INVESTIGATION OF STRENGTH CRITERIA FOR CONSTRUCTIONS WITH COMPLEX STRUCTURES IN URBAN PLANNING

SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 88 INVESTIGATION OF STRENGTH CRITERIA FOR CONSTRUCTIONS WITH COMPLEX STRUCTURES IN URBAN PLANNING Sh.A. Anarova1, M.N. Samidov2, S.B. Qarshiyeva3 Professor, Tashkent University of Information Technologies named after Muhammad alKhwarizmi1 PhD student, Tashkent University of Information Technologies named after Muhammad alKhwarizmi2 PhD student, Tashkent University of Information Technologies named after Muhammad alKhwarizmi3 https://doi.org/10.5281/zenodo.17468737 Abstract. In the article, the strength criteria of constructions with complex fractal structures in urban planning are investigated. The fractal configuration of structures used in modern architecture and construction leads to complex mechanical behavior during deformation and displacement processes. Using fractal geometry and mathematical modeling methods, the stress and strain states of such structures are analyzed. Furthermore, classical strength theories such as those of Galileo, Mariotte, Tresca, Coulomb, Mohr, and Davidenkov–Fridman are reconsidered for media with fractal properties. In the study, the distribution of stresses in complex geometric shapes is modeled based on the energy criterion, as well as the maximum normal and shear stress criteria. The obtained results demonstrate that the findings can be applied to assess the stability of fractal structures and to design safe and durable urban construction facilities. Keywords: fractal structure, urban planning, strength criteria, Mohr theory, deformation, stress, fracture mechanics, plastic flow, fractal geometry, stable construction. Introduction One of the main challenges in the current development of urban planning is ensuring the strength and stability of structures with complex configurations. In particular, the fractal architecture of modern structural forms and construction materials – characterized by selfsimilarity and multi-dimensionality – complicates the prediction of their mechanical behavior using conventional theories. This necessitates a deeper investigation of the complexities that arise in displacement processes, force distribution, and deformation mechanisms. The use of fractal geometry and its mathematical representations makes it possible to more accurately describe the displacement and load-bearing behavior of complex structures in urban planning. This, in turn, is a crucial factor in the design, construction, and operation of safe and long-lasting buildings and facilities [1, 2]. Conventional methods are often based on ideally smooth or simple geometric shapes, which fail to provide sufficient accuracy when calculating stresses and deformations in complex fractal surfaces and structures. As a result, projects developed on the basis of such approaches may not yield the expected outcomes under real conditions, potentially leading to technical failures, accidents, and even disasters. Therefore, it is essential to adapt modern scientific and technical approaches, including mathematical models and algorithms, to fractal structures in order to correctly simulate the real behavior of urban constructions. In this regard, the main focus is on determining and modeling the SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 89 strength parameters of displacement processes while taking into account the fractal characteristics of the structures [3, 4, 5]. In this scientific study, the main objective is to investigate the displacement processes of fractal-structured objects, conduct an in-depth analysis of stress and strain distribution on their surfaces, and develop new mathematical models for assessing their strength. These models not only possess a theoretical foundation but can also be applied in practical construction processes, including the design and building of engineering structures. Strength-related issues are among the key challenges in designing fracture processes and evaluating the stability of fractal objects used in urban planning. The fracture of solid bodies is such a complex and multifaceted phenomenon that, to date, there is no single universally applicable strength theory in practice. Nevertheless, due to the relevance of the problem, studies in this field are of great importance. As a result, a distinct scientific discipline known as fracture mechanics has emerged. Initially, a solid body was considered an absolutely homogeneous and isotropic medium. Such a model was studied by N. Wiener based on the “black box” principle: an external signal (impact) is applied to the system, and the output signal (response) is recorded. The objective of the study is to determine a quantitative relationship between these signals, which represents a phenomenological approach. In relation to the fracture process, this involves determining a certain function of the main stress components and comparing it with a strength criterion. This criterion is usually associated with the simplest stress states, such as tension, compression, or shear. 1 2 3 , , .()Fk     (1) The historically first form of such an approach is Galileo’s maximum normal stress criterion, which states: “Fracture occurs when the maximum normal stress reaches its critical value.”: . i max   (2) This criterion provides satisfactory results in the case of uniaxial stress conditions or volumetric tension when the three principal stresses differ in magnitude, particularly for the failure of brittle materials. The widely used “strength limits” in practice (such as tensile and compressive strength, etc.) are also based on this criterion. With the advancement of scientific knowledge, other criteria were later proposed. Maximum tensile strain criterion (Mariotte’s criterion). According to this criterion, a body fails when its relative deformation exceeds the following value: . i max   (3) This criterion is applicable only in cases where the brittle failure of fractal objects used in urban planning occurs through separation (fracture). According to Hooke’s law, under elastic deformation: 1 2 3 [( /)] iE         and / max max E   here  - Poisson's ratio. So, the criterion can be written as follows: 1 2 3 – .[ ( )] max       (4) Maximum tensile stress criterion (Coulon–Treska criterion): SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 90 . i max   (5) In a complex stress state, the tangential stress reaches a maximum in the region at an angle of 45 to the line of action of the normal stresses. 13 /2() i     and /2 max  then: 13 –. max     (6) This criterion is valid for plastic bodies and is essentially a condition for the occurrence of plastic flow. Energy criterion. Destruction occurs when the potential deformation energy accumulated in the body reaches a maximum value: . i max AA (7) If the expression for the transformed potential energy in the three-dimensional case is 2 ()/2AE   , then the energy can be written as:: max 2 2 2 2 1 2 2 3 1 3 ( .) ( ) ( ) 2                (8) Considering that here is ()/2 i j ij     , the decay condition can be expressed as follows:: 2 2 2 12 23 31 2 .( ) max        (9) As can be seen from this expression, it is close to the Coulomb criterion and is primarily suitable for the degradation of plastic materials. The Coulon theory is based on the following hypothesis: the resistance of a body to sliding along a given surface  (the shear stress) is equal to the product of the adhesion force s  and the normal stress  perpendicular to the surface, expressed by the angle of internal friction: . сtg      (10) The main essence of the Mohr theory of strength is that the destruction of a body occurs due to the combined action of normal and shear stresses. These stresses are interconnected, calculated using vector sums, and can be described by maximum stress circles. There is a specific stress circle corresponding to each stress state. Therefore, there is a family of stress circles constructed for different stress states. The curve surrounding the circles of ultimate stresses is called the Mohr envelope or strength passport. This line consists of a set of points reflecting the maximum stress state of the structure. According to Mohr theory, the actual envelope of the stress circles is always nonlinear (i.e., in the form of a curve). This curve grows monotonically, the abscissa axis is relatively symmetrical, the stresses are closed in the tension zone and open in the compression zone. It has been found experimentally [7] that, depending on the type of fractal objects used in urban planning, the Mohr line is represented by the following parabolic or hyperbolic curves, in some cases by a cycloid or a combination of a straight line added to the cycloid. The equation of the hyperbola is written as follows: 3 28 max 22, x xa      (11) here max  - maximum shear resistance of structures with fully closed cracks; p x   - the amount of normal stresses relative to the coordinate origin; a - a parameter of the shape of the curve that represents the difficulty of closing cracks in structures. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 91 For solid fractal objects used in urban planning, a curved (curved) line is represented by the equation of a parabola [3-7]:   , p K     (12) where K - is the shape parameter of the curved line. The trailer size is determined from the following condition 0   , then , cp K   (13) The tangent of the angle of internal friction is determined from the first derivative of the equation of the inclined line at the coordinates   ;0 c     of the point, that is:   /2 . c arctg K   (14) The curve shape parameter depends on the structure of the K rock, primarily its fracture. One possible expression for the parameter is given by the Richter equation [3-7]:   22 p p p cж cж K            (15) According to Mohr's theory, there are two types of fracture of solids: • brittle fracture - occurs as a result of cracks; • plastic (smooth) fracture - occurs under the influence of sliding stresses. Developing this idea, N.N. Davidenkov and Ya.B. Fridman [3, 5, 6] developed a generalized theory of strength, which determines the nature of the fracture of a body depending on the ratio of the main normal stresses. At present, it can be considered as firmly established that fracture by shear alone is impossible in principle. Tangential stresses that cause plastic deformation weaken the bonds between atoms only in certain sections, and final fracture occurs due to cracks. Therefore, the state characteristic m for real materials is nonlinear. Phenomenological theories of strength prove to be very useful for engineering calculations and are widely used in design. However, they do not take into account the physics of the processes of decay of bodies in any way. Considering the body at the atomic level allows us to theoretically determine the stress required for the rupture of the body, a certain fraction of S containing n particles bound by the interaction force f0, i.e. the theoretical strength (0) 0 / pf N S   . With this calculation, it turns out that the theoretical strength, as a rule, is several times higher than the stresses required for the destruction of the body. For the first time, this contradiction was resolved for brittle materials (glass) by A. A. Griffiths [3-8]. Based on the theoretical studies of G.V. Kolosov, who determined the laws of stress concentration around the crack, Griffiths formulated a unified theory of the destruction of brittle bodies. Considering the elongation of a plate of the same thickness as the crack as a model, he showed that the destruction of a body is determined by the growth of a single “main” crack. Due to the concentration of stresses at its mouth, the development of a crack requires much less energy than the destruction of a perfectly flawless body. The formation of a crack is accompanied by the release of elastic energy: 22 . E L AE   (16) The rate of elastic energy release as the crack grows: SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 92 2 2, E T dA GdL E   (17) is called the crack driving force. Its measurement (N/m) corresponds to the force per unit length of the crack. The released energy is spent on the formation of new surfaces as the crack grows, since work must be done against the surface tension forces. We denote the specific surface energy of the body by es. Since when a crack is formed, two new surfaces, each with an area of 2L×1 (the curvature of the crack edges can be neglected), energy must be expended for this: 4 Ss A Le (18) It should be noted that A. Griffiths [3-8] defined the es index in his paper as “the surface tension of the material”. The change in this energy with increasing crack length is called the resistance to crack growth: 4 ss dA Re dL  (19) Crack growth condition (Griffiths criterion): T GR (20) Taking into account expressions (17) and (19): 2 2 4s eE   (21) The value of R depends only on the specific surface energy of es and this property, i.e. the body constant. In it, equation (21) describes the relationship between the stress and the critical length of a crack that can grow at a given stress. Although stresses are accumulated in each of the many cracks in a real body, failure begins with the growth of a single “critical” crack. The critical crack length is given by equation (21): 2 4 2, s kp Ee L   (22) From this formula it follows that the force can be reduced to maintain the critical length of the crack in the body. In fact, this means that the external load is no longer needed. The plastic energy stored in the body when the crack reaches the critical length is sufficient to complete the corrosion process. Thus, the Griffiths theory shows the destructive nature of brittle corrosion, the huge accelerations in the crack growth, which cannot stop the crack growth process if it has already passed a critical point. The corrosion stress (tensile strength) corresponding to the Griffiths criterion is determined by the following formula: 2s p kp Ee L   (23) This formula, like all previous calculations, is valid for the plane stress state of the body. In the case of plane deformation, its transverse component must be taken into account, that is, Poisson's ratio ν. Then the above formula takes the form   2 2 1 s p kp Ee L   (24) Griffiths theory still remains the main tool for research in the field of fracture mechanics, since it accurately reflects the physics of the processes. However, the quantitative estimates of the theory (Griffiths criteria) are consistent with experimental data only for thin amorphous bodies. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 93 For crystalline materials, the fracture stresses should be much higher than predicted by the theory. In addition, the es indicator obtained by A. Griffiths for the material constant is not in fact the same. Numerous experimental studies show that the value of es is significantly affected by the nature and speed of loading of bodies, the environment, etc. The mechanism of plastic deformation of solids due to shear stresses was discovered by J. Taylor in 1934. He first put forward the idea of \u200b\u200bthe existence of linear defects in crystals - dislocations, which can interact with each other. Later, the mechanism of dislocation propagation was described (Frank-Reed, Ziner, Cottrell, Balaf-Gilman, Orovan-Straw models [9-10]). The connection of these phenomena with the destruction of crystals was established by the Hungarian physicist E. Orovan [11, 12], the author of the dislocation theory of plastic deformation. He showed that additional energy is spent on plastic (irreversible) deformation of the crack edges, which often significantly exceeds the specific surface energy of the bodies. Indeed, amorphous bodies that do not have dislocations break up so that the fragments stick together with high accuracy (for example, a glass object can be glued together), while the fracture surface of the stone undergoes plastic deformation, which does not allow the fragments to collect and stick together. To numerically account for this mechanism, Orovan proposes to introduce a new quantity eр – the specific energy of plastic deformations – into the Griffiths criterion (equation 25), in which:   2sp p kp E e e L    (25) For urban construction objects, the value of ep is 2-3 orders of magnitude greater than es, therefore, the elastic component of es is often completely excluded from the formula due to its small size. However, there is no theoretical means of estimating the value of the specific energy of elastic deformation. Therefore, it is determined only elastically by destroying samples with an artificially created cross section that simulates a crack. In this regard, in engineering practice, the force approach to fracture mechanics is more often used. This elastic is the stress intensity coefficient proposed by J. Irwin [6]. When a load is applied to a body with a crack, the edges of this crack shift relative to each other. Based on the superposition principle of the theory of linear elasticity [10], this shift can be expressed as follows. Depending on the nature of the stresses, cracks of three types or genera can develop: separation (I), transverse (II) and longitudinal (III) displacement. The most important characteristic of a crack is the stress intensity coefficient. For type I cracks: I KL   (26) The development of cracks of types II and III is determined by the tensile stresses  , where: II KL   (27) The stress intensity factor relates the strength properties of a loaded body to the crack length and is a material constant. Additional stresses at the end of the crack: 2/Lr   (28) or taking into account expression (26): 2/ I KL   (29). Thus, any cracks loaded to the same KI value for a given body will have the same stress field. For example, two cracks of size 4L and L will have the same stress fields if the first crack is SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 10 OCTOBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 94 loaded to  , and the second to 2  . It is in this case that the stress intensity coefficient will be the same. If the 2L crack plate is eroded under tension   , then the stress intensity coefficient will have a critical (limiting) value: . IC p KL   (30) The development of type II and III cracks is determined by tensile stresses, therefore , IIC CDB KL   (31) where CDB  - is the shear strength of the body. Griffiths criterion (23) - equation takes the following form, taking into account the expression for the stress intensity coefficient: 2 2 / 4 IC s K E e (32) then 2 IC s K Ee const (33). In this case (as a property of a body), the stress intensity coefficient is called the erosion viscosity or fracture toughness. Its magnitude can be used to estimate the specific surface energy of the body: 2/2 IC s K E e (34) Unlike the Griffiths model in a real body, a main crack can have an arbitrary orientation. If the angle of deviation of the crack normal from the load line of action is  , then in the crack plane, the stress intensity coefficients 2 cos and cos sin y xy        are then determined by the formulas [10]: 2 cos I KL     (35) cos sin II KL      (36). To experimentally determine the stress intensity coefficients, the method of cuts, known from the course "Resistance of Materials", is usually used. At the same time, it is technically most convenient to use the three-point beam technique of artificially cut beams simulating a crack. Griffiths's theory, corrected by Irwin-Orovan, accurately describes the mechanism of damage to building structures. Conclusion. 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