CN117951771A - Method for rapidly determining position of construction support of continuous steel box girder bridge oriented to optimal stress - Google Patents
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Abstract
A rapid determination method for the position of a continuous steel box girder bridge construction bracket facing to optimal stress relates to the technical field of bridge construction. For a three-span continuous steel box girder bridge, selecting a pier top section as an internal force calculation section, respectively carrying out discretization treatment on the continuous girder in a bracket-free state and a bracket-equipped state, constructing a balance equation, constructing a unit stiffness matrix and a total stiffness matrix of each span, solving solid-to-end force caused by non-node load, converting a solid-to-end force coordinate system, solving continuous girder node load, solving a continuous girder node displacement matrix, solving rod end force, and finally determining the optimal erection position of the bracket by utilizing the change rate. Based on the continuous beam mechanics simplified analysis model, the change of the internal force of the main beam before and after the support is removed in the continuous beam construction process can be effectively reduced, and the construction efficiency and safety are improved.
Description
Technical Field
The invention relates to the technical field of bridge construction, in particular to a method for quickly determining the position of a continuous steel box girder bridge construction bracket facing optimal stress.
Background
Along with the increasing complexity of the steel structure bridge, the construction technology is continuously improved, and in the development process of the steel structure bridge, the traditional method for setting up a full framing gradually loses the dominant position due to the requirement on the construction efficiency. At present, more common steel structure bridge construction methods comprise a pushing construction method, a hoisting construction method, a cantilever assembly method and the like. For pushing construction, a spacious and flat site condition is required, so that the method is difficult to apply in urban environment; for the hoisting construction method and the cantilever assembly method, the economical efficiency and the safety of bridge construction are comprehensively considered, and a large number of brackets are used, so that the construction period is prolonged, and the construction cost is increased. However, in the actual construction process, a part of the support is usually erected as an auxiliary construction means, so that the type and the position of the support are critical to the overall construction efficiency. Proper bracket placement can significantly improve the efficiency of engineering progress, but improper bracket placement can introduce unnecessary uncertainty and even compromise construction safety. Thus, the brackets still play an important role in the construction of continuous steel box girder bridges.
When searching the optimal layout position of the temporary support, the internal force change of the main beams before and after the support is removed must be considered. On the one hand, the freedom of the continuous steel box girder will increase after the removal of the brackets, allowing for greater deformations and displacements. This may result in changes in the overall shape of the bridge, such as flexing and torsion. Such deformation may result in additional internal forces, particularly in the short term after removal of the stent. On the other hand, removal of the brackets may result in redistribution of internal forces, including axial forces, bending moments, shearing forces, and the like. Internal forces may be transferred from the seat location to other parts of the bridge, resulting in uneven stress conditions. If the stress of the continuous steel box girder exceeds the load-bearing capacity of its material after the bracket is removed, it may cause a decrease in structural durability and safety. Therefore, before the construction brackets are erected, it is necessary to find a suitable position and to perform detailed structural analysis and evaluation of the internal force changes before and after the bracket is installed and removed, so as to ensure that the bridge can safely withstand the new internal force distribution. The three-span continuous steel box girder bridge is a typical steel structure bridge, and a rapid and accurate method for determining the position of a continuous steel box girder bridge construction bracket is needed at present, so that more reliable guarantee is provided for bridge safety and traffic efficiency.
Disclosure of Invention
Aiming at the defects existing in the background technology, the invention provides the rapid determination method for the construction bracket position of the continuous steel box girder bridge facing the optimal stress, which is based on a simplified analysis model of continuous girder mechanics, can effectively reduce the internal force change of the girder before and after the bracket is dismantled in the continuous girder construction process, and improves the construction efficiency and safety.
In order to achieve the above purpose, the invention adopts the following technical scheme: the method for rapidly determining the position of the continuous steel box girder bridge construction bracket facing to the optimal stress comprises the following steps:
step one: internal force calculation section determination
Selecting a pier top section as an internal force calculation section aiming at a three-span continuous steel box girder bridge;
Step two: discretizing the continuous beam in the state of no support
Taking four fulcrums as nodes, sequentially numbering one end of a continuous beam as a node A, a node B, a node C and a node D from the other end of the continuous beam, dispersing the continuous beam into three rod units based on the node positions, sequentially numbering the three rod units as a rod unit AB, a rod unit BC and a rod unit CD, determining the initial end node and the final end node of each rod unit according to the numbering directions, establishing an integral coordinate system with consistent directions and three unit coordinate systems, wherein F represents the node load under the integral coordinate system, delta represents the node displacement, i represents the linear rigidity under the unit coordinate system, P represents the rod end force, delta represents the rod end displacement, in the bracket-free calculation process, an upper corner mark (a) represents the initial end node, (B) represents the final end node, lower corner marks AB, BC and CD respectively represent corresponding rod units, A, B, C and D respectively represent corresponding nodes;
step three: constructing equilibrium equations in a stent-free state
In the unit coordinate system, the balance equation of the rod end force P and the rod end displacement δ is as follows:
the balance condition of the node displacement delta under the integral coordinate system and the rod end displacement delta under the unit coordinate system is established as follows:
The balance condition of the node load F under the whole coordinate system and the rod end force P under the unit coordinate system is established as follows:
and then the balance equation of the node load F and the node displacement delta can be obtained as follows:
step four: constructing a unit stiffness matrix and a total stiffness matrix of each span in a bracket-free state
According to the balance equation of the rod end force P and the rod end displacement delta, the rod end displacement delta is taken as an unknown quantity, and the linear stiffness i is taken as an influence coefficient, and then the stiffness matrix of each rod unit is calculated by the following formula:
according to a balance equation of the node load F and the node displacement delta, taking the node displacement delta as an unknown quantity and the line stiffness i as an influence coefficient, calculating the total stiffness matrix by the following formula:
Wherein,
Wherein E represents the elastic modulus of the continuous beam material, I represents the section moment of inertia of the continuous beam, and L represents the span length of the continuous beam;
Step five: solving the solid end force caused by non-node load in the bracket-free state
For the continuous beam in the construction stage, the non-node loads are all uniform loads, the uniform load is q, and the fixed end forces of the initial end node and the tail end node of the rod unit under the unit coordinate system are respectivelyAnd/>The solid force matrix under the unit coordinate system is listed as follows:
Wherein P' represents the solid end force under the unit coordinate system;
Step six: solid end force coordinate system conversion under bracket-free state
The solid end force under the unit coordinate system is converted into the solid end force under the whole coordinate system, and the conversion formula is as follows:
Wherein P ' represents the solid end force under the whole coordinate system, T represents the coordinate transformation matrix, and the solid end force P ' under the whole coordinate system is equal to the solid end force P ' under the whole coordinate system because the whole coordinate system is consistent with the direction of the unit coordinate system and the coordinate transformation matrix is the unit matrix;
Step seven: solving continuous beam node load in bracket-free state
Considering that there are two solid end forces of the end node of the rod unit AB and the beginning node of the rod unit BC at the node B, there are two solid end forces of the end node of the rod unit BC and the beginning node of the rod unit CD at the node C, the node load matrix of the continuous beam is listed as follows:
Step eight: solving displacement matrix of continuous beam node in bracket-free state
The original stiffness equation of the bridge structure is:
F＝KΔ
The node displacement matrix is:
step nine: solving the rod end force in the bracket-free state
The continuous beam is stressed by non-node load in the construction stage, and the rod end force of each rod unit can be disassembled into the sum of the fixed end force under the unit coordinate system and the rod end force generated by the node load under the integral coordinate system, as shown in the following formula:
P＝P'+kejΔej
Where k ejΔej represents the rod end force generated by the node load, J=1, 2,3, and for Δ, its subscripts 1,2,3,4 correspond to subscripts A, B, C, D in the previous derivation process, respectively;
step ten: discretizing the continuous beam in the state of supporting frame
In order to simplify calculation, considering that the bracket provides vertical support, the three-span continuous beam is considered as a four-span continuous beam uniformly loaded, the sum of span lengths L 1 and L 2 of two spans in the middle of the three-span continuous beam is equal to the span length L of the continuous beam in a bracket-free state, a node O is added between a node B and a node C, the continuous beam is divided into four rod units based on the node position, the rod unit BC is divided into a rod unit BO and a rod unit OC, in the calculation process of the bracket, an upper corner mark (a) is used for representing a starting end node, (B) is used for representing an end node, lower corner marks AB, BO, OC and CD are used for representing corresponding rod units respectively, and A, B, O, C and D are used for representing corresponding nodes respectively;
step eleven: constructing a balance equation in a stent state
In the unit coordinate system, the balance equation of the rod end force P and the rod end displacement δ is as follows:
the balance condition of the node displacement delta under the integral coordinate system and the rod end displacement delta under the unit coordinate system is established as follows:
The balance condition of the node load F under the whole coordinate system and the rod end force P under the unit coordinate system is established as follows:
and then the balance equation of the node load F and the node displacement delta can be obtained as follows:
Step twelve: the span length L of the midspan in the stented state is divided into L 1 and L 2, so that:
according to the balance equation of the rod end force P and the rod end displacement delta, taking the rod end displacement delta as an unknown quantity and taking the linear stiffness i as an influence coefficient, the stiffness matrix of each rod unit is expressed as follows:
according to a balance equation of the node load F and the node displacement delta, taking the node displacement delta as an unknown quantity and the line stiffness i as an influence coefficient, calculating the total stiffness matrix by the following formula:
Step thirteen: solving the solid end force caused by non-node load in the state of a bracket
Considering the influence of stent introduction, the solid-end force matrix under the unit coordinate system is listed as follows:
The solid end force P 'under the unit coordinate system is considered to be equal to the solid end force P' under the whole coordinate system;
step fourteen: solving the load of the continuous beam node in the state of the bracket
Considering that two solid end forces of a rod unit AB end node and a rod unit BO beginning end node exist at a node B, two solid end forces of a rod unit BO end node and a rod unit OC beginning end node exist at a node O, two solid end forces of a rod unit OC end node and a rod unit CD beginning end node exist at a node C, the node load matrix of the continuous beam is listed as follows:
Fifteen steps: solving displacement matrix of continuous beam node under bracket state
Since the total stiffness matrix and the node load matrix are known, in combination with the original stiffness equation of the bridge structure, the node displacement matrix is expressed as:
step sixteen: solving the rod end force in the state of supporting
The node displacement corresponding to each node is obtained through the node displacement matrix, and the rod end force of each rod unit is calculated as follows in consideration of the influence of the bracket introduction:
P＝P'+kejΔej
in the method, in the process of the invention, And for delta, subscripts 1,2,3,4,5 correspond to subscripts A, B, O, C, D, respectively, of the previous derivation process;
Seventeenth step: determining the optimal erection position of the bracket
For the continuous beam under the action of the bracket, under the condition that the span length L of the midspan is fixed, the different values of L 1 and L 2 can cause the change of the internal force of the girder at the section of the pier top, so the bracket is erected according to the symmetrical position during construction, the position of the bracket at one side is determined, the position of the bracket at the other side can be correspondingly obtained, the change rate S is calculated according to the change of the internal force of the girder at the section of the pier top under the action of the bracket, the optimal erection position of the bracket is determined according to the value of L 1 and L 2 corresponding to the minimum change rate, and the change rate is calculated as follows:
Wherein S represents the change rate, M 1 represents the girder internal force at the pier top section in the non-bracket state, and M 2 represents the girder internal force at the pier top section in the bracket state.
Compared with the prior art, the invention has the beneficial effects that: the invention reasonably simplifies the boundary conditions of the three-span continuous steel box girder bridge in the state of the bracket, analyzes the internal force of the main girder by utilizing a matrix displacement method, unifies the rigidity matrix and the displacement matrix of the main girder in different supporting states in form, improves the calculation efficiency, has universal applicability, has practical value for the three-span continuous steel box girder bridge with equal span and unequal span, can calculate the internal force of the main girder in the state of the bracket supporting state only by bringing each span into the corresponding matrix according to actual engineering, and provides a new implementation thought for the determination of the construction bracket position of the continuous steel box girder bridge.
Drawings
FIG. 1 is a schematic view of the discrete structure of a three-span continuous steel box girder bridge of the present invention in a bracket-free state;
fig. 2 is a schematic view of the discrete structure of the three-span continuous steel box girder bridge in the state of the bracket according to the present invention.
Detailed Description
The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention, and it is apparent that the described embodiments are only some embodiments of the invention, but not all embodiments, and all other embodiments obtained by those skilled in the art without making creative efforts based on the embodiments of the present invention are all within the protection scope of the present invention.
As shown in fig. 1-2, the method for quickly determining the position of the construction bracket of the continuous steel box girder bridge facing the optimal stress comprises the following steps:
step one: internal force calculation section determination
Selecting a pier top section as an internal force calculation section aiming at a three-span continuous steel box girder bridge;
Step two: discretizing the continuous beam in the state of no support
With reference to fig. 1, four fulcrums are used as nodes, and the continuous beam is sequentially numbered as a node a, a node B, a node C and a node D from one end to the other end, is discretized into three rod units based on the node positions, and is sequentially numbered as a rod unit AB, a rod unit BC and a rod unit CD, and the start node and the end node of each rod unit are determined according to the numbering direction.
And establishing an integral coordinate system and three unit coordinate systems, wherein the integral coordinate system takes a node A as an origin, takes a node D direction as a positive direction of an x axis, takes a vertical downward direction of the node A as a positive direction of a y axis, and the unit coordinate systems respectively take a starting node of each rod unit as the origin, take a terminal node direction as a positive direction of the x axis and take a vertical downward direction of the starting node as a positive direction of the y axis. In the overall coordinate system, F represents the node load, Δ represents the node displacement, in the unit coordinate system, i represents the line stiffness, P represents the rod end force, and δ represents the rod end displacement. In the calculation process without a bracket, an upper corner mark (a) is used for representing a start node, (b) is used for representing an end node, lower corner marks AB, BC and CD are used for respectively representing corresponding rod units, A, B, C and D are used for respectively representing corresponding nodes;
step three: constructing equilibrium equations in a stent-free state
In the unit coordinate system, the balance equation of the rod end force P and the rod end displacement δ is as follows:
the balance condition of the node displacement delta under the integral coordinate system and the rod end displacement delta under the unit coordinate system is established as follows:
The balance condition of the node load F under the whole coordinate system and the rod end force P under the unit coordinate system is established as follows:
and then the balance equation of the node load F and the node displacement delta can be obtained as follows:
step four: constructing a unit stiffness matrix and a total stiffness matrix of each span in a bracket-free state
According to the balance equation of the rod end force P and the rod end displacement delta, the rod end displacement delta is taken as an unknown quantity, and the linear stiffness i is taken as an influence coefficient, and then the stiffness matrix of each rod unit is calculated by the following formula:
according to a balance equation of the node load F and the node displacement delta, taking the node displacement delta as an unknown quantity and the line stiffness i as an influence coefficient, calculating the total stiffness matrix by the following formula:
Wherein,
Wherein E represents the elastic modulus of the continuous beam material, I represents the section moment of inertia of the continuous beam, and L represents the span length of the continuous beam;
Step five: solving the solid end force caused by non-node load in the bracket-free state
For the continuous beam in the construction stage, the non-node loads are all uniform loads, the uniform load is set as q, any point is selected to solve the solid end force, the distance from the point to the initial end node of each rod unit is m, and under the condition that the node load is not considered, the general formula for solving the solid end force of each rod unit under the unit coordinate system is as follows:
The above formula is suitable for solving the internal force at any point in the rod unit, and for the rod end force, only the initial end node and the final end node of the rod unit are considered, so that m has only two values of 0 and L, and the fixed end forces of the initial end node and the final end node of the rod unit under the unit coordinate system are respectively And/>Correspondingly, the solid-to-end force matrix under the unit coordinate system is listed as follows: /(I)
Wherein P' represents the solid end force under the unit coordinate system;
Step six: solid end force coordinate system conversion under bracket-free state
Through the relation between the unit coordinate system and the whole coordinate system, the solid end force under the unit coordinate system is converted into the solid end force under the whole coordinate system, and the conversion formula is as follows:
wherein P ' represents the solid end force under the whole coordinate system, T represents the coordinate transformation matrix, the form of the coordinate transformation matrix is only related to the included angle between the whole coordinate system and the unit coordinate system, the included angle is 0 because the directions of the two coordinate systems are consistent, and the corresponding coordinate transformation matrix is the unit matrix, so that the solid end force P ' under the unit coordinate system and the solid end force P ' under the whole coordinate system can be considered to be equal in the follow-up deduction;
Step seven: solving continuous beam node load in bracket-free state
On the basis of completing the conversion of the solid end forces, considering that two solid end forces of a rod unit AB end node and a rod unit BC beginning end node exist at a node B, two solid end forces of a rod unit BC end node and a rod unit CD beginning end node exist at a node C, the node load matrix of the continuous beam is listed as follows:
Step eight: solving displacement matrix of continuous beam node in bracket-free state
The original stiffness equation of the bridge structure is:
F＝KΔ
The original stiffness equation of the bridge structure can be transformed into:
step nine: solving the rod end force in the bracket-free state
The continuous beam is stressed by non-node load in the construction stage, and the rod end force of each rod unit can be disassembled into the sum of the fixed end force under the unit coordinate system and the rod end force generated by the node load under the integral coordinate system, as shown in the following formula:
P＝P'+kejΔej
Where k ejΔej represents the rod end force generated by the node load, J=1, 2,3, and for Δ, its subscripts 1,2,3,4 correspond to subscripts A, B, C, D in the previous derivation process, respectively;
step ten: discretizing the continuous beam in the state of supporting frame
In connection with fig. 2, for simplicity of calculation, it is considered that the bracket provides vertical support, which is equivalent to cutting off and arranging the brackets in the midspan of the continuous beam, and the three-span continuous beam is considered as a four-span continuous beam uniformly loaded, so that the sum of span lengths L 1 and L 2 of the middle two spans is equal to the span length L of the midspan of the continuous beam in the bracket-free state. Adding a node O between a node B and a node C, dispersing the continuous beam into four rod units based on the node position, dividing the rod unit BC into a rod unit BO and a rod unit OC, in the calculation process of the bracket, using an upper corner mark (a) to represent a starting end node, (B) to represent an ending end node, using lower corner marks AB, BO, OC and CD to represent corresponding rod units respectively, and A, B, O, C and D to represent corresponding nodes respectively;
step eleven: constructing a balance equation in a stent state
In the unit coordinate system, the balance equation of the rod end force P and the rod end displacement δ is as follows:
the balance condition of the node displacement delta under the integral coordinate system and the rod end displacement delta under the unit coordinate system is established as follows:
The balance condition of the node load F under the whole coordinate system and the rod end force P under the unit coordinate system is established as follows:
and then the balance equation of the node load F and the node displacement delta can be obtained as follows:
step twelve: cell stiffness matrix and total stiffness matrix of each span in bracket state
The linear stiffness i of each rod end member is considered to be equal in the non-stented state, whereas the span length L of the midspan in the stented state is divided into L 1 and L 2, and thus can be considered as:
according to the balance equation of the rod end force P and the rod end displacement delta, taking the rod end displacement delta as an unknown quantity and taking the linear stiffness i as an influence coefficient, the stiffness matrix of each rod unit is expressed as follows:
/>
according to a balance equation of the node load F and the node displacement delta, taking the node displacement delta as an unknown quantity and the line stiffness i as an influence coefficient, calculating the total stiffness matrix by the following formula:
Step thirteen: solving the solid end force caused by non-node load in the state of a bracket
The derivation process of the solid end force of each rod unit under the non-bracket state is analogized, and the solid end force matrix under the unit coordinate system is listed as follows in consideration of the influence of bracket introduction:
The directions of the whole coordinate system and the unit coordinate system in the bracket state are consistent with those in the bracket-free state, so that the coordinate transformation matrix T is also an identity matrix, and the solid end force P 'in the unit coordinate system can be considered to be equal to the solid end force P' in the whole coordinate system;
step fourteen: solving the load of the continuous beam node in the state of the bracket
Considering that two solid end forces of a rod unit AB end node and a rod unit BO beginning end node exist at a node B, two solid end forces of a rod unit BO end node and a rod unit OC beginning end node exist at a node O, two solid end forces of a rod unit OC end node and a rod unit CD beginning end node exist at a node C, the node load matrix of the continuous beam is listed as follows:
Fifteen steps: solving displacement matrix of continuous beam node under bracket state
Since the total stiffness matrix and the node load matrix are known, in combination with the original stiffness equation of the bridge structure, the node displacement matrix can be expressed as:
step sixteen: solving the rod end force in the state of supporting
Substituting the corresponding L, L 1 and L 2 into a node displacement matrix to obtain the node displacement corresponding to each node, and comparing the derivation process of the rod end force of each rod unit in a bracket-free state, wherein the rod end force of each rod unit is calculated as follows in consideration of the influence of bracket introduction:
P＝P'+kejΔej
in the method, in the process of the invention, And for delta, subscripts 1,2,3,4,5 correspond to subscripts A, B, O, C, D, respectively, of the previous derivation process;
Seventeenth step: determining the optimal erection position of the bracket
For the continuous beam under the action of the bracket, under the condition that the span length L of the midspan is fixed, the different values of L 1 and L 2 can cause the change of the internal force of the girder at the section of the pier top, so that the bracket is erected according to the symmetrical position during construction, the position of the bracket at the other side can be correspondingly obtained only by determining the position of the bracket at one side, the change rate S is calculated according to the change of the internal force of the girder at the section of the pier top under the action of the bracket or not for simplifying calculation, the optimal erection position of the bracket is determined by finding out the values of L 1 and L 2 corresponding to the minimum change rate, and the change rate calculation formula is as follows:
Wherein S represents the change rate, M 1 represents the girder internal force at the pier top section in the non-bracket state, and M 2 represents the girder internal force at the pier top section in the bracket state.
Examples
Taking AK0+196.61 sections of overpass in iron high-speed five-common-western interchange overpass engineering as an example, the method is used for rapidly determining the optimal erection position of the three-span continuous steel box girder bridge bracket, and specifically comprises the following steps:
S1, aiming at the overpass, taking the pier top section of the overpass as an internal force calculation section;
S2, dispersing the continuous beam into three rod units by using a node A, a node B, a node C and a node D, and sequentially numbering the three rod units as a rod unit AB, a rod unit BC and a rod unit CD, so as to establish an integral coordinate system and three unit coordinate systems. The elastic modulus E= 206000000kN/m 2 of the main beam, the cross-sectional area A of the main beam=0.45 m 2, the cross-sectional moment of inertia I=1.72 m 4 of the main beam, the span length L=50m of the main beam, the main beam material is Q355 steel, the volume weight gamma=78.5 kN/m 3 of the main beam material is adopted, and the uniform load q=35.50 kN of the main beam material is uniformly distributed. For the continuous beam in the bracket-free state, the linear rigidity i of each rod unit is consistent, and the specific numerical value is calculated according to the following formula:
And S3, under a unit coordinate system, a balance equation of the rod end force P and the rod end displacement delta is as follows:
the balance condition of the node displacement delta under the integral coordinate system and the rod end displacement delta under the unit coordinate system is as follows:
the balance condition of the node load F under the integral coordinate system and the rod end force P under the unit coordinate system is as follows:
the equilibrium equation for the node load F and the node displacement delta is as follows:
s4, according to a balance equation of the rod end force P and the rod end displacement delta, the stiffness matrix of each rod unit is as follows:
according to the balance equation of the node load F and the node displacement delta, the total stiffness matrix is as follows:
S5, for the continuous beam in the construction stage, the non-node loads are uniformly distributed, and the fixing end force of each rod unit under the unit coordinate system is as follows:
the solid force matrix under the unit coordinate system is as follows:
/>
S6, because the direction of the whole coordinate system is consistent with that of the unit coordinate system when the coordinate system is constructed, and the coordinate transformation matrix is the identity matrix, in the process of transforming the solid end force under the unit coordinate system into the solid end force under the whole coordinate system, the solid end force under the whole coordinate system is as follows:
S7, on the basis of finishing the solid end force conversion, considering that two solid end forces, namely a rod unit AB end node and a rod unit BC beginning end node, exist at a node B, and two solid end forces, namely a rod unit BC end node and a rod unit CD beginning end node, exist at a node C, and combining the relation between the solid end force and the node load after the continuous beam is scattered, the node load matrix is as follows:
S8, according to an original stiffness equation of the bridge structure, carrying out a total stiffness matrix K and a node load matrix F, wherein the node displacement matrix is as follows:
s9, the left pier top section is correspondingly provided with a node B, and the rod end force of the rod unit BC is as follows:
in the bracket-free state, the bending moment value corresponding to the node B of the left pier top section is as follows:
MB＝M1＝-36823.34kN·m
S10, for simplifying calculation, considering that the bracket provides vertical support, which is equivalent to cutting off and arranging the support in the middle span of the continuous beam, and considering that the three-span continuous beam is a four-span continuous beam uniformly loaded, so that the sum of span lengths L 1 and L 2 of the middle two spans is equal to the span length L of the middle span of the continuous beam in a bracket-free state. Adding a node O between the node B and the node C, and dispersing the continuous beam into four rod units based on the node position, wherein the rod unit BC is divided into a rod unit BO and a rod unit OC;
And S11, under a unit coordinate system, a balance equation of the rod end force P and the rod end displacement delta is as follows:
the balance condition of the node displacement delta under the integral coordinate system and the rod end displacement delta under the unit coordinate system is as follows:
the balance condition of the node load F under the integral coordinate system and the rod end force P under the unit coordinate system is as follows:
the equilibrium equation for the node load F and the node displacement delta is as follows:
s12, the linear rigidity i of each rod end member in the bracket state is as follows:
According to the balance equation of rod end force P and rod end displacement delta, each rod unit stiffness matrix is expressed as:
according to the balance equation of the node load F and the node displacement delta, the total stiffness matrix is expressed as:
s13, for the continuous beam in the construction stage, the non-node loads are uniformly distributed, and the fixed end force of each rod unit under the unit coordinate system is as follows:
the solid force matrix under the unit coordinate system is as follows:
S14, considering that two solid end forces of a rod unit AB end node and a rod unit BO beginning end node exist at a node B, two solid end forces of a rod unit BO end node and a rod unit OC beginning end node exist at a node O, two solid end forces of a rod unit OC end node and a rod unit CD beginning end node exist at a node C, and the node load matrix of the continuous beam is listed as follows:
s15, according to an original stiffness equation of the bridge structure, carrying out a total stiffness matrix K and a node load matrix F, wherein the node displacement matrix is as follows:
Since the sum of L 1 and L 2 is l=50m, in order to reduce the unknown parameter, L 1 is represented by the unknown parameter x, and L 2 can be replaced by 50-x to be carried into calculation, and the displacements of node a and node B are respectively represented as:
S16, the left pier top section is correspondingly provided with a node B, and the rod end force of the rod unit AB is as follows:
in the state of the bracket, the bending moment value corresponding to the node B of the left pier top section is as follows:
S17, calculating the change rate of the change of the internal force of the main beam at the section of the pier top, wherein the change rate is as follows:
And defining the range of x as [0,50], carrying out iterative calculation on S in the MATLAB program, wherein x corresponding to the S minimum value is the optimal position of one side bracket, and finally determining the optimal position L 2 = 38.62m of the bracket symmetrically arranged with x = L 1 = 11.38 m.
It will be evident to those skilled in the art that the invention is not limited to the details of the foregoing illustrative embodiments, and that the present invention may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. Any reference sign in a claim should not be construed as limiting the claim concerned.
Furthermore, it should be understood that although the present disclosure describes embodiments, not every embodiment is provided with a separate embodiment, and that this description is provided for clarity only, and that the disclosure is not limited to the embodiments described in detail below, and that the embodiments described in the examples may be combined as appropriate to form other embodiments that will be apparent to those skilled in the art.
Claims (1)
1. The method for rapidly determining the position of the continuous steel box girder bridge construction bracket facing to the optimal stress is characterized by comprising the following steps of: the method comprises the following steps:
step one: internal force calculation section determination
Selecting a pier top section as an internal force calculation section aiming at a three-span continuous steel box girder bridge;
Step two: discretizing the continuous beam in the state of no support
Taking four fulcrums as nodes, sequentially numbering one end of a continuous beam as a node A, a node B, a node C and a node D from the other end of the continuous beam, dispersing the continuous beam into three rod units based on the node positions, sequentially numbering the three rod units as a rod unit AB, a rod unit BC and a rod unit CD, determining the initial end node and the final end node of each rod unit according to the numbering directions, establishing an integral coordinate system with consistent directions and three unit coordinate systems, wherein F represents the node load under the integral coordinate system, delta represents the node displacement, i represents the linear rigidity under the unit coordinate system, P represents the rod end force, delta represents the rod end displacement, in the bracket-free calculation process, an upper corner mark (a) represents the initial end node, (B) represents the final end node, lower corner marks AB, BC and CD respectively represent corresponding rod units, A, B, C and D respectively represent corresponding nodes;
step three: constructing equilibrium equations in a stent-free state
In the unit coordinate system, the balance equation of the rod end force P and the rod end displacement δ is as follows:
the balance condition of the node displacement delta under the integral coordinate system and the rod end displacement delta under the unit coordinate system is established as follows:
The balance condition of the node load F under the whole coordinate system and the rod end force P under the unit coordinate system is established as follows:
and then the balance equation of the node load F and the node displacement delta can be obtained as follows:
step four: constructing a unit stiffness matrix and a total stiffness matrix of each span in a bracket-free state
According to the balance equation of the rod end force P and the rod end displacement delta, the rod end displacement delta is taken as an unknown quantity, and the linear stiffness i is taken as an influence coefficient, and then the stiffness matrix of each rod unit is calculated by the following formula:
according to a balance equation of the node load F and the node displacement delta, taking the node displacement delta as an unknown quantity and the line stiffness i as an influence coefficient, calculating the total stiffness matrix by the following formula:
Wherein,
Wherein E represents the elastic modulus of the continuous beam material, I represents the section moment of inertia of the continuous beam, and L represents the span length of the continuous beam;
Step five: solving the solid end force caused by non-node load in the bracket-free state
For the continuous beam in the construction stage, the non-node loads are all uniform loads, the uniform load is q, and the fixed end forces of the initial end node and the tail end node of the rod unit under the unit coordinate system are respectivelyAnd/>The solid force matrix under the unit coordinate system is listed as follows:
Wherein P' represents the solid end force under the unit coordinate system;
Step six: solid end force coordinate system conversion under bracket-free state
The solid end force under the unit coordinate system is converted into the solid end force under the whole coordinate system, and the conversion formula is as follows:
Wherein P ' represents the solid end force under the whole coordinate system, T represents the coordinate transformation matrix, and the solid end force P ' under the whole coordinate system is equal to the solid end force P ' under the whole coordinate system because the whole coordinate system is consistent with the direction of the unit coordinate system and the coordinate transformation matrix is the unit matrix;
Step seven: solving continuous beam node load in bracket-free state
Considering that there are two solid end forces of the end node of the rod unit AB and the beginning node of the rod unit BC at the node B, there are two solid end forces of the end node of the rod unit BC and the beginning node of the rod unit CD at the node C, the node load matrix of the continuous beam is listed as follows:
Step eight: solving displacement matrix of continuous beam node in bracket-free state
The original stiffness equation of the bridge structure is:
F＝KΔ
The node displacement matrix is:
step nine: solving the rod end force in the bracket-free state
The continuous beam is stressed by non-node load in the construction stage, and the rod end force of each rod unit can be disassembled into the sum of the fixed end force under the unit coordinate system and the rod end force generated by the node load under the integral coordinate system, as shown in the following formula:
P＝P'+kejΔej
Where k ejΔej represents the rod end force generated by the node load, J=1, 2,3, and for Δ, its subscripts 1,2,3,4 correspond to subscripts A, B, C, D in the previous derivation process, respectively;
step ten: discretizing the continuous beam in the state of supporting frame
In order to simplify calculation, considering that the bracket provides vertical support, the three-span continuous beam is considered as a four-span continuous beam uniformly loaded, the sum of span lengths L 1 and L 2 of two spans in the middle of the three-span continuous beam is equal to the span length L of the continuous beam in a bracket-free state, a node O is added between a node B and a node C, the continuous beam is divided into four rod units based on the node position, the rod unit BC is divided into a rod unit BO and a rod unit OC, in the calculation process of the bracket, an upper corner mark (a) is used for representing a starting end node, (B) is used for representing an end node, lower corner marks AB, BO, OC and CD are used for representing corresponding rod units respectively, and A, B, O, C and D are used for representing corresponding nodes respectively;
step eleven: constructing a balance equation in a stent state
In the unit coordinate system, the balance equation of the rod end force P and the rod end displacement δ is as follows:
the balance condition of the node displacement delta under the integral coordinate system and the rod end displacement delta under the unit coordinate system is established as follows:
The balance condition of the node load F under the whole coordinate system and the rod end force P under the unit coordinate system is established as follows:
and then the balance equation of the node load F and the node displacement delta can be obtained as follows:
step twelve: cell stiffness matrix and total stiffness matrix of each span in bracket state
The span length L of the midspan in the stented condition is divided into L 1 and L 2, so:
according to the balance equation of the rod end force P and the rod end displacement delta, taking the rod end displacement delta as an unknown quantity and taking the linear stiffness i as an influence coefficient, the stiffness matrix of each rod unit is expressed as follows:
according to a balance equation of the node load F and the node displacement delta, taking the node displacement delta as an unknown quantity and the line stiffness i as an influence coefficient, calculating the total stiffness matrix by the following formula:
Step thirteen: solving the solid end force caused by non-node load in the state of a bracket
Considering the influence of stent introduction, the solid-end force matrix under the unit coordinate system is listed as follows:
The solid end force P 'under the unit coordinate system is considered to be equal to the solid end force P' under the whole coordinate system;
step fourteen: solving the load of the continuous beam node in the state of the bracket
Considering that two solid end forces of a rod unit AB end node and a rod unit BO beginning end node exist at a node B, two solid end forces of a rod unit BO end node and a rod unit OC beginning end node exist at a node O, two solid end forces of a rod unit OC end node and a rod unit CD beginning end node exist at a node C, the node load matrix of the continuous beam is listed as follows:
Fifteen steps: solving displacement matrix of continuous beam node under bracket state
Since the total stiffness matrix and the node load matrix are known, in combination with the original stiffness equation of the bridge structure, the node displacement matrix is expressed as:
step sixteen: solving the rod end force in the state of supporting
The node displacement corresponding to each node is obtained through the node displacement matrix, and the rod end force of each rod unit is calculated as follows in consideration of the influence of the bracket introduction:
P＝P'+kejΔej
in the method, in the process of the invention, And for delta, subscripts 1,2,3,4,5 correspond to subscripts A, B, O, C, D, respectively, of the previous derivation process;
Seventeenth step: determining the optimal erection position of the bracket
For the continuous beam under the action of the bracket, under the condition that the span length L of the midspan is fixed, the different values of L 1 and L 2 can cause the change of the internal force of the girder at the section of the pier top, so the bracket is erected according to the symmetrical position during construction, the position of the bracket at one side is determined, the position of the bracket at the other side can be correspondingly obtained, the change rate S is calculated according to the change of the internal force of the girder at the section of the pier top under the action of the bracket, the optimal erection position of the bracket is determined according to the value of L 1 and L 2 corresponding to the minimum change rate, and the change rate is calculated as follows:
Wherein S represents the change rate, M 1 represents the girder internal force at the pier top section in the non-bracket state, and M 2 represents the girder internal force at the pier top section in the bracket state.
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| CN102493354A (en) * | 2011-12-23 | 2012-06-13 | 中铁大桥局股份有限公司 | Whole-unit incremental launching construction method for high-speed railway prestressed concrete continuous beam |
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