CN108416116B - Method for determining arch crown weight during arch springing enlarged section reinforcing parabolic arch - Google Patents

Method for determining arch crown weight during arch springing enlarged section reinforcing parabolic arch Download PDF

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CN108416116B
CN108416116B CN201810138805.0A CN201810138805A CN108416116B CN 108416116 B CN108416116 B CN 108416116B CN 201810138805 A CN201810138805 A CN 201810138805A CN 108416116 B CN108416116 B CN 108416116B
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concrete
parabola
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杨雨厚
罗月静
黎力韬
鞠玉财
于孟生
岑贞志
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Guangxi Transportation Research and Consulting Co Ltd
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Abstract

The invention discloses a method for determining arch crown pressing weight when an arch springing increases a section and reinforces a parabolic arch, which can quickly, accurately and real-timely determine arch crown pressing weight under the current arch springing concrete pouring amount by determining a one-to-one corresponding relation between the arch springing concrete pouring amount and the arch crown pressing weight according to a design drawing and a construction scheme, and the analytic formula directly reflects the mechanical relation between the arch crown flexibility value and each parameter of the structure, thereby being beneficial to the control of an engineer on reinforcement design and the construction scheme, being used for determining the arch crown pressing weight under different pouring amounts when the arch springing pours concrete, and ensuring the structure safety in the reinforcement construction of an arch bridge. Compared with the most common finite element method at present, the method is simple to operate and easy to implement, can be used in cooperation with a scientific calculator, can be generally applicable to similar engineering conditions by editing a calculation program, does not need to establish a finite element numerical model for each bridge, and saves a large amount of time and resources.

Description

Method for determining arch crown weight during arch springing enlarged section reinforcing parabolic arch
Technical Field
The invention belongs to the field of bridge structure reinforcing construction, and particularly relates to a method for determining arch crown pressure weight when an arch springing increases a section and reinforcing a parabolic arch.
Background
The arch is a structure and a building and is widely popular with people. However, with the improvement of the social industrial level, the increase of traffic load, the influence of factors such as low original design standard and degradation of the material condition of the bridge, and the like, a plurality of arch bridges constructed in the past become four or five types of bridges, and the bridges are urgently required to be maintained, reinforced or dismantled and reconstructed. From the economic benefit analysis, for the bridge which can be transformed, the method of maintenance and reinforcement is a better choice.
The method for increasing the cross section by wrapping concrete outside the arch springing is one of common reinforcing methods for arch bridges, and can effectively improve the bending resistance bearing capacity, the shearing resistance bearing capacity and the rigidity of arch ribs and the bearing capacity and the rigidity of the normal cross section of the arch ribs. For the arch bridge reinforced by the method of increasing the section, the most adverse working condition is that in the pouring process of arch springing concrete, because the reinforcing layer does not form rigidity at the moment, the reinforcing layer can not participate in stress, and the dead load is increased, corresponding measures are generally needed in the process. The arch crown is most concerned and solved when the arch crown is poured with concrete, and the common solution in construction is to weigh the arch crown, wherein the water tank weight is the most common way. However, as the amount of the pouring square of the arch springing concrete increases, the arch crown weight also changes, and the purpose of weighting by using the water tank is to conveniently adjust the size of the weight in real time, namely, the arch crown weight is dynamically adjusted and has an optimal value matched with the amount of the pouring square of the concrete, and the condition that the arch crown weight is too large or too small is very unfavorable for controlling the deformation of the arch crown and the safety of the whole structure.
Disclosure of Invention
The invention aims to solve the technical problem of providing a method for determining the arch crown weight when an arch springing increases the section and is reinforced by a parabola, which is simple to operate and easy to realize, and is used for determining the arch crown weight under different pouring amounts when concrete is poured on the arch springing, so that the structural safety in arch bridge reinforcement construction is ensured.
In order to solve the technical problems, the invention adopts the following technical scheme:
a method for determining the arch crown pressure weight during the reinforcing of the parabolic arch with the arch springing enlarged section includes such steps as designing the design drawing and construction organization to obtain the parameters of structure, replacing the parabola with the suspension cable, and building the arch crown pressure weight G according to the principle that the arch crown deflection is not changed during the reinforcing of the arch springing enlarged sectionpConcrete volume V poured in reinforcing with section increased by two side arch feetcOne by oneCorresponding relation formula, and controlling arch top pressure weight G when pouring concrete according to the relation formula in real timep(ii) a The method is suitable for the constant-section parabola hingeless arch.
The method for determining the arch crown weight when the arch springing enlarged section reinforces the parabola arch comprises the following steps:
firstly, obtaining various parameter values including a span l, a rise f and a horizontal dip angle of a tangent line of a cross section of an arch springing from a design drawing and a construction organization scheme
Figure BDA0001575952050000027
Thickness t and width w of arch springing enlarged section part and volume weight gamma of arch springing pouring concretecTensile and compressive stiffness EA and bending stiffness EI of the arch section;
secondly, replacing the arch axis of the original bridge by the suspension cable approximately, and obtaining an arch constant a of the equivalent suspension cable by utilizing a dichotomy;
thirdly, establishing arch top pressure weight G on the basis that arch top deflection is not changed in arch foot enlarged section reinforcement constructionpConcrete volume V poured in reinforcing with section increased by two side arch feetcAccording to the one-to-one corresponding relation formula, the arch top pressure weight G is controlled in real time when the concrete is poured on the arch springingp。
The corresponding relation is as follows:
Figure BDA0001575952050000021
wherein v (x) is the deflection of the section of the dome when unit force is applied to the section x of the parabola arch; under a coordinate system which takes the vault as an origin (0, 0), the radial direction of the arch as an x axis and the rise direction as a y axis, the calculation formula is
v(x)＝f(x)-k1f1(x)-k2f2(x)-k3f3(x) (ii) a v (0) represents the deflection of the section of the parabola arch vault when unit force is applied to the section; v (x) the values of the parameters in the calculation are calculated according to the following equation:
Figure BDA0001575952050000022
Figure BDA0001575952050000023
Figure BDA0001575952050000024
Figure BDA0001575952050000025
k1＝c1/δ11
k2＝c2/δ22
k3＝c3/δ33
Figure BDA0001575952050000026
Figure BDA0001575952050000031
Figure BDA0001575952050000032
Figure BDA0001575952050000033
Figure BDA0001575952050000034
Figure BDA0001575952050000035
aiming at the problems of arch crown weight control during arch springing of the reinforcing parabolic arch with the enlarged cross section of the existing arch springing, the inventor establishes a method for determining arch crown weight during arch springing of the reinforcing parabolic arch with the enlarged cross section of the arch springing. Compared with the most common finite element method at present, the method is simple to operate and easy to implement, can be used in cooperation with a scientific calculator, can be generally applicable to similar engineering conditions by editing a calculation program, does not need to establish a finite element numerical model for each bridge, and saves a large amount of time and resources.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Fig. 2 is a flowchart of the calculation of the equivalent parabola-replacing suspension wire arch constant a.
Fig. 3 is a schematic diagram of the deflection calculation of an arbitrary section i of the arch when a unit force acts on the section x of the parabolic hingeless arch.
Fig. 4 is a schematic diagram of calculation of arch top pressure weight in reinforcing construction of an enlarged section of a parabolic hingeless arch springing.
Detailed Description
The method for determining the arch crown pressure weight when the arch springing increases the section and reinforces the parabolic arch is suitable for the constant-section parabolic hingeless arch and specifically comprises the following steps:
firstly, obtaining various parameter values including a span l, a rise f and a horizontal dip angle of a tangent line of a cross section of an arch springing from a design drawing and a construction organization scheme
Figure BDA0001575952050000036
Thickness t and width w of arch springing enlarged section part and volume weight gamma of arch springing pouring concretecTensile and compressive stiffness EA and flexural stiffness of arch sectionEI；
Secondly, replacing the arch axis (namely parabola) of the original bridge by the suspension wire approximation, and obtaining the arch constant a of the equivalent suspension wire by utilizing a dichotomy;
thirdly, establishing arch top pressure weight G on the basis that arch top deflection is not changed in arch foot enlarged section reinforcement constructionpConcrete volume V poured in reinforcing with section increased by two side arch feetc(two-side arch foot concrete symmetrical pouring, V)cTotal amount of concrete poured on both sides) of the arch springing, and controlling the arch top pressure weight G when pouring concrete on the arch springing in real time according to the corresponding relationp。
The corresponding relation is as follows:
Figure BDA0001575952050000041
wherein v (x) is the deflection of the section of the dome when unit force is applied to the section x of the parabola arch; in a coordinate system with the dome as an origin (0, 0), the x-axis (positive to the left) in the radial direction of the arch, and the y-axis (positive to the down) in the sagittal direction, the calculation formula is v (x) f (x) -k1f1(x)-k2f2(x)-k3f3(x) (ii) a v (0) represents the deflection of the section of the parabola arch vault when unit force is applied to the section; v (x) the values of the parameters in the calculation are calculated according to the following equation:
Figure BDA0001575952050000042
Figure BDA0001575952050000043
Figure BDA0001575952050000044
Figure BDA0001575952050000045
k1＝c1/δ11
k2＝c2/δ22
k3＝c3/δ33
Figure BDA0001575952050000046
Figure BDA0001575952050000047
Figure BDA0001575952050000051
Figure BDA0001575952050000052
Figure BDA0001575952050000053
Figure BDA0001575952050000054
the above method is also applicable to variable cross-section parabolas without hinges, except that the section moment of inertia I is a function of x.
For ease of understanding, the key second and third steps are described in detail below.
For a parabolic arch, because the integral along the arch axis does not have an explicit expression, see formula (1), a straight line, a circular arc line or a suspension line is generally adopted to approximate a substitute parabola when calculating deformation, internal force and the like, the integrals of the approximate curves have concise expressions, see formulas (2) to (4) respectively
Figure BDA0001575952050000055
∫sds＝x (2)
Figure BDA0001575952050000056
∫sds＝sinh(x/a)/a (4)
In the expressions (1) to (4), s is an arch axis integration path; y' is the first derivative of the parabola arch axis equation; x is a coordinate with the origin of the coordinate positioned at the vault along the span direction; l is the span; f is rise; r is the radius of the arc arch; a is the suspension wire arch parameter.
Compared with a straight line and a circular arc line, the suspension cable line is closer to a parabola, so that the suspension cable line is adopted to approximately replace the parabola. Under a coordinate system taking the vault as an origin, the suspension line equation is as follows:
Figure BDA0001575952050000057
as long as the parameter a is adjusted to enable the newly constructed curve to be as close as possible to the original arch axis curve, the integral along the parabola arch axis can obtain an explicit approximate solution with higher precision. The solution for parameter a may utilize a dichotomy. From the parabolic arch boundary conditions, it is easy to know that the following holds:
Figure BDA0001575952050000061
from this, the equation can be derived
Figure BDA0001575952050000062
The value of parameter a can be obtained by solving equation (7) by dichotomy, and the flow chart is shown in fig. 2. For convenience, the a values corresponding to the common vector-span ratio under the unit span are listed, and are detailed in table 1.
TABLE 1 value of parameter a corresponding to the common rise-to-span ratio at unit span
Rise-to-span ratio 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10
a 0.421166 0.537160 0.655863 0.776289 0.897848 1.020178 1.143053 1.266324
On the basis of equivalently replacing a parabolic arch by a suspension cable arch, deducing a deflection calculation formula of the arch crown section when unit force acts on the section x of the parabolic arch:
(1) computing thinking
And (3) solving the deflection influence line by adopting a static method, acting the unit force P on any section b, and solving the deflection influence line of the vault section. According to the elastic center method, the unarticulated arch under the unit force is converted into a static structure with redundant force, as shown in fig. 3, and the parabolic equation y is 4fx2/l2The tensile and compressive stiffness of the arch section is EA, the bending stiffness is EI,
Figure BDA0001575952050000064
the included angle between the tangent line of the arch axis and the horizontal line, the surplus force including bending moment x1Axial force x2Shear force x3Elastic center distance from dome is ys. For the dome section deflection, a dummy load was applied at this location and the internal forces of the base structure at all loads are listed in table 2.
TABLE 2 internal force of basic structure under all loads
Figure BDA0001575952050000063
(2) Computing the remaining power of the project
The premise of solving the deflection of the arch is to calculate the redundant force of the basic structure (figure 3) under the action of unit force according to the principle of basic mechanics. The suspension line is used to replace the parabola, and the simplification of the curve integral according to the formula (4) can obtain the redundancy calculation formula, and the solving process and the formula are shown in table 3.
TABLE 3 calculation of proud power
Figure BDA0001575952050000071
Note: delta1p、Δ2p、Δ3pIs based on the fact that unit force acts on the left half-span, and when unit force acts on the right half-span, delta1pAnd Δ2pStill take the corresponding value of left half span, Δ3pTake-1 times the corresponding left half span.
(3) Solving for dome deflection values
As shown in FIG. 2, the termThe deflection delta of the point O when the unit force and the redundant force act on the point B of the structureoIs composed of
ΔO＝ΔOp+x1ΔO1+x2ΔO2+x3ΔO3 (8)
The meanings of the parameters and the calculation formula in the formula (8) are shown in Table 4.
TABLE 4 calculation of vault deflection value
Figure BDA0001575952050000081
From the equations (8), tables 3 and 4, it is found that the deflection v (x) of the dome section when a unit force is applied to the parabola arch section x is:
v(x)＝f(x)-k1f1(x)-k2f2(x)-k3f3(x) (9)
in the formula:
Figure BDA0001575952050000082
Figure BDA0001575952050000083
Figure BDA0001575952050000084
Figure BDA0001575952050000091
k1＝c1/δ11 (14)
k2＝c2/δ22 (15)
k3＝c3/δ33 (16)
wherein the coefficient k1、k2And k3The values of the parameters in the calculation formula are calculated according to the following formula:
Figure BDA0001575952050000092
Figure BDA0001575952050000093
Figure BDA0001575952050000094
Figure BDA0001575952050000095
Figure BDA0001575952050000096
Figure BDA0001575952050000097
(4) example verification
In order to verify the calculation accuracy of the arch crown deflection derived by the method, 2 equal-section parabolas are taken as an example, the method is respectively calculated by adopting a finite element method and the method, and the calculation error of the formula is judged by taking the finite element analysis result as a reference.
California, arch span 117.5m, rise 22.158m, cross-sectional tensile and compressive stiffness 2329804.38kN, flexural stiffness 9552.1994kNm2The arch acts on unit concentrated load (1 kN); example 2, rise 39.5m, other parameters are the same as example l. The comparison of the values of the camber of the dome cross-section by the formula calculation and finite element calculation of the present invention is shown in Table 5.
TABLE 52 comparison of typical values of the examples
Figure BDA0001575952050000101
Note: the deflection in the table is positive upward and negative downward.
As can be seen from Table 5, the maximum difference between the formula calculation and the finite element analysis result is not more than 0.40%, and the formula calculation of the constant-section parabolic hingeless arch crown deflection has very high engineering precision. On the basis, the vault deflection h in the concrete pouring process is respectively calculated according to the attached figure 4cVault deflection h under the action of vault ballastpSee formula (23) and formula (24). To simplify the calculations, the dome weight is reduced to a concentrated force on the dome.
Figure BDA0001575952050000102
hp＝Gpv(0) (24)
The arch crown deflection is kept unchanged in the arch springing section-enlarging reinforcing construction, namely the following formula is always established in the arch springing concrete pouring process:
hc＝hp (25)
then, the dome pressure weight GpConcrete volume V poured in reinforcing with section increased by two side arch feetc(two-side arch foot concrete symmetrical pouring, V)cThe total amount of concrete poured on both sides) has the following correspondence:
Figure BDA0001575952050000103
where v (x) is the deflection of the section of the dome at a parabolic arch section x with a unit force applied. v (0) represents the deflection of the section of the parabola arch when unit force is applied to the section.
Dome pressure weight G according to formula (26)pConcrete volume V poured in reinforcing with section increased by two side arch feetcThe one-to-one corresponding relation can realize the real-time control of the arch crown counter weight in the process of pouring concrete by enlarging the cross section of the arch springing.

Claims (1)

1. Arch foot increasing cross sectionA method for determining the arch crown pressing weight of a fixed parabola arch is characterized by obtaining various structural parameter values according to design drawings and construction organization design, using a suspension cable to approximately replace a parabola, and establishing arch crown pressing weight G on the basis that arch crown deflection does not change in arch springing section-increasing reinforcing constructionpConcrete volume V poured in reinforcing with section increased by two side arch feetcAccording to the one-to-one corresponding relation formula, the arch top pressure weight G is controlled in real time when the concrete is poured on the arch springingp(ii) a The method is suitable for the constant-section parabola hinge-free arch; the method specifically comprises the following steps:
firstly, obtaining various parameter values including a span l, a rise f and a horizontal dip angle of a tangent line of a cross section of an arch springing from a design drawing and a construction organization scheme
Figure FDA0003113943480000016
Thickness t and width w of arch springing enlarged section part and volume weight gamma of arch springing pouring concretecTensile and compressive stiffness EA and bending stiffness EI of the arch section;
secondly, replacing the arch axis of the original bridge by the suspension cable approximately, and obtaining an arch constant a of the equivalent suspension cable by utilizing a dichotomy;
thirdly, establishing arch top pressure weight G on the basis that arch top deflection is not changed in arch foot enlarged section reinforcement constructionpConcrete volume V poured in reinforcing with section increased by two side arch feetcAccording to the one-to-one corresponding relation formula, the arch top pressure weight G is controlled in real time when the concrete is poured on the arch springingp；
The corresponding relation is as follows:
Figure FDA0003113943480000015
wherein v (x) is the deflection of the section of the dome when unit force is applied to the section x of the parabola arch; in a coordinate system with the arch as an origin (0, 0), the radial direction along the arch as an x-axis, and the sagittal direction as a y-axis, the calculation formula is v (x) f (x) -k1f1(x)-k2f2(x)-k3f3(x) (ii) a v (0) represents the deflection of the section of the parabola arch vault when unit force is applied to the section; v (x) the values of the parameters in the calculation are calculated according to the following equation:
Figure FDA0003113943480000011
Figure FDA0003113943480000012
Figure FDA0003113943480000013
Figure FDA0003113943480000014
k1＝c1/δ11
k2＝c2/δ22
k3＝c3/δ33
Figure FDA0003113943480000021
Figure FDA0003113943480000022
Figure FDA0003113943480000023
Figure FDA0003113943480000024
Figure FDA0003113943480000025
Figure FDA0003113943480000026
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