A method for determining the cable strand force of the anchor span in the completed state of a suspension bridge

Abstract

The present invention discloses a method for determining the cable force of the anchor span of a suspension bridge in the completed bridge state, which includes the following steps: First, clarify the positional relationship and overall alignment between the cable strands of the anchor span and the saddle; then determine the cable force distribution pattern where the cable forces of each cable strand in the anchor span at the tangent point with the saddle are equal; next, conduct a segmented study on the cable strands of the anchor span, and successively determine the explicit expressions of the elevation difference, transverse distance, and longitudinal distance of each section of the cable strand. Establish a control equation set based on elevation difference closure, transverse distance closure, longitudinal distance closure, saddle force balance, identical cable strand axial forces, and a certain cable strand slope, and use the method of solving by linear programming to solve the relevant parameters of each cable strand, so as to determine the cable force of the cable strands in the anchor span of the suspension bridge. Based on the force analysis and geometric coordination relationship of the cable strands and the saddle in the anchor span, the present invention is convenient to calculate, has strong practicability, and can be used in the design stage of the anchor span part of the suspension bridge.

Description

Technical Field

[0001] The present invention relates to the field of bridge analysis theory, and particularly to a method for determining the cable strand force of the anchor span in the completed bridge state of a suspension bridge. Background Art

[0002] Due to its reasonable mechanical properties, strong spanning ability, and beautiful appearance structure, the suspension bridge has become one of the preferred bridge types for extra-long-span bridges. The completion and opening to traffic of the 1915 Çanakkale Bridge in Turkey has updated the suspension bridge span record to 2023 m. The Shiziyang Bridge with a main span of 2180 m and the Zhangjinggao Yangtze River Bridge with a main span of 2300 m are under construction in China. It can be seen that with the development of design theory, calculation methods, and construction techniques, more long-span suspension bridges will be built.

[0003] At present, the main cables of suspension bridges are mostly constructed by the PPWS method. Such main cables are composed of several cable strands anchored to the anchor blocks. During the design, construction, and service life of a suspension bridge, it is crucial to accurately calculate and control the cable strand tension. During the design process of a suspension bridge, the catenary exact analytical solution is used to calculate the main cable. During the calculation of the completed bridge state, the main cables of the middle and side spans are calculated as a whole. The anchor span is the transition section between the main cable system and the anchorage system. The main cable passes through the deflection and dispersion of the saddle to divide one main cable into multiple cable strands and anchor them to the anchor block. Therefore, the anchor span needs to be calculated strand by strand considering the strand dispersion. Although the anchor span only accounts for a very small part of the entire structure of a suspension bridge, due to its relationship with the rationality of the design of the saddle and the anchorage system and the accuracy of the calculation of the cutting length of the main cable strands, the analysis of the anchor span cable strands is an important content in the design of a suspension bridge. The analysis of the anchor span cable strands not only needs to calculate the spatial orientation of the cable strands to provide an accurate design basis for the design of the saddle and the anchor block, but more importantly, it needs to accurately calculate the anchoring tension of the cable strands in the completed bridge state to ensure the safety of the overall structure of the suspension bridge and the alignment and internal forces in the completed bridge state meet the design requirements.

[0004] During the analysis process of the anchor span cable strands, the positional relationship between the saddle and the cable strands will inevitably be involved. The saddle is a complex spatial body composed of multiple arcs with different radii and having a flat bend, and it can support the main cable system. The main cable strands can smoothly change the direction and disperse after passing through the saddle, so as to divide one main cable into multiple cable strands and anchor them to the anchor block. Therefore, attention should be paid to the fact that the cable strands are dispersed when analyzing the anchor span cable strands. After passing through the flat bend and vertical bend in the saddle groove, the cable strands leave the saddle and form a spatial catenary under the action of gravity. It can be seen that the anchor span cable strands are dispersed spatial cable strands with a relatively complex alignment, and the analysis and calculation process of the anchor span cable strands is very complicated.

[0005] At present, when designing a suspension bridge, the calculation of the cable strands in the anchor span is generally simplified, and the cable strands are still combined and treated as a single main cable. This treatment does not consider the differences in the spatial positions of the cable strands, making it difficult to accurately calculate the cable force distribution of the cable strands in the anchor span during the construction stage or in the completed bridge state, bringing blindness to the construction and control and reducing the safety factor of the main cable. Even if some studies consider the geometric position of the cable strands in detail, the iterative algorithm is not easy to converge and the accuracy is difficult to guarantee. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to provide a method for determining the cable force of the cable strands in the anchor span of a suspension bridge in the completed bridge state in view of the above-mentioned deficiencies of the prior art. The method for determining the cable force of the cable strands in the anchor span of a suspension bridge in the completed bridge state can conveniently and quickly calculate the cable force of the cable strands in the anchor span of a suspension bridge, with high calculation accuracy and low cost.

[0007] To solve the above technical problem, the technical solution adopted by the present invention is:

[0008] A method for determining the cable force of the cable strands in the anchor span of a suspension bridge in the completed bridge state, comprising the following steps.

[0009] Step 1: Define the direction of the cable strands in the anchor span: After the main cable in the side span passes through the saddle groove of the cable saddle, it is divided into n cable strands in the anchor span through the turning and dispersion of the cable saddle and is anchored to the anchor block.

[0010] The bottom of the saddle groove of the cable saddle has a horizontal bend arc and a vertical bend arc from the side span to the anchor span direction.

[0011] The main cable in the side span enters the cable saddle in a tangential manner, and the tangent point is the main cable tangent point S, and the angle between the tangent line and the longitudinal bridge direction is α.

[0012] The i-th cable strand in the anchor span leaves the horizontal bend arc in a tangential manner, and the tangent point is the horizontal bend tangent point P. The angle between the horizontal bend tangent line at the horizontal bend tangent point P and the longitudinal bridge direction is γ i ; the angle between the vertical bend tangent line at the horizontal bend tangent point P and the longitudinal bridge direction is β i ′.

[0013] The i-th cable strand in the anchor span leaves the vertical bend arc in a tangential manner, and the tangent point is the vertical bend tangent point Q, and the angle between the tangent line and the longitudinal bridge direction is β i 。

[0014] The i-th cable strand in the anchor span forms an anchoring point A on the anchor block; where 1 ≤ i ≤ n.

[0015] Step 2: Determine the cable force distribution pattern of the cable strands in the anchor span; on the premise that the actions of the cable strands in the anchor span and the main cable in the side span on the cable saddle satisfy the balance condition of the cable saddle, assume that the cable force T i at the tangent point of each cable strand and the cable saddle is equal.

[0016] Step 3. Segmental study of the anchor span cable strands: Each anchor span cable strand is divided into three segments, namely the AQ segment, the QP segment, and the PS segment; the elevation difference, the transverse distance, and the longitudinal distance of the three segments of each anchor span cable strand are all expressed as functions of H ai , a i , l ai , γ i , β i ′ and β i ; where H ai is the horizontal component force of the i-th anchor span cable strand; a i is the catenary parameter of the i-th anchor span cable strand; l ai is the longitudinal distance of the i-th anchor span cable strand from the anchorage point A to the vertical bend tangent point Q.

[0017] Step 4. Determine 5n basic unknowns: Since β i is a function of ai, therefore, H ai , a i , l ai , γ i and β i ′, a total of 5n unknowns are all regarded as basic unknowns.

[0018] Step 5. Determine 5n control equations: According to the elevation closure, the transverse distance closure, the longitudinal distance closure, the same axial force, the certain slope of the AP segment cable strand, and the overall force balance of the dispersion saddle of each anchor span cable strand, 5n control equations including the 5n basic unknowns determined in Step 4 are established.

[0019] Step 6. Solve the 5n basic unknowns: Use the programming solver method to solve the 5n control equations established in Step 5, so as to obtain the values of the 5n basic unknowns.

[0020] Step 7. Determine the cable force T i of the anchor span cable strand: Substitute the basic unknowns a i and H ai corresponding to any one of the anchor span cable strands obtained by solving in Step 6 into the following calculation formula:

[0021] T i = H ai / cosβ i

[0022] Where:

[0023] β i = tan -1 (sinha i )

[0024] Thus, the cable force T i of the anchor span cable strand is obtained.

[0025] In Step 3, the transverse distance Δy of the AQ section of the i-th cable strand in the anchor span i and the elevation difference Δz i are respectively:

[0026] Δy i = l ai tanγ i (3 - 5)

[0027]

[0028] Where:

[0029]

[0030] In the formula, c i is a parameter of the catenary equation, unit: m; q is the self-weight per unit length of the cable strand in the anchor span, kN / m.

[0031] In Step 1, the horizontal bending circular arc of the cable saddle is located on the left or right side of each saddle groove of the cable saddle; the vertical bending circular arc of the cable saddle is located on the vertical plane of each saddle groove of the cable saddle; assuming that the vertical bending circular arc has four sections from the side span to the anchor span direction, which are marked as Circular Arc 1, Circular Arc 2, Circular Arc 3 and Circular Arc 4 in sequence; among them, the horizontal bending tangent point P and the vertical bending tangent point Q are both located on Circular Arc 4, and the vertical bending tangent point Q is closer to the anchorage point A; then in Step 3, the elevation difference ΔZ QPi , longitudinal distance ΔX QPi and transverse distance ΔY QPi of the QP section of the i-th cable strand in the anchor span are respectively:

[0032] ΔZ QPi = r4(cosβ i ' - cosβ i ) (3 - 7)

[0033] ΔX QPi = r4(sinβ i - sinβ i ') (3 - 10)

[0034] ΔY QPi = r4(sinβ i - sinβ i ')tanγ i (3 - 11)

[0035] In the formula, r4 is the radius of Circular Arc 4 in the cable saddle.

[0036] In Step 3, for simplifying the calculation, the elevation difference, transverse distance and longitudinal distance of the PS section of the i-th cable strand in the anchor span are converted into the elevation difference ΔZ PJi , transverse distance ΔXPJi and the longitudinal distance ΔY PJi , specifically, they are as follows:

[0037] ΔZ PJi = -(ΔZ a,1 +ΔZ a,2 +ΔZ a,3 +ΔZ ai,4 ) + Z C1 -Z J (3 - 12)

[0038] ΔX PJi = (ΔX a,1 +ΔX a,2 +ΔX a,3 +ΔX ai,4 ) + X C1 -X J (3 - 13)

[0039]

[0040] In the formula, ΔZ a,1 , ΔZ a,2 , ΔZ a,3 and ΔZ ai,4 are the elevation differences between the center of the first arc and the center of the second arc, the center of the second arc and the center of the third arc, the center of the third arc and the center of the fourth arc, and the center of the fourth arc and the tangent point P of the flat bend, respectively;

[0041] ΔX a,1 , ΔX a,2 , ΔX a,3 and ΔX ai,4 are the horizontal distances between the center of the first arc and the center of the second arc, the center of the second arc and the center of the third arc, the center of the third arc and the center of the fourth arc, and the center of the fourth arc and the tangent point P of the flat bend, respectively.

[0042] X C1 and Z C1 are the abscissa and ordinate of the center C1 of the first arc in the global coordinate system, respectively, and both are known values.

[0043] X J , Y J and Z J are the abscissa, vertical coordinate, and ordinate of the rotation center J of the saddle in the global coordinate system, respectively, and both are known values.

[0044] Y C2 is the vertical coordinate of the center C2 of the flat bend arc in the global coordinate system, and it is a known value;

[0045] r5 is the radius of the flat bend arc; φ is the inclination angle of the saddle in the completed bridge state, and it is a known value.

[0046] ΔZai,4 and ΔX ai,4 are both functions of the basic unknown β i ′, and the specific expressions are as follows:

[0047] ΔZ ai,4 = r4cosβ i ′(3 - 14d)

[0048] ΔX ai,4 = r4sinβ i ′(3 - 15d).

[0049] In step 5, the method for determining the 5n control equations includes the following steps:

[0050] Step 51: According to the three conditions of the elevation closure, transverse distance closure, and longitudinal distance closure of the n anchor - span cable strands, establish the following 3n control equations:

[0051] Δz i +ΔZ QPi +ΔZ PJi =ΔZ AJi (5 - 1)

[0052] Δy i +ΔY QPi +ΔY PJi =ΔY AJi (5 - 2)

[0053] l ai +ΔX QPi +ΔX PJi =ΔX AJi (5 - 3)

[0054] In the formula, ΔZ AJi , ΔY AJi and ΔX AJi are respectively the elevation difference, transverse distance, and longitudinal distance between the i - th anchor - span cable strand at the anchorage point A and the rotation center J of the saddles, and are all known values.

[0055] Step 52: According to the equality of the moments of the cable force at the tangent point of the anchor - span cable strand and the cable force at the tangent point of the side - span main cable about the rotation center point, establish the following 1 control equation:

[0056] ΣM ai +M P =M s (5 - 4)

[0057] Where:

[0058] ∑M ai =∑(T i cosβ icosγ i ·e 1,i +T i sinβ i ·e 2,i ) (5-5)

[0059] M s =H·e3 + Htanα·e4 (5-6)

[0060] M P =G·e5 (5-7)

[0061] Wherein, M ai is the bending moment of the cable force of the i-th anchor span cable strand on the saddle, M s is the bending moment of the main cable force in the side span on the saddle, M P is the bending moment generated by the self-weight of the saddle.

[0062] e 1,i and e 2,i are respectively the moment arms of the horizontal component force and the vertical component force of the i-th anchor span cable strand at the vertical bending tangent point Q with respect to the rotation center J of the saddle, and are both functions of the basic unknown β i .

[0063] e3 and e4 are respectively the moment arms of the horizontal component force and the vertical component force of the main cable in the side span at the main cable tangent point S with respect to the rotation center J of the saddle, and are both known fixed values.

[0064] e5 is the gravity moment arm of the saddle and is a known fixed value.

[0065] Step 53: According to the cable force distribution pattern of the anchor span, establish the following n - 1 control equations:

[0066]

[0067] Wherein, T1, T2, T3, T i-1 , T n-1 and T n are respectively the cable forces of the 1st, 2nd, 3rd, (i - 1)-th, i-th, (n - 1)-th and n-th anchor span cable strands.

[0068] Step 54: According to the constant slope of the cable strand in the AP section, establish the following n control equations:

[0069]

[0070] Among them:

[0071] ΔY APi =Δy i +ΔY QPi (5-13)

[0072]

[0073] In the formula, ΔY APi is the transverse distance of the i-th cable strand in the anchor span from the anchor point A to the tangent point P of the horizontal curve.

[0074] ΔX APi is the longitudinal distance of the i-th cable strand in the anchor span from the anchor point A to the tangent point P of the horizontal curve.

[0075] ΔX C2J is the longitudinal distance of the i-th cable strand in the anchor span from the center C2 of the horizontal curve arc to the rotation center J of the cable saddle, which is a known value.

[0076] In step 52, the expressions of e 1,i and e 2,i are respectively:

[0077]

[0078]

[0079] Where:

[0080] ΔZ ai,4 ′ = r4cosβ i (5 - 9)

[0081] ΔX ai,4 ′ = r4sinβ i (5 - 10)

[0082] In the formula, ΔZ ai,4 ′ and ΔX ai,4 ′ are respectively the height difference and the horizontal distance between the center of the four - center arc and the vertical curve tangent point Q.

[0083] R1 and R2 are respectively the distances from the IP point of the cable saddle to the center C1 of the first arc and the rotation center J of the cable saddle.

[0084] And are respectively the angles between the connecting lines of the IP point of the cable saddle with the center C1 of the first arc and the rotation center J of the cable saddle and the vertical line.

[0085] In step 6, the method of using the Solver method to solve 5n basic unknowns includes the following steps:

[0086] Step 61: Establish an error - function equation f j : The error - function equation f j is the difference between the left - hand side and the right - hand side of the j - th control equation in step 5; where, 1 ≤ j ≤ 5n.

[0087] Step 62: Establish the following objective function:

[0088]

[0089] Step 63: On the premise of making the objective function established in Step 62 infinitely close to 0, use the nonlinear GRG method to perform programming and solution on the objective function, so as to obtain the values of 5n basic unknowns at one time.

[0090] The present invention has the following beneficial effects: The present invention can quickly and accurately solve the cable force of the cable strands in the anchor span of a suspension bridge in practice; it is completely based on the geometric coordination of the anchor span structure and the static force analysis of the saddle, with clear ideas and more definite physical meanings, and has strong versatility and practicability, and can be used in the design stage of the anchor span part of a suspension bridge. Description of the Drawings

[0091] Figure 1 Shows a three-dimensional simulation schematic diagram of the main cable in the side span being dispersed and turned by the saddle.

[0092] Figure 2 Is a simplified elevation view of the saddle in a specific embodiment.

[0093] Figure 3 Is a simplified plan view of the saddle in a specific embodiment.

[0094] Figure 4 Is an elevation view of the cable strand position in a specific embodiment.

[0095] Figure 5 Is a plan view of the cable strand position in a specific embodiment.

[0096] Figure 6 Is an elevation view of the coordinate system in a specific embodiment.

[0097] Figure 7 Is a plan view of the coordinate system in a specific embodiment.

[0098] Figure 8 Is a schematic diagram of the geometric relationship of the saddle in a specific embodiment.

[0099] Figure 9 Is a schematic diagram of the force exerted by the cable strand on the saddle in a specific embodiment.

[0100] Figure 10 Is a schematic diagram of the relevant parameters of the connection line between the IP point of the saddle and the center C1 of the first arc in a specific embodiment.

[0101] Figure 11 Is a schematic diagram of the relevant parameters of the connection line between the IP point of the saddle and the rotation center J of the saddle in a specific embodiment.

[0102] Figure 12Schematic diagram of the gravity action of the cable saddle and its lever arm in a specific embodiment. Specific embodiments

[0103] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific preferred embodiments.

[0104] In the description of the present invention, it should be understood that the orientation or positional relationship indicated by terms such as "left side", "right side", "upper part", "lower part", etc. is based on the orientation or positional relationship shown in the accompanying drawings, and is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation. "First", "second", etc. do not represent the importance of the components, so they cannot be understood as limitations to the present invention. The specific dimensions adopted in this embodiment are only for illustrating the technical solution and do not limit the protection scope of the present invention.

[0105] As Figure 1 shown, after the main cable of the side span passes through the saddle groove of the cable saddle, through the turning and dispersion of the cable saddle, it is divided into n anchor span cable strands and anchored to the anchor block.

[0106] As Figure 2 and Figure 3 shown, the bottom of the saddle groove of the cable saddle has a horizontal bend arc and a vertical bend arc from the side span to the anchor span direction.

[0107] As Figure 3 shown, the horizontal bend arc is located on the left or right side of each saddle groove of the cable saddle; the radius of the horizontal bend arc is set to r5.

[0108] As Figure 2 shown, the vertical bend arc is located on the vertical surface of each saddle groove of the cable saddle; in this embodiment, it is assumed that the vertical bend arc has four segments from the side span to the anchor span direction, which are sequentially marked as arc one, arc two, arc three, and arc four. The radii of arc one, arc two, arc three, and arc four are r1, r2, r3, and r4 respectively, and are all known values. As an alternative, the vertical bend arc of the cable saddle can also be divided into three segments or five segments, etc. Further, the cable saddle has a rotation center J.

[0109] Figure 2 In, θ0 represents the angle between the radius of the first segment of the cable saddle (side span side) and the vertical line, rad; θ1, θ2, θ3, and θ4 respectively represent the central angles corresponding to arc one, arc two, arc three, and arc four.

[0110] A method for determining the force of the anchor span cable strand in the completed state of a suspension bridge includes the following steps.

[0111] Step 1: Determine the direction of the anchor span cable strand

[0112] As Figure 4 and Figure 5As shown, the main cable of the side span enters the saddle tangentially, and the tangent point is the main cable tangent point S. The included angle between the tangent line and the longitudinal bridge direction is α. Figure 4 Point U in Figure 4 indicates that the wire strands start to bend horizontally to the left and right from point U.

[0113] The i-th wire strand of the anchor span leaves the horizontal bending arc tangentially, and the tangent point is the horizontal bending tangent point P. The included angle between the horizontal bending tangent line at the horizontal bending tangent point P and the longitudinal bridge direction is γ i ; the included angle between the vertical bending tangent line at the horizontal bending tangent point P and the longitudinal bridge direction is β i ′.

[0114] The i-th wire strand of the anchor span leaves the vertical bending arc tangentially, and the tangent point is the vertical bending tangent point Q. The included angle between the tangent line and the longitudinal bridge direction is β i .

[0115] The i-th wire strand of the anchor span forms an anchorage point A on the anchor block; where 1 ≤ i ≤ n.

[0116] The above-mentioned horizontal bending tangent point P and vertical bending tangent point Q are both located on arc four, and the vertical bending tangent point Q is closer to the anchorage point A.

[0117] Step 2: Determine the cable force distribution pattern of the anchor span wire strands

[0118] It should be ensured that the actions of the anchor span wire strands and the main cable of the side span on the saddle meet the balance conditions of the saddle, that is, the moments of the cable forces at the tangent points of the anchor span wire strands and the cable forces at the tangent point of the main cable of the side span about the center point of the rotation axis are equal. On this premise, the reasonable distribution pattern of the anchor span cable force is that the cable forces T i at the tangent points of each wire strand with the saddle are equal.

[0119] Step 3: Conduct a sectional study on the anchor span wire strands

[0120] Each wire strand of the anchor span is divided into three sections, namely the AQ section, the QP section, and the PS section; the height differences, transverse bridge distances, and longitudinal bridge distances of the three sections of each wire strand of the anchor span are all expressed as expressions about H ai , a i , l ai , γ i , β i ′ and β i ; where H ai is the horizontal component force of the i-th wire strand of the anchor span; a i is the catenary parameter of the i-th wire strand of the anchor span; l ai is the longitudinal bridge distance of the i-th wire strand of the anchor span from the anchorage point A to the vertical bending tangent point Q.

[0121] Before the study, a coordinate system is established first, and the coordinate system includes a global coordinate system and a local coordinate system of the anchor span wire strands.

[0122] As Figure 6 and Figure 7As shown in the figure, the overall coordinate system: with the rotation center J of the spreader saddle as the origin, the X-axis is horizontally to the left along the longitudinal bridge direction, the Z-axis is vertically downward, and the Y-axis is horizontally outward along the transverse bridge direction.

[0123] The local coordinate system of the i-th anchor span cable strand: with the vertical bending tangent point Q as the origin, the x-axis is horizontally to the left along the longitudinal bridge direction, the z-axis is vertically downward, and the y-axis is horizontally outward along the transverse bridge direction.

[0124] Step 31. In the local coordinate system of the i-th anchor span cable strand, the linear equation expression of the first segment of the i-th anchor span cable strand is

[0125] y = x tanγ i (3-1)

[0126]

[0127] Where:

[0128]

[0129] In the formula, γ i is the flat bending tangent angle of the i-th anchor span cable strand at the spreader saddle, in rad, that is, the angle between the plane where the i-th anchor span cable strand is located after passing through the flat bending tangent point P and the xoz plane; when the split cable strand is on the positive half-axis of the y-axis, γ i takes a positive value, otherwise it takes a negative value.

[0130] a i 、b i and c i are all parameters of the catenary equation. a i is dimensionless and is the basic unknown quantity; b i and c i both have the dimension of m.

[0131] H ai is the horizontal component force of the i-th anchor span cable strand, unit: kN; q is the self-weight per unit length of the anchor span cable strand, kN / m.

[0132] According to the boundary conditions: when x = 0, z = 0, then there is:

[0133] b i =-c i cosh a i (3-3)

[0134] Substituting Equation (3-3) into Equation (3-2) gives

[0135]

[0136] The transverse distance Δy i and the elevation difference Δz i of the AQ section of the i-th anchor span cable strand are respectively:

[0137] Δy i = l ai tanγ i (3 - 5)

[0138]

[0139] Step 32. Under the overall coordinate system of the anchor - span cable strands, the elevation difference ΔZ QPi of the QP section of the i - th anchor - span cable strand is expressed as:

[0140] ΔZ QPi = r4(cosβ i ' - cosβ i ) (3 - 7)

[0141] According to the geometric relationship at the catenary tangent point, there is

[0142]

[0143] β i = tan -1 (sinha i ) (3 - 9)

[0144] The longitudinal distance ΔX QPi of the QP section of the i - th anchor - span cable strand is expressed as:

[0145] ΔX QPi = r4(sinβ i - sinβ i ) (3 - 10)

[0146] The transverse distance ΔY QPi of the QP section of the i - th anchor - span cable strand is expressed as

[0147] ΔY QPi = r4(sinβ i - sinβ i ')tanγ i (3 - 11)

[0148] Step 33. It is rather troublesome to determine the tangent point S of the main cable on the side - span side of the third section PS of the anchor - span cable strand. Therefore, the relevant calculations of the elevation difference, transverse distance, and longitudinal distance of the third section PS of the anchor - span cable strand will be converted into the elevation difference, transverse distance, and longitudinal distance between the flat - bend tangent point P and the rotation center J of the saddle.

[0149] Under the overall coordinate system of the anchor - span cable strands, the elevation difference ΔZ PJi , transverse distance ΔX PJi between the flat - bend tangent point P and the rotation center J of the saddle are respectively expressed as:

[0150] ΔZ PJi = -(ΔZ a,1 + ΔZ a,2 + ΔZ a,3 + ΔZ ai,4 ) + Z C1 - Z J (3 - 12)

[0151] ΔX PJi = (ΔX a,1 + ΔX a,2 + ΔX a,3 + ΔX ai,4 ) + X C1 - X J (3 - 13)

[0152] Wherein, ΔZ a,1 , ΔZ a,2 , ΔZ a,3 and ΔZ ai,4 are respectively the height differences between the center of the first arc and the center of the second arc, between the center of the second arc and the center of the third arc, between the center of the third arc and the center of the fourth arc, and between the center of the fourth arc and the tangent point P of the flat bend.

[0153] ΔX a,1 , ΔX a,2 , ΔX a,3 and ΔX ai,4 are respectively the horizontal distances between the center of the first arc and the center of the second arc, between the center of the second arc and the center of the third arc, between the center of the third arc and the center of the fourth arc, and between the center of the fourth arc and the tangent point P of the flat bend.

[0154] and are respectively the abscissa and ordinate of the center C1 of the first arc in the global coordinate system, both of which are known values.

[0155] X J 、Y J and Z J are respectively the abscissa, vertical coordinate and ordinate of the rotation center J of the loose cable saddle in the global coordinate system, all of which are known values.

[0156] As Figure 8 shown, the expressions of ΔZ a,1 , ΔZ a,2 , ΔZ a,3 , ΔZ ai,4 , ΔX a,1 , ΔX a,2 , ΔX a,3 and ΔX ai,4 are respectively:

[0157] ΔZ a,1=(r1 - r2)cos(θ0 + θ1) (3 - 14a)

[0158] ΔZ a,2 =(r2 - r3)cos(θ0 + θ1 + θ2) (3 - 14b)

[0159] ΔZ a,3 =(r3 - r4)cos(θ0 + θ1 + θ2 + θ3) (3 - 14c)

[0160] ΔZ ai,4 =r4cosβ i ′ (3 - 14d)

[0161] ΔX a,1 =(r1 - r2)sin(θ0 + θ1) (3 - 15a)

[0162] ΔX a,2 =(r2 - r3)sin(θ0 + θ1 + θ2) (3 - 15b)

[0163] ΔX a,3 =(r3 - r4)sin(θ0 + θ1 + θ2 + θ3) (3 - 15c)

[0164] ΔX ai,4 =r4sinβ i ′ (3 - 15d)

[0165] Taking the center C2 of the flat - curved circular arc as the origin of the plane coordinate system, with the x - axis horizontally to the left along the longitudinal bridge direction and the y - axis horizontally outward along the transverse bridge direction, the elliptical linear equation of the third section of the cable - strand in the anchor - span is

[0166]

[0167] a = r5 (3 - 17a)

[0168] b = r5·cosφ (3 - 17b)

[0169] where r5 is the radius of the flat - curved circular arc, unit: m; r5 takes a positive value when the cable - strand is on the positive y - axis, and a negative value otherwise; φ is the inclination angle of the saddle at the completed - bridge state, a known value, unit: rad.

[0170] Deriving the ellipse gives

[0171]

[0172] So there is

[0173]

[0174] where x Piand y Pi are the abscissa and ordinate of the flat bending tangent point P in the elliptical plane coordinate system, respectively.

[0175] By combining Equation (3-16) and Equation (3-19), we can obtain

[0176]

[0177]

[0178] In the overall coordinate system of the anchor span cable strand, the transverse distance ΔY between the flat bending tangent point P and the rotation center J of the saddle PJi has the following expression:

[0179]

[0180] Y C2 is the vertical coordinate of the center C2 of the flat bending circular arc in the overall coordinate system, which is a known value.

[0181] Step 4. Determine 5n basic unknowns: Since β i is a function of ai, therefore, H ai , ai, l ai , γ i and β i ′, a total of 5n unknowns are all regarded as basic unknowns.

[0182] Step 5. Determine 5n control equations: According to the elevation closure, transverse distance closure, longitudinal distance closure, same axial force, constant slope of the cable strand in the AP section, and overall force balance of the saddle for each anchor span cable strand, 5n control equations containing the 5n basic unknowns determined in Step 4 are established.

[0183] The determination method of the above 5n control equations preferably includes the following steps.

[0184] Step 51. According to the three conditions of elevation closure, transverse distance closure, and longitudinal distance closure of n anchor span cable strands, establish the following 3n control equations:

[0185] Δz i +ΔZ QPi +ΔZ PJi =ΔZ AJi (5-1)

[0186] Δy i +ΔY QPi +ΔY PJi =ΔY AJi (5-2)

[0187] l ai +ΔX QPi +ΔXPJi = ΔX AJi (5 - 3)

[0188] Wherein, ΔZ AJi , ΔY AJi and ΔX AJi are respectively the elevation difference, the transverse distance and the longitudinal distance between the i-th anchor span cable strand at the anchorage point A and the rotation center J of the saddles, and all are known values.

[0189] Step 52: According to the equality of the moments of the cable force at the tangent point of the anchor span cable strand and the cable force at the tangent point of the side span main cable about the rotation center of the shaft, establish the following 1 control equation:

[0190] ∑M ai + M P = M s (5 - 4)

[0191] Wherein:

[0192] ∑M ai = ∑(T i cosβ i cosγ i ·e 1,i + T i sinβ i ·e 2,i ) (5 - 5)

[0193] M s = H·e3 + Htanα·e4 (5 - 6)

[0194] M P = G·e5 (5 - 7)

[0195] Wherein, M ai is the bending moment of the i-th anchor span cable strand about the saddle, M s is the bending moment of the side span main cable force about the saddle, M P is the bending moment generated by the self-weight of the saddle.

[0196] e 1,i and e 2,i are respectively the lever arms of the horizontal component force and the vertical component force of the i-th anchor span cable strand at the vertical bending tangent point Q with respect to the rotation center J of the saddle, and are both functions of the basic unknown β i .

[0197] e3 and e4 are respectively the lever arms of the horizontal component force and the vertical component force of the side span main cable at the main cable tangent point S with respect to the rotation center J of the saddle, and are both known fixed values.

[0198] e5 is the gravity lever arm of the saddle and is a known fixed value.

[0199] The horizontal component force H of the i-th cable strand of the anchor span at the vertical bending tangent point Q ai cosγ i and the vertical component force H ai tanβ i , the horizontal component force H and the vertical component force Htanα of the side-span main cable at the main cable tangent point S are respectively as Figure 9 shown, then the expressions of the above e 1,i 、e 2,i 、e3 and e4 are respectively:

[0200]

[0201]

[0202]

[0203]

[0204]

[0205] Furthermore:

[0206] ΔZ ai,4 ′ = r4cosβ i (5 - 9)

[0207] ΔX ai,4 ′ = r4sinβ i (5 - 10)

[0208] In the formula, ΔZ ai,4 ′ and ΔX ai,4 ′ are respectively the height difference and the horizontal distance between the center of the circular arc with four centers and the vertical bending tangent point Q.

[0209] R1 and R2 are respectively the distances from the IP point of the cable saddle to the center C1 of the first circular arc and the rotation center J of the cable saddle, as Figure 10 、 11 shown.

[0210] R3 is the distance from the center of gravity of the cable saddle to the rotation center J, as Figure 12 shown.

[0211] and are respectively the angles between the connecting lines of the IP point of the cable saddle and the center C1 of the first circular arc, the rotation center J of the cable saddle and the vertical line, as Figure 10 、 11 shown.

[0212] is the angle between the connecting line of the center of gravity of the cable saddle and the rotation center J point and the vertical line, as Figure 12 shown.

[0213] Step 53: According to the cable force distribution pattern of the anchor span, establish the following n - 1 control equations:

[0214]

[0215] wherein, T1, T2, T3, T i-1 , T n-1 and T n are the cable forces of the 1st, 2nd, 3rd, (i - 1)th, ith, (n - 1)th, and nth cable strands of the anchor span respectively.

[0216] Step 54: According to the constant slope of the cable strand in the AP section, establish the following n control equations:

[0217]

[0218] wherein:

[0219] ΔY APi =Δy i +ΔY QPi (5 - 13)

[0220]

[0221] wherein, ΔY APi is the transverse distance of the ith cable strand of the anchor span from the anchorage point A to the tangent point P of the flat bend.

[0222] ΔX APi is the longitudinal distance of the ith cable strand of the anchor span from the anchorage point A to the tangent point P of the flat bend.

[0223] ΔX C2J is the longitudinal distance of the ith cable strand of the anchor span from the center C2 of the flat bend arc to the rotation center J of the cable saddle, which is a known value.

[0224] Step 6: Solve the 5n basic unknowns: Use the method of solving by linear programming to solve the 5n control equations established in Step 5, so as to obtain the values of the 5n basic unknowns.

[0225] Among them, the method of using the method of solving by linear programming to solve the 5n basic unknowns includes the following steps:

[0226] Step 61: Establish the error function equation f j : The error function equation f j is the difference between the left and right ends of the equal sign in the jth control equation in Step 5; wherein, 1 ≤ j ≤ 5n.

[0227] Step 62: Establish the following objective function:

[0228]

[0229] Step 63: On the premise of making the objective function established in Step 62 infinitely close to 0, use the non-linear GRG method to perform programming and solution on the objective function, so as to obtain the values of 5n basic unknowns at one time.

[0230] Step 7: Determine the cable force T of the anchor span cable strand i : Substitute the basic unknowns a i and H ai corresponding to any one of the anchor span cable strands obtained by solving in Step 6 into the following calculation formula:

[0231] T i = H ai / cosβ i

[0232] Where:

[0233] β i = tan -1 (sinha i )

[0234] Thus, the cable force T of the anchor span cable strand is obtained i .

[0235] Based on the force analysis and geometric coordination relationship of the anchor span cable strand and the dispersion saddle, the present invention is convenient for calculation, has strong practicability, and can be used in the design stage of the anchor span part of the suspension bridge.

[0236] 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 belong to the protection scope of the present invention.

Claims

1. A method for determining the cable force of the anchor span of a completed suspension bridge, characterized in that: It includes the following steps: Step 1. Determine the alignment of the anchor span strands: After the main cable in the side span passes through the saddle groove of the cable spreading saddle, it is divided into n anchor span strands through the turning and dispersion of the cable spreading saddle and is anchored to the anchor block; The bottom of the saddle groove of the cable spreading saddle has a horizontal curve arc and a vertical curve arc from the side span to the anchor span direction; The main cable in the side span enters the cable spreading saddle in a tangent manner, and the tangent point is the main cable tangent point S, and the included angle between the tangent line and the longitudinal bridge direction is α; The i-th cable strand of the anchor span departs from the horizontal curve arc in a tangential manner, and the tangent point is the horizontal curve tangent point P. The angle between the horizontal curve tangent at the horizontal curve tangent point P and the longitudinal bridge direction is γ i ; the angle between the vertical curve tangent at the horizontal curve tangent point P and the longitudinal bridge direction is β i '; The i-th cable strand of the anchor span departs from the vertical curve arc in a tangential manner, and the tangent point is the vertical curve tangent point Q, and the included angle between the tangent line and the longitudinal bridge direction is β i ; the i-th cable strand of the anchor span forms an anchorage point A on the anchor block; where, 1 ≤ i ≤ n; Step 2. Determine the cable force distribution pattern of the anchor span cable strands; on the premise that the actions of the anchor span cable strands and the side span main cables on the saddle meet the balance condition of the saddle, assume that the cable force T of each anchor span cable strand at the tangent point with the saddle i is equal; Step 3. Segmental study of the anchor span strands: Each anchor span strand is divided into three segments, namely the AQ segment, the QP segment, and the PS segment; the elevation difference, the transverse distance, and the longitudinal distance of the three segments of each anchor span strand are all expressed as expressions regarding H ai , a i , l ai , γ i , β i ′ and β i ; where H ai is the horizontal component force of the i-th anchor span strand; a i is the catenary parameter of the i-th anchor span strand; l ai is the longitudinal distance of the i-th anchor span strand from the anchorage point A to the vertical bend tangent point Q Step 4. Determine 5n basic unknowns: Since β i is a function of a i , thus, all 5n unknowns of H ai , a i , l ai , γ i and β i ′ are taken as basic unknowns; Step 5. Determine 5n control equations: According to the elevation closure, transverse bridge distance closure, longitudinal bridge distance closure, same axial force, constant slope of the strand in the AP section, and overall force balance of the cable spreading saddle for each anchor span strand, 5n control equations regarding 5n basic unknowns determined in Step 4 are established; Step 6. Solve the 5n basic unknowns: Use the Solver method to solve the 5n control equations established in Step 5, so as to obtain the values of the 5n basic unknowns; Step 7: Determine the cable force T of the anchor span cable strand i : Substitute the basic unknowns a i and H ai corresponding to any one of the anchor span cable strands obtained in Step 6 into the following calculation formula: T i = H ai / cosβ i Wherein: β i = tan -1 (sinha i ) Thus, the cable force T of the anchor span cable strands is obtained. i .

2. The method for determining the cable strand force of the anchor span in the completed state of a suspension bridge according to claim 1, wherein: In Step 3, the cross-bridge distance Δy i and the elevation difference Δz i of the AQ section of the i-th cable strand of the anchor span are respectively: Δy i = l ai tanγ i (3 - 5) Wherein: where c i is a parameter of the catenary equation, unit: m; q is the self-weight per unit length of the cable strands in the anchor span, kN / m.

3. The method for determining the cable strand force of the anchor span of a completed suspension bridge according to claim 2, characterized in that: In Step 1, the flat bending arc of the cable saddle is located on the left or right side of each saddle groove of the cable saddle; the vertical bending arc of the cable saddle is located on the vertical plane of each saddle groove of the cable saddle; it is assumed that the vertical bending arc has four sections from the side span to the anchor span direction, which are sequentially marked as arc one, arc two, arc three, and arc four; among them, the flat bending tangent point P and the vertical bending tangent point Q are both located on arc four, and the vertical bending tangent point Q is closer to the anchorage point A; then in Step 3, the elevation difference ΔZ QPi , longitudinal bridge direction distance ΔX QPi and transverse bridge direction distance ΔY QPi are respectively: ΔZ QPi = r4(cosβ′ i - cosβ i )(3 - 7) ΔX QPi = r4(sinβ i - sinβ′ i )(3 - 10) ΔY QPi = r4(sinβ i - sinβ′ i )tanγ i (3-11) In the formula, r4 is the radius of the fourth arc in the cable spreading saddle.

4. The method for determining the cable strand force of the anchor span of a completed suspension bridge according to claim 3, characterized in that: In step 3, for the sake of simplifying the calculation, the elevation difference, the distance in the transverse direction of the bridge, and the distance in the longitudinal direction of the bridge of the PS section of the i-th cable strand of the anchor span are converted into the elevation difference ΔZ PJi between the flat bend tangent point P and the rotation center J of the saddle PJi , the distance ΔX in the transverse direction of the bridge PJi , and the distance ΔY in the longitudinal direction of the bridge, specifically as follows: where ΔZ a,1 , ΔZ a,2 , ΔZ a,3 and ΔZ ai,4 are the height differences between the center of the first arc and the center of the second arc, the center of the second arc and the center of the third arc, the center of the third arc and the center of the fourth arc, and the center of the fourth arc and the tangent point P of the flat bend, respectively; ΔX a,1 , ΔX a,2 , ΔX a,3 and ΔX ai,4 are respectively the horizontal distances between the center of the first arc and the center of the second arc, between the center of the second arc and the center of the third arc, between the center of the third arc and the center of the fourth arc, and between the center of the fourth arc and the flat bend tangent point P; and are respectively the abscissa and ordinate of the center C1 of the first arc in the overall coordinate system, both of which are known values; X J , Y J and Z J are the horizontal coordinate, vertical coordinate and ordinate of the rotation center J of the loose cable saddle in the global coordinate system, all of which are known values; is the vertical coordinate of the center C2 of the flat bending circular arc in the global coordinate system, known value; r5 is the radius of the horizontal curve arc; φ is the inclination angle of the cable spreading saddle in the completed bridge state, which is a known value.

5. The method for determining the cable strand force of the anchor span of a completed suspension bridge according to claim 4, characterized in that: ΔZ ai,4 and ΔX ai,4 are both functions of the basic unknown β i ′, and the specific expressions are as follows: ΔZ ai,4 = r4cosβ i ′ (3 - 14d) ΔX ai,4 = r4sinβ i ′ (3 - 15d).

6. The method for determining the cable strand force of the anchor span of a completed suspension bridge according to claim 4, wherein: In Step 5, the method for determining the 5n control equations includes the following steps: Step 51. According to the three conditions of elevation closure, transverse bridge distance closure, and longitudinal bridge distance closure for n anchor span strands, establish the following 3n control equations: Δz i +ΔZ QPi +ΔZ PJi =ΔZ AJi (5-1) Δy i +ΔY QPi +ΔY PJi =ΔY AJi (5 - 2) l ai +ΔX QPi +ΔX PJi =ΔX AJi (5 - 3) where ΔZ AJi , ΔY AJi and ΔX AJi are respectively the elevation difference, the transverse distance and the longitudinal distance between the i-th cable strand of the anchor span and the rotation center J of the cable saddle, and are all known values; Step 52. According to the equality of the moments of the cable force at the tangent point of the anchor span strand and the cable force at the main cable tangent point in the side span about the center point of the rotation axis, establish the following 1 control equation: ∑M ai +M P =M s (5 - 4) Wherein: ΣM ai = Σ(T i cosβ i cosγ i ·e 1,i + T i sinβ i ·e 2,i ) (5-5) M s = H·e3 + Htanα·e4 (5-6) M P = G·e5 (5-7) where M ai is the bending moment of the cable force of the i-th cable strand in the anchor span on the saddle, M s is the bending moment of the main cable force in the side span on the saddle, M P is the bending moment generated by the self-weight of the saddle; e 1,i and e 2,i are the lever arms of the horizontal and vertical component forces of the i-th anchor cable strand at the vertical bending tangent point Q with respect to the rotation center J of the saddle, respectively, and both are functions of the basic unknown quantity β i ; e3 and e4 are respectively the lever arms of the horizontal component force and the vertical component force of the main cable in the side span at the main cable tangent point S with respect to the rotation center J of the cable spreading saddle, both of which are known fixed values; e5 is the lever arm of the gravity of the cable spreading saddle, which is a known fixed value; Step 53. According to the cable force distribution pattern of the anchor span, establish the following n - 1 control equations: where T1, T2, T3, T i-1 , T n-1 and T n are the cable forces of the 1st, 2nd, 3rd, (i - 1)th, ith, (n - 1)th, and nth cable strands of the anchor spans respectively; Step 54. According to the constant slope of the strand in the AP section, establish the following n control equations: Wherein: ΔY APi = Δy i + ΔY QPi (5 - 13) Where, ΔY APi is the transverse distance of the i-th cable strand of the anchor span from the anchorage point A to the tangent point P of the flat bend; ΔX APi is the longitudinal distance of the i-th cable strand of the anchor span from the anchorage point A to the tangent point P of the flat bend; ΔX C2J It is the longitudinal bridge distance from the center C2 of the flat bend circular arc to the rotation center J of the cable saddle for the i-th anchor span cable strand, which is a known value.

7. The method for determining the cable strand force of the anchor span in the completed state of a suspension bridge according to claim 6, wherein: In step 52, e 1,i and e 2,i are respectively expressed as: Wherein: ΔZ ai,4 ′ = r4cosβ i (5-9) ΔX ai,4 ′ = r4sinβ i (5 - 10) where ΔZ ai,4 ' and ΔX ai,4 ' are the elevation difference and the horizontal distance between the four centers of the circular arc and the vertical bending tangent point Q, respectively; R1 and R2 are respectively the distances from the IP point of the cable spreading saddle to the center C1 of the first arc and the rotation center J of the cable spreading saddle; and are respectively the angles between the connecting lines of the IP point of the saddle and the center C1 of the first arc, and the rotation center J of the saddle and the vertical line.

8. The method for determining the cable strand force of the anchor span of a completed suspension bridge according to claim 6, wherein: In Step 6, the method for solving the 5n basic unknowns by using the Solver method includes the following steps: Step 61, establish the error function equation f j : The error function equation f j is the difference between the left and right sides of the equal sign in the j-th control equation in Step 5; where 1 ≤ j ≤ 5n; Step 62. Establish the following objective function: Step 63. On the premise of making the objective function established in Step 62 infinitely close to 0, use the non - linear GRG method to perform programming and solving on the objective function, so as to obtain the values of the 5n basic unknowns at one time.

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