CALCULATION METHODOLOGY OF THE EFFECT OF STATIC LOADS OF CIRCULAR CROSS-SECTION HYDROTECHNICAL TUNNELS

АННОТАЦИЯ

В этой статье рассматривается метод расчета влияния статических нагрузок на гидротехнические тоннели круглого сечения, которые широко используются в системах управления водными ресурсами. Существуют следующие основные элементы статических нагрузок: собственный вес конструкции, давление грунта, гидростатическое давление и временные нагрузки от оборудования. Методология включает в себя теоретическое моделирование, аналитические расчеты и численные методы для оценки реакции конструкции и распределения нагрузок. С учетом различных типов грунтовых условий были определены параметры, влияющие на деформацию и внутренние усилия в круглых тоннелях под действием статических нагрузок. В данном расчетном методе круглые тоннели разбиваются на два полукруга. Сначала в каждом полукруге определяется интенсивность внешних нагрузок. Затем в произвольных сечениях этих полукругов получаются выражения сил, действующих вдоль и поперек. Результаты исследования показали, что форма круглого сечения способствует равномерному распределению нагрузок, что приводит к снижению концентрации напряжений в отдельных участках и повышению надежности тоннеля. Результаты дают возможность оптимизировать проектирование гидротехнических сооружений и снизить риск аварийных ситуаций.

ABSTRACT

This paper discusses a method for calculating the effect of static loads on circular hydraulic tunnels, which are widely used in water management systems. The main elements of static loads are the structure's own weight, soil pressure, hydrostatic pressure, and live loads from equipment. The methodology includes theoretical modeling, analytical calculations, and numerical methods to evaluate the structure's response and load distribution. Taking into account different types of soil conditions, the parameters affecting the deformation and internal forces in circular tunnels under static loads were determined. In this calculation method, circular tunnels are divided into two semicircles. First, the intensity of external loads is determined in each semicircle. Then, expressions for forces acting along and across are obtained in arbitrary sections of these semicircles. The results of the study showed that the circular cross-section shape promotes uniform load distribution, which leads to a decrease in stress concentration in individual areas and an increase in the reliability of the tunnel. The results make it possible to optimize the design of hydraulic structures and reduce the risk of emergency situations.

Ключевые слова: гидротехнический туннель, полукруг, статическая нагрузка, активное и реактивное давление, сдвигающая и продольная сила.

Keywords: hydraulic tunnel, semicircle, static load, active and reactive pressure, shear and longitudinal force.

Introduction. Circular hydraulic tunnels play an important role in water supply, drainage and water management systems [1-3]. Consideration of static loads acting on such structures is key to ensuring their reliability and durability. This article proposes a calculation method for static loads on hydraulic tunnels, taking into account the specific characteristics of their operation [4-8]. Static loads are loads that act on a structure over a long period of time and change slowly. These include [9-11]:

• the structure's own weight;
• soil pressure;
• hydrostatic water pressure;
• temporary loads (e.g. from equipment).

Circular hydraulic tunnels are characterized by high strength and resistance to deformation [12-16]. Their shape allows for uniform distribution of loads, which reduces stress concentration in individual sections of the structure.

This study utilizes a quantitative approach to analyze the effects of static loads on circular cross-section hydrotechnical tunnels [17-22]. The calculation method of the impact of static loads on circular hydraulic tunnels ensures the reliability and safety of structures [23-29]. The main stages include determining the parameters of the structure, analyzing static loads, summing up the effects and calculating stresses. The use of this method helps optimize the design of hydraulic structures and prevent emergency situations.

Methods. The methodology comprises theoretical modeling, analytical calculations, and simulation techniques to evaluate load distributions and structural responses under various conditions [30-33]. The calculation methodology is grounded in the principles of structural mechanics and hydrostatics. Theoretical models is developed to account for various static load components acting on circular tunnels, including [34-39]:

Self-Weight of the Structure: The gravitational force exerted by the tunnel structure itself.

Soil Pressure: The lateral and vertical pressures exerted by surrounding soil masses, influenced by factors such as soil type, moisture content, and depth.

Hydrostatic Pressure: The pressure exerted by water, which varies with depth and must be considered in conjunction with groundwater levels.

Temporary Loads: Loads from construction activities and equipment that may affect the structural integrity during and post-construction [40-44].

The analytical framework is employ equilibrium equations and deformation compatibility conditions to derive the internal forces and deformations of tunnel sections. The calculations is involve [45-51]:

Load Projections: Determining the projections of hydrostatic and ground pressures onto the relevant axes [52-54].

Resultant Forces: Using equilibrium principles to establish relationships among the forces acting on tunnel segments.

Support Reactions: Calculating support reactions based on established load conditions and structural connections [55, 56].

Results and Discussion. Circular cross-section tunnels are constructed in the field of hydraulic engineering, especially as part of water discharge systems from reservoirs, primarily in rocky terrains. If such a tunnel's route passes through an unstable structural ground environment, it is excavated around it in an arched rectangular shape, and the gap between the excavation line and the tunnel ring is filled with concrete mortar (Figure 1). However, when a circular cross-section tunnel passes through solid ground, the excavation line also follows the edge of the circle, and there is no need to fill a large amount of cement mortar between the soil and the tunnel ring. During the construction of the tunnel, two or four joint seams are placed. Two seams are along the horizontal central line of the circular cross-section tunnel, and four seams, symmetrically above and below the horizontal central axis relative to the vertical central axis as shown in Figure 1. These seams connect the individual parts of the tunnel in both flexible and rigid ways. The endpoints of different parts of the circular cross-section tunnel are connected with two types of flexible joints and one rigid support type, enabling separate calculations of these parts under static loads, which allows for the calculation of all parts of the tunnel ring.

In pressurized hydraulic mode, the calculation method for the static effects of the upper element of circular cross-section tunnels is similar to the calculation method for the arch element of a non-pressurized hydraulic tunnel in rectangular form. That is, the deformation and internal forces of the upper AB element of the circular tunnel section can be calculated using principles derived from the schemes shown in Figure 1, whether the outer points of this tunnel section are flexible or rigidly connected. Taking into account distances from the tunnel's peak point to the groundwater level (ho) and the natural surface level (Ho), the calculation for static effects of the CD element on the lower side of the section must consider irregularly intensive loads along the radius line relative to the symmetrical central axis. This load represents the sum of projections of hydrostatic pressure along the radius line and the active and reactive pressure loads of the ground on this line (Rf axis) (Figure 1):

                                                                 (1)

Her:  - represents the intensity of hydrostatic pressure load (due to groundwater) considered in the cross-section.

                                                                  (2)

- the projection of the active pressure load of the ground onto the ordinate axis Kp:

                                          (3)

q_r (φ)- the projection of the load created by the reactive resistance of the ground environment onto the ordinate axis Kf and can be determined according to the Vinkler model as follows:

                     (4)

m₂ - the stiffness indicator of the ground located below the groundwater level, [t/m⁴]; Having stable settlement, the reactions of supports on the upper and lower parts of the tunnel section are determined under the condition of being equal to each other.

The values of the intensity of the active pressure load of the ground on the characteristic upper, middle, and lower points of the circular cross-section tunnel are determined by the following expressions:

According to the principle of equilibrium of deformation and internal forces, for the isolated CD element at the bottom of the tunnel, the resultant reaction force (N_C=N_D) can be determined under the condition that the sum of all forces acting on this element and their respective projections onto its axis equals zero:

                                                          (5)

In equation (3), considering the expression of q_w (φ), the force N4 is determined as follows:

                              (6)

The sum of the projections K_f of all forces is expressed as follows:

                                     (7)

From equation (3), the force Q_φ is found as follows:

                            (8)

If we find the reaction force ND, we get the following formula:

                                (9)

Since the support reaction N_A for the upper half-ring of the tunnel can be found in advance using the formula (6), the parameter W_C can be found from the following formula (9) from the condition N_D =N_A.

After the final reaction forces are known, in order to determine the longitudinal N_φ and shear Q_φ forces generated in the tunnel ring in the section forming an angle φ with the vertical central axis of the tunnel, it is necessary to use the conditions that the sum of the projections of all forces onto the Kt and Kf axes is equal to zero. For this, the effect of all forces on the half of the lower CD ring of the tunnel should be considered.

Projections of loads and effects on the Kt axis are the sum (Figure 1):

                                           (10)

Figure 1. Scheme for determining static load s acting along the full cross-section of a circular cross-section tunnel

                                          (11)

Equations (1) to (9) provide a detailed representation of the forces and conditions governing the static loads. Each equation is utilized to derive the necessary parameters, including hydrostatic load intensity, active and reactive ground pressures, and support reactions.

Conclusion. In conclusion, the proposed methodology for calculating static loads on circular cross-section hydrotechnical tunnels addresses the unique structural and environmental challenges these tunnels face. By accounting for primary static load sources—such as self-weight, soil pressure, hydrostatic water pressure, and temporary equipment loads—this approach aims to enhance the accuracy and reliability of load distribution predictions. The circular geometry of these tunnels plays a crucial role in evenly distributing loads, thus reducing localized stress concentrations and enhancing structural integrity. Additionally, by considering the variability in soil and groundwater conditions and the specific placement of seams and joints, the model provides a detailed framework for anticipating deformation and internal forces under static conditions. This methodology also integrates principles from rock mechanics and hydraulic engineering, bridging critical aspects of tunnel design to support long-term durability in diverse geological settings. The detailed equilibrium calculations for the upper and lower tunnel elements facilitate targeted analysis of each section, taking into account both flexible and rigid connections. Such comprehensive load analysis is essential for ensuring that circular hydraulic tunnels can withstand prolonged pressures and maintain functionality within water supply, drainage, and management systems.

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