Method for calculating stress of skewback abutment base

Abstract

The present application relates to a kind of skewback type abutment stress calculation method, the force and bending moment of abutment are converted, including axial pressure N, longitudinal shear force Fx, horizontal shear Fz, longitudinal bending moment Mz, transverse bending moment Mx and torque My, then according to stress balance, force and moment are distributed in the abutment slope and its vertical imaginary plane, i.e. the stress of any point on the abutment slope and imaginary plane can be directly calculated using the stress formula of eccentric compression rectangular section, abutment slope, imaginary plane and base horizontal plane form right triangle geometric relationship, can be calculated according to the geometric projection relationship, vertical stress balance and bending moment balance, the base stress of any point on base horizontal plane, greatly simplify the calculation process, the method is solved according to the most direct direction of force transmission path, the solution result is also the actual stress of abutment base, not component force, and the calculation method is simple and applicable.

Description

Technical Field

[0001] This invention relates to the field of bridge engineering technology, and in particular to a method for calculating the base stress of a sloping arch abutment, as well as the arch abutment and the bridge. Background Technology

[0002] Due to the limitations of mountainous terrain, the construction of long-span arch bridges is becoming increasingly common. The arch abutment plays a crucial role in the overall load-bearing system of the arch bridge, its importance being self-evident. Among the choices for arch abutment types, there are mainly two categories: spread foundations and pile foundations. Because the arch abutment also bears a significant amount of horizontal force transmitted from the arch ring, pile foundation arch abutments need to be designed as inclined piles or a combination of horizontal and vertical piles. However, the construction of inclined and horizontal piles is relatively complicated, so many arch bridge builders prefer spread foundation arch abutments.

[0003] Since the arch abutment is mainly subjected to oblique thrust, from the perspective of stress and economy, arch bridge designers are increasingly adopting the sloping-back type enlarged foundation arch abutment. The size of the enlarged foundation arch abutment is closely related to the magnitude of the base stress, but the enlarged foundation arch abutment and the foundation often have multiple contact surfaces, and the magnitude of its base stress is difficult to calculate accurately. Currently, commonly used calculation methods include the method in the "Pier and Foundation Design Manual", the deformation compatibility method, and the solid finite element method.

[0004] The method in the "Abutment and Foundation" design manual is to calculate the horizontal plane by projecting the enlarged foundation vertically onto the bottom surface of the foundation, without considering the influence of the back of the foundation on the ground. The horizontal thrust on the arch seat is balanced by the friction force of the bottom surface. Because the influence of the back of the foundation on the arch seat is ignored, the base stress calculated by this method and the arch seat size designed accordingly are often too large.

[0005] The deformation coordination method projects the arch abutment onto both the horizontal and vertical planes simultaneously. The rotation of the arch abutment is a rigid body rotation around the intersection of the projection planes. Based on the force equilibrium conditions and the deformation coordination of the foundation bottom and back surfaces, the base stress on the two projection planes can be calculated. This method is a good calculation method when the base has not become detached. However, this method does not consider the influence of lateral loads on the base stress, and when the base becomes detached, the deformation coordination conditions for rotation are difficult to obtain.

[0006] The solid finite element method is a method for solving the base stress using finite element software. As long as the boundary conditions are properly simulated, this method can calculate the base stress relatively accurately. However, the contact between the foundation and the arch foundation is still difficult to simulate accurately, and the finite element modeling process is relatively complicated. Summary of the Invention

[0007] The purpose of this invention is to provide a method for calculating the base stress of a sloping arch seat, as well as an arch seat and a bridge, in order to address the problems existing in the prior art.

[0008] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0009] In a first aspect, the present invention provides a method for calculating the base stress of a sloping arch abutment, comprising the following steps:

[0010] S1. The forces and moments transmitted by the arch ring and the self-weight of the arch ring, which are borne by the arch seat, are converted to the inclined surface of the arch seat. The converted forces and moments include axial pressure N, longitudinal shear force Fx, horizontal shear force Fz, longitudinal bending moment Mz, transverse bending moment Mx, and torque My.

[0011] S2. Taking the inclined surface of the arch as the first force-bearing surface and the imaginary surface perpendicular to the first force-bearing surface as the second force-bearing surface, the converted force and moment are distributed to the first and second force-bearing surfaces through force balance. The first force-bearing surface bears a total of one axial compressive force N1, longitudinal bending moment M1, and transverse bending moment T1, while the second force-bearing surface bears a total of one axial compressive force N2, longitudinal bending moment M2, and transverse bending moment T2.

[0012] S3. The stress at any point (x, y) on stress surface one and stress surface two is calculated using the stress formula for an eccentrically compressed rectangular section:

[0013]

[0014] In the formula: N is the axial compressive force, A is the cross-sectional area, M is the longitudinal bending moment, and I... xx T is the longitudinal moment of inertia, and I is the transverse bending moment. yy The moment of inertia in the transverse direction of the bridge;

[0015] S4. The projection plane of the base horizontal plane, i.e., the third stress surface, is a part of the first stress surface and the second stress surface. Based on the vertical force balance and bending moment balance, the axial compressive force N3, longitudinal bending moment M3, and transverse bending moment T3 on the third stress surface are:

[0016] N3 = N1·COS(θ) + N2·SIN(θ)

[0017] M3 = M1 + M2

[0018] T3 = T1·COS(θ) + T2·SIN(θ)

[0019] The stress at a point (x, y) on the stress-bearing surface is:

[0020]

[0021] S5. Substituting the relationships between forces and moments on each stress surface, as well as the geometric relationships between the three stress surfaces, into the above equation, we obtain the following stress relationship between a point on the three stress surfaces and its projection point:

[0022] σ3=σ1cos 2(θ)+σ2sin 2 (θ)

[0023] Based on the above formula, the base stress at any point on the horizontal plane of the arch base can be calculated according to the stress on the projection plane.

[0024] As a preferred technical solution of the present invention, in step S2, the distribution ratio of the longitudinal bending moment Mz on the first and second force surfaces is obtained according to the deformation coordination relationship of the rigid body rotation of the arch seat. The ratio is the cube of the side length L1 of the first force surface to the cube of the side length L2 of the second force surface.

[0025] As a preferred technical solution of the present invention, after step S3, if there is a stress void at any point on each stress surface, calculate the stress redistribution after the stress void is resolved, and then proceed to step S4.

[0026] As a preferred technical solution of the present invention, some arch supports are designed with a serrated stepped surface on the back slope. The axial pressure N3, longitudinal bending moment M3, and transverse bending moment T3 at the centroid are obtained by summing the stresses on the slope corresponding to a step. Then, based on the force balance and the deformation coordination relationship that the rotation of the horizontal and vertical surfaces of the steps should be a rigid body rotation, the axial pressure and bending moment on the horizontal and vertical step surfaces are obtained respectively. Thus, the stress relationship of a point on the slope corresponding to its projection point is obtained as follows:

[0027] Horizontal step surface

[0028]

[0029] Vertical step surface

[0030]

[0031] Incline σ3=σ 3N +σ 3M +σ 3T

[0032] In the formula σ iN σ iM σ iT These represent the stresses generated by the axial pressure, longitudinal bending moment, and transverse bending moment at a point on the inclined plane corresponding to the step.

[0033] Secondly, the present invention also provides an arch base, which is designed using the method for calculating the base stress of a sloping arch base as described in any of the above claims.

[0034] Thirdly, the present invention also provides a bridge, including the arch base described above.

[0035] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:

[0036] The present invention discloses a method for calculating the base stress of a sloping arch abutment. This method converts the forces and moments borne by the abutment into lateral loads such as horizontal shear force Fz, longitudinal bending moment Mz, and torque My. Then, based on force equilibrium, the distribution of forces and moments is performed on the sloping surface of the abutment and an imaginary surface perpendicular to it. The stress at any point on the sloping surface and the imaginary surface can be directly calculated using the stress formula for an eccentrically compressed rectangular section. The sloping surface, the imaginary surface, and the horizontal plane of the base form a right-angled triangle geometric relationship. Based on the geometric projection relationship, vertical force equilibrium, and bending moment equilibrium, the stress on the horizontal plane of the base can be calculated. The method greatly simplifies the calculation process by determining the base stress at any point. It solves for the base stress of the arch abutment in the direction of the most direct force transmission path, and the result is the actual stress of the arch abutment base, not the component force. The calculation method is simple and applicable. Through a more accurate base stress algorithm, the design of enlarged foundation arch abutments can be carried out. Under the premise of ensuring safety, the arch abutment design can be made more economical and save engineering costs. At the same time, the slope reinforcement problem can be considered based on the base stress, making the design of the arch abutment and slope reinforcement more economical and reasonable. This method can also be applied to multi-stage small-step enlarged foundation arch abutments. Attached Figure Description

[0037] Figure 1 This is a schematic diagram of the forces acting on the inclined back arch seat;

[0038] Figure 2 This is a schematic diagram of the elevation for calculating the base stress of the inclined arch abutment;

[0039] Figure 3 A schematic diagram of the projection surface for calculating the base stress of the inclined back arch.

[0040] Figure 4 A schematic diagram for stress calculation on the horizontal base surface;

[0041] Figure 5 This is a schematic diagram for calculating the stress on the step surface. Detailed Implementation

[0042] The present invention will now be described in detail with reference to the accompanying drawings.

[0043] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0044] Example 1

[0045] like Figures 1 to 4 As shown, the method for calculating the base stress of a sloping arch abutment according to the present invention includes the following steps:

[0046] S1. The forces and bending moments transmitted by the arch ring and the self-weight of the arch ring, which are borne by the arch abutment, are transferred to the inclined surface of the arch abutment. After the transfer, as follows: Figure 1 As shown, the forces and moments include axial pressure N, longitudinal shear force Fx, horizontal shear force Fz, longitudinal bending moment Mz, transverse bending moment Mx, and torque My. These forces and moments are in equilibrium with the supporting force and frictional force provided by the base.

[0047] Since the friction force on the arch seat is mostly static friction and intersects with the axial pressure on the other contact surface, the influence of friction force is usually ignored in general calculations, and the calculation results are on the safe side.

[0048] S2, such as Figure 2 As shown, the entire supporting surface of the arch (i.e., the inclined surface of the arch mentioned above) is considered as an inclined back force-bearing surface one and an imaginary force-bearing surface two perpendicular to it. According to the force balance, force-bearing surface one bears the load of... Figure 1 The axial compressive force N, part of the longitudinal bending moment Mz, and the transverse bending moment Mx in the load-bearing surface two bear the load. Figure 1 The longitudinal shear force Fx, partial longitudinal bending moment Mz, and torque My are included.

[0049] The distribution ratio of the longitudinal bending moment Mz on the first and second force surfaces can be obtained from the deformation coordination relationship of the rigid body rotation of the arch seat. It is the ratio of the cube of the side length L1 of the first force surface to the cube of the side length L2 of the second force surface. The horizontal shear force Fz is often small and its influence can be ignored.

[0050] After distributing the forces and moments, surface one experiences a total of one axial compressive force N1, a longitudinal bending moment M1, and a transverse bending moment T1, while surface two experiences a total of one axial compressive force N2, a longitudinal bending moment M2, and a transverse bending moment T2. Figure 2 As shown.

[0051] S3. The stress at any point (x, y) on force-bearing surface one and force-bearing surface two can be calculated using the stress formula for an eccentrically compressed rectangular section:

[0052]

[0053] Where: N is the axial compressive force; A is the cross-sectional area; M is the longitudinal bending moment; I xx T is the longitudinal moment of inertia; T is the transverse bending moment; I yy The moment of inertia is in the transverse direction of the bridge.

[0054] S4. The stress at any point on each load-bearing surface can be calculated using the above formula. If there is stress detachment, the stress redistribution after detachment can be calculated by referring to the paper "Base Axial Pressure of Bidirectional Eccentrically Compressed Rectangular Foundation" (Du Minggan).

[0055] S5. Since the main contact surface between the arch abutment and the foundation is the first stress-bearing surface, the actual stress on the inclined contact surface is the calculated stress on the corresponding first stress-bearing surface. The actual stress on the horizontal plane of the arch abutment base can be obtained from the stress on the two projection planes, such as... Figure 4 As shown, the projection plane of the base horizontal plane, i.e., the force-bearing surface three, is a part of the force-bearing surface one and the force-bearing surface two. The resultant force of the stress on the force-bearing surface one is the axial compressive force N1, the longitudinal bending moment M1, and the transverse bending moment T1. The resultant force of the stress on the force-bearing surface two is the axial compressive force N2, the longitudinal bending moment M2, and the transverse bending moment T2. According to the vertical force balance and bending moment balance, the axial compressive force N3, the longitudinal bending moment M3, and the transverse bending moment T3 on the force-bearing surface three are:

[0056] N3 = N1·COS(θ) + N2·SIN(θ)

[0057] M3 = M1 + M2

[0058] T3 = T1·COS(θ) + T2·SIN(θ)

[0059] The stress at a point (x, y) on the stress-bearing surface is:

[0060]

[0061] S6. Substituting the relationships between forces and moments on each stress surface, as well as the geometric relationships between the three stress surfaces, into the above equation, we can obtain the following stress relationship between a point on the three stress surfaces and its projection point:

[0062] σ3=σ1cos 2 (θ)+σ2sin 2 (θ)

[0063] Based on the above formula, the base stress at any point on the horizontal plane of the arch base can be calculated from the stress on the projection plane.

[0064] This embodiment describes a method for calculating the base stress of a sloping arch abutment. It transforms the forces and moments borne by the abutment, including lateral loads such as horizontal shear force Fz, longitudinal bending moment Mz, and torque My. Then, based on force equilibrium, it distributes the forces and moments on the sloping surface of the abutment and an imaginary surface perpendicular to it. The stress at any point on the sloping surface and the imaginary surface can be directly calculated using the stress formula for an eccentrically compressed rectangular section. The sloping surface, the imaginary surface, and the horizontal plane of the base form a right-angled triangle geometric relationship. The stress can be calculated based on the geometric projection relationship, vertical force equilibrium, and bending moment... The method of balancing and calculating the base stress at any point on the horizontal plane of the base greatly simplifies the calculation process. This method solves for the base stress of the arch abutment in the direction of the most direct force transmission path, and the result is the actual stress of the base of the arch abutment, rather than the component force. The calculation method is simple and applicable. Through a more accurate base stress algorithm, the arch abutment of the enlarged foundation can be designed. Under the premise of ensuring safety, the arch abutment can be designed to be more economical and save engineering costs. At the same time, the slope reinforcement problem can be considered based on the base stress, making the design of the arch abutment and slope reinforcement more economical and reasonable.

[0065] Example 2

[0066] The method for calculating the base stress of the inclined arch abutment described in this invention, such as... Figure 5 As shown, some arch supports have their back slopes designed as serrated steps. The stress on the horizontal and vertical steps is related to the stress on the corresponding slopes.

[0067] By summing the stresses on the inclined plane corresponding to a step, we can obtain the axial pressure N3, longitudinal bending moment M3, and transverse bending moment T3 at the centroid. Then, based on the force balance and the deformation coordination relationship that the rotation of the horizontal and vertical planes of the step should be a rigid body rotation, we can obtain the axial pressure and bending moment on the horizontal and vertical step surfaces respectively.

[0068] Finally, by re-deriving the process from Example 1, the stress relationship between a stress at a point on the inclined plane and its projection point is obtained as follows:

[0069] Horizontal step surface

[0070]

[0071] Vertical step surface

[0072]

[0073] Incline σ3=σ 3N +σ 3M +σ 3T

[0074] In the formula σ iN σ iM σ iTLet θ be the stress generated by the axial pressure, longitudinal bending moment, and transverse bending moment at a point on the inclined surface corresponding to the step. From the above formula, it can be seen that the stress components on the horizontal or vertical step surface differ only in that caused by the longitudinal bending moment, while the other two are equal. When the angle θ is 45°, the stresses are completely equal.

[0075] When the step length is small, the bending moment on the corresponding inclined plane is often small. Therefore, it can be approximated that the normal stress at a point on the inclined plane is equal to the stress at the corresponding projection point on the step surface.

[0076] The method in this embodiment solves for the stress in the arch base according to the direction of the most direct force transmission path. The solution result is the actual stress in the arch base, rather than the component force. The calculation method is simple and applicable.

[0077] Example 3

[0078] The arch seat described in this invention is designed using the stress calculation method for the base of the inclined back arch seat as described in Example 1 or Example 2.

[0079] Example 4

[0080] The bridge described in this invention includes an arch as described in Example 3.

[0081] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for calculating the base stress of a sloping arch abutment, characterized in that, Includes the following steps: S1. The forces and moments transmitted by the arch ring and the self-weight of the arch ring, which are borne by the arch seat, are converted to the inclined surface of the arch seat. The converted forces and moments include axial pressure N, longitudinal shear force Fx, horizontal shear force Fz, longitudinal bending moment Mz, transverse bending moment Mx, and torque My. S2. Taking the inclined surface of the arch as the first force-bearing surface and the imaginary surface perpendicular to the first force-bearing surface as the second force-bearing surface, the converted force and moment are distributed to the first and second force-bearing surfaces through force balance. The first force-bearing surface bears a total of one axial compressive force N1, longitudinal bending moment M1, and transverse bending moment T1, while the second force-bearing surface bears a total of one axial compressive force N2, longitudinal bending moment M2, and transverse bending moment T2. S3. The stress at any point (x, y) on stress surface one and stress surface two is calculated using the stress formula for an eccentrically compressed rectangular section: In the formula: N is the axial compressive force, A is the cross-sectional area, M is the longitudinal bending moment, and I... xx T is the longitudinal moment of inertia, and I is the transverse bending moment. yy The moment of inertia in the transverse direction of the bridge; S4. The projection plane of the base horizontal plane, i.e., the third stress surface, is a part of the first stress surface and the second stress surface. Based on the vertical force balance and bending moment balance, the axial compressive force N3, longitudinal bending moment M3, and transverse bending moment T3 on the third stress surface are: N3 = N1·COS(θ) + M2·SIN(θ) M3 = M1 + M2 T3 = T1·COS(θ) + T2·SIN(θ) The stress at a point (x, y) on the stress-bearing surface is: S5. Substituting the relationships between forces and moments on each stress surface, as well as the geometric relationships between the three stress surfaces, into the above equation, we obtain the following stress relationship between a point on the three stress surfaces and its projection point: σ3=σ l cos 2 (θ)+σ2sin 2 (i) Based on the above formula, the base stress at any point on the horizontal plane of the arch base can be calculated according to the stress on the projection plane.

2. The method for calculating the base stress of a sloping arch abutment according to claim 1, characterized in that, In step S2, based on the deformation coordination relationship of the rigid body rotation of the arch seat, the distribution ratio of the longitudinal bending moment Mz on the first and second force surfaces is obtained as the cube of the side length L1 of the first force surface to the cube of the side length L2 of the second force surface.

3. The method for calculating the base stress of a sloping arch abutment according to claim 1, characterized in that, After step S3, if there is stress voiding at any point on each stress surface, calculate the stress redistribution after stress voiding, and then proceed to step S4.

4. The method for calculating the base stress of a sloping arch abutment according to any one of claims 1-3, characterized in that, Some arch supports have their backslopes designed as serrated steps. By summing the stresses on the slope corresponding to a step, we can obtain the axial pressure N3, longitudinal bending moment M3, and transverse bending moment T3 at the centroid. Then, based on the force balance and the deformation compatibility relationship that the rotation of the horizontal and vertical planes of the steps should be as rigid body rotation, we can obtain the axial pressure and bending moment on the horizontal and vertical step surfaces respectively. Thus, the stress relationship of a point on the slope corresponding to its projection point is as follows: Horizontal step surface Vertical step surface Inclined plane σ3 = σ 3N +σ 3M +σ 3T In the formula σ iN σ iM σ iT These represent the stresses generated by the axial pressure, longitudinal bending moment, and transverse bending moment at a point on the inclined plane corresponding to the step.

5. An arch base, characterized in that, The design is carried out using the stress calculation method for the base of the inclined back arch as described in any one of claims 1-4.

6. A bridge, characterized in that, Includes the arch seat as described in claim 5.

Images