CN108170913B - Cable-stayed bridge cable parameter calculation method based on graph method and bisection method - Google Patents

Cable-stayed bridge cable parameter calculation method based on graph method and bisection method Download PDF

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CN108170913B
CN108170913B CN201711369929.1A CN201711369929A CN108170913B CN 108170913 B CN108170913 B CN 108170913B CN 201711369929 A CN201711369929 A CN 201711369929A CN 108170913 B CN108170913 B CN 108170913B
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cable
graph
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stayed bridge
bisection
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CN108170913A (en
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李锦华
宋金博
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East China Jiaotong University
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Abstract

The invention provides a cable-stayed bridge cable parameter calculation method based on a graphic method and a bisection method, which is used for avoiding the random defect of assuming the initial value interval of the tangential slope of a cable at the beam end in the calculation and solving process, drawing a function graph of the linear shape of the cable and the tangential slope of the cable at the beam end through the graphic method, quickly determining an effective value interval from the graph, combining a bisection program, more accurately solving the tangential slope of the cable at the beam end in a cable linear equation, and further solving related cable parameters such as cable length, tangential slope of the cable at the tower end, tensile force of the cable at the beam end and the tower end and the like. The calculation result of the parameters of the stay cable of the bridge on the Yangtze river highway by the method shows that the calculation of the parameters of the stay cable of the bridge based on the graph method and the bisection method is simpler, more visual and more accurate. In addition, the method can also control the calculation precision according to the actual engineering needs, so that the calculation is more accurate and the efficiency is higher.

Description

Cable-stayed bridge cable parameter calculation method based on graph method and bisection method
Technical Field
The invention belongs to the field of bridge engineering, and particularly relates to a method for calculating parameters of a cable-stayed bridge cable based on a graph method and a bisection method.
Background
The stay cable is a force transmission component of the cable-stayed bridge, and transmits live loads (vehicle loads and crowd loads) acting on a bridge floor, dead weight and dead load of the bridge and the like to the bridge tower, then to the bridge pier and the foundation and finally to the foundation. The design of the stay cable occupies a very important position in the design of the whole cable-stayed bridge, and the installation of the anchor backing plate and the stay cable guide pipe is directly influenced by the determination of the static linear shape of the stay cable. Therefore, it is necessary to accurately determine the cable alignment.
The method for researching the static linear shape of the stay cable mainly adopts a catenary theory. The static linear shape derived by adopting the catenary principle has more complex expression but higher precision. And (3) based on expressions of related parameters such as cable line shape, beam end cable tangent slope, cable length and the like which are derived in detail by a catenary theory, and calculating related parameters of the cable-stayed bridge. In the calculation process, an initial value interval of the tangential slope of the stay cable at the beam end needs to be assumed first, and iterative operation is carried out. However, arbitrarily assuming the initial value range will bring disadvantages in computational efficiency.
Disclosure of Invention
The invention aims to provide a method for calculating parameters of a cable-stayed bridge cable based on a graph method and a bisection method, which aims to overcome the defect that the initial value interval of the tangential slope of a beam-end cable is assumed in the calculation and solving process, and realize the effective calculation of the cable-stayed bridge cable parameters, which is simpler, more visual and more accurate.
The invention is realized in the way, and a method for calculating the parameters of a cable-stayed bridge cable based on a graph method and a bisection method comprises the following steps:
step (S1) of drawing a stay cable shape function by a graphic method
Figure DEST_PATH_IMAGE001
Finding a function image and
Figure 282421DEST_PATH_IMAGE002
then determines the left and right two nearest to the intersection point
Figure DEST_PATH_IMAGE003
And assume the initial value
Figure 540840DEST_PATH_IMAGE004
、
Figure DEST_PATH_IMAGE005
The upper and lower limits of the dichotomy are defined.
Step (S2) of taking the median of the upper and lower limits according to the characteristics of the dichotomy
Figure 258260DEST_PATH_IMAGE006
And substituting the obtained value into a stay cable shape function to obtain
Figure DEST_PATH_IMAGE007
And then compared
Figure 122311DEST_PATH_IMAGE008
And a magnitude relation of 0.
Step (S3), if
Figure DEST_PATH_IMAGE009
When the zero value is in the interval [ 2 ]
Figure 280891DEST_PATH_IMAGE004
,
Figure 634512DEST_PATH_IMAGE010
]In the meantime, make
Figure 308070DEST_PATH_IMAGE005
=
Figure 38128DEST_PATH_IMAGE010
. If it is
Figure DEST_PATH_IMAGE011
When the zero value is in the interval [ 2 ]
Figure 51215DEST_PATH_IMAGE010
,
Figure 991051DEST_PATH_IMAGE005
]In between, order
Figure 276539DEST_PATH_IMAGE004
=
Figure 420075DEST_PATH_IMAGE010
. The steps (S2) and (S3) are repeated.
Step (S4) when
Figure 881144DEST_PATH_IMAGE012
(precision allowed value), let
Figure DEST_PATH_IMAGE013
Is a function
Figure 248671DEST_PATH_IMAGE014
In the interval
Figure 427980DEST_PATH_IMAGE004
,
Figure 234262DEST_PATH_IMAGE005
]The zero solution of (a), is the slope of the a terminal,and solve the tension of the A end of the stay cableT A Slope of B terminalk B B end pull forceT B And total cable lengthS。
In the above step, the function of the shape of the stay cable
Figure DEST_PATH_IMAGE015
Tension at end AT A Slope of the B terminalk B B end pull forceT B And total cable lengthSRespectively as follows:
Figure 221940DEST_PATH_IMAGE016
Figure DEST_PATH_IMAGE017
Figure 494790DEST_PATH_IMAGE018
Figure DEST_PATH_IMAGE019
Figure 689623DEST_PATH_IMAGE020
in the formula:
Figure 706121DEST_PATH_IMAGE021
is a vertical component of the pulling force of the end A of the inhaul cable; ch () is a hyperbolic cosine function; arsh () is an inverse hyperbolic sine function; sh () is a hyperbolic sine function;
Figure 610623DEST_PATH_IMAGE022
mass density of the guy cable unit length;
Figure DEST_PATH_IMAGE023
is the acceleration of gravity;
Figure 319953DEST_PATH_IMAGE024
the A end of the stay cable is cut and pulled obliquely;
Figure 801750DEST_PATH_IMAGE025
the horizontal projection distance of the two ends of the stay cable is shown;
Figure 621938DEST_PATH_IMAGE026
is the vertical projection distance of the two ends of the inhaul cable.
The invention achieves the following beneficial effects: (1) in the aspect of the complexity of the method, the effective initial value range of the slope of the cable-stayed bridge cable can be visually and conveniently determined by a graph method; (2) in the aspect of timeliness of the method, the initial value range of the stayed cable of the cable-stayed bridge is determined by the graph method, so that the method is visual, convenient and effective, and the invalid initial range caused by any value is avoided, so that the processing time of the invalid initial range is removed, and the calculation efficiency of the method is improved; (3) in the aspect of the calculation precision of the method, the calculation precision can be controlled according to the actual engineering requirements, so that the calculation is more accurate and the efficiency is higher.
Drawings
FIG. 1 is a flow chart of the steps of a cable-stayed bridge cable parameter calculation method based on a graph method and a bisection method;
FIG. 2 is a guy cable layout drawing from pier No. 3 to pier No. 6 of the Changjiang river highway bridge of the copper rail in the embodiment of the invention
FIG. 3 shows an A1 cable k according to an embodiment of the present invention A -F relation diagram
FIG. 4 shows an A2 cable k according to the embodiment of the present invention A -F relation diagram
FIG. 5 shows an A3 cable k according to an embodiment of the present invention A -F relation diagram
FIG. 6 shows an A16 cable k according to the embodiment of the present invention A -F relation diagram
FIG. 7 shows a J1 cable k according to an embodiment of the present invention A -F relation diagram
FIG. 8 shows a J2 cable k according to the embodiment of the present invention A -F is offIs a drawing
FIG. 9 shows a J3 cable k according to the embodiment of the present invention A -F relation diagram
FIG. 10 shows a J16 cable k according to the embodiment of the present invention A -F relation diagram
FIG. 11 shows a J22 cable k according to the embodiment of the present invention A -F relation diagram
FIG. 12 shows a J26 cable k according to an embodiment of the present invention A -F relation diagram
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention provides a method for calculating parameters of a cable-stayed bridge cable based on a graph method and a bisection method, which comprises the following steps as shown in figure 1:
step (S1) of drawing a stay cable shape function by a graphic method
Figure 380947DEST_PATH_IMAGE001
Finding a function image and
Figure 854653DEST_PATH_IMAGE002
then determines the left and right two nearest to the intersection point
Figure 699113DEST_PATH_IMAGE003
And is assumed to be the initial value
Figure 57413DEST_PATH_IMAGE004
、
Figure 61141DEST_PATH_IMAGE005
The upper and lower limits of the dichotomy are defined.
In S1, according to the vertical component of the pulling force of the A end of the stay cable
Figure 578185DEST_PATH_IMAGE021
Of the length of the cableMass density
Figure 34575DEST_PATH_IMAGE022
Acceleration of gravity
Figure 196566DEST_PATH_IMAGE023
Horizontal projection distance of two ends of the stay cable
Figure DEST_PATH_IMAGE027
And vertical projection distance
Figure 133429DEST_PATH_IMAGE026
Drawing a stay cable shape function by a graph method
Figure 355463DEST_PATH_IMAGE001
The graph can preliminarily visually and conveniently determine the effective initial value interval of the slope of the cable-stayed bridge cable by a graphic method
Figure 174514DEST_PATH_IMAGE004
,
Figure 140196DEST_PATH_IMAGE028
]This range will serve as the initial upper and lower limits of the dichotomy.
Step (S2) of taking the median of the upper and lower limits according to the characteristics of the dichotomy
Figure 852937DEST_PATH_IMAGE006
And substituting the function into a stay cable shape function to obtain
Figure 449135DEST_PATH_IMAGE007
And then compared
Figure 283711DEST_PATH_IMAGE008
And a magnitude relation of 0.
In S2, the
Figure 380980DEST_PATH_IMAGE029
And
Figure DEST_PATH_IMAGE030
respectively substituted into the stay cable shape function to obtain the corresponding
Figure 26856DEST_PATH_IMAGE014
Value, i.e.
Figure 918589DEST_PATH_IMAGE031
And
Figure DEST_PATH_IMAGE032
. Due to the fact that
Figure 243391DEST_PATH_IMAGE004
，
Figure 550875DEST_PATH_IMAGE005
Is to take two values around zero, and thus exists
Figure 847996DEST_PATH_IMAGE033
。
Step (S3), if
Figure 176209DEST_PATH_IMAGE009
When the zero value is in the interval [ 2 ]
Figure 191569DEST_PATH_IMAGE004
,
Figure 302745DEST_PATH_IMAGE010
]In between, order
Figure 844585DEST_PATH_IMAGE005
=
Figure 759013DEST_PATH_IMAGE010
. If it is
Figure 261669DEST_PATH_IMAGE011
When the zero value is in the interval [ 2 ]
Figure 910957DEST_PATH_IMAGE010
,
Figure 307303DEST_PATH_IMAGE005
]In the meantime, make
Figure 852685DEST_PATH_IMAGE004
=
Figure 373796DEST_PATH_IMAGE010
. Repeating the steps (S2) and (S3).
Step (S4) when
Figure 154670DEST_PATH_IMAGE012
(precision allowed value), let
Figure 546468DEST_PATH_IMAGE013
I.e. as a function
Figure 793910DEST_PATH_IMAGE014
In the interval
Figure 802317DEST_PATH_IMAGE004
,
Figure 386883DEST_PATH_IMAGE005
]The zero solution of (A) is the slope of the end A, and the tension of the end A of the stay cable is solvedT A Slope of B terminalk B B end pull forceT B And total cable lengthS。
The invention relates to an application case of a cable-stayed bridge cable parameter calculation method based on a graph method and a bisection method, which comprises the following steps: take the established highway bridge of the Yangtze river of the Yangshan mountain in Tongling City, anhui province as an example.
The Changjiang river highway bridge of Tongling was built in 1991 in 12 months and in 1995 in 26 months. The bridge is a prestressed reinforced concrete double-tower cable-side cable-stayed bridge, the total length of the bridge is 2592 meters, the total length of a main bridge is 1152m, 7 holes of 80+90+190+432+190+90+80m are continuously arranged, and the bridge consists of a double-tower cable-side prestressed concrete cable-stayed bridge with a main span of 432m and a continuous T-shaped rigid frame side span. The bridge tower adopts an H-shaped door type structure, the section of the box is a box, the height of the tower is 153.65m, the stay cables are arranged in a sector shape, each sector is provided with 26 pairs of cables, and the cable distance is 8 m. The arrangement of the guy cables on the piers No. 3 to No. 6 of the bridge is shown in figure 2. For comparative study, 10 cables A1, A2, A3, a16, J1, J2, J3, J16, J22 and J26 were selected for case testing, and the basic known parameters of each cable are shown in table 1.
TABLE 1 relevant parameters of each stay cable of the great bridge of the Changjiang river, copper Ling
Tab.1 The cable related parameters of Tongling Yangtze river highway bridge
Figure 161416DEST_PATH_IMAGE035
TABLE 2 slope of each cable beam endk A Is taken as a value interval
Tab.2 The region taking value of beam-end slope k A
A1 A2 A3 A16 J1 J2 J3 J16 J22 J26
k A Value range [4,4.5] [2.5,3] [2,2.5] [0.5,1] [4,4.5] [2.5,3] [2,2.5] [0.5,1] [0,1] [0,1]
In the process of solving the cable parameters, an initial value interval of the tangential slope of the cable at the beam end needs to be assumed, and iterative operation is carried out. In order to avoid invalid assumed intervals, a function graph of each stay cable line shape and the slope of the beam-end stay cable tangent line is drawn through a graphic method, and effective value intervals of the slope of each stay cable beam-end tangent line are determined. Because the included angle between the beam end guy cable and the horizontal direction is more than 0 degree, namely
Figure DEST_PATH_IMAGE038
>0. Only when the span of the cable-stayed bridge reaches 10000m, the tangential slope of the cable at the beam end can be equal to zero. At present, the longest span of the cable-stayed bridge just breaks through kilometers but not reaches ten thousand meters, so that the cable-stayed bridge,k A may take zero, i.e. a lower limit ofk A And =0. The upper limit value can be the slope of the chord line of the guy cable nearest to the bridge tower
Figure 154091DEST_PATH_IMAGE039
=4.44, take 4.5. Because the closer to the guy cable of the bridge tower, the larger the slope of the tangent line of the beam end is, and the actual position of the guy cable can deviate from the chord line under the influence of gravity, the guy cable is more stable, and the guy cable is more stable and reliable in usek A May be taken to be at an upper limitk A =4.5, whose actual position is lower than the chord line. Using graphical means to draw the cordsk A The diagram F is shown in FIGS. 3 to 12, and the accurate bisection solution can be further carried out according to the diagram
Figure 542347DEST_PATH_IMAGE040
The value interval of the tangent slope of each cable beam end is obtained, and the result is shown in table 2. The effective reduction of the value interval can reduce the iteration times of the bisection method and is beneficial to improving the calculation efficiency. Taking A1 cable as an example, the determination can be carried out by a graphic method
Figure 174317DEST_PATH_IMAGE041
The range of the interval of the accurate solution is [4, 4.5 ]]. The method for calculating the parameters of the cable-stayed bridge cable based on the graph method and the bisection method can calculate 10 -3 Of precision
Figure 232402DEST_PATH_IMAGE042
And the solution only needs to be iterated for 9 times. Therefore, the combination of the graph method and the dichotomy can not only improve the precision, but also control the iteration times of high-precision calculation, and effectively improve the calculation efficiency.
TABLE 3 static force solution of each stayed-cable of Changjiang river highway bridge in copper tomb
Tab.3 The static solution of each cable of Tongling Yangtze river highway bridge
Figure 808877DEST_PATH_IMAGE044
Note: a is the target value, B is the calculated value of the existing method, and C is the calculated value of the method of the invention
TABLE 4 Beam endsTension forceT A Tension force of tower endT B
Tab.4 The beam-end tension T A and the pylons-end tension T B
Figure 141770DEST_PATH_IMAGE046
Note: b is the calculated value of the existing method, and C is the calculated value of the method of the invention
The results of the calculation of the respective cable-stayed parameters of the cable-stayed bridge are shown in tables 3 and 4. Considering the influence of the table space, the case only carries out error comparison calculation on the slope of the beam end tangent in the table. From the calculation of the A-B error and the A-C error in Table 3, it can be seen that: the iteration method adopted by the existing method is used for solving the problem that the maximum error of the slope of the tangent line of the beam end is stay J22, the error value reaches 4.67%, the minimum error is stay J1, and the error is 0.41%. The maximum error value of the method is 0.05% of the stay J3, the error value is very small and can be almost ignored, and the minimum error reaches 0%, and the error of most stays is 0%. It can be seen that the graphic method adopted by the method of the invention is combined with the dichotomy, and the precision is very high. As can be seen from the error calculation in table 4: the beam end guy cable tension force of the existing method has a certain error with the beam end guy cable tension force calculated by the method, the maximum error of the beam end of the J22 guy cable is 2.27%, and the maximum error of the tower end is 5.41%, which is caused by the fact that the existing method has a large error in calculating the tangent slope of the beam end of the J22 guy cable. The existing method carries out iterative operation by selecting an initial value, and random selection of the initial value can make the iterative operation invalid or obtain a result with a large error to a certain extent, so the existing method has certain limitation and uncertainty.
The invention provides a cable-stayed bridge cable parameter calculation method based on a graph method and a dichotomy, which can avoid the random defect of an initial value interval of a cable-stayed bridge end cable tangent slope, draws a function graph of a cable line shape and the beam end cable tangent slope through the graph method, quickly determines an effective value interval from the graph, combines a dichotomy program, can more accurately solve the beam end cable tangent slope in a cable line equation, and further solves related cable parameters such as cable length, tower end cable tangent slope, beam end and tower end cable tension. The calculation result of the parameters of the stay cable of the bridge on the Yangtze river highway by the method shows that the calculation of the parameters of the stay cable of the bridge based on the graph method and the bisection method is simpler, more visual and more accurate. In addition, the method can also control the calculation precision according to the actual engineering requirements, so that the calculation is more accurate and the efficiency is higher.
The above description is intended to be illustrative of the preferred embodiment of the present invention and should not be taken as limiting the invention, but rather, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention.

Claims (3)

1. A cable-stayed bridge cable parameter calculation method based on a graph method and a bisection method is characterized by comprising the following steps:
step (S1) of drawing a stay cable shape function by a graph method
Figure 479808DEST_PATH_IMAGE001
Finding a function image and
Figure 402765DEST_PATH_IMAGE002
then two of the left and right nearest to the intersection are determined
Figure 899474DEST_PATH_IMAGE003
And assume the initial value
Figure 32515DEST_PATH_IMAGE004
、
Figure 210556DEST_PATH_IMAGE005
As the upper and lower limits of the dichotomy;
step (S2) of taking upper and lower limits according to the characteristics of the dichotomyMedian value of (2)
Figure 486554DEST_PATH_IMAGE006
And substituting the function into a stay cable shape function to obtain
Figure 52968DEST_PATH_IMAGE007
And then compared
Figure 656994DEST_PATH_IMAGE008
A magnitude relationship with 0;
step (S3), if
Figure 638725DEST_PATH_IMAGE009
When the zero value is in the interval [ 2 ]
Figure 270695DEST_PATH_IMAGE004
,
Figure 46890DEST_PATH_IMAGE010
]In between, order
Figure 403791DEST_PATH_IMAGE005
=
Figure 736683DEST_PATH_IMAGE010
；
If it is
Figure 410110DEST_PATH_IMAGE011
When the zero value is in the interval [ 2 ]
Figure 685102DEST_PATH_IMAGE010
,
Figure 76769DEST_PATH_IMAGE005
]In between, order
Figure 134724DEST_PATH_IMAGE004
=
Figure 741286DEST_PATH_IMAGE010
；
Repeating steps (S2), (S3);
step (S4) when
Figure 311813DEST_PATH_IMAGE012
(precision allowed value), let
Figure 738247DEST_PATH_IMAGE013
Is a function
Figure 927788DEST_PATH_IMAGE014
In the interval
Figure 575808DEST_PATH_IMAGE004
,
Figure 677756DEST_PATH_IMAGE005
]The zero solution of (A) is the slope of the end A, and the tension of the end A of the stay cable is solvedT A Slope of the B terminalk B B end tensionT B And total cable lengthS。
2. The method for calculating parameters of a cable-stayed bridge cable based on the graph method and the bisection method as claimed in claim 1, wherein in the step (S1), the graph method is used to draw a cable-shape function graph of the cable-stayed bridge cable, and the initial interval of the slope of the cable-stayed bridge cable is determined rapidly, intuitively and effectively and is used as the initial upper and lower limits of the bisection method.
3. The cable-stayed bridge cable parameter calculation method based on graphic method and dichotomy as claimed in claim 1, wherein in the step (S4), since the dichotomy is adopted in combination, the calculation accuracy can be controlled according to the actual needs of the project
Figure 106332DEST_PATH_IMAGE015
So that the cable-stayed bridgeCable parameter calculation is more accurate, and efficiency is higher.
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