CN117763668A - Method for determining minimum cross-sectional area of rope required by rope-catenary arch combined structure - Google Patents
Method for determining minimum cross-sectional area of rope required by rope-catenary arch combined structure Download PDFInfo
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Abstract
The invention discloses a method for determining the minimum cross-sectional area of a rope required by a rope-catenary arch combined structure, which comprises the following steps: establishing a cable-arch combined structure model, establishing a basic equation, obtaining a standardized fitting type of arch foot hogging moment ratio of the reinforced cable-catenary arch and the catenary arch to be reinforced according to model analysis, and deducing a minimum cross-sectional area evaluation formula of the cable; collecting or actually measuring structural parameters of the catenary arch bridge to be reinforced, calculating the sagittal ratio, and determining the ratio of the negative bending moment of the arch foot required by the reinforced cable-catenary arch and the catenary arch to be reinforced and the cable constraint positionξ 1 Substituting the ratio of the hogging moment of the arch feet of the arch bridge before and after reinforcement into fitting calculation to obtain the axial rigidity ratio t of the cable-arch which correspondingly meets the design requirement; based on the calculated cable-arch axial stiffness ratio t and the minimum cross-sectional area of the cableThe product value formula can be used for solving the minimum cross-sectional area of the required cable. The invention has the advantages of accurate calculation result, wide applicability, convenient calculation and construction material saving.
Description
Technical Field
The invention belongs to the technical field of bridge reinforcement, and particularly relates to a method for determining the minimum cross-sectional area of a rope required by a rope-catenary arch combined structure.
Background
The arch bridge has the characteristics of strong crossing capability, material saving, simple construction and the like, so that the arch bridge is widely applied to valleys, large rivers and deep ditches in the middle and western part of China by designers, however, the damaged arch bridge cannot meet the normal operation requirements along with the increase of traffic volume, service life and material aging. Because the main arch ring of the arch bridge is a bending component, the arch ring cracks are increased due to overlarge bending moment, and particularly, the section resistance is reduced due to the development of arch foot cracks, so that the structural bearing capacity is reduced, and the safety of the arch bridge structure is endangered. If the old dangerous bridges are removed and rebuilt in large quantity, huge capital and manpower and material resources are consumed; if the reinforcement and transformation are carried out on the dangerous old arch bridge, only ten percent to thirty percent of the construction of the new arch bridge are needed, the reinforcement and transformation of a huge amount of dangerous old arch bridges can be seen, the requirements of modern transportation can be met, and the method has good economic benefit and social significance.
The common reinforcement methods of the arch bridge comprise passive reinforcement and active reinforcement, wherein the passive reinforcement mainly comprises reinforcement methods of increasing the section of a main arch ring, adjusting the constant load of a building on the arch and the like, and the active reinforcement mainly comprises reinforcement methods of applying external prestress, changing a structural stress system and the like.
The existing reinforcement method is mainly passive reinforcement, for example, application of UHPC in masonry arch bridge reinforcement, published by Wang Zongshan et al, adopts ultra-high performance concrete (UHPC) to reinforce a main arch ring, so that the rigidity and bearing capacity of an arch bridge structure are obviously improved, but simply increasing the cross section size of the arch ring can cause the arch cross section to apply foam concrete to a double arch bridge of a Yangtze river bridge in Nanjing, so that a good reinforcement effect is obtained. However, simply increasing the sectional size of the arch ring causes the oversize of the original arch, and further aggravates the weight of the whole bridge structure and the burden of the lower structure, which is not economical and attractive; as another example, zhang Lili et al, disclose a comparative analysis of a solid bridge test before and after reinforcing a double arch bridge steel plate, which discusses the influence of a light-weight adhesive steel plate on the bearing capacity of an arch bridge structure, consider that the adhesive steel plate has a good reinforcing effect, but the reinforcing effect depends on the adhesive performance of adhesive to a great extent, and the steel plate exposed to the outside accelerates corrosion, so that the problem of excessively high later maintenance cost exists.
In the method for applying external prestress in active reinforcement, for example, song et al, published in analysis of external prestress rope internal force of diagonal web member, an external prestress technology of a diagonal web member rope structure is adopted for reinforcing a concrete beam, and a calculation formula of rope deformation increment and a calculation formula of rope increment in a combined structure are deduced based on a rope structure theoretical model, so that the hogging moment of the upper edge of a arch foot of an arch bridge is effectively reduced, but the problem is that prestress can be lost along with the passage of time, the material strength utilization rate of a reinforcing member is weakened, and the reinforcing effect of a bridge structure is further reduced.
The structural system method for active reinforcement, such as a cable-arch structure, is used for jointly stressing the cable and the arch to reduce the bending moment of the arch and increase the span, and has been studied in the model calculation of the bridge and the application of the bridge. The method for determining the section of the inhaul cable for reinforcing the arch springing of the upper-bearing arch bridge disclosed in the patent application number CN202211409020.5 comprises the following steps: (1) Respectively establishing a stress model of an original arch structure and a stress model of a reinforcing structure after the inhaul cable is arranged; (2) Obtaining the position of the least adverse load of the negative bending moment of the arch foot of the original arch structure stress model and the reinforcing structure stress model, and applying the same load; (3) Deducing an arch springing bending moment expression of an original arch structure stress model and a reinforcement structure stress model; (4) Fitting an expression y (t) of the arch springing hogging moment reduction amplitude y relative to the stiffness ratio t; (5) And establishing a relation between the reduction amplitude of the bending resistance bearing capacity of the arch springing section and the reduction amplitude of the hogging moment of the arch springing after the inhaul cable is arranged. The method for determining the section of the inhaul cable is deduced on the basis of the arch bridge reinforcement technology of the construction cable-arch structure, however, the inhaul cable is horizontally arranged, one ends of the two ropes are respectively connected with bridge abutments at two ends, the other ends of the two ropes are connected with the arch crown, the rigidity ratio t is only used as an independent variable in the deduction process, the method cannot be universally applied to various arch bridge reinforcement projects with different cable anchoring positions, errors exist in the solving method, and the precision can be further improved.
Disclosure of Invention
The object of the present invention is to solve the above mentioned technical problems and to provide a method for determining the minimum cross-sectional area of a rope required for a rope-catenary arch combined structure.
In order to achieve the above purpose, the present invention adopts the following technical scheme:
the method for determining the minimum cross-sectional area of the rope required by the rope-catenary arch combined structure comprises the following steps of:
step one, a stress model of a cable-catenary arch combined structure is established, an arch is assumed to be a catenary arch with a uniform cross section, a basic equation is established and simplified based on an elastic center method, an approximate curve integration method is adopted to deduce an arch bending moment influence line analytic type of the reinforced cable-catenary arch and the catenary arch to be reinforced, and the accuracy of the analytic type is verified through finite element model software;
step two, based on the deduced arch bending moment influence line analytic style, combining the primary cable-arch design parameters, keeping the sagittal ratio 0.20 and the arch axis coefficient 2.0 unchanged, and respectively taking the cable-arch axial rigidity ratio and the cable constraint position as a first factor variable and a second factor variable to obtain the arch foot hogging moment ratio M of the cable-catenary arch after reinforcement and the catenary arch to be reinforced L /M L,0 Is a normalized fit of (correlation coefficient r=0.9985):
wherein: m is M L To strengthen the negative bending moment of the arch foot of the rear cable-catenary arch, M L,0 A negative bending moment for the arch springing of the catenary arch to be reinforced; t is the ratio of cable-arch axial stiffness; zeta type toy 1 Is the constraint position of the cable, and the calculation formula is xi 1 ＝2x 1 /L，x 1 Is the abscissa of the intersection point of the cable and the arch, L is the arch span,representing xi 1 To the power of 1.845;
it should be noted that the correlation coefficient R is an index used in statistics to measure the closeness between a set of data and its fitted curve, and the value ranges from 0 to 1, and that a value closer to 1 indicates a higher degree of fit between the data and the fitted curve, and vice versa.
Wherein the initially-planned design parameters are obtained according to the existing catenary arch bridge data, the initially-planned sagittal-span ratio is 0.20, the arch axis coefficient is 2.0, the value of the cable-arch axial rigidity ratio t is between 0.02 and 0.1, and the cable constraint position xi 1 0 to 0.8;
step three, obtaining a minimum cross-sectional area evaluation formula of the cable:
let the cable-arch axial stiffness ratio t=e c A c EA, minimum cross-sectional area of cable obtained by variation, evaluate formula A c ＝EAt/E c Wherein: EA is the axial rigidity of the arch, E is the elastic modulus of the arch material, A is the cross-sectional area of the arch, E c A c Representing the axial stiffness of the cable E c Represents the modulus of elasticity of the material of the rope, A c Represents the cross-sectional area of the rope, where t=0 represents the catenary arch where the rope is not disposed;
step four, collecting or actually measuring structural parameters of the catenary arch bridge to be reinforced, including arch span, sagittal height, material properties of the cable and the like; calculating the sagittal ratio and the corresponding cable constraint position xi 1 And determining the ratio M of the hogging moment of the arch springing required by the design L /M L,0 According to the design parameters, the corresponding meeting design requirements are obtainedThe calculated cable-arch axial rigidity ratio t;
and fifthly, according to the cable-arch axial rigidity ratio t calculated in the step four, calculating the minimum cross-sectional area of the required cable according to a minimum cross-sectional area evaluation formula calculation formula of the third cable.
Further description of the method for determining the minimum cross-sectional area of the rope required for the rope-catenary arch combined structure according to the present invention: the process of establishing the basic equation in the first step is as follows:
firstly, a calculation diagram of a cable-arch combined structure under the action of vertical moving load is established, an arch is divided into a left-right symmetrical cantilever arch by a force method, an elastic center simplification system is adopted to simplify a force method equation into a regular equation which is easy to solve, and a catenary expression adopted based on an arch axis is adopted:
wherein: m is an arch axis coefficient; f is the sagittal height; ζ is the relative axis of the arch, and ζ=2x/L, where x is the abscissa of the arch axis and L is the span of the arch; the hash () is a hyperbolic cosine function; catenary equation variables
Establishing a basic equation for a basic system of a cable-catenary arch combined structure:
wherein: delta ij To be constantly changed in position, delta iP For load deflection, the elastic center of the basic structure under the independent action of unit force or unit external load is respectively represented along X j Displacement in direction X j (j=1, 2, 3) is the dome redundancy, where X 1 Is a bending moment, X 2 Is axial force X 3 Is shear force; x is X j (j=4) is the cable force.
Further description of the method for determining the minimum cross-sectional area of the rope required for the rope-catenary arch combined structure according to the present invention: the process for deducing the force analytic formula in the cable-catenary arch in the step one is as follows:
based on the basic equation, considering the axial deformation of the arch, expanding arch axis curve differential by utilizing a Taylor formula based on an elastic center theory and a numerical integration method, and deducing an arch bending moment analytic type of the reinforced cable-catenary arch under the vertical moving load based on the superposition principle according to different moving load acting positions, cable constraint positions and cable force zero coordinates:
wherein: x is X i (i=1, 2, 3) is the dome redundancy, where X 1 Is a bending moment, X 2 Is axial force X 3 Is shear force; x is X 4 Is the cable force; x and y are the abscissa and ordinate of the arch axis respectively; y is s Is the rigid arm length; x is x P The coordinates of the vertical moving load P; x is x 1 The abscissa of the cable constraint position is; x is x 0 Is the zero abscissa of the cable force; alpha is the rope constraint inclination angle; l is the arch span.
As a further explanation of the method for determining the minimum cross-sectional area of the cable required by the cable-catenary arch combined structure according to the present invention, the process for deriving the catenary arch internal force analysis in the first step is as follows:
the cable no longer acting as a constraint on the elastic arch, i.e. cable force X 4 Zero, cause delta ij And delta iP Zero, and deducing arch bending moment analysis type of the catenary arch to be reinforced under the vertical moving load by combining the internal force analysis type of the cable-catenary arch:
wherein: x is X i (i=1, 2, 3) is the dome redundancy, where X 1 Is a bending moment, X 2 Is axial force X 3 Is shear force; x and y are the abscissa and ordinate of the arch axis respectively; y is s Is the rigid arm length; x is x p Is the coordinates of the vertically moving load P.
Further description of the method for determining the minimum cross-sectional area of the rope required for the rope-catenary arch combined structure according to the present invention: in the first step, the accuracy of the analysis type is verified by finite element model software, specifically:
taking a catenary arch bridge with a certain constant section as an example, establishing a finite element model through ANSYS, comparing the deduced internal force results, comparing the relative error between the analytic formula calculation value and the finite element result, and observing the fitting degree of the internal force curve of the analytic formula and the finite element solution.
Further description of the method for determining the minimum cross-sectional area of the rope required for the rope-catenary arch combined structure according to the present invention: the axial rigidity ratio t of the cable-catenary arch is 0.02-0.10, and the cable is restrained at the position xi 1 The value is between 0 and 0.8.
The invention has the advantages that:
(1) In the basic assumption of the calculation model, the arch is a catenary arch with a uniform cross section, and under the action of external load, the cable and the arch are slightly deformed and are not damaged by large deformation; deducing bending moment analysis type of the arch structure by using an elastic center method and an approximate curve integration method, and performing comparison verification with actual literature data to basically coincide with the actual data; the finite element software is used for verifying the analytic expression, and the accuracy of the analytic expression is proved, so that the method has the advantages of accurate calculation result, reduction of interference of external factors on the calculation result and more convenience and rapidness in the calculation process;
(2) The invention deduces the evaluation formula of the minimum cross-section area of the cable, and the axial rigidity ratio t and the cable constraint position xi 1 The two variables are set, so that the influence process of different cable structure parameters on the calculation of the negative bending moment of the arch foot can be intuitively and simply reflected, the calculation mode of the invention is suitable for various engineering applications with different cable constraint positions, the application range is wider, and the practicability is improved;
(3) The invention obtains the evaluation method of the minimum cross-sectional area of the cable by deducing the fitting, and further obtains the minimum cross-sectional area required by the reinforcement by using the cable-catenary arch combined structure, thereby saving the construction cost.
Drawings
Fig. 1 is a schematic structural arrangement of the cable-catenary arch combination structure of the present invention.
FIG. 2 is a schematic diagram of a force analysis of a cable-catenary arch combination structure.
FIG. 3 is a graph showing the change in bending moment characterization values of a cable-catenary arch combined structure with different cable-arch axial stiffness ratios and cable constraint positions.
FIG. 4 is a graph of cable force versus cable constraint position for a cable-catenary arch combination structure.
In fig. 1, 1-arch, 2-cable.
Detailed Description
The specific analysis of the stress mechanism and the mechanical characteristics of the cable-catenary arch combined structure is as follows:
the upper bearing arch bridge is a multi-time hyperstatic space structure, the arch is a main bearing member, the over-arch building is generally considered as a force transfer member, and the restraint between the over-arch building and the arch enables the deflection and bending moment of the main arch ring to be reduced to a certain extent. The combined action of the building on the arch and the arch is not considered, and a 'reserve' is provided for the safety of the arch bridge to a certain extent. Based on the analysis, the arch bridge structure is simplified into a hingeless calculation model, and the following basic assumption is made for obtaining a better analysis result:
(1) The cable is ideal flexible, and is not subjected to bending moment and compression;
(2) The material used for the cords and arches only considers the wire elasticity;
(3) The arch is a catenary arch with a uniform cross section, and under the action of external load, the cable and the arch are slightly deformed and are not damaged by large deformation.
As shown in fig. 2, A, C denotes a crown, B denotes a dome, E (E ') denotes a cable-arch intersection, and one end anchoring point D (D') of the cable is equal in height to the dome B. Dividing the arch into bilateral symmetry cantilever arches by using a force method, wherein the redundant force of the arch is X i (i=1, 2, 3), the cable force is X 4 Simplifying the force equation into a simple force equation by adopting an elastic center simplification systemAnd solving a regular equation. The span of the arch is L, the sagittal height is f, and the length of the rigid arm is y s Let the coordinate of the vertical moving load P be x P The coordinates of the cable constraint position are x 1 The rope constraint dip angle is alpha,is the horizontal inclination angle of the tangent line at any point K on the arch axis. Under the condition that the prestress is not considered, according to the deformation characteristics of the arch structure, under the action of moving load, the deformation increment of the semi-arch cable where the load is positioned is a positive increment, and the cable force is X 4 The method comprises the steps of carrying out a first treatment on the surface of the The deformation increment of the semi-arch cable at the other side is a negative increment, and the cable exits the work, namely the cable force is zero. For ease of calculation, the relative coordinates ζ=2x/L are taken, where the moving load is relative to the coordinates ζ P ＝2x P Relative coordinates of cable constraint position ζ 1 ＝2x 1 /L。
The arch axis is a catenary, and the expression is:
wherein: m is an arch axis coefficient; the hash () is a hyperbolic cosine function; catenary equation variables
The basic equation is established for the basic system shown in fig. 2:
wherein: delta ij To be constantly changed in position, delta iP For load deflection, the elastic center of the basic structure under the independent action of unit force or unit external load is respectively represented along X j Displacement in direction X j (j=1, 2, 3) is the dome redundancy, where X 1 Is a bending moment, X 2 Is axial force X 3 Is shear force; x is X j (j=4) is the cable force.
The displacement of the elastic center in the basic structure consists of arch shaft deformation and cable deformation. According to the method, the axial deformation of the arch (without considering shear deformation) is considered, the arch axis curve differential is developed by utilizing a Taylor formula based on an elastic center theory and a numerical integration method, and arch bending moment analysis type of the reinforced cable-catenary arch under the vertical moving load is deduced according to the difference of a moving load acting position, a cable constraint position and a cable force zero point coordinate based on a superposition principle:
wherein: x is X i (i=1, 2, 3) is the dome redundancy, where X 1 Is a bending moment, X 2 Is axial force X 3 Is shear force; x is X 4 Is the cable force; x and y are the abscissa and ordinate of the arch axis respectively; y is s Is the rigid arm length; x is x P The coordinates of the vertical moving load P; x is x 1 The abscissa of the cable constraint position is; x is x 0 Is the zero abscissa of the cable force; alpha is the rope constraint inclination angle; l is the arch span.
Wherein, the constraint deflection or constraint rigidity of the ropes at the two sides of the arch is zero to cause the rope force X 4 When the cable is zero, the cable does not have a constraint function on the elastic arch, namely the cable is equivalent to a constant-section catenary arch without the cable, and the cable force X 4 Delta caused by ij And delta iP And the bending moment analysis type of the catenary arch to be reinforced under the vertical moving load can be obtained by combining the basic equation with the bending moment analysis type of the catenary arch to be reinforced being 0.
Wherein: x is X i (i=1, 2, 3) is the dome redundancy, where X 1 Is a bending moment, X 2 Is axial force X 3 Is shear force; x and y are the abscissa and ordinate of the arch axis respectively; y is s Is the rigid arm length; x is x p Is the coordinates of the vertically moving load P.
According to the analysis, the stress mechanisms of the catenary arch and the rope-catenary arch are compared, and the force transmission path of the catenary arch is to transmit external load to arch legs at two sides along an arch axis; stress mechanism of cable-catenary arch: the cable and the arch bear external load together by changing the force transmission path, one part of the external load is transmitted to arch feet at two sides along the arch axis, and the other part of the external load is transmitted to the cable; secondly, the stress mode of the arch structure is improved, and the cable and the catenary arch are stressed together, so that the force in the arch is redistributed, the cable force provides moment opposite to external load for the arch foot section, the bending moment of the arch is reduced, and the bending state of the arch tends to be in a 'shaft center compression' state.
In order to verify the accuracy of the analytical formula of the arch bending moment, taking a catenary arch bridge with a certain constant section as an example, establishing a finite element model through ANSYS, comparing the deduced internal force results, and comparing the relative errors of the analytical formula calculation value and the finite element results, wherein the results show that the accuracy of the analytical formula and the internal force curve fitting degree of the finite element solution is high, and the accuracy of the analytical formula and the finite element solution is proved.
Based on arch bending moment analysis type deduced by the invention, the section of the arch springing is taken as a key attention object, and the influence of the cable constraint position and the cable-arch axial rigidity ratio on the arch internal force under the action of a lane is analyzed. According to bridge and culvert design specifications, uniformly distributing load q=10.5 kN/m and concentrated load P=280 kN, and only applying one concentrated load.
According to the existing catenary arch bridge data, the sagittal ratio gamma is 1/8-1/4, the arch axis coefficient m is 1.167-2.814, and the cable axial rigidity is 2% -10% of the arch axial rigidity. For a bridge with a span of 30m in a single hole, design parameters of a bridge with a span of 20-40 m can be referred, meanwhile, the application range of the parameters in actual engineering is considered and the parameters are properly extended to ensure the universality of analysis results, and the value ranges of the parameters are as follows: selecting a common sagittal ratio gamma of an arch bridge to be 0.20, an arch axis coefficient m to be 2.0, a planned cable-arch axial rigidity ratio t to be 0.02-0.10 and a cable constraint position xi 1 0 to 0.8.
In order to quantitatively evaluate the influence of the design parameters on the in-arch force, combining the initial cable-arch design parameters, keeping the sagittal ratio of 0.20 and the arch axis coefficient of 2.0 unchanged, taking the cable-arch axial rigidity ratio and the cable constraint position as a first dependent variable and a second dependent variable respectively, wherein the cable-arch axial rigidity ratio t takes 0.02, 0.04, 0.06, 0.08 and 0.10 respectively, analyzing the influence of the cable-arch axial rigidity ratio and the cable constraint position on the change of the in-arch force based on the arch bending moment influence line deduced herein to simplify the sensitivity parameters, and calculating the result as shown in fig. 3, and analyzing the following rules:
(1) under different rope constraint positions, the reduction amplitude of the hogging moment of the arch foot increases in a nonlinear way along with the increase of the axial rigidity ratio t of the rope to the arch, and the t increases from 0.02 to 0.10 and xi 1 The larger the arch springing hogging moment is, the larger the reduction amplitude is, and the maximum reduction amplitude is 63.7%.
(2) The cable force increases in a nonlinear way along with the increase of the axial rigidity ratio of the cable to the arch; zeta type toy 1 The increase rate of the cable force from 0 to 0.6 decreases step by step, at ζ 1 =0.6 to 0.8 vice versa. In the cable restraining position xi 1 In the range of=0 to 0.8, the cable force increases and decreases, when ζ 1 Maximum value is reached when=0.6, which means that the rope force increases at a rate and its value is Cheng Fanbi, the rope force increases at a rate greater as the rope force decreases, when t increases from 0.02 to 0.10, and ζ 1 At=0.6, the cable force increases by a maximum of 1.8 times.
(3) Therefore, the axial rigidity ratio of the cable and the arch has an important effect on the force in the arch, and as the axial rigidity ratio of the cable and the arch is changed, the arch springing bending moment is changed to different degrees, the arch springing bending moment is reduced, and the bearing capacity and the safety storage of the arch are improved.
Preferably, the ratio M of the negative bending moment of the arch foot of the reinforced cable-catenary arch to the catenary arch to be reinforced is analyzed, wherein the sagittal ratio is 0.20 L /M L,0 Axial stiffness ratio t and cable constraint position xi of cable-arch 1 The change relation between them can deduce the ratio M of hogging moment of arch foot L /M L,0 The normalized fit of (correlation coefficient r=0.9985):
wherein: m is M L To strengthen the negative bending moment of the arch foot of the rear cable-catenary arch, M L,0 A negative bending moment for the arch springing of the catenary arch to be reinforced; t is the ratio of cable-arch axial stiffness; zeta type toy 1 Is a ropeConstraint position, calculated as xi 1 ＝2x 1 /L，x 1 The abscissa of the cable constraint position is; l is the arch span; zeta type toy 1 1.845 Representing xi 1 To the power of 1.845;
and then according to the cable constraint position, the cable-arch axial rigidity ratio which meets the design requirement correspondingly can be calculated, and the required minimum cable cross-sectional area is further calculated.
The invention will be further described with reference to the drawings and examples.
Example 1:
the method for determining the minimum cross-sectional area of the rope required by the rope-catenary arch combined structure comprises the following steps of:
step one, a stress model of a cable-catenary arch combined structure is established, an arch rib is assumed to be a catenary arch with a uniform cross section, a basic equation is established and simplified based on an elastic center method, an approximate curve integration method is adopted to deduce an arch bending moment influence line analytic formula of the reinforced cable-catenary arch and the catenary arch to be reinforced, and the accuracy of the analytic formula is verified through finite element model software;
step two, based on the deduced arch bending moment influence line analytic style, combining the primary cable-arch design parameters, keeping the sagittal ratio 0.20 and the arch axis coefficient 2.0 unchanged, and respectively taking the cable-arch axial rigidity ratio and the cable constraint position as a first dependent variable and a second dependent variable to obtain the arch foot hogging moment ratio M of the reinforced cable-catenary arch and the catenary arch to be reinforced L /M L,0 Is a normalized fit of (correlation coefficient r=0.9985):
wherein: m is M L To strengthen the negative bending moment of the arch foot of the rear cable-catenary arch, M L,0 A negative bending moment for the arch springing of the catenary arch to be reinforced; t is the ratio of cable-arch axial stiffness; zeta type toy 1 Is the constraint position of the cable, and the calculation formula is xi 1 ＝2x 1 /L，x 1 Is the abscissa of the intersection point of the cable and the arch, L is the arch span, and ζ 1 1.845 Representing xi 1 To the power of 1.845;
wherein the initially-planned design parameters are obtained according to the existing catenary arch bridge data, the initially-planned sagittal-span ratio is 0.20, the arch axis coefficient is 2.0, the value of the cable-arch axial rigidity ratio t is between 0.02 and 0.1, and the cable constraint position xi 1 0 to 0.8;
step three, obtaining an evaluation formula of the minimum cross-sectional area of the cable:
let the cable-arch axial stiffness ratio t=e c A c Variable cable cross-sectional area evaluation formula A c ＝EAt/E c Wherein: EA is the axial rigidity of the arch, E is the elastic modulus of the arch material, A is the cross-sectional area of the arch, E c A c Representing the axial stiffness of the cable E c Represents the modulus of elasticity of the material of the rope, A c Represents the cross-sectional area of the rope, where t=0 represents the catenary arch where the rope is not disposed;
step four, collecting or actually measuring structural parameters of the catenary arch bridge to be reinforced, including arch span, sagittal height, material properties of the cable and the like; calculating the sagittal ratio and the corresponding cable constraint position xi 1 And determining the ratio M of the hogging moment of the arch springing required by the design L /M L,0 Substituting the design parameters into a fitting mode to obtain the cable-arch axial rigidity ratio t which meets the design requirements correspondingly;
and fifthly, according to the axial stiffness ratio t of the cable-arch calculated in the step four, calculating the minimum cross-sectional area of the required cable according to a calculation formula of the minimum cross-sectional area evaluation formula of the cable in the step three.
Example 2:
this embodiment differs from embodiment 1 only in that: the process of establishing the basic equation in the first step is as follows:
firstly, a calculation diagram of a cable-arch combined structure under the action of vertical moving load is established, an arch is divided into a left-right symmetrical cantilever arch by a force method, an elastic center simplification system is adopted to simplify a force method equation into a regular equation which is easy to solve, and a catenary expression adopted based on an arch axis is adopted:
wherein: m is an arch axis coefficient; f is the sagittal height; ζ is the relative coordinates and ζ=2x/L, where L is the arch span; the hash () is a hyperbolic cosine function; catenary equation variables
Establishing a basic equation for a basic system of a cable-catenary arch combined structure:
wherein: delta ij To be constantly changed in position, delta iP For load deflection, the elastic center of the basic structure under the independent action of unit force or unit external load is respectively represented along X j Displacement in direction X j (j=1, 2, 3) is the dome redundancy, where X 1 Is a bending moment, X 2 Is axial force X 3 Is shear force; x is X j (j=4) is the cable force.
Example 3:
this embodiment differs from embodiment 2 only in that: the step one is to deduce the analytical process of the bending moment of the reinforced cable-catenary arch, which comprises the following steps:
based on the basic equation, considering the axial deformation of the arch rib, expanding arch axis curve differential by utilizing a Taylor formula based on an elastic center theory and a numerical integration method, and based on a superposition principle, according to different moving load acting positions, cable constraint positions and cable force zero coordinates, performing arch bending moment analysis on the reinforced cable-catenary arch under vertical moving load:
wherein: x is X i (i=1, 2, 3) is the dome redundancy, where X 1 Is a bending moment, X 2 Is axial force X 3 Is shear force; x is X 4 Is the cable force; x and y are the abscissa and ordinate of the arch axis respectively；y s Is the rigid arm length; x is x P The coordinates of the vertical moving load P; x is x 1 The abscissa of the cable constraint position is; x is x 0 Is the zero abscissa of the cable force; alpha is the rope constraint inclination angle; l is the arch span.
Example 4:
this embodiment differs from embodiment 3 only in that: the process for deducing the bending moment analysis type of the catenary arch to be reinforced in the first step comprises the following steps:
the cable no longer acting as a constraint on the elastic arch, i.e. cable force X 4 Zero, cause delta ij And delta iP All are zero, and the bending moment analytic formula of the constant-section catenary arch under the vertical moving load is deduced by combining the bending moment analytic formulas of the cable-catenary arch:
wherein: x is X i (i=1, 2, 3) is the dome redundancy, where X 1 Is a bending moment, X 2 Is axial force X 3 Is shear force; x and y are the abscissa and ordinate of the arch axis respectively; y is s Is the rigid arm length; x is x p Is the coordinates of the vertically moving load P.
Example 5:
this embodiment differs from embodiment 4 only in that: in the first step, the accuracy of the analysis type is verified by finite element model software, specifically:
taking a catenary arch bridge with a certain constant section as an example, establishing a finite element model through ANSYS, comparing the deduced internal force results, comparing the relative error between the analytic formula calculation value and the finite element result, and observing the fitting degree of the internal force curve of the analytic formula and the finite element solution.
Example 6:
as described above, the method for determining the minimum cross-sectional area of the cable required by the cable-catenary arch combined structure preferably has the value of the ratio t of the axial rigidity of the cable-arch of the cable-catenary arch between 0.02 and 0.10 and the cable constraint position xi 1 The value is between 0 and 0.8.
Example 7:
hollow type equal-section catenary hyperbolic arch bridge with certain span 30m, rise of the arch is 6m, the arch is a concrete rectangular section, and the axial rigidity EA=1.08x10 10 And N, the resistance of the arch bridge when the arch foot section does not crack is-523.6 kN.m, the arch bridge generates cracks on the arch foot section along with oversized traffic, service life increase and vehicle overload, and the calculated arch foot section resistance is-338.2 kN.m. When the reinforcement method of arranging the cables at the two sides of the arch is adopted, the resistance reduced part of the arch foot section can be compensated by the mode of sharing the external load by the cables, and then M is known L ＝-338.2kN·m，M L,0 -523.6 kn.m, when the cable constrains position ζ 1 When=0.6, the substitution fitting is as follows:
solving that t is more than or equal to 0.046, and if the cable adopts an epoxy steel strand, knowing the elastic modulus E of the cable material c ＝1.95×10 11 N/m 2 Equation A is evaluated according to the cross-sectional area of the cable c ＝EAt/E c Thus, A can be obtained c ≥2575mm 2 It has been shown that at least 14 steel strands with nominal diameters of 17.8mm (1 x 7) are required to meet the load-bearing design requirements.
It is to be understood that the above-described embodiments are merely illustrative of the invention and are not intended to limit the practice of the invention. Other variations or modifications of the above teachings will be apparent to those of ordinary skill in the art; it is not necessary here nor is it exhaustive of all embodiments; and obvious variations or modifications thereof are contemplated as falling within the scope of the present invention.
Claims (6)
1. The method for determining the minimum cross-sectional area of the rope required by the rope-catenary arch combined structure is characterized by comprising the following steps of:
step one, a stress model of a cable-catenary arch combined structure is established, an arch is assumed to be a catenary arch with a uniform cross section, a basic equation is established and simplified based on an elastic center method, an approximate curve integration method is adopted to deduce an arch bending moment influence line analytic type of the reinforced cable-catenary arch and the catenary arch to be reinforced, and the accuracy of the analytic type is verified through finite element model software;
step two, based on the deduced arch bending moment influence line analytic style, combining the primary cable-arch design parameters, keeping the sagittal ratio 0.20 and the arch axis coefficient 2.0 unchanged, and respectively taking the cable-arch axial rigidity ratio and the cable constraint position as a first factor variable and a second factor variable to obtain the arch foot hogging moment ratio M of the cable-catenary arch after reinforcement and the catenary arch to be reinforced L /M L,0 Is a normalized fit of (correlation coefficient r=0.9985):
wherein: m is M L To strengthen the negative bending moment of the arch foot of the rear cable-catenary arch, M L,0 A negative bending moment for the arch springing of the catenary arch to be reinforced; t is the ratio of cable-arch axial stiffness; zeta type toy 1 Is the constraint position of the cable, and the calculation formula is xi 1 ＝2x 1 /L，x 1 Is the abscissa of the intersection point of the cable and the arch, L is the arch span, and ζ 1 1.845 Representing xi 1 To the power of 1.845;
step three, obtaining a minimum cross-sectional area evaluation formula of the cable:
let the cable-arch axial stiffness ratio t=e c A c EA, minimum cross-sectional area of cable obtained by variation, evaluate formula A c ＝EAt/E c Wherein: EA is the axial rigidity of the arch, E is the elastic modulus of the arch material, A is the cross-sectional area of the arch, E c A c Representing the axial stiffness of the cable E c Represents the modulus of elasticity of the material of the rope, A c Represents the cross-sectional area of the rope, where t=0 represents the catenary arch where the rope is not disposed;
step four, collecting or actually measuring structural parameters of the catenary arch bridge to be reinforced, including arch span, sagittal height and material properties of the cable; calculating the sagittal ratio and the corresponding cable constraint position xi 1 And determining the ratio M of the hogging moment of the arch springing required by the design L /M L,0 Substituting the design parameters into a fitting mode to obtain the cable-arch axial rigidity ratio t which meets the design requirements correspondingly;
and fifthly, according to the calculated cable-arch axial rigidity ratio t in the step four, the minimum cross-sectional area of the required cable can be obtained by using a minimum cross-sectional area evaluation formula of the cable in the step three.
2. The method for determining the minimum cross-sectional area of a rope required for a rope-catenary arch composite structure according to claim 1, wherein the establishing a basic equation in the first step is:
firstly, a calculation diagram of a cable-arch combined structure under the action of vertical moving load is established, an arch is divided into a left-right symmetrical cantilever arch by a force method, an elastic center simplification system is adopted to simplify a force method equation into a regular equation which is easy to solve, and a catenary expression adopted based on an arch axis is adopted:
wherein: m is an arch axis coefficient; f is the sagittal height; ζ is the relative axis of the arch, and ζ=2x/L, where x is the abscissa of the arch axis and L is the span of the arch; the hash () is a hyperbolic cosine function; catenary equation variables
Establishing a basic equation for a basic system of a cable-catenary arch combined structure:
wherein: delta ij To be constantly changed in position, delta iP For load deflection, the elastic center of the basic structure under the independent action of unit force or unit external load is respectively represented along X j Displacement in direction X j (j=1, 2, 3) is the dome redundancy, where X 1 Is a bending moment, X 2 Is axial force X 3 Is shear force; x is X j (j=4) is the cable force.
3. The method for determining the minimum cross-sectional area of a cable required by a cable-catenary arch combined structure according to claim 2, wherein the step one of deriving the bending moment resolution of the cable-catenary arch is as follows:
based on the basic equation, considering the axial deformation of the arch, expanding arch axis curve differential by utilizing a Taylor formula based on an elastic center theory and a numerical integration method, and deducing an arch bending moment analytic type of the reinforced cable-catenary arch under the vertical moving load based on the superposition principle according to different moving load acting positions, cable constraint positions and cable force zero coordinates:
wherein: x is X i (i=1, 2, 3) is the dome redundancy, where X 1 Is a bending moment, X 2 Is axial force X 3 Is shear force; x is X 4 Is the cable force; x and y are the abscissa and ordinate of the arch axis respectively; y is s Is the rigid arm length; x is x P Is the abscissa of the vertically moving load P; x is x 1 The abscissa of the cable constraint position is; x is x 0 Is the zero abscissa of the cable force; alpha is the rope constraint inclination angle; l is the arch span.
4. A method for determining the minimum cross-sectional area of a rope required for a rope-catenary arch combined structure according to claim 3, wherein the step one of deriving the bending moment resolution of the catenary arch comprises:
the cable no longer acting as a constraint on the elastic arch, i.e. cable force X 4 Zero, cause delta ij And delta iP All are zero, and the internal force analytic expression of the catenary arch to be reinforced under the vertical moving load is deduced by combining the internal force analytic expression of the cable-catenary arch:
wherein: x is X i (i=1, 2, 3) is the dome redundancy, where X 1 Is a bending moment, X 2 Is axial force X 3 Is shear force; x and y are the abscissa and ordinate of the arch axis respectively; y is s Is the rigid arm length; x is x p Is the abscissa of the vertically moving load P.
5. The method for determining the minimum cross-sectional area of a cable required by a cable-catenary arch combined structure according to claim 1, wherein the verification of the accuracy of the resolution by the finite element model software in the first step is specifically:
taking a catenary arch bridge with a certain constant section as an example, establishing a finite element model through ANSYS, comparing the deduced internal force results, comparing the relative error between the analytic formula calculation value and the finite element result, and observing the fitting degree of the internal force curve of the analytic formula and the finite element solution.
6. The method for determining the minimum cross-sectional area of a cable required by a cable-catenary arch combined structure according to claim 1, wherein the ratio t of the axial rigidity of the cable-arch of the cable-catenary arch is 0.02 to 0.10, and the cable constraint position ζ 1 The value is between 0 and 0.8.
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