CN113255029B - Method for determining structural deformation and internal force of suspension bridge under action of live load - Google Patents

Method for determining structural deformation and internal force of suspension bridge under action of live load Download PDF

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CN113255029B
CN113255029B CN202110417931.1A CN202110417931A CN113255029B CN 113255029 B CN113255029 B CN 113255029B CN 202110417931 A CN202110417931 A CN 202110417931A CN 113255029 B CN113255029 B CN 113255029B
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main cable
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张文明
常佳琦
刘钊
王景全
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Southeast University
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Abstract

The application discloses a method for determining structural deformation and internal force of a suspension bridge under the action of live load, which comprises the following steps: firstly, determining basic unknowns for expressing the state of a deformed structure; then, a control equation set is established according to the stress-free length conservation of each section of main cable, the coordination of the force and deformation of each suspender, the closure of each span and height difference and the stress balance of the main girder; then, sequentially analyzing the deformation of the main cable, the deformation of the main beam, the relation between the main cable and the main beam and the deformation of the bridge tower, and expressing the non-basic unknown quantity in the control equation set as a function of the basic unknown quantity; and then solving a control equation set by using a planning solving method, obtaining all values of the basic unknown quantity at one time, and finally determining the deformation and the internal force of the suspension bridge structure under the action of the live load by using the values of the basic unknown quantity. The method considers the rigidity contribution of the main beam, the lateral movement of the bridge tower, the longitudinal drift of the main beam and the inclination and the extension of the suspender, and can accurately determine the structural deformation and the internal force of the suspension bridge under the action of live load.

Description

Method for determining structural deformation and internal force of suspension bridge under action of live load
Technical Field
The application belongs to the field of bridge analysis theory, and particularly relates to a method for determining structural deformation and internal force of a suspension bridge under the action of live load.
Background
The suspension bridge structure is light and has reasonable stress, so that the suspension bridge has incomparable spanning capacity. The number and scale of construction of suspension bridges is growing worldwide. The suspension bridge is used as a long-span flexible structure, and is easier to generate large deformation or deflection under the action of live load. The response of suspension bridges under vertical live load is a key element of long-term research. The calculation theory is subjected to three stages of elasticity theory, deflection theory and limited displacement theory in the development process.
The elastic theory does not consider the contribution of constant load to vertical rigidity and the nonlinear influence of displacement, so the designed suspension bridge often has clumsy bridge towers and too high stiffening beams. Compared with the elastic theory, the deflection theory considers the vertical displacement of the main cable and the stiffening girder under the live load and the resistance of the constant load to the live load. To date, deflection theory is still an important calculation theory for the live load response of suspension bridges. However, the theory of deflection assumes a parabolic shape for the main cable under constant load and simplifies the discrete booms into a continuously distributed film, failing to take into account the inclination, elongation and longitudinal displacement of the stiffeners under live load, which are all inconsistent with reality, which can significantly affect the results. In addition, the deflection theory shows that the main cable stiffness contributes to the resistance to deformation much more than the main beam stiffness. As such, only the contribution of the main cable is considered and the main beam is ignored in many subsequent analytical analyses. However, with further development of suspension bridges, particularly large-span railway suspension bridges, the rigidity of the main girder needs to be great according to the use requirements. It is not appropriate to ignore the stiffness contribution of the main beams, which needs to be taken into account in the calculation.
With the development of the finite element method, the finite displacement theory is gradually developed into a powerful analysis tool of the large-span suspension bridge, the influence of various factors can be contained, the actual structure is simulated more truly, and the result is more reliable. The calculation of the response of the suspension bridge under live load can also be realized based on a finite element platform. However, rod units are typically used in finite element models to simulate segments of a main cable, such as Link units in ANSYS, which would not take into account geometric nonlinearities inside each segment of the main cable. In addition, the model building process is complex, finite element analysis mainly depends on matrix operation, and a user is difficult to grasp explicit analysis expression in the calculation process, so that the physical meaning is not clear.
In order to accurately calculate the deformation and the internal force of the suspension bridge structure under the action of the live load, a calculation method is required to be invented.
Disclosure of Invention
The application aims to: aiming at the problems, the application provides a calculation method for structural deformation and internal force of a suspension bridge under the action of live load, which takes the rigidity contribution of a main beam, the lateral movement of a bridge tower, the longitudinal drift of the main beam, the inclination and the extension of a suspender into consideration and has definite physical meaning.
The technical scheme is as follows: in order to achieve the technical purpose, the application adopts the following technical scheme:
a method for determining structural deformation and internal force of a suspension bridge under the action of live load comprises the following steps:
(1) Determining a basic unknown quantity for expressing the state of the deformed structure;
(2) Establishing a control equation set according to the stress-free length conservation of each section of main cable, the coordination of the force and deformation of each suspender, the closure of each span and height difference and the stress balance of the main girder;
(3) Analyzing the deformation of the main cable, and expressing the non-basic unknown quantity related to the deformation of the main cable in the control equation set as a function of the basic unknown quantity;
(4) Analyzing the girder deformation, and expressing the non-basic unknown quantity related to the girder deformation in the control equation set as a function of the basic unknown quantity;
(5) Analyzing the relation between the main cable and the main beam, and expressing the non-basic unknown quantity related to the relation between the main cable and the main beam in the control equation set as a function of the basic unknown quantity;
(6) Analyzing the bridge tower deformation, and expressing the non-basic unknown quantity related to the bridge tower deformation in the control equation set as a function of the basic unknown quantity;
(7) Solving a control equation set by using a planning solving method, and obtaining all values of the basic unknown quantity at one time;
(8) And determining the structural deformation and the internal force of the suspension bridge under the action of the live load by using the value of the basically unknown quantity.
Further, the basic unknowns used in the step (1) to express the deformed structural states include: left side cross main cable horizontal projection length L L Horizontal projection length l of main cable of each section of main span 1 ～l n+1 Horizontal projection length L of main cable of right side span R Left-hand cross main cable catenary parameter a L Main cable catenary parameter a of main span head section 1 Catenary parameter a of main cable on right side R Left side span main cable horizontal force H L Horizontal force H of main cable at main head section of main span 1 Horizontal force H of main cable of right side span R Axial force P of each boom 1 ～P n Longitudinal rigid body displacement delta of main beam QU Branch reaction force variation delta R at left end of main beam Q Branch reaction force variation delta R at right end of main beam U Where the subscript n denotes the number of booms, a total of 2n+12 basic unknowns.
Further, the specific steps for creating the control equation set in the step (2) are as follows:
(2.1) stress-free Length conservation of Main Cables of the segments
S L ＝S′ L
S i ＝S′ i ,1≤i≤n+1
S R ＝S′ R
Wherein S is L ' is the unstressed length of the left-hand span main cable in the initial state, S i 'is the main cable unstressed length of the ith section of catenary in the main span range in the initial state, S' R Is the unstressed length of the main cable on the right in the initial state, all of which are known amounts; s is S L Is the unstressed length of the left-side span main cable after deformation, S i Is the stress-free length of the main cable of the ith section of catenary in the main span range after deformation, S R Is the unstressed length of the main cable of the right span after deformation;
(2.2) force and deformation coordination of the respective booms
Wherein E is h And A h The modulus of elasticity and the cross-sectional area of the boom; ΔP i Is the increment of the axial force of the ith boom, delta P i ＝P i -P′ i ，P i For the axial force of the i-th boom after deformation, P i ' is the axial force of the ith boom in the initial state, which is a known amount; l's' h,i Is the initial length of the ith boom,wherein (1)>The heights of the upper and lower hanging points of the ith hanging rod in the initial state are all known quantities; Δl h,i Is the elongation of the ith boom, deltal h,i ＝l h,i -l′ h,i ，l h,i Is the length of the ith hanging rod after deformation, < >>Wherein X is Oi 、X Gi Is the ith root after deformationLongitudinal bridge coordinates of upper and lower hanging points of the hanger rod, < ->The elevation of the upper and lower hanging points of the ith hanging rod after deformation;
(2.3) closure of spans and height differences
L L ＝L′ L +Δ B
L R ＝L′ R +Δ C
Δh L ＝Y A′ -Y B′
Δh R ＝Y C′ -Y D′
Wherein, the left side spans the horizontal projection length L 'of the main cable in the initial state' L Horizontal projection length L 'of main cable of main span' M Horizontal projection length L 'of main cable is striden on right side' R All are known quantities, and in the initial state, the elevation Y of the left side span anchoring point A ', the left tower vertex B ', the right tower vertex C ', and the right side span anchoring point D A′ 、Y B′ 、Y C′ 、Y D′ Are all known amounts; Δh L Is the height difference delta h of the left end point of the deformed left-side span main cable i Is the height difference delta h of the left and right end points of the main cable of the i-th section of catenary of the main span after deformation R Is the height difference of the left end point and the right end point of the main cable of the right side span after deformation; delta B And delta C Is the offset of the top of the left bridge tower and the right bridge tower to the midspan after deformation;
(2.4) stress balance of the Main girder
Wherein F is the active load concentration force, P xF,i And P yF,i The horizontal component force variation and the vertical component force variation of the ith boom force caused by the live load F are respectively, d F Is the horizontal distance from the action point of the live load F to the left end of the main beam, d i Is P yF,i L is the length of the main beam, the magnitude (F) and the position (d) of the active load concentration force F ) Is a known parameter determined by the operating conditions to be studied.
Further, analyzing the deformation of the main cable in the step (3), and controlling the non-essential relation of the deformation of the main cable in the equation set
The specific steps for expressing the unknowns as a function of the basic unknowns are as follows:
(3.1) the difference in elevation Δh of the left and right end points of the main span i-th segment catenary main cable i The expression is as follows:
wherein, c i ＝-H i /q，H i Is the horizontal component force kN of the ith section main cable under the combined action of constant load and active load; q is the dead weight of each linear meter of the main cable, kN/m; a, a i Is a parameter of the catenary equation, l i The horizontal projection length of the main cable is the i-th section of the main span;
main cable unstressed length S of main span ith section catenary i The expression is as follows:
wherein E is c And A c Modulus of elasticity of main cables respectivelyAnd cross-sectional area;
(3.2) any lifting point O under the combined action of constant load and active load i The coordinates of (c) are expressed as:
wherein X, Y represents the longitudinal and vertical coordinates, X, respectively, of a point in the global coordinate system B’ 、Y B’ Is a known quantity of the initial state;
and (3.3) expressing the recurrence relation of the main cable horizontal force H and the catenary parameter a between the main span adjacent catenaries under the combined action of constant load and active load as follows:
H i +P i sinθ i ＝H i+1
wherein P is i kN is the axial force of the ith boom; θ i The vertical dip angle of the ith boom;
(3.4) the left side spans the height difference Δh of the left and right end points of the Main Cable L And stress free length S L The expression is as follows:
wherein, c L ＝-H L /q，H L Is the horizontal component force kN of the left side span main cable under the combined action of constant load and active load; a, a L Is a parameter of the catenary equation; l (L) L Is a left side span main partThe horizontal projection length of the cable;
the right side spans the height difference deltah of the left and right end points of the main cable R And stress free length S R The expression is as follows:
wherein, c R ＝-H R /q，H R Is the horizontal component force kN of the right side span main cable under the combined action of constant load and active load; a, a R Is a parameter of the catenary equation; l (L) R Is the horizontal projection length of the main cable on the right side.
Further, the step (4) of analyzing the deformation of the main beam, and expressing the non-substantially unknown quantity related to the deformation of the main beam in the control equation set as a function of the substantially unknown quantity includes the following specific steps:
(4.1) calculation model considering Main girder
Cross-inner receiving F 1 ～F n+1 N+1 vertical concentrated forces act, and the left and right counter forces are respectively marked as F 0 And F n+2 ，x i Ith concentration force F i The horizontal distance from the action point of the main beam to the left end of the main beam is divided into n+2 sections according to the action position of the concentrated force, and the deflection on the ith section is expressed as follows:
wherein E is b And I b The elastic modulus and the bending rigidity of the main beam are respectively, and w (x) is positive in downward displacement; c (C) 1,1 And C 1,2 Is the constant coefficient of the ith section, and the constant coefficients of adjacent sections have the following recurrence relation
And has
And (4.2) comparing the girder after the live load action with a girder calculation model, and solving girder deformation:
delta R is considered when calculating girder deflection Q 、P yF,1 ～P yF,n F and DeltaR U N+3 vertical acting forces are added, the n+3 vertical acting forces are sequentially arranged according to the sequence from left to right of the acting point, and then compared with the main beam calculation model substituted in the step (4.1), the deflection w (x) response of the main beam under live load is calculated;
further, any one of the hanging points G after the live load action i The coordinates of (c) are expressed as:
wherein d i The vertical bridge direction and the vertical coordinate of the ith lower hanging point in the initial state are the horizontal distance from the ith lower hanging point to the left end (Q' point) of the main beamIs a known quantity.
Further, in the step (5), the relationship between the main cable and the main beam is analyzed, and the specific step of expressing the non-basically unknown quantity related to the relationship between the main cable and the main beam in the control equation set as a function of the basically unknown quantity is as follows:
(5.1) in the initial State, the magnitude of the boom axial force is P i 'because the boom is vertical, the lower suspension point is subjected to a horizontal force P' x,i And vertical force P' y,i The expression is as follows:
P′ x,i ＝0
P′ y,i ＝P′ i
(5.2) in the deformed state, the magnitude of the boom axial force becomes P i And the direction of the boom force has a vertical inclination angle theta i At this time, the horizontal force P applied to the lower suspension point x,i And vertical force P y,i The expression is as follows:
P x,i ＝P i sinθ i
P y,i ＝P i cosθ i
in θ i From the position of the lifting points in the global coordinate system, i.e
(5.3) horizontal component Change amount P of i-th boom force by live load F xF,i And vertical component force variation amount P yF,i The expression is as follows:
P xF,i ＝P x,i -P′ x,i
P yF,i ＝P y,i -P′ y,i 。
further, the bridge tower deformation is analyzed in step (6), and the non-substantially unknown quantity related to the bridge tower deformation in the control equation set is expressed as a function of the substantially unknown quantity in the following manner:
offset delta of the tower top of the deformed left bridge tower and the bridge tower to span B And delta C The expression is as follows:
Δ B ＝(H 1 -H L )δ B
Δ C ＝(H n+1 -H R )δ C
in delta B 、δ C The compliance coefficients for the left and right pylons are known.
Further, in the step (7), the control equation set is solved by using a planning solution method, and the specific way of obtaining all values of the basically unknown quantity at a time is as follows:
step (2) establishes a system of 2n+12 control equations, steps (3) to (7) express all parameters in the control equations with 2n+12 basic unknowns determined in step (1), deform each control equation, shift the term on the right end of the equal sign to the left end, rewrite to the functional form f () =0, and combine all deformed control equations into one objective function as follows:
and (3) planning and solving the objective function by using a planning and solving method, and solving the 2n+12 values of the basic unknown quantity in the step (1), so that the 2n+12 control equations in the step (2) are simultaneously established.
Further, in the step (8), the specific way of determining the deformation and the internal force of the suspension bridge structure under the action of the live load by using the value of the basically unknown quantity is as follows:
bringing the value of the basically unknown quantity obtained in the step (7) back to the step (3) to determine the deformation and the internal force of the main cable under the action of the live load;
bringing the basically unknown value obtained in the step (7) back to the step (4) to determine the deformation and the internal force of the main girder under the action of the live load;
bringing the value of the basically unknown quantity obtained in the step (7) back to the step (5) to determine the deformation and the internal force of the boom under the action of the live load;
and (3) returning the value of the basically unknown quantity obtained in the step (7) to the step (6) to determine the deformation and the internal force of the bridge tower under the action of the live load.
The beneficial effects are that:
the application adopts the technical scheme and has the following beneficial effects: the application considers the influence of girder rigidity contribution, bridge tower side shift, girder longitudinal drift, inclination and extension of the suspender, and is more close to engineering practice; the solution is carried out by means of the basic unknowns and the control equations with the same quantity, the thought is clear, the physical meaning is more clear, and the method is convenient for engineers to apply.
Drawings
FIG. 1 is a schematic view of a main beam in a constant load condition in an embodiment;
FIG. 2 is a schematic view of a main cable in a constant load state according to an embodiment;
FIG. 3 is a schematic diagram of a main span main cable under the combined action of constant load and live load in an embodiment;
FIG. 4 is a schematic diagram of a side span main cable under the combined action of constant load and live load in an embodiment;
FIG. 5 is a schematic diagram of a computational model of a primary beam in an exemplary embodiment;
FIG. 6 is a schematic view of a main beam under the combined action of constant load and live load in an embodiment;
FIG. 7 is a schematic illustration of the main cable and main beam relationship at the boom in an exemplary embodiment;
FIG. 8 is a schematic illustration of a pylon under constant and live loads in an embodiment.
Detailed Description
The present application is further illustrated below in conjunction with specific embodiments, it being understood that these embodiments are meant to be illustrative of the application and not limiting the scope of the application, and that modifications of the application, which are equivalent to those skilled in the art to which the application pertains, fall within the scope of the application defined in the appended claims after reading the application.
The application relates to a method for determining structural deformation and internal force of a suspension bridge under the action of live load, which comprises the following steps:
(1) Determining a basic unknown quantity for expressing the state of the deformed structure;
(2) Establishing a control equation set according to the stress-free length conservation of each section of main cable, the coordination of the force and deformation of each suspender, the closure of each span and height difference and the stress balance of the main girder;
(3) Analyzing the deformation of the main cable, and expressing the non-basic unknown quantity related to the deformation of the main cable in the control equation set as a function of the basic unknown quantity;
(4) Analyzing the girder deformation, and expressing the non-basic unknown quantity related to the girder deformation in the control equation set as a function of the basic unknown quantity;
(5) Analyzing the relation between the main cable and the main beam, and expressing the non-basic unknown quantity related to the relation between the main cable and the main beam in the control equation set as a function of the basic unknown quantity;
(6) Analyzing the bridge tower deformation, and expressing the non-basic unknown quantity related to the bridge tower deformation in the control equation set as a function of the basic unknown quantity;
(7) Solving a control equation set by using a planning solving method, and obtaining all values of the basic unknown quantity at one time;
(8) And determining the structural deformation and the internal force of the suspension bridge under the action of the live load by using the value of the basically unknown quantity.
The specific implementation mode is as follows:
the first step: a basic unknown quantity is determined that is used to represent the state of the deformed structure. The basic unknowns used to express the state of the deformed structure include: left side cross main cable horizontal projection length L L Horizontal projection length l of main cable of each section of main span 1 ～l n+1 Horizontal projection length L of main cable of right side span R Left-hand cross main cable catenary parameter a L Main cable catenary parameter a of main span head section 1 Catenary parameter a of main cable on right side R Left side span main cable horizontal force H L Horizontal force H of main cable at main head section of main span 1 Horizontal force H of main cable of right side span R Axial force P of each boom 1 ～P n Longitudinal rigid body displacement delta of main beam QU Branch reaction force variation delta R at left end of main beam Q Branch reaction force variation delta R at right end of main beam U Where the subscript n denotes the number of booms, a total of 2n+12 basic unknowns.
And a second step of: and establishing a control equation set. The equation can be established by utilizing the stress-free length conservation of each section of main cable, the coordination of the force and deformation of each suspender, the closure of each span and height difference and the stress balance of the main girder:
(1) Stress-free length conservation of main cables of each section
S L ＝S′ L
S i ＝S′ i ,1≤i≤n+1
S R ＝S′ R
Wherein S' L Is the unstressed length of the left-hand span main cable in the initial state, S' is the ith section of catenary in the main span range in the initial stateIs the stress-free length of the main cable of S' R Is the unstressed length of the main cable on the right in the initial state, all of which are known amounts; s is S L Is the unstressed length of the left-side span main cable after deformation, S i Is the stress-free length of the main cable of the ith section of catenary in the main span range after deformation, S R Is the unstressed length of the main cable of the right span after deformation.
(2) Force and deformation coordination of each suspender
Wherein E is h And A h The modulus of elasticity and the cross-sectional area of the boom; ΔP i Is the increment of the axial force of the ith boom, delta P i ＝P i -P′ i ，P i For the axial force of the i-th boom after deformation, P i ' is the axial force of the ith boom in the initial state, a known amount (as shown in FIG. 1); l's' h,i Is the initial length of the ith boom,wherein (1)>The heights of the upper and lower hanging points of the ith hanging rod in the initial state are all known quantities; Δl h,i Is the elongation of the ith boom, deltal h,i ＝l h,i -l′ h,i ，l h,i Is the length of the ith hanging rod after deformation, < >>Wherein (1)> Is the longitudinal bridge coordinate of the upper and lower hanging points of the ith suspender after deformation, +.>Is the elevation of the upper and lower hanging points of the ith hanging rod after deformation.
(3) Closure of spans and height differences
L L ＝L′ L +Δ B
L R ＝L′ R +Δ C
Δh L ＝Y A′ -Y B′
Δh R ＝Y C′ -Y D′
In the initial state, the left side spans the horizontal projection length L 'of the main cable' L Horizontal projection length L 'of main cable of main span' M Horizontal projection length L 'of main cable is striden on right side' R All are known quantities, and in the initial state, the elevation Y of the left side span anchoring point A ', the left tower vertex B ', the right tower vertex C ', and the right side span anchoring point D A′ 、Y B′ 、Y C′ 、Y D′ Are known amounts (as shown in fig. 2); Δh L Is the height difference delta h of the left end point of the deformed left-side span main cable i Is the height difference delta h of the left and right end points of the main cable of the i-th section of catenary of the main span after deformation R Is the height difference of the left end point and the right end point of the main cable of the right side span after deformation; delta B And delta C Is the offset of the top of the left bridge tower and the right bridge tower to the midspan after deformation.
(4) Stress balance of girder
Wherein F is the active load concentration force, P xF,i And P yF,i The horizontal component force variation and the vertical component force variation of the ith boom force caused by the live load F are respectively, d F Is the horizontal distance from the action point of the live load F to the left end of the main beam, d i Is P yF,i L is the length of the main beam, the magnitude (F) and the position (d) of the active load concentration force F ) Is a known parameter determined by the operating conditions to be studied.
And a third step of: the main cable deformation is analyzed, and the non-basic unknown quantity related to the main cable deformation in the control equation set is expressed as a function of the basic unknown quantity.
(1) Analyzing the main span main cable (shown in figure 3) under the combined action of constant load and active load, and the height difference delta h of the left end point and the right end point of the i-th section catenary main cable of the main span i The expression is as follows:
wherein, c i ＝-H i /q，H i Is the horizontal component force kN of the ith section main cable under the combined action of constant load and active load; q is the dead weight of each linear meter of the main cable, kN/m; a, a i Is a parameter of the catenary equation, l i The horizontal projection length of the main cable is the i-th section of the main span;
main cable unstressed length S of main span ith section catenary i The expression is as follows:
wherein E is c And A c The modulus of elasticity and the cross-sectional area of the main cable, respectively.
(2) Constant load and live load combinationUnder the action of any upper hanging point O i The coordinates of (c) are expressed as:
wherein X, Y represents the longitudinal and vertical coordinates, X, respectively, of a point in the global coordinate system B’ 、Y B’ Is a known quantity of the initial state.
(3) The recurrence relation of the main cable horizontal force H and the catenary parameter a between the main span adjacent catenaries under the combined action of the constant load and the active load can be expressed as follows:
H i +P i sinθ i ＝H i+1
wherein P is i kN is the axial force of the ith boom; θ i Is the vertical dip angle of the ith boom.
(4) Analyzing the side span main cable (shown in figure 4) under the combined action of constant load and active load, and the height difference delta h of the left end point and the right end point of the left side span main cable L And stress free length S L The expression is as follows:
wherein, c L ＝-H L /q，H L Is the horizontal component force kN of the left side span main cable under the combined action of constant load and active load; a, a L Is a parameter of the catenary equation; l (L) L Level of main cable for left side spanProjection length;
the right side spans the height difference deltah of the left and right end points of the main cable R And stress free length S R The expression is as follows:
wherein, c R ＝-H R /q，H R Is the horizontal component force kN of the right side span main cable under the combined action of constant load and active load; a, a R Is a parameter of the catenary equation; l (L) R Is the horizontal projection length of the main cable on the right side.
Fourth step: analyzing the girder deformation and expressing an unknown quantity related to the girder deformation in the control equation set as a function of the unknown quantity.
(1) Considering the girder calculation model, as shown in fig. 5,
cross-inner receiving F 1 ～F n+1 N+1 vertical concentrated forces act, and the left and right counter forces are respectively marked as F 0 And F n+2 ，x i Ith concentration force F i The horizontal distance from the action point of the main beam to the left end of the main beam is divided into n+2 sections according to the action position of the concentrated force, and the deflection on the ith section is expressed as follows:
wherein E is b And I b The elastic modulus and the bending rigidity of the main beam are respectively, and w (x) is positive in downward displacement; c (C) 1,1 And C 1,2 Is the constant coefficient of the ith section, and the constant coefficients of adjacent sections have the following recurrence relation
And has
(2) The girder after live load action is compared by using a girder calculation model, and as shown in fig. 6, the girder deformation is solved:
delta R is considered when calculating girder deflection Q 、P yF,1 ～P yF,n F and DeltaR U N+3 vertical acting forces are added, the n+3 vertical acting forces are sequentially arranged according to the sequence from left to right of the acting point, and then compared with the main beam calculation model substituted in the step (4.1), the deflection w (x) response of the main beam under live load is calculated;
further, any one of the hanging points G after the live load action i The coordinates of (c) are expressed as:
wherein d i The vertical bridge direction and the vertical coordinate of the ith lower hanging point in the initial state are the horizontal distance from the ith lower hanging point to the left end (Q' point) of the main beamIs a known quantity.
Fifth step: and analyzing the relation between the main cable and the main beam, and expressing the non-basic unknown quantity related to the relation between the main cable and the main beam in the control equation set as a function of the basic unknown quantity.
The main cable and main beam relationship at the boom before and after deformation was analyzed as shown in fig. 7:
(1) In the initial state, the axial force of the boom is of the magnitude P i 'because the boom is vertical, the lower suspension point is subjected to a horizontal force P' x,i And vertical force P' y,i The expression is as follows:
P′ x,i ＝0
P′ y,i ＝P′ i
(2) In the deformed state, the magnitude of the boom axial force becomes P i And the direction of the boom force has a vertical inclination angle theta i At this time, the horizontal force P applied to the lower suspension point x,i And vertical force P y,i The expression is as follows:
P x,i ＝P i sinθ i
P y,i ＝P i cosθ i
in θ i From the position of the lifting points in the global coordinate system, i.e
(3) Horizontal component force variation amount P of ith boom force caused by live load F xF,i And vertical component force variation amount P yF,i The expression is as follows:
P xF,i ＝P x,i -P′ x,i
P yF,i ＝P y,i -P′ y,i 。
sixth step: analyzing the bridge tower deformation and expressing the non-substantial unknowns related to the bridge tower deformation in the control equation set as a function of the substantial unknowns.
Analyzing the deformed bridge tower (shown in fig. 8), and the offset delta of the top of the deformed bridge tower to the midspan direction B And delta C The expression is as follows:
Δ B ＝(H 1 -H L )δ B
Δ C ＝(H n+1 -H R )δ C
in delta B 、δ C The compliance coefficients for the left and right pylons are known.
Seventh step: and solving a control equation set by using a planning solving method, and obtaining all values of the basic unknown quantity at one time.
Step (2) establishes a system of 2n+12 control equations, steps (3) to (7) express all parameters in the control equations with 2n+12 basic unknowns determined in step (1), deform each control equation, shift the term on the right end of the equal sign to the left end, rewrite to the functional form f () =0, and combine all deformed control equations into one objective function as follows:
and (3) planning and solving the objective function by using a planning and solving method, and solving the 2n+12 values of the basic unknown quantity in the step (1), so that the 2n+12 control equations in the step (2) are simultaneously established.
Eighth step: and determining the structural deformation and the internal force of the suspension bridge under the action of the live load by using the value of the basically unknown quantity.
Bringing the value of the basically unknown quantity obtained in the step (7) back to the step (3) to determine the deformation and the internal force of the main cable under the action of the live load;
bringing the value of the basically unknown quantity obtained in the step (7) back to the step (4) to determine the deformation and the internal force of the main girder under the action of the live load;
bringing the value of the basic unknown quantity obtained in the step (7) back to the step (5) to determine the deformation and the internal force of the boom under the action of the live load;
bringing the substantially unknown value obtained in step (7) back to step (6) allows the deformation and internal forces of the pylon under live load to be determined.

Claims (5)

1. The method for determining the structural deformation and the internal force of the suspension bridge under the action of live load is characterized by comprising the following steps of:
(1) Determining a basic unknown quantity for expressing the state of the deformed structure;
(2) Establishing a control equation set according to the stress-free length conservation of each section of main cable, the coordination of the force and deformation of each suspender, the closure of each span and height difference and the stress balance of the main girder;
(3) Analyzing the deformation of the main cable, and expressing the non-basic unknown quantity related to the deformation of the main cable in the control equation set as a function of the basic unknown quantity;
(4) Analyzing the girder deformation, and expressing the non-basic unknown quantity related to the girder deformation in the control equation set as a function of the basic unknown quantity;
(5) Analyzing the relation between the main cable and the main beam, and expressing the non-basic unknown quantity related to the relation between the main cable and the main beam in the control equation set as a function of the basic unknown quantity;
(6) Analyzing the bridge tower deformation, and expressing the non-basic unknown quantity related to the bridge tower deformation in the control equation set as a function of the basic unknown quantity;
(7) Solving a control equation set by using a planning solving method, and obtaining all values of the basic unknown quantity at one time;
(8) Determining the structural deformation and the internal force of the suspension bridge under the action of the live load by using a value of the basic unknown quantity;
the basic unknowns used to express the state of the deformed structure in step (1) include: left side cross main cable horizontal projection length L L Horizontal projection length l of main cable of each section of main span 1 ～l n+1 Horizontal projection length L of main cable of right side span R Left-hand cross main cable catenary parameter a L Main cable catenary parameter a of main span head section 1 Catenary parameter a of main cable on right side R Horizontal force H of main cable of left side span L Horizontal force H of main cable at main head section of main span 1 Horizontal force H of main cable of right side span R Axial force P of each boom 1 ～P n Longitudinal rigid displacement delta of main beam QU Branch reaction force variation delta R at left end of main beam Q Branch reaction force variation delta R at right end of main beam U Wherein the subscript n represents the number of booms, a total of 2n+12 basic unknowns;
the specific steps for establishing the control equation set in the step (2) are as follows:
(2.1) stress-free Length conservation of Main Cables of the segments
S L ＝S L ′
S i ′S i ′,1≤i≤n+1
S R ＝S′ R
Wherein S is L ' left side span in initial StateStress-free length of main cable S i 'is the main cable unstressed length of the ith section of catenary in the main span range in the initial state, S' R Is the unstressed length of the main cable on the right in the initial state, all of which are known amounts; s is S L Is the unstressed length of the left-side span main cable after deformation, S i Is the stress-free length of the main cable of the ith section of catenary in the main span range after deformation, S R Is the unstressed length of the main cable of the right span after deformation;
(2.2) force and deformation coordination of the respective booms
Wherein E is h And A h The modulus of elasticity and the cross-sectional area of the boom; ΔP i Is the increment of the axial force of the ith boom, delta P i ＝P i -P i ′，P i For the axial force of the i-th boom after deformation, P i ' is the axial force of the ith boom in the initial state, which is a known amount; l's' h,i Is the initial length of the ith boom,wherein (1)> The heights of the upper and lower hanging points of the ith hanging rod in the initial state are all known quantities; Δl h,i Is the elongation of the ith boom, deltal h,i ＝l h,i -l′ h,i ，l h,i Is the length of the ith hanging rod after deformation, < >>Wherein (1)>Is the longitudinal bridge coordinate of the upper and lower hanging points of the ith hanging rod after deformation, +.>The elevation of the upper and lower hanging points of the ith hanging rod after deformation;
(2.3) closure of spans and height differences
L L ＝L′ L +Δ B
L R ＝L′ R +Δ C
Δh L ＝Y A′ -Y B′
Δh R ＝Y C ′-Y D′
Wherein, the left side spans the horizontal projection length L 'of the main cable in the initial state' L Horizontal projection length L 'of main cable of main span' M Horizontal projection length L 'of main cable is striden on right side' R All are known quantities, and in the initial state, the elevation Y of the left side span anchoring point A ', the left tower vertex B ', the right tower vertex C ', and the right side span anchoring point D A′ 、Y B′ 、Y C′ 、Y D′ Are all known amounts; Δh L Is the height difference delta h of the left end point of the deformed left-side span main cable i Is the height difference delta h of the left and right end points of the main cable of the i-th section of catenary of the main span after deformation R Is the height difference of the left end point and the right end point of the main cable of the right side span after deformation; delta B And delta C Is the offset of the top of the left bridge tower and the right bridge tower to the midspan after deformation;
(2.4) stress balance of the Main girder
Wherein F is the active load concentration force, P xF,i And P yF,i The horizontal component force variation and the vertical component force variation of the ith boom force caused by the live load F are respectively, d F Is the horizontal distance from the action point of the live load F to the left end of the main beam, d i Is P yF,i L is the length of the main beam, the magnitude (F) and the position (d) of the active load concentration force F ) Is a known parameter determined by the operating conditions to be studied;
analyzing the girder deformation in the step (4), and expressing the non-basically unknown quantity related to the girder deformation in the control equation set as a function of the basically unknown quantity comprises the following specific steps:
(4.1) calculation model considering Main girder
Cross-inner receiving F 1 ～F n+1 N+1 vertical concentrated forces act, and the left and right counter forces are respectively marked as F 0 And F n+2 ，x i Ith concentration force F i The horizontal distance from the action point of the main beam to the left end of the main beam is divided into n+2 sections according to the action position of the concentrated force, and the deflection on the ith section is expressed as follows:
wherein E is b And I b The elastic modulus and the bending rigidity of the main beam are respectively, and w (x) is positive in downward displacement; c (C) 1,1 And C 1,2 Is the constant coefficient of the ith section, and the constant coefficients of adjacent sections have the following recurrence relation
And hasC 1,2 ＝0；
And (4.2) comparing the girder after the live load action with a girder calculation model, and solving girder deformation:
delta R is considered when calculating girder deflection Q 、P yF,1 ～P yF,n F and DeltaR U N+3 vertical acting forces are added, the n+3 vertical acting forces are sequentially arranged according to the sequence from left to right of the acting point, and then compared with the main beam calculation model substituted in the step (4.1), the deflection w (x) response of the main beam under live load is calculated;
further, any one of the hanging points G after the live load action i The coordinates of (c) are expressed as:
wherein d i The horizontal distance from the ith lower suspension point to the point Q' at the left end of the main beam is the vertical bridge direction and the vertical coordinate of the ith lower suspension point in the initial stateIs a known quantity;
analyzing the bridge tower deformation in step (6), expressing the non-substantial unknowns associated with the bridge tower deformation in the set of control equations as a function of the substantial unknowns in the set of control equations in the following manner:
offset delta of the tower top of the deformed left bridge tower and the bridge tower to span B And delta C The expression is as follows:
Δ B ＝(H 1 -H L )δ B
Δ C ＝(H n+1 -H R )δ C
in delta B 、δ C The compliance coefficients for the left and right pylons are known.
2. The method for determining structural deformation and internal force of a suspension bridge under live load according to claim 1, wherein the step (3) of analyzing the deformation of the main cable, the specific step of expressing the non-basic unknown quantity related to the deformation of the main cable in the control equation set as a function of the basic unknown quantity is as follows:
(3.1) the difference in elevation Δh of the left and right end points of the main span i-th segment catenary main cable i The expression is as follows:
wherein, c i ＝-H i /q，H i Is the horizontal component force kN of the ith section main cable under the combined action of constant load and active load; q is the dead weight of each linear meter of the main cable, kN/m; a, a i Is a parameter of the catenary equation, l i The horizontal projection length of the main cable is the i-th section of the main span;
main cable unstressed length S of main span ith section catenary i The expression is as follows:
wherein E is c And A c The elastic modulus and the cross-sectional area of the main cable are respectively;
(3.2) any lifting point O under the combined action of constant load and active load i The coordinates of (c) are expressed as:
wherein X, Y represents the longitudinal and vertical coordinates, X, respectively, of a point in the global coordinate system B’ 、Y B’ Is a known quantity of the initial state;
and (3.3) expressing the recurrence relation of the main cable horizontal force H and the catenary parameter a between the main span adjacent catenaries under the combined action of constant load and active load as follows:
H i +P i sinθ i ＝H i+1
wherein P is i kN is the axial force of the ith boom; θ i The vertical dip angle of the ith boom;
(3.4) the left side spans the height difference Δh of the left and right end points of the Main Cable L And stress free length S L The expression is as follows:
wherein, c L ＝-H L /q，H L Is the horizontal component force kN of the left side span main cable under the combined action of constant load and active load; a, a L Is a parameter of the catenary equation; l (L) L The horizontal projection length of the left side span main cable;
the right side spans the height difference deltah of the left and right end points of the main cable R And stress free length S R The expression is as follows:
wherein, c R ＝-H R /q，H R Is the horizontal component force kN of the right side span main cable under the combined action of constant load and active load; a, a R Is a parameter of the catenary equation; l (L) R Is the horizontal projection length of the main cable on the right side.
3. The method for determining structural deformation and internal force of a suspension bridge under a live load according to claim 1, wherein the specific steps of analyzing the relationship between the main cable and the main beam in the step (5) and expressing the non-basically unknown quantity related to the relationship between the main cable and the main beam in the control equation set as a function of the basically unknown quantity are as follows:
(5.1) in the initial State, the magnitude of the boom axial force is P i 'because the boom is vertical, the lower suspension point is subjected to a horizontal force P' x,i And vertical force P' y,i The expression is as follows:
P′ x,i ＝0
P′ y,i ＝P i ′
(5.2) in the deformed state, the magnitude of the boom axial force becomes P i And the direction of the boom force has a vertical inclination angle theta i At this time, the horizontal force P applied to the lower suspension point x,i And vertical force P y,i The expression is as follows:
P x,i ＝P i sinθ i
P y,i ＝P i cosθ i
in θ i From the position of the lifting points in the global coordinate system, i.e
(5.3) byHorizontal component force variation amount P of ith boom force caused by live load F xF,i And vertical component force variation amount P yF,i The expression is as follows:
P xF,i ＝P x,i -P′ x,i
P yF,i ＝P y,i -P′ y,i 。
4. the method for determining structural deformation and internal force of a suspension bridge under the action of live load according to claim 1, wherein the specific way of solving the control equation set by using the programming solving method in the step (7) is as follows:
step (2) establishes a system of 2n+12 control equations, steps (3) to (7) express all parameters in the control equations with 2n+12 basic unknowns determined in step (1), deform each control equation, shift the term on the right end of the equal sign to the left end, rewrite to the functional form f () =0, and combine all deformed control equations into one objective function as follows:
and (3) planning and solving the objective function by using a planning and solving method, and solving the 2n+12 values of the basic unknown quantity in the step (1), so that the 2n+12 control equations in the step (2) are simultaneously established.
5. The method of claim 1, wherein the step (8) of determining the deformation and internal force of the suspension bridge under the action of the live load is performed by using a value of a substantially unknown quantity as follows:
bringing the value of the basically unknown quantity obtained in the step (7) back to the step (3) to determine the deformation and the internal force of the main cable under the action of the live load;
bringing the basically unknown value obtained in the step (7) back to the step (4) to determine the deformation and the internal force of the main girder under the action of the live load;
bringing the value of the basically unknown quantity obtained in the step (7) back to the step (5) to determine the deformation and the internal force of the boom under the action of the live load;
and (3) returning the value of the basically unknown quantity obtained in the step (7) to the step (6) to determine the deformation and the internal force of the bridge tower under the action of the live load.
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CN108491635A (en) * 2018-03-26 2018-09-04 东南大学 A kind of suspension bridge sunpender power and main cable alignment combined calculation method
CN109271733A (en) * 2018-10-09 2019-01-25 东南大学 A kind of suspension bridge sky cable installs linear and cable saddle initial makeup location determination method
CN109815588A (en) * 2019-01-24 2019-05-28 东南大学 A kind of main cable alignment construction method of the three-tower suspension bridge of main spans such as not

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108491635A (en) * 2018-03-26 2018-09-04 东南大学 A kind of suspension bridge sunpender power and main cable alignment combined calculation method
CN109271733A (en) * 2018-10-09 2019-01-25 东南大学 A kind of suspension bridge sky cable installs linear and cable saddle initial makeup location determination method
CN109815588A (en) * 2019-01-24 2019-05-28 东南大学 A kind of main cable alignment construction method of the three-tower suspension bridge of main spans such as not

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