. The advantages of the MESC are that its implementation requires small computer resources and does not use the procedure for refining discrete models of composite bodies. The use of MESC in strength calculations of bodies with a micro-inhomogeneous regular structure shows its effectiveness.
1. Equivalent strength conditions and equivalent strength structures. Suppose two elastic structures V1 and V2 have the same shape, geometrical dimensions, fixings and static loading, but differ in elasticity modulus.
Suppose strength conditions n1, n2 are given for the safety factors, respectively of structures V1, V2
n\ < n < n\.
nl < n2 < nb :
(1) (2)
where na, n2a > 1 ; na, n2a, nb, nb - are given; safety factors n1 ( n2 ) complie with the precise solution of elasticity theory, built for structures V1 ( V2 ).
For structures V1, V2 the following two definitions are introduced:
Definition 1. Fulfillment of conditions (2) for the coefficient n2 implies fulfillment of conditions (1) for the coefficient n1 and vice versa, if the fulfillment of conditions (1) for the coefficient n1 implies the fulfillment of conditions (2) for the coefficient n2 , then the strength conditions (1), (2) will be called equivalent strength conditions for structures V2 , V1 , respectively.
Definition 2. Suppose the structures V1, V2, for which respectively condition (2), (1) is equivalent to strength conditions do not collapse under the same operating conditions. Then the structures V1, V2 will be called strength equivalent.
In practice, the equivalence in strength of structures V1, V2 means that V2 structure can be used instead of a working structure V1, and vice versa. It should be noted that of the two structures equivalent in strength, it is advisable to use such a structure that is more technologically advanced in manufacturing, meets the specified technical requirements and more cost effective for manufacturing and operation.
2. Provisions of the method of equivalent strength conditions MESC are used to calculate the strength of structures (bodies) that satisfy the following:
Provision 1. Linearly elastic three-dimensional isotropic homogeneous bodies and bodies with an inhomo-geneous, micro-inhomogeneous regular structure, which consist of plastic materials, have smooth boundaries and static loading are considered. The body loading functions are smooth functions. Solid boundaries do not degenerate into points.
Provision 2. Composite bodies consist of isotropic homogeneous bodies of different modulus, connections between which are ideal, that is, on common boundaries of homogeneous bodies of different modulus, the functions of displacements and stresses are continuous.
Provision 3. Displacements, deformations and stresses of heterogeneous isotropic homogeneous bodies correspond to the Cauchy relations and Hooke's law of the three-dimensional linear problem of elasticity theory [25]. Equivalent stresses for bodies are determined according to the 4th theory of strength [1].
Provision 4. The maximum equivalent stress of the basic discrete model of a composite body (which consists of a first-order FE of the cube shape, takes into account the inhomogeneous structure of the composite body and generates a three-dimensional uniform mesh) shows a
small difference with the exact solution. It should be noted that due to the convergence of the FEM, such basic discrete models for composite bodies always exist.
Provision 5. For the typical dimensions of a composite body and its regular cell, the condition d / B << 1 is fulfilled, where d is the maximum typical size of the regular cell of the composite body, B is the minimum typical size of the composite body.
It should be noted that positions 4, 5, as a rule, are fulfilled for bodies with micro-inhomogeneous regular structure.
3. The main theorem of the method of equivalent strength conditions. Without losing shared judgments, we consider bodies with an inhomogeneous regular fibrous structure, which are widely used in practice. The MESC is based on the following theorem:
Theorem. Suppose the strength conditions of the form 3 are given to the safety factor of a composite body n0 (fibrous structure).
n < no < n2, (3)
where n1, n2 - are given, n1 > 1, n0 =aT / ct0 , aT -fiber yield stress, ct0 - the maximum equivalent stress of the body V0 , which corresponds to the exact solution of the problem of the elasticity theory, constructed for the body V0 .
Then there is such an isotropic homogeneous body Vb and such a number p > 0 (equivalence coefficient) that
if the body Vb safety factor nb satisfies the corrected equivalent strength conditions
pn1 ■< nb <-pn2
(4)
1 -8« " 1 + 8a' then, safety factor n0 of the structure V0 meets the strength requirements (3), where nb =aT / ab, ab - the
maximum equivalent stress of the body Vb, which corresponds to the approximate solution of the theory of elasticity problem, constructed for the body Vb ,
5 < ni-ni
(5)
8a - upper bound on relative error, Sb pressure ab
of body Vb, |Sb | <5a .
Deduction.
First, let us prove the existence of equivalent strength conditions for linearly elastic composite bodies. Suppose an elastic homogeneous isotropic body Vb and a composite body V0 have the same shape, size, fixation and loading, but differ in elastic moduli. Suppose the elastic moduli of the body Vb and fiber be the same. The safety factors n0, n° respectively bodies V0, Vb are found
by the formulas
ст0
(6)
(7)
CT
T
n
0
CT
T
n=
CT
where ctt - fiber yield strength [1-3]; a° - maximum
equivalent body stress Vb, corresponding to the exact solution of the elasticity theory problem.
Suppose coefficient n0 meets the requirements (3). Applying (6) to (3) we obtain
ctt
ni < — < n2 •
There is a number p > 0 .
P =-?r-
Considering (9) in (8), we obtain
Applying (7) in (10), we obtain Pn1 < nb < Pn2 .
(8)
(9)
(10)
(11)
|SJ<Sa< ca =
Щ + «2
(12)
where n1, n2 - are given; n1 > 1, n2 > n1, Sb - relative stress error ab, i. e.
From (13) it follows that ab = (1 + Sb ) a^. Hence obtain
«0 = (1 + §b )nb . (14)
Let us note that in (12) Ca < 1. Suppose S0 is such that S0 = | Sb |. Then due to (12) obtain
0<§0 = |Sb| <8a< 1. (15)
Assuming in (14) consecutively §b =-§0, Sb =S0, apply coefficients
«i = (1 -§0)"b , "2r = (1 +§0)"b , (16)
Then due to (14), (16) obtain
nb = n, or nb = n2 •
Apply coefficients n1d , nf according to formulas
nf = (1 -Sa )nb , nd2 = (1 + §a )nb . (18)
(17)
So, the safety factor nb of an isotropic homogeneous body Vb satisfies conditions (11). Conversely, suppose body Vb safety factor nb satisfy the strength conditions (11). Applying (7) in (11) considering (9), we obtain paT
pn1 < —— < pn2. Whence, taking into account (6), fol-a0
lows the fulfillment of the strength conditions for the safety factor n0 of the composite body V0 (3). It is
shown that each coefficient nb e (pn1, pn2) corresponds to a single coefficient n0 e (nj, n2) found by formula (6), and vice versa. Further limiting cases are considered. Suppose nb = pn1 . Using relation (7) in the latter equation we obtain paT / ct0 = pn1. Whence, taking into account (6) it follows n0 = n1. Similarly, one can show that if nb = pn2, then n0 = n2. Suppose n0 = n1. Using (6), (9) in the latter equation, we obtain ctt / ab = pn1. Now then, taking into account (7), it follows that nb = pn1. Similarly, one can show that if n0 = n2, then nb = pn2. Hence it follows that conditions (11), according to Definition 1, are equivalent strength conditions for a body V0 .
Suppose for the body Vb the maximum equivalent stress has been defined as ab, corresponding to the approximate solution of the elasticity theory problem, such that
Due to 0 < §a < 1, nb > 0, from (18) it follows that
d d n2 ^ n1 •
(19)
Equivalent strength conditions that take into account stress error, i. e., corrected equivalent strength conditions (4) are presented in the form
P«1 (1 + §a) < nb (1 - §2) < pn2 (1 - §a), (20) where nb = aT / ab, aT - fiber yield strength.
Suppose for coefficient nb strength conditions are met (20), i. e. suppose pn1 < (1 - § )nb, (1 + §g )nb < pn2.
Hence for the coefficient n1d, n, , taking into account (18), (19) inequation is done
Pn1 < n1d < Щ < Pn2 •
(21)
Sb =-
(13)
Comparing (16), (18) with respect to (15), equations nf < n1r, n2 < nd follow. Hence, considering that according to (16) n[ < n2, we obtain
nd < n[ < n2 < nd . (22)
Then, due to (21), (22) inequations are done
p«1 < n1 < n2 < P«2 . (23)
From (23) taking into account (17), i. e. from meeting for the body Vb safety factor nb (corresponding to the
approximate solution) of the corrected equivalent strength conditions (20), that is (4), it follows that strength conditions (11 ) for the safety factor nb of the body Vb (corresponding to the exact solution) are met, therefore, satisfying the given strength conditions (3) for the safety factor n0 of the composite body V0 (corresponding to the exact solution). Constraints on the parameter §a are found from the assumption of strength conditions existence (4), i. e. suppose inequation pn1(1 + §a ) < pn2(1 -§a ) is done. Whence it follows that
where 5a - upper bound for error 5b.
S < с =
a a .
(24)
CT
CT
CT
It should be noted that, since n2 > n1 > 1, then from (24) it follows that 0 < C« < 1. If 8« = C« , then the range for varying values of the coefficient n0 is zero, which is difficult to perform in practice. Now then 8« < C« , it is possible to meet the equivalent strength conditions (11) for the coefficient nb0 applying corrected equivalent strength conditions (4) and the approximate solution that generates an error 8b for the stress ab that | 8b | <8« . Note that meeting conditions (11) implies the fulfillment of the specified strength conditions (3). The theorem is proved.
Note that it follows from the theorem that if the safety factor nb of the body Vb satisfies the corrected equivalent strength conditions (4), then this means that the error 8b of the maximum equivalent stress ab of the body Vb is not greater than 8«, i. e. 18b |< 8« .
4. Procedures for implementing the method of equivalent strength conditions. Implementation of the MESC is reduced to construction of equivalent strength conditions (4) applying the MFEM, that is, to determination of the equivalence coefficient p, and to determination
of the maximum equivalent stress CTb with an error 15b | < 5„
nb = CTT / CTb
is determined by the formula (9), i. e.
p = -
for the body V b The coefficient p
(25)
Without losing shared judgments, for convenience and clarity of presentation, we will consider the basic procedures for the implementation of MESC using the example of calculating the strength of a composite beam (body) V0 with dimensions H x L x H , where H = 128h, L = 1536h , h - is given, Fig. 1. The body V0 is reinforced with continuous longitudinal fibers of constant cross-section with dimensions h x h . The fibers have the same modulus of elasticity. When y = 0 the body is fixed and has loading qz (x, y) on the surface z = H . The in-homogeneous structure of the body V0 is represented by regular cells G0 with 8h x 8h x 8h size, fig. 2, the sections
of 16 fibers are painted over. It is believed [26] that if the fiber thickness is less than 0.5 mm, then such fibers form a micro-inhomogeneous structure. Suppose L = 600 мм, H = 50 мм , then h = 0.3906 мм . In this case, the body V0 has a micro-inhomogeneous regular structure.
It should be noted that since the filling factor of the composite body V0 is small (equal to 0.25), it is difficult to determine the effective elastic moduli for the body V0 . The case when the filling coefficient is close to one was considered in [23].
Suppose the strength conditions (3) are given for the safety factor n0 of the composite body V0 . The basic discrete V0 body model R0 consists of finite elements (FE) of the 1st order of a cube shape with a side h [6], in which a three-dimensional SSS is realized, accounting for
the inhomogeneous structure of the beam and generates a basic uniform mesh with a step h with dimension 129 x 1537 x 129 .
Fig. 1. The characteristic sizes of the beam (body) V0 Рис. 1. Характерные размеры балки (тела) V0
Fig. 2. Regular cell (body) G0 Рис. 2. Регулярная ячейка (тело) G0
Fig. 2 shows the basic grid G0 of a regular dimension cell 9 x 9 x 9; i, j, к = 1,...,9 . The model R0 has N0 = 76681728 nodal unknown FEM, system tape width of FEM equations is b0 = 50316 . The basic model R0 takes into account the micro-inhomogeneous structure of the body V0 with high dimension, therefore we can assume that this model satisfies position 4. However, it is difficult to apply the discrete model R0 in calculations, since the implementation of the FEM for the R0 model requires essential computer resources.
According to the MESC, introduced is an isotropic homogeneous body Vb such that the bodies Vb, V0 have the same shape, dimensions, specified fixing and loading, but differ in elastic moduli. The elastic moduli of the body Vb are equal to the elastic moduli of the body V0 fiber. For the body Vb we define a discrete model V.., which consists of an FE V(en) of the 1st order of a cube shape with a side hn [6] and has a uniform mesh with a
step hn with dimension n|
W v ».(nK,»
<n
2
n3
where
n(n) = 8n +1, n2n) = 12 X 8n +1,
n(n) =
= 8n +1, n = 1,2,3,....
(26)
CT
0
CT
The steps of the fine mesh of the model V.00 along the axeses Ox, Oy, Oz equal h(xn) = H /(8n), h{") = L /(96n), hZn) = H /(8n). Since L = 12H , then hn = hXn) = hyyn) = hZn). Due to (26) we obtain hn = Pnh , where Pn - scale factor, Pn = 16/n , n = 1,2,3,.... Under n = 1,...,15 we have Pn > 1, i. e. hn > h . Under n ^ 16 we have Pn ^ 1, p16 = 1, h16 = h . Discrete model V^ of a finite number of bodies of the same shape G0 with dimensions 8hn x 8hn x 8hn , n = 1,2,3,.... The body and the regular cell G0 have the same shape (cube shape), but differ in characteristic dimensions.
Let us introduce a composite body G° (cube shaped) with dimensions 8hn x 8hn x 8hn . Suppose the composite body G0 consist of FE V(en) cube-shaped with the side hn . The composite body Gn0 is of fibrous structure, the same number of fibers (16 longitudinal fibers with a square cross section hn x hn, the distance between the fibers equals hn ) and the same mutual arrangement as in the regular cell G0 (the cell G0 has 16 with dimensions h x h, the distance between them equals h, fig. 2). n = 1,2,3,.... Inhomogeneous structure in the composite
body g0 is taken into account using FE V(en). Fibers and matrices of the bodies Gn0 , G0 have the same elastic moduli. The bodies Gn0 , G0 in fact differ only in scale, they can formally be written as G^ = (Pn )3 G0. Under n = 16 we obtain p16 = 1, i. e. G106 = G0.
Using the bodies Gn0 instead of the bodies Gnb in the discrete model Vnb we obtain a composite discrete model n = 1,2,3,..., which accounts for inhomogeneous
R0
To reduce the dimensions of the models V^, R° MgFE are used [8-22]. Since the models R06, Vb6 have the same high dimension as the basic discrete body model R0, which has 76681728 nodal unknown FEM, we believe that the maximum equivalent stress a°6 (stress a^) of the model R106 (model V1b6 ) differs a little from the
exact stress a0 (a°). Therefore, we assume a0 =aj06,
a0 ab
ab = a16 .
We find the equivalence coefficient p by formula (25) accounting for the latter 2 equations, i. e.
0 , b
(27)
0b
structure. Composite body G0 is, in fact, a regular cell for the model №, n = 1,2,3,.... Discrete model R° has the same uniform grid with step hn and dimensions Vnb . Under n = 16 the discrete models Vb6, R106 and R0 have the same shape, characteristic size and dimensions. Since G16 = G0, then under n = 16 models R106 and R0 coincide, i. e. R106 = R0. Thus, the discrete models Vn0, R° possess the same shape , characteristic size and dimensions, the same fixing and loading, like a body (beam) V0, but differ only in elastic moduli n = 1,2,3,.... It is important to note the following:
1. Dimensions of discrete models Vn0 , Rn0 under n = 1,...,15, due to (26), are less than the dimensions of the basic discrete model R0 of a composite body V0.
2. When constructing composite discrete models {Ro°}05=
2 , the procedure of grinding composite discrete models is not applied.
Taking into account in the formula pn = an / a where an ( an ) is the maximum equivalent stress of the model R° (model), which at n ^ 16 we have an ^ aj°6, abn ^ a^ , due to (27) we have pn ^ p at n ^ 16. Suppose pn quickly converge to p . Let the value §n =| p« - p«-1 | /p« be small, where then we accept hat p = pn. Applying the found coefficient p and parameter §a ( §a specified and satisfies condition (5)) n1 and n2 specified in representation (4), we determine the corrected equivalent strength conditions, which accounts for the stress error. Suppose abn quickly converge to a^. Let the small value §«« =| a^ -a«-1 | /a« and | §bn | < §a, where §bn is the relative voltage error, a« §a is given, §a < Ca n = 2,3,...,. Then we accept that ab =abn , i. e., the maximum equivalent body Vb stress ab is found. Suppose the found safety factor nb (where nb =aT / ab, i. e. nb =aT / abn ) of an isotropic homogeneous body Vb (corresponding to an approximate solution) satisfy the constructed equivalent strength conditions (4). Then the safety factor n0 of the composite body V0 (which corresponds to the exact solution) satisfies the given strength conditions (3).
When calculating the composite bodies strength according to MESC, it is advisable to use MgFE [24]. In this case, the implementation of MESC requires small computer resources.
5. Application of the corrected equivalent strength conditions in the calculations of composite bodies with a certain type of loading. The calculations given below show that the corrected equivalent strength conditions (4), constructed for a specific body loading, can be used in the strength calculations of a composite body V0 (fig. 1), for which a certain type of loading is specified.
In [24], an example of a cantilever beam V0 (fig. 1) strength analysis according to MESC using three-mesh FE is considered in detail. The beam is reinforced with longitudinal fibers. The regular cell of the beam is shown in fig. 2. Under y = 0 , u = v = w = 0, i. e in the xOz
plane the beam is fixed. For the safety factor n0 of the beam, the given strength conditions have the form
1.3 < n0 < 3.2.
(28)
In the calculations of the beam the following data were used:
h = 0.3906 ; ctt = 5; Ev = 10 , Ec = 1,
v c =vv = 0.3, qz = 0.0018, (29)
where Ec, Ev (vc, vv) - Young's moduli (Poisson's ratios) of the binder and fibers, respectively, ctt is the yield stress qz of the fiber, the load acts on the surface z = H , 0.5L < y < L , fig. 1.
The equivalence factorp for the composite beam V0 is determined using the procedure described above. Discrete models V^, Rn0, n = 9, 11, 12 are constructed using 3sFE (the construction procedure of which is described in detail in [24]) on the basis of basic regular partitions, respectively of dimensions: 73 x 865 x 73, 89 x1057 x 89
and 97 x 1153 x 97 . The coefficients pn are found by the formula pn =CTn / abn , where an , abn are the maximum equivalent stresses, respectively of the models R, Vt, n = 9, 11, 12. As a result of calculations we get: p9 = 3.002, pjj = 3.000, p12 = 2.999. The relative errors for the found coefficients p9, pn, p12 are
Sj(%) = 100%x|p„ - p91/pu = = 100 %x | 3.002 -3.000 | /3.000 = 0.066 %,
S2(%) = 100 %x | p^2 - pn|/A2 = = 100 %x | 3.000 -2.999 | /2.999 = 0.033 %.
Since p9 > pu > p12 and S2 is the smallest value, we consider, equivalent coefficient equals p = p12 = 2.999. Applying to (4) Sa = 0.15, n1 = 1.3 , n2 = 3.2 , we obtain the corrected equivalent strength conditions expressed in terms of the equivalence coefficient p
1.5288p < nb < 2.7805p . (30)
Applying to (30) p = 2.999, we obtain the following corrected equivalent strength conditions 4.584 < nb < 8.339, which in practice, in order to take
into account the error of computer calculations, is used in the following modified form
4.65 < nb < 8.25. (31)
Table 1 shows the results of calculations for five loadings ql of the beam V0, for which the equivalence coefficients pn are found, where x, y, z are the coordinates of the points of the beam surface, on which a constant load qnz, n = 1,...,5 is applied. Loads q"z, n = 1, 4, 5 provide direct bending of the beam, loads q2z, q3z - oblique bending of the beam. The relative error 8n (%) for the equivalence coefficient pn , presented in table, is determined by the formula
8n (%) = 100% x | p - pn |/ p , (32) where p = 2.999, n = 1,...,5 .
Analysis of the calculation results shows that the equivalence coefficients pn, n = 1.5 differ from the equivalence coefficient p = 2.999 by small values, which are 0.35 % less (see formula (32), tab. 1). According to (30), the corrected equivalent strength conditions for the equivalence coefficient pn, n = 1,...,5 have the form
1.5288pn < nb < 2.7805pn .
(33)
Since the coefficients pn, n = 1,...,5 have minor difference with p (see formula (32), fig. 1), then equivalent strength conditions (33) will differ a little from the equivalent strength conditions (30); moreover, we have
1.5288pn < 4.65 < nb < 8.25 < 2.7805pn , (34) where n = 1,...,5 .
Fulfillment of (34) implies that the equivalence coefficients pn, n = 1.5 , in fact, generate corrected equivalent strength conditions (31).
Consequently, the results of the calculations show that when calculating the strength of a composite beam V0
under the action of piecewise constant loads qzn on the surface z = H , n = 1,...,5 it is possible to use the corrected equivalent strength conditions (31) constructed for a beam V0 with loading qz = 0.0018 on the surface 0.5Z < y < L , z = H i. e., constructed using the equivalence coefficient p = 2.999.
The results of calculations of the beam V0
0
n x y z qn pn 5n (%)
1 0 < x < H 0 < y < L H 0.0078 2.997 0.066 %
2 0 < x < H/2 0 < y < L/2 H 0.543 2.991 0.267 %
3 0 < x < H/2 0 < y < L H 0.125 2.989 0.333 %
4 0 < x < H 0,998L < y < L H 2.8000 2.999 0.000 %
5 0 < x < H 0 < x < H 0 < y < L/2 0,5L < y < L H 0.0145 0.0034 2.994 0.167 %
Now then, if a piecewise constant load qz acts on the upper surface of the beam V0, which provides direct or oblique bending of the beam, then when calculating the strength of the beam V0 , you can use the corrected
equivalent strength conditions (31).
Given in [24] example of calculating the strength of a cantilever beam (having a micro-inhomogeneous regular fibrous structure) using the MESC shows its high efficiency.
Conclusion. The method of equivalent strength conditions is proposed for calculating the static strength of elastic bodies with an inhomogeneous, micro-inhomogeneous regular structure under given strength conditions. The proposed method is implemented applying FEM using multigrid finite elements and is limited to calculation of isotropic homogeneous bodies strength using equivalent strength conditions that account for solution errors. In the process of implementation, the method of equivalent strength conditions requires little time or computer resources and is exceptionally effective when calculating the strength of bodies that have a micro-inhomogeneous regular fibrous structure.
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Ма^ееу А. D., 2020
Matveev Alexander Danilovich - Cand. Sc., associate Professor, senior researcher; Institute of computational modeling SB RAS. E-mail: mtv241@mail.ru.
Матвеев Александр Данилович - кандидат физико-математических наук, доцент, старший научный сотрудник; Институт вычислительного моделирования СО РАН. E-mail: mtv241@mail.ru.
