An analytical method for the thrust influence line of a two-hinged arch with variable cross-section

Abstract

The present invention discloses an analytical method for calculating the thrust influence line of a two-hinged arch with a variable cross-section, belonging to the technical field of analytical solution calculation for two-hinged arch thrust influence lines. Based on the force method equation and the Ritter formula, the present invention configures the arch axis curve and the variable cross-section to derive a practical analytical solution for the thrust influence line of a two-hinged arch with a variable cross-section parabola and a catenary. This analytical solution can be used to obtain a relatively accurate numerical value for the thrust influence line, enabling rapid assessment of the thrust influence line in engineering projects and providing guidance for design.

Description

Technical Field

[0001] The present invention relates to the technical field of analytical solution calculation of a two-hinge arch thrust influence line, and in particular to an analytical method for a variable-section two-hinge arch thrust influence line. Background Art

[0002] Two-hinged arches have no bending moment effect at the arch foot, are less affected by concrete creep, shrinkage, and temperature, and place lower demands on the foundation structure. Therefore, two-hinged arches are widely used in actual arch bridge engineering. Due to the unique characteristics of arch structures, thrust is a key mechanical characteristic, and this thrust can be used to determine the state of the arch structure. Currently, the only way to obtain the thrust influence line of an arch is through model simulation, and analytical solutions for this effect are still lacking. Therefore, an analytical method for the thrust influence line of a two-hinged arch with a variable cross-section is proposed. Summary of the Invention

[0003] The technical problem to be solved by the present invention is how to calculate the analytical solution of the thrust influence line of a parabolic two-hinged arch with variable cross-section and a catenary two-hinged arch, and provides an analytical method for the thrust influence line of a two-hinged arch with variable cross-section.

[0004] The present invention solves the above technical problems through the following technical solutions, which include the following steps:

[0005] S1: Establish a two-hinged arch basic system and construct the corresponding load displacement and self-displacement expressions through the Ritter formula and two arch axis equations, namely the parabola equation and the catenary equation;

[0006] S2: List the force method equations according to the force method principle;

[0007] S3: Derive the analytical solution of the thrust influence line of the variable cross-section parabola and catenary two-hinged arch.

[0008] Furthermore, in step S1, the Ritter formula is as follows:

[0009]

[0010] The parabolic self-variation calculation formula is as follows:

[0011] δ 11 =16f 2 L / 15EI0

[0012] The calculation formula for parabolic load displacement is as follows:

[0013]

[0014] The calculation formula of the catenary self-deflection is as follows:

[0015] δ 11 =[f2 L / 4kEI0(m 2 -2m+1)](8km 2 +8e -k m-8e k me -2k +e 2k +4k)

[0016] The calculation formula for the load displacement corresponding to the catenary type is as follows:

[0017]

[0018] Among them, L is the half span length of the arch structure, n is the coefficient of variation of the arch rib section, f is the sagittal height, I0 and h0 are the arch crown moment of inertia and arch crown section height respectively. is the horizontal angle of the arch section, m is the arch axis coefficient, x p is the horizontal distance from the concentrated load to the arch top, and x is the horizontal distance from any section of the arch structure to the arch top.

[0019] Furthermore, in step S2, the bending moment generated by the unit force is expressed as follows:

[0020] M1=-(fy)

[0021] The bending moment generated by external load is expressed as follows:

[0022] When -L<x<x p hour,

[0023] When x p When <x<L,

[0024] Among them, M1 is the bending moment generated by the unit force in any section of the basic structure, M p is the bending moment generated by the load in the basic structure.

[0025] Furthermore, in step S2, the calculation formula of the load displacement in the lossless state is as follows:

[0026]

[0027] Where E is the elastic modulus, I is the moment of inertia, and x is the horizontal distance from any section of the arch structure to the arch top;

[0028] The calculation formula of the self-deflection in the lossless state is as follows:

[0029]

[0030] in,

[0031] Furthermore, in step S3, the analytical solution of the thrust influence line of the variable-section parabola two-hinged arch is specifically:

[0032]

[0033] Furthermore, in step S3, the analytical solution of the thrust influence line of the variable-section catenary two-hinged arch is specifically:

[0034]

[0035] Compared with the existing technology, the present invention has the following advantages: the analytical method for the thrust influence line of a variable-section two-hinged arch is based on the force method equation and the Ritter formula, and the arch axis curve and the variable section are set to derive the analytical solution of the thrust influence line of the variable-section parabola and catenary two-hinged arch with practical application value. Through this analytical solution, a relatively accurate analytical solution value of the thrust influence line can be obtained, which can achieve the effect of quickly evaluating the thrust influence line in engineering and is also of guiding significance in design. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 1 is a flow chart of a method for analyzing the thrust influence line of a variable-section two-hinge arch in the first embodiment of the present invention;

[0037] Figure 2 Schematic diagram of the basic system of two hinged arches in embodiment 1 of the present invention;

[0038] Figure 3 This is the finite element model of the two-hinge arch structure in the second embodiment of the present invention;

[0039] Figure 4 It is a schematic diagram of the parabolic arch axis with five different rise-to-span ratios in the second embodiment of the present invention.

[0040] FIG5( a ) is a comparison diagram of the thrust influence line calculated by analytical solution and finite element analysis for a parabolic two-hinge arch with a variable cross-section and a span ratio of 1 / 4 in Example 2 of the present invention;

[0041] FIG5( b ) is a comparison diagram of the analytical solution calculation and finite element results of the thrust influence line of the variable cross-section parabolic two-hinge arch at a rise-to-span ratio of 1 / 5 in Example 2 of the present invention;

[0042] FIG5( c ) is a comparison diagram of the analytical solution calculation and finite element results of the thrust influence line of the variable cross-section parabolic two-hinged arch at a rise-to-span ratio of 1 / 6 in Example 2 of the present invention;

[0043] FIG5( d ) is a comparison diagram of the thrust influence line calculated by analytical solution and finite element analysis results for a parabolic two-hinge arch with a variable cross-section and a span ratio of 1 / 7 in Example 2 of the present invention;

[0044] FIG5( e ) is a comparison diagram of the thrust influence line calculated by analytical solution and finite element analysis results for a parabolic two-hinge arch with a variable cross-section and a span ratio of 1 / 8 in Example 2 of the present invention;

[0045] FIG5( f ) is a comparison diagram of the thrust influence line calculated by the analytical solution and the finite element analysis results of the variable-section catenary two-hinge arch at a rise-to-span ratio of 1 / 4 in Example 2 of the present invention;

[0046] 5( g ) is a comparison diagram of the thrust influence line analytical solution calculation and finite element results of the variable cross-section catenary two-hinge arch at a rise-to-span ratio of 1 / 5 in Example 2 of the present invention;

[0047] 5( h ) is a comparison diagram of the thrust influence line calculated by analytical solution and finite element analysis results for the variable-section catenary two-hinge arch at a rise-to-span ratio of 1 / 6 in Example 2 of the present invention;

[0048] FIG5(i) is a comparison diagram of the thrust influence line calculated by analytical solution and finite element analysis results of the variable cross-section catenary two-hinge arch at a rise-to-span ratio of 1 / 7 in Example 2 of the present invention;

[0049] FIG5(j) is a comparison diagram of the thrust influence line analytical solution calculation and finite element results of the variable-section catenary two-hinge arch at a span ratio of 1 / 8 in Example 2 of the present invention. DETAILED DESCRIPTION

[0050] The following is a detailed description of an embodiment of the present invention. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process. However, the protection scope of the present invention is not limited to the following embodiment.

[0051] Example 1

[0052] like Figure 1 As shown, this embodiment provides a technical solution: a method for analyzing the thrust influence line of a variable-section two-hinge arch, comprising the following steps:

[0053] (1) Establish the basic system of two-hinged arches (see Figure 2 ), the corresponding load displacement and self displacement expressions are constructed through the Ritter formula and two arch axis equations, namely the parabola equation and the catenary equation;

[0054] In this step, the Ritter formula is as follows:

[0055]

[0056] The corresponding expressions of load and self-variation are as follows:

[0057] Table 1 Expressions of inflection and self-inflection

[0058]

[0059] Among them, L is the half span length of the arch structure, n is the coefficient of variation of the arch rib section, f is the sagittal height, I0 and h0 are the arch crown moment of inertia and arch crown section height respectively. is the horizontal angle of the arch section, Where m is the arch axis coefficient, x p is the horizontal distance from the concentrated load to the arch top, and x is the horizontal distance from any section of the arch structure to the arch top.

[0060] Load displacement Δ 1p : displacement in the X1 direction caused by load P; self-deflection δ 11 : The displacement along the X1 direction caused by the unit force X1=1 is often called the flexibility coefficient.

[0061] (2) List the force method equations based on the force method principle;

[0062] In this step, the bending moment generated by unit force is expressed as follows:

[0063] M1=-(fy)

[0064] The bending moment generated by external load is expressed as follows:

[0065] When -L<x<x p hour,

[0066] When x p When <x<L,

[0067] Among them, M1 is the bending moment generated by the unit force in any section of the basic structure, M p is the bending moment generated by the load in the basic structure.

[0068] In this step, the calculation formula of the load displacement in the lossless state is as follows:

[0069]

[0070] Where E is the elastic modulus, I is the moment of inertia, and x is the horizontal distance from any section of the arch structure to the arch top;

[0071] The calculation formula of the self-deflection in the lossless state is as follows:

[0072]

[0073] in,

[0074] (3) The analytical solution expressions for the thrust influence lines of the variable cross-section parabola and catenary two-hinged arch are derived.

[0075] In this step, we first substitute the two arch axis expressions, namely the parabolic and catenary expressions, based on the force method equation. At the same time, we substitute the Ritter formula into the force method equation to deduce:

[0076] Parabolic self-variation calculation formula: δ 11 =16f 2 L / 15EI0;

[0077] Parabolic load displacement calculation formula:

[0078] The calculation formula of catenary self-deflection: δ 11 =[f 2 L / 4kEI0(m 2 -2m+1)](8km 2 +8e -k m-8e k me -2k +e 2k +4k);

[0079] The calculation formula of catenary load displacement is:

[0080] Secondly, through the arch structure thrust formula F H =-Δ 1p / δ 11 ;

[0081] The analytical solution of the thrust influence line of a parabolic two-hinged arch with variable cross-section can be derived:

[0082]

[0083] Analytical solution of the thrust influence line of a variable-section catenary two-hinged arch:

[0084]

[0085] Example 2

[0086] In order to verify the accuracy of the analytical solution calculation method of the present invention, the finite element models of the variable cross-section parabola two-hinged arch structure and the variable cross-section catenary two-hinged arch structure are established, as shown in Figure 2. Figure 3 shown.

[0087] When establishing the model, the formula for the variable section height at any parabola section is as follows:

[0088]

[0089] The formula for the variable section height at any section of the catenary is as follows:

[0090]

[0091] The thrust influence line values ​​obtained from finite element simulations were compared with those obtained analytically using this method to investigate the practical accuracy of the analytical solution for the thrust influence line. The model is a two-hinged arch structure with a span of 50.934 m, a mid-span section height of 1 m, and an arch rib section width of 1 m. The arch axis coefficient m is 1.988, the arch thickness variation coefficient n is 0.4, and k is 1.31.

[0092] For this embodiment, five span ratios of 1 / 4, 1 / 5, 1 / 6, 1 / 7, and 1 / 8 are taken for both parabola and catenary arch axis types for simulation verification. Figure 4 As shown. Take the arch as the origin, horizontally to the left is positive, vertically downward is positive, starting from 25m on the right semi-axis, divide the unit into 0.5m units until 25m on the left semi-axis. The horizontal distance of each node from the arch is the load position x p .

[0093] The comparison between the analytical solution calculation results of the present invention and the finite element calculation results is shown in the following table.

[0094] Table 2 Comparison of analytical solution calculation and finite element calculation results

[0095]

[0096] (Note: The thrust measurement point is at the arch foot position)

[0097] As can be seen from the table above, when comparing the loads applied to seven typical sections for each rise-to-span ratio and listing the internal force data for the typical sections, the analytical solution obtained by the present invention is less affected by the line type. Under the same rise-to-span ratio and different line types, the difference in the thrust influence line values ​​is small. When comparing the same line type with different rise-to-span ratios, when the load acts at mid-span, the deviation between the analytical solution calculation and the finite element calculation results is the smallest. The maximum deviation between the analytical solution calculation and the finite element calculation results is within 8.5%, both of which are negative deviations (not considering the influence of axial force). The calculation deviation near the support is greater than that of the mid-span section.

[0098] Comparison of the two sets of curves in Figures 5(a)-(e) and 5(f)-(j) shows that for variable-section parabolic two-hinged arches and catenary two-hinged arches at different rise-to-span ratios, the maximum deviation between the analytical solution and the finite element results is 8.425% (for the case of a variable-section parabolic two-hinged arch with a 1 / 4 rise-to-span ratio), indicating a high degree of agreement between the curves. Comparison of the five sets of curves in Figures 5(a) and 5(f), 5(b) and 5(g), 5(c) and 5(h), 5(d) and 5(i), and 5(e) and 5(j) shows that for two-hinged arch structures with the same rise-to-span ratio but different linear shapes, the maximum thrust influence line at the arch crown is close when the analytical solution is compared with the finite element results, with a maximum deviation of 2.637% (for the case of a variable-section catenary two-hinged arch with a 1 / 4 rise-to-span ratio).

[0099] By comparing the analytical solution with the finite element calculation results, it is shown that the deviation of the thrust influence line at the arch crown is the smallest, and the deviation of the thrust influence line at the arch foot is larger. The analysis found that since the thrust influence line value at the arch crown is large, the relative deviation is small, while the thrust influence line value at the arch foot is small, so the relative deviation is large.

[0100] In summary, the analytical method for the thrust influence line of the variable-section two-hinged arch in the above embodiment proposes an analytical solution calculation method for the influence line of the variable-section parabola and catenary two-hinged arch. Compared with the finite element results, its analysis deviation is within 8.5%, which meets the engineering accuracy requirements. It can realize the rapid calculation of the thrust influence line value in bridge analysis, inspection and reinforcement, and can also serve as the basis for the design of the arch seat foundation strength of the arch structure under the action of moving loads.

[0101] Although the embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and are not to be construed as limitations on the present invention. A person skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention.

Claims

1. An analytical method for the thrust influence line of a variable cross-section two-hinged arch, characterized in that: The following steps are involved: S1: Establish a two-hinged arch basic system and construct the corresponding load displacement and self-displacement expressions through the Ritter formula and two arch axis equations, namely the parabola equation and the catenary equation; S2: List the force method equations according to the force method principle; S3: Derive the analytical solution of the thrust influence line of the variable cross-section parabola and catenary two-hinged arch; In step S1, the Ritter formula is as follows: The parabolic self-variation calculation formula is as follows: d 11 =16f 2 L / 15EI0 The calculation formula for parabolic load displacement is as follows: The calculation formula of the catenary self-deflection is as follows: δ 11 =[f 2 L / 4kEI0(m 2 -2m+1)](8km 2 +8e -k m-8e k m-e -2k +e 2k +4k) The calculation formula for the load displacement corresponding to the catenary type is as follows: Among them, L is the half span length of the arch structure, n is the coefficient of variation of the arch rib section, f is the sagittal height, I0 and h0 are the arch crown moment of inertia and arch crown section height respectively. is the horizontal angle of the arch section, m is the arch axis coefficient, x p is the horizontal distance from the concentrated load to the arch top, and x is the horizontal distance from any section of the arch structure to the arch top; In step S2, the bending moment generated by the unit force is expressed as follows: M1=-(fy) The bending moment generated by external load is expressed as follows: When -L<x<x p hour, When x p When <x<L, Among them, M1 is the bending moment generated by the unit force in any section of the basic structure, M p is the bending moment generated by the load in the basic structure; In step S2, the calculation formula of the load displacement in the lossless state is as follows: Where E is the elastic modulus, I is the moment of inertia, and x is the horizontal distance from any section of the arch structure to the arch top; The calculation formula of the self-deflection in the lossless state is as follows: in, 2. The analytical method for the thrust influence line of a variable cross-section two-hinge arch according to claim 1, characterized in that: In step S3, the analytical solution of the thrust influence line of the variable-section parabola two-hinged arch is specifically:

3. The analytical method for the thrust influence line of a variable cross-section two-hinge arch according to claim 2, characterized in that: In step S3, the analytical solution of the thrust influence line of the variable-section catenary two-hinged arch is specifically:

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