Method for quickly determining live load response of suspension bridge

Abstract

The application discloses a kind of quick determination methods of suspension bridge live load response, including steps 1, establishing the analogy relationship of main cable linear shape and simply supported beam bending moment;Step 2, determine the constant load state parameter: constant load main cable linear shape, constant load main cable horizontal component and constant load main cable unstressed length;Step 3, establish live load response analytical expression: including the analytical expression of main beam deflection or main cable deflection, and the analytical expression of main beam rotation angle;Step 4, obtain live load response result: first, according to the conservation of main cable unstressed length, determine the increment of main cable horizontal force, and then obtain live load response.The application can quickly and accurately solve the main beam deflection and rotation angle response of suspension bridge under the action of live load in practice;Consider the stiffness of main beam, can more truly reflect the structural characteristics;Based on second-order differential equation operation, can give concise analytical solution, physical meaning is also more clear, has strong universality and practicality, is convenient for engineer application.

Description

Technical Field

[0001] This invention relates to the field of bridge theoretical analysis, and in particular to a method for rapidly determining the live load response of a suspension bridge. Background Technology

[0002] Suspension bridges, as one of the bridge types with the strongest span capacity, are widely used in the global transportation sector, and their development over the past decade has shown two trends. Firstly, the span of suspension bridges is continuously increasing. The preliminary design of the Shiziyang Bridge (main span 2180m) has passed review, and construction of the Zhangjinggao Yangtze River Bridge (main span 2300m) has begun. In the future, more ultra-long-span suspension bridges with spans exceeding 2000m will appear worldwide. Secondly, ultra-long-span suspension bridges are beginning to be used in railway and high-speed rail operations. China has already built the world's first long-span high-speed railway suspension bridge—the Wufengshan Yangtze River Bridge. Furthermore, the world's first long-span dedicated railway suspension bridge—the Jinsha River Bridge—has completed its main structural construction. Undoubtedly, suspension bridges will have even broader engineering application prospects in the future.

[0003] As is well known, suspension bridges are a type of cable-stayed bridge, a flexible structure. The main girder is suspended from the main cable by suspenders, and the main cable transfers the load to the bridge towers and anchorages. Under live loads, the flexible suspension bridge will undergo significant deformation, a matter of great concern to scholars in the field. Accurately calculating the deformation response under live loads is crucial for understanding the structural characteristics and is an important foundation for the safe design of suspension bridges.

[0004] In the calculation of live load deformation of suspension bridges, one of the most important theories is deflection theory. All related research is based on the equilibrium differential equations of the main cable and main girder elements. However, solving fourth-order differential equations is quite complex, often requiring numerical methods to obtain approximate solutions. If the order of the differential equations used for solving these equations can be reduced, explicit analytical solutions can be easily obtained, significantly reducing the difficulty of the solution process.

[0005] Some scholars have discovered a parallel between the main cable alignment and the bending moment of an equivalent simply supported beam, offering a simple and effective solution method. Specifically, the main cable alignment under load can be obtained by dividing the bending moment of a simply supported beam of the same span under the same load by the horizontal force of the main cable. However, related studies have neglected the influence of the main girder, failing to consider its stiffness in resisting and distributing live loads. Railway operations demand high track smoothness, requiring strict limitations on the deformation of the main girder under live loads to prevent excessive track deformation from affecting train safety. Therefore, railway suspension bridges require high main girder stiffness. The two railway suspension bridges mentioned earlier both use steel truss girders, with the Wufengshan Yangtze River Bridge's main girder reaching a height of 16m, exhibiting considerable stiffness. Therefore, ignoring the stiffness contribution of the main girder is inappropriate and must be considered in analytical calculations. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to provide a rapid method for determining the live load response of a suspension bridge, which, while considering the contribution of the main girder stiffness, accurately and quickly solves the deformation response of the suspension bridge under live load.

[0007] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0008] A rapid method for determining the live load response of a suspension bridge includes the following steps.

[0009] Step 1: Establish the analogy between the main cable profile and the bending moment of a simply supported beam: Based on the vertical equilibrium differential equations of the main cable and the simply supported beam, the profile of the main cable under load is obtained by dividing the bending moment generated by the simply supported beam of the same span under the same load by the horizontal component of the main cable.

[0010] Step 2: Determine the dead load state parameters: The dead load state parameters include the dead load main cable shape y. q0 Horizontal component of constant load main cable H q0 and the stress-free length L of the constant load main cable u The constant load state parameters can be obtained by analyzing the load applied to the main cable.

[0011] Step 3: Establish the analytical expression for live load response: The analytical expression for live load response includes the main beam deflection w. b Or main cable deflection w c The analytical expressions for w and the main beam rotation angle θ; where w b =w c , θ=w b A concentrated live load P is applied to the main girder of a suspension bridge, causing deformation in both the main cable and the main girder. Based on the cable's shape before and after deformation caused by the concentrated live load P, the deflection w of the main girder is obtained. b Regarding the increase in horizontal force on the main cable The parsing expression.

[0012] Step 4: Obtain the live load response results: First, determine the increment of the horizontal force on the main cable based on the conservation of the stress-free length of the main cable. Next, Substituting the values ​​into the analytical expression for the live load response established in step 3, we obtain the live load response w. b =w c The analytical results for θ.

[0013] In step 1, under the same load q(x), the main cable alignment and the bending moment distribution of a simply supported beam have the following analogy:

[0014]

[0015] In the formula, y(x) is the main cable shape under the load q(x).

[0016] M(x) is the bending moment of a simply supported beam with the same span under load q(x).

[0017] H is the horizontal component of the main cable.

[0018] In step 2, the horizontal component of the constant load main cable The expression is:

[0019]

[0020] In the formula, f0 is the sag of the main cable.

[0021] q0 is the uniformly distributed dead load, which includes the self-weight of the main beam and the self-weight of the main cable.

[0022] L is the main cable span.

[0023] Constant load main cable alignment The expression is:

[0024]

[0025] In the formula, x is the distance from the point of application of the constant load to the left end point.

[0026] In step 3, the deflection w of the main beam b Or main cable deflection w c The parsing expression:

[0027]

[0028] in:

[0029]

[0030]

[0031] In the formula, x p This is the distance from the point of application of the concentrated live load P to the left endpoint.

[0032] x is any point on the main beam within the entire span.

[0033] E b It is the elastic modulus of the main beam.

[0034] I b The vertical bending moment of inertia of the main beam.

[0035] A and B are intermediate variables.

[0036] C1, C2, C3, and C4 are constant coefficients.

[0037] The solution methods for C1, C2, C3 and C4 include the following steps.

[0038] Step 3-1: Establish second-order constant coefficient differential equations: Based on the cable shape before and after deformation caused by concentrated live load P, establish the following two second-order constant coefficient differential equations:

[0039]

[0040] In the formula, The increment of the membrane force of the suspender generates the bending moment for the main cable and other simply supported beams.

[0041] Step 3-2: Introduce four boundary conditions, specifically:

[0042]

[0043]

[0044]

[0045]

[0046] In the formula, They represent the ranges in [0, x], x ... p ] and [x p Take x within the interval [L]. p The value of .

[0047] Step 3-3, Solve for C1, C2, C3 and C4: Solve for C1, C2, C3 and C4 by combining the two second-order constant coefficient differential equations from Step 3-1 and the four boundary conditions from Step 3-2.

[0048] Main cable horizontal force increment The method for determining [the value] includes the following steps.

[0049] Step 4-1: Calculate the stress-free length L of the main cable under constant load. u The specific calculation formula is as follows:

[0050]

[0051] In the formula, E c It is the elastic modulus of the main cable.

[0052] A c This refers to the cross-sectional area of ​​the main cable.

[0053] Step 4-2: Calculate the slope of the main cable profile under the combined action of dead and live loads. The specific calculation formula is as follows:

[0054]

[0055] Step 4-3: Calculate the stress-free length L of the main cable under constant live load. u The specific calculation formula is as follows:

[0056]

[0057] Step 4-4, Solve According to the conservation of the stress-free length of the main cable, that is:

[0058] L′ u =L u

[0059] By combining the formulas from steps 4-1 to 4-3, the solution can be obtained.

[0060] The present invention has the following beneficial effects: it can quickly and accurately solve the deflection and rotation response of the main beam of a suspension bridge under live load in practice; it takes into account the stiffness of the main beam, which can more realistically reflect the structural characteristics; it is based on the second-order differential equation, which can give a concise analytical solution with clearer physical meaning, and has strong versatility and practicality, making it easy for engineers to use. Attached Figure Description

[0061] Figure 1 This is a schematic diagram simulating the main cable alignment under an arbitrary distributed load q(x).

[0062] Figure 2 This is a schematic diagram simulating the bending moment of a simply supported beam under an arbitrary distributed load q(x).

[0063] Figure 3 This is a schematic diagram of the main cable under constant load in a specific embodiment.

[0064] Figure 4 This is a schematic diagram of the main cable in its deformed state in a specific embodiment.

[0065] Figure 5 This is a schematic diagram of the main beam in the deformed state in a specific embodiment. Detailed Implementation

[0066] The present invention will now be described in further detail with reference to the accompanying drawings and specific preferred embodiments.

[0067] The present invention will be further illustrated below with reference to specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. After reading the present invention, any modifications of the present invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.

[0068] like Figure 1 As shown, a method for rapidly determining the live load response of a suspension bridge includes the following steps.

[0069] Step 1: Establish the analogy between the main cable alignment and the bending moment of a simply supported beam.

[0070] Based on the vertical equilibrium differential equations of the main cable and the simply supported beam, the bending moment generated by the simply supported beam of the same span under the same load is divided by the horizontal component of the main cable to obtain the cable's shape under load. This is the theoretical basis for subsequent calculations of the main cable's shape.

[0071] like Figure 1 and Figure 2 As shown, under the same load q(x), the bending moment distribution of the main cable and the simply supported beam have the following analogy:

[0072]

[0073] In the formula, y(x) is the main cable shape under the load q(x).

[0074] M(x) is the bending moment of a simply supported beam with the same span under load q(x).

[0075] H is the horizontal component of the main cable.

[0076] Step 2: Determine the dead load state parameters: The dead load state parameters include the dead load main cable profile. Horizontal component of constant load main cable and the stress-free length L of the constant load main cable u The constant load state parameters can be obtained by analyzing the load applied to the main cable.

[0077] Under constant load conditions, the constant load acting on the main cable is uniformly distributed; therefore, the main cable is a parabola, such as... Figure 3 As shown.

[0078] Under the same uniformly distributed load, the bending moment generated in a simply supported beam with span L is... This can be expressed as:

[0079]

[0080] In the formula, q0 is the uniformly distributed dead load, which includes the self-weight of the main beam and the self-weight of the main cable; L is the main cable span.

[0081] Therefore, the main cable profile under constant load conditions This can be expressed as:

[0082]

[0083] In the formula, x is the distance from the point of application of the constant load to the left end point;

[0084] The horizontal component of the main cable under constant load is calculated using the following formula:

[0085]

[0086] In the formula, f0 is the sag of the main cable.

[0087] The stress-free length L of the main cable under constant load u It can be calculated by subtracting the elastic elongation from the length of the parabola:

[0088]

[0089] In the formula, E c and A c These are the elastic modulus and cross-sectional area of ​​the main cable, respectively.

[0090] Since the main beam under constant load is uniformly supported by the suspender membrane, it can be approximated as being in a stress-free state. Therefore, it is not necessary to focus on the stress on the main beam under constant load.

[0091] Step 3: Establish the analytical expression for live load response: The analytical expression for live load response includes the main beam deflection w. b Or main cable deflection w c The analytical expressions for w and the main beam rotation angle θ; where w b =w c , When a concentrated live load P is applied to the main girder of a suspension bridge, both the main cable and the main girder deform. Based on the shape of the main cable before and after deformation caused by the concentrated live load P, the deflection w of the main girder is obtained. b Regarding the increase in horizontal force on the main cable The parsing expression.

[0092] The aforementioned main beam deflection w b Or main cable deflection w c The parsing expression:

[0093]

[0094] in:

[0095]

[0096]

[0097] In the formula, x p This is the distance from the point of application of the concentrated live load P to the left endpoint.

[0098] x is any point on the main beam within the entire span.

[0099] E bIt is the elastic modulus of the main beam.

[0100] I b The vertical bending moment of inertia of the main beam.

[0101] A and B are intermediate variables.

[0102] C1, C2, C3 and C4 are constant coefficients. The preferred solution method includes the following steps.

[0103] Step 3-1: Establish a second-order differential equation with constant coefficients.

[0104] When the main girder is subjected to a concentrated live load P, both the main cable and the main girder deform, and the tension of the suspender membrane also changes, such as... Figure 4 and Figure 5 As shown.

[0105] q h The increase in the tension of the suspender membrane caused by live load has equal magnitude and opposite direction of effect on the main cable and main beam. Therefore, the increase in the suspender membrane force also has equal magnitude and opposite direction of bending moments on the main cable and the equivalent simply supported beam and main beam, respectively.

[0106]

[0107] In the formula, The bending moment generated in the main beam by the increment of the membrane force of the hanger; The increase in the membrane force of the suspender rod results in the bending moment generated by the main cable and other simply supported beams.

[0108] Deformed main cable profile It can be obtained from the current bending moment and horizontal force, that is

[0109]

[0110] In the formula, This represents the increase in the horizontal force component of the main cable caused by live load.

[0111] Vertical deformation of the main cable caused by live load w c This can be obtained from the main cable profile before and after deformation:

[0112]

[0113] As mentioned earlier, under constant load, the self-weight of the main beam is balanced by the uniformly distributed tension of the suspender membrane. Therefore, the deformed main beam is actually a simply supported beam with no self-weight, subjected to concentrated live load and increased tension of the suspender membrane within its span; the bending moment M of the deformed main beam is... b It can be obtained by adding the two parts:

[0114]

[0115] In the formula, M b,P The bending moment of the main beam caused by the concentrated live load is a piecewise function; x p This is the distance from the point of application of the live load to the left endpoint.

[0116] For the main beam, the bending moment M b and deflection w b The following relationship exists between them:

[0117] M b =-E b I b w b "

[0118] In the formula, E b and I b These are the elastic modulus and vertical bending moment of inertia of the main beam, respectively.

[0119] Since the vertical deformation of the main beam and the main cable is the same, the above formula can be further written in the following form:

[0120] M b =-E b I b w c "

[0121] Expanding, we get:

[0122]

[0123] This actually establishes two second-order differential equations with constant coefficients, with the general solution as follows:

[0124]

[0125] In the formula, The increment of the membrane force of the suspender generates the bending moment for the main cable and other simply supported beams.

[0126] Step 3-2: Introduce four boundary conditions, specifically:

[0127]

[0128]

[0129]

[0130]

[0131] In the formula, They represent the ranges in [0, x], x ... p ] and [x p Take x within the interval [L]. p The value of .

[0132] Step 3-3, Solve for C1, C2, C3 and C4: Solve for C1, C2, C3 and C4 by combining the two second-order constant coefficient differential equations from Step 3-1 and the four boundary conditions from Step 3-2.

[0133] The four constant coefficients can be solved:

[0134]

[0135]

[0136]

[0137]

[0138] This provides an understanding of the bending moment generated by the increased membrane force of the suspender in a simply supported beam, such as the main cable. The parsing expression.

[0139] Taking the first derivative of the deflection, we obtain the analytical expression for the rotation angle of the main beam:

[0140]

[0141] Step 4: Obtain the live load response results: First, determine the increment of the horizontal force on the main cable based on the conservation of the stress-free length of the main cable. The above derivation has only one unknown, namely the increase in the horizontal component of the main cable force caused by live load. It remains constant throughout the entire main cable and is independent of x, so it is treated as a constant during differentiation; then, Substituting the values ​​into the analytical expression for the live load response established in step 3, we obtain the live load response w. b =w c The analytical results for θ.

[0142] The above-mentioned horizontal force increment of the main cable The method for determining [the value] includes the following steps.

[0143] Step 4-1: Calculate the stress-free length L of the main cable under constant load. u The specific calculation formula is shown in step 2.

[0144] Step 4-2: Calculate the slope of the main cable profile under the combined action of dead and live loads. The specific calculation formula is as follows:

[0145]

[0146] Step 4-3: Calculate the stress-free length L of the main cable under constant live load. u The specific calculation formula is as follows:

[0147]

[0148] Step 4-4, Solve According to the conservation of the stress-free length of the main cable, that is:

[0149] L′ u =L u

[0150] By combining the formulas from steps 4-1 to 4-3, the solution can be obtained.

[0151] Once the solution is obtained based on the conservation of the stress-free length of the main cable... By obtaining the value of , we can obtain the deflection and rotation values ​​at any point on the main beam within the entire span.

[0152] The preferred embodiments of the present invention have been described in detail above. However, the present invention is not limited to the specific details in the above embodiments. Within the scope of the technical concept of the present invention, various equivalent transformations can be made to the technical solutions of the present invention, and these equivalent transformations all fall within the protection scope of the present invention.

Claims

1. A method for rapidly determining the live load response of a suspension bridge, characterized in that: Includes the following steps: Step 1: Establish the analogy between the main cable profile and the bending moment of the simply supported beam: Based on the vertical equilibrium differential equations of the main cable and the simply supported beam, the profile of the main cable under load is obtained by dividing the bending moment generated by the simply supported beam of the same span under the same load by the horizontal component of the main cable. Step 2: Determine the dead load state parameters: The dead load state parameters include the dead load main cable profile. Horizontal component of constant load main cable and the stress-free length L of the constant load main cable u The constant load state parameters can be obtained by analyzing the load applied to the main cable. Constant load main cable alignment The expression is: In the formula, x is the distance from the point of application of the constant load to the left end point; Horizontal component of constant load main cable The expression is: In the formula, f0 is the sag of the main cable; q0 is the uniformly distributed dead load, including the self-weight of the main beam and the self-weight of the main cable; L is the main cable span; Step 3: Establish the analytical expression for live load response: The analytical expression for live load response includes the main beam deflection w. b Or main cable deflection w c The analytical expressions for w and the main beam rotation angle θ; where w b =w c , θ=w b ′; where w b ′ is the main beam deflection w b The derivative; When a concentrated live load P is applied to the main girder of a suspension bridge, both the main cable and the main girder deform. Based on the shape of the main cable before and after deformation caused by the concentrated live load P, the deflection w of the main girder is obtained. b Regarding the increase in horizontal force on the main cable The parsing expression; Main beam deflection w b Or main cable deflection w c The parsing expression: in: In the formula, x p The distance from the point of application of the concentrated live load P to the left end point; x is any point on the main beam within the entire span; E b The elastic modulus of the main beam; I b The vertical bending moment of inertia of the main beam; A and B are intermediate variables; C1, C2, C3, and C4 are constant coefficients; Step 4: Obtain the live load response results: First, determine the increment of the horizontal force on the main cable based on the conservation of the stress-free length of the main cable. Next, Substituting the values ​​into the analytical expression for the live load response established in step 3, we obtain the live load response w. b =w c The analytical results for θ.

2. The method for rapidly determining the live load response of a suspension bridge according to claim 1, characterized in that: In step 1, under the same load q(x), the main cable alignment and the bending moment distribution of a simply supported beam have the following analogy: In the formula, y(x) is the main cable shape under load q(x); M(x) is the bending moment of a simply supported beam with the same span under load q(x); H is the horizontal component of the main cable.

3. The method for rapidly determining the live load response of a suspension bridge according to claim 1, characterized in that: The solution methods for C1, C2, C3, and C4 include the following steps: Step 3-1: Establish second-order constant coefficient differential equations: Based on the cable shape before and after deformation caused by concentrated live load P, establish the following two second-order constant coefficient differential equations: In the formula, The increase in the membrane force of the suspender rod generates the bending moment for the main cable and other simply supported beams; Step 3-2: Introduce four boundary conditions, specifically: In the formula, They represent the ranges in [0, x], x ... p ] and [x p Take x within the interval [L]. p The value; Step 3-3, Solve for C1, C2, C3 and C4: Solve for C1, C2, C3 and C4 by combining the two second-order constant coefficient differential equations from Step 3-1 and the four boundary conditions from Step 3-2.

4. The method for rapidly determining the live load response of a suspension bridge according to claim 1, characterized in that: Main cable horizontal force increment The method for determining this includes the following steps: Step 4-1: Calculate the stress-free length L of the main cable under constant load. u The specific calculation formula is as follows: In the formula, E c The elastic modulus of the main cable; A c The cross-sectional area of ​​the main cable; For the constant load main cable configuration The derivative; Step 4-2: Calculate the slope of the main cable profile under the combined action of dead and live loads. The specific calculation formula is as follows: Step 4-3: Calculate the stress-free length L of the main cable under constant live load. u The specific calculation formula is as follows: Step 4-4, Solve According to the conservation of the stress-free length of the main cable, that is: L' u =L u By combining the formulas from steps 4-1 to 4-3, the solution can be obtained.

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