CN106503383B - A kind of interim inverted arch or the stress and deformation analysis method of stull - Google Patents
A kind of interim inverted arch or the stress and deformation analysis method of stull Download PDFInfo
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Abstract
The present invention relates to Tunnel Engineering fields, and in particular to a kind of interim inverted arch or the stress and deformation analysis method of stull.The analysis method is the following steps are included: (1) establishes the elastic foundation Curved Beams model of interim inverted arch (stull)；(2) curve differential equation of interim inverted arch (stull) is established, and is solved to obtain the solution of differential equation general type；(3) interim inverted arch (stull) internal force of arbitrary section and deformation expression formula when both ends act on identical shearing, moment of flexure and axle power are acquired；(4) it obtains the constituent parts displacement at steelframe arch springing needed for interim inverted arch (stull) and steelframe integrated solution and carries to conjugate；(5) redundant force at steelframe vault and arch springing is acquired；(6) determine that arch steelframe acts on shearing, moment of flexure and the axle power value at interim inverted arch (stull) both ends；(7) amount of deflection, corner, moment of flexure, shearing and the subgrade reaction value for obtaining interim inverted arch (stull) arbitrary section under the load action that arch steelframe transmits, are finally completed interim inverted arch (stull) stress and deformation analysis.
Description
One, technical field:
The present invention relates to Tunnel Engineering fields, and in particular to the stress and deformation analysis side of a kind of interim inverted arch or stull
Method.
Two, background technique:
Interim inverted arch is set or stull is a kind of soft stratum tunnel commonly country rock large deformation control measure.Setting is interim
Supporting construction is closed in inverted arch or stull in time, can not only inhibit the convergent deformation of country rock rapidly, but also can share just Zhi Gang
The foundation load of arching foot is to reduce arch sedimentation and the ground settlement in tunnel.At present about interim inverted arch or the application of stull
Report is existing very much, some researchs has also been carried out in function and effect, but existing research conclusion has only generally been affirmed and temporarily faced upward
Encircle the function and effect in terms of inhibiting country rock convergent deformation, and shares steelframe arch springing foundation load about interim inverted arch or stull
And the aspect equally to merit attention in the construction such as reduction tunnel subsidence, existing research are not related to.In addition, China is existing
Associated tunnel design specification do not provide the mechanics model and corresponding force analysis method of interim inverted arch or stull, lead
Cause interim inverted arch or stull that can only choose design parameter by rule of thumb in design.With constantly gushing for large span soft stratum tunnel
Existing, interim inverted arch or stull will be used largely.In view of the interim inverted arch in tunnel or stull and permanent inverted arch construction technology and institute
The difference of supported condition, in the case where lacking construction experience, seek it is a kind of suitable for interim inverted arch or the stress of stull with
Deformation analysis method provides theoretical foundation as the task of top priority for design.
Three, summary of the invention
It is of the invention that the stress and deformation analysis method of a kind of interim inverted arch or stull are provided.
To achieve the above object, the technical solution adopted by the present invention are as follows: a kind of interim inverted arch or the stress and deformation of stull
Analysis method, it is characterised in that: the method and step are as follows:
Step 1): the elastic foundation Curved Beams model of interim inverted arch or stull is established；
Step 2): the elastic foundation Curved Beams model of the interim inverted arch or stull established according to step 1), establish interim
Inverted arch or the curve differential equation of stull are solved and obtain interim inverted arch or stull amount of deflection, moment of flexure, shearing, axle power and section turn
The general expression at angle；
Step 3): interim inverted arch or stull amount of deflection, moment of flexure, shearing, axle power and the sectional twisting angle obtained according to step 2)
General expression, it is any when both ends act on identical shearing, moment of flexure and axle power using interim inverted arch or stull is further acquired
The internal force and deformation expression formula in section；
Step 4): according to interim inverted arch obtained by step 3) or stull when identical shearing, moment of flexure and axle power are born in both ends
The internal force and deformation expression formula of arbitrary section, further obtain steelframe needed for interim inverted arch or stull and steelframe integrated solution
Constituent parts displacement and load displacement at arch springing；
Step 5): displacement is conjugated and carried according to the constituent parts at steelframe arch springing obtained by step 4), solves to obtain using force method
Redundant force at steelframe vault and arch springing；
Step 6): the redundant force at steelframe vault and arch springing obtained according to step 5) is determined and is acted on temporarily
Inverted arch or the shearing at stull both ends, moment of flexure and axle power value；
Step 7): shearing, moment of flexure and the axle power value and step of interim inverted arch or stull both ends that step 6) obtains are utilized
3) the interim inverted arch obtained or the stull internal force of arbitrary section and change when both ends are respectively subjected to identical shearing, moment of flexure and axle power
Shape expression formula obtains under the load action that arch steelframe transmits the amount of deflection of interim inverted arch or stull arbitrary section, corner, curved
Square, shearing and subgrade reaction value are finally completed the stress and deformation analysis of interim inverted arch or stull.
The interim inverted arch or stull elastic foundation curved Beam Model that the step 1) is established should meet the following conditions:
A: temporarily inverted arch or stull is the Curved Beams on elastic foundation, and it is false that suffered radial direction subgrade reaction obeys Wen Keer
It is fixed；
B: temporarily inverted arch or stull are elastomer, obey Hooke's law；
C: interim inverted arch or stull thickness compare arc radius very little, belong to thin beam.
Compared with prior art, the invention has the advantages that and effect:
The present invention combines the Specific construction technique of interim inverted arch or stull, by considering arch steelframe and interim inverted arch inverted arch
Or stull and arch springing underlie ground common carrying and compatible deformation, provide a kind of suitable for interim inverted arch or the stress of stull
With deformation analysis method, theoretical foundation is provided for interim inverted arch or stull design, avoids interim inverted arch or stull in design
Design parameter and existing blindness are only chosen by rule of thumb.
Four, Detailed description of the invention:
Fig. 1 is interim inverted arch or stull mechanics model and its force analysis method flow chart；
Fig. 2 is interim inverted arch or stull mechanics model schematic diagram；
Fig. 3 is interim inverted arch or micro- section of stull of stress balance schematic diagram；
Fig. 4 is that interim inverted arch or stull both ends act on identical moment of flexure schematic diagram；
Fig. 5 is that interim inverted arch or stull both ends act on identical shearing schematic diagram；
Fig. 6 is that interim inverted arch or stull both ends act on identical axle power schematic diagram；
Fig. 7 is the force method primary structure schematic diagram of steelframe and interim inverted arch or stull；
Five, specific embodiment
Technical solution of the present invention is described in further detail below with reference to specific embodiment:
A kind of interim inverted arch or the stress and deformation analysis method and step of stull are as follows:
Step 1): the elastic foundation Curved Beams model of interim inverted arch or stull is established；
Step 2): the elastic foundation Curved Beams model of the interim inverted arch or stull established according to step 1), establish interim
Inverted arch or the curve differential equation of stull are solved and obtain interim inverted arch or stull amount of deflection, moment of flexure, shearing, axle power and section turn
The general expression at angle；
Step 3): interim inverted arch or stull amount of deflection, moment of flexure, shearing, axle power and the sectional twisting angle obtained according to step 2)
General expression, it is any when both ends act on identical shearing, moment of flexure and axle power using interim inverted arch or stull is further acquired
The internal force and deformation expression formula in section；
Step 4): according to interim inverted arch obtained by step 3) or stull when identical shearing, moment of flexure and axle power are born in both ends
The internal force and deformation expression formula of arbitrary section, further obtain steelframe needed for interim inverted arch or stull and steelframe integrated solution
Constituent parts displacement and load displacement at arch springing；
Step 5): displacement is conjugated and carried according to the constituent parts at steelframe arch springing obtained by step 4), solves to obtain using force method
Redundant force at steelframe vault and arch springing；
Step 6): the redundant force at steelframe vault and arch springing obtained according to step 5) is determined and is acted on temporarily
Inverted arch or the shearing at stull both ends, moment of flexure and axle power value；
Step 7): shearing, moment of flexure and the axle power value and step of interim inverted arch or stull both ends that step 6) obtains are utilized
3) the interim inverted arch obtained or the stull internal force of arbitrary section and change when both ends are respectively subjected to identical shearing, moment of flexure and axle power
Shape expression formula obtains under the load action that arch steelframe transmits the amount of deflection of interim inverted arch or stull arbitrary section, corner, curved
Square, shearing and subgrade reaction value are finally completed the stress and deformation analysis of interim inverted arch or stull.
The interim inverted arch or stull elastic foundation curved Beam Model that the step 1) is established should meet the following conditions:
A: temporarily inverted arch or stull is the Curved Beams on elastic foundation, and it is false that suffered radial direction subgrade reaction obeys Wen Keer
It is fixed；
B: temporarily inverted arch or stull are elastomer, obey Hooke's law；
C: interim inverted arch or stull thickness compare arc radius very little, belong to thin beam.
Embodiment:
As shown in Figure 1, initially setting up the elastic foundation curved Beam Model of interim inverted arch (stull), end is passed by arch steelframe
The moment M passedj, shearing QjWith axle power Nj, while the counter-force by the ground that underlies, as shown in Fig. 2, and meet the following conditions:
(1) interim inverted arch (stull) is the Curved Beams on elastic foundation, and it is false that suffered radial direction subgrade reaction obeys Wen Keer
It is fixed；
(2) interim inverted arch (stull) is elastomer, obeys Hooke's law；
(3) interim inverted arch (stull) thickness compares arc radius very little, belongs to thin beam.
Under arc-coordinate system, any micro- section that radius of curvature is r is intercepted from interim inverted arch (stull),WithShearing Q and Q+dQ, moment M and M+dM, axle power N and N+dN and subgrade reaction pdx are accompanied by section respectively, each power
Positive direction is as shown in Figure 3.According to the stress balance and micro- section of equalising torque of diametrical direction and tangential direction, can obtain as follows
Three equilibrium equations:
DM=Qdx (3)
By p=Kb0Y,Formula (1)~(3) are substituted into respectively, can be obtained
Formula (3) seeks first derivative to x again, has
Formula (5) are substituted into formula (6), can be obtained
Using formula (3), formula (7) is further indicated that are as follows:
The differential equation of elastic curve when to beam flexure
Single order and three order derivatives are asked respectively, are had
Formula (10) are first substituted into formula (8), then again can obtain formula (8) substitution formula (11):
Above formula is fundamental differential when interim inverted arch (stull) is considered as elastic foundation Curved Beams.Make substitution of variable,
It enablesThen formula (12) can further indicate that are as follows:
The general solution of formula (13) is represented by
Wherein:
Formula (14) are substituted into formula (1)~(3) and formula (9), the moment of flexure in interim inverted arch (stull) any section can be obtained, shear and
Axle power expression formula is as follows:
If at coordinate originThe corner in section is θ0, then the corner in any section of interim inverted arch (stull) can indicate
Are as follows:
Since the radius r of interim inverted arch (stull) is larger, generally it is all satisfiedIt can be considered flat curved beam.Have at this time,
α*≈β*≈rλ*=ρ, whereinCorrespondingly, differential equation when flexure (formula (9)) can approximate representation are as follows:
It can further obtain
And then it can be obtained by formula (4):
If taking symmetry axis is vertical reference axis, have at symmetrical shaft section:WithBy formula
(14) it can be further indicated that with (16) are as follows: α*C2+β*C3=0, β*C2-α*C3=0, C can be obtained2=C3=0.
So, the general expression of amount of deflection can be simplified shown as when interim inverted arch (stull) is flat curved beam:
The general expression of moment of flexure can be simplified shown as when interim inverted arch is flat curved beam:
The general expression that interim inverted arch shears when being flat curved beam can be simplified shown as:
The general expression of axle power can be simplified shown as when interim inverted arch is flat curved beam:
N=rKb0C0 (25)
In a symmetrical case, in sectionThere is θ at place0=0 and u0=0, and then formula (23) substitution formula (18) can must be faced
When inverted arch be flat curved beam when sectional twisting angle general expression are as follows:
Below according to interim inverted arch (stull) amount of deflection, moment of flexure, shearing, axle power and sectional twisting angle general expression, into one
Step acquires interim inverted arch (stull) internal force of arbitrary section and deformation expression when both ends act on identical shearing, moment of flexure and axle power
Formula.
(1) both ends act on identical moment of flexure
When interim inverted arch (stull) both ends act on identical moment of flexure, as shown in figure 4, according to both ends boundary condition:It can obtain:
And then it can obtain:
(2) both ends act on identical shearing
When interim inverted arch (stull) both ends act on identical shearing, as shown in figure 5, according to both ends boundary condition:It can obtain:
And then it can obtain:
(3) both ends act on identical axle power
When interim inverted arch (stull) both ends act on identical axle power, as shown in fig. 6, according to both ends boundary condition:WithIt can obtain:C1=0, C4=0.
And then it can obtain:
Interim inverted arch (stull) is used as flat curved beam, according to the above analysis, has identical moment M when both ends act on simultaneouslyj, shearing
QjWith axle power NjWhen, the unknowm coefficient in formula (22)~(26) can be acquired using principle of stacking are as follows:
Since steelframe is flexible fastening at interim inverted arch (stull) both ends, in the entirety of steelframe and interim inverted arch (stull)
In solution procedure, the compatibility of deformation of the two junction must be considered.Interim inverted arch is as flat curved beam, and axle power is to self-deformation
Influence it is very small and can ignore, the horizontal displacement and vertical displacement of each arch springing of steelframe can approximation be taken as interim inverted arch (stull) end
The horizontal component and vertical component of portion's amount of deflection.In view of this, according to interim inverted arch (stull) when identical shearing, moment of flexure etc. are born in both ends
The internal force and deformation expression formula of arbitrary section, can be obtained at steel arch foot needed for interim inverted arch (stull) and steelframe integrated solution
Constituent parts displacement and carry displacement are as follows:
Wherein have for shallow embedding no-bias tunnel
Have for deep tunnel
The β1L、β1R、u1LAnd u1RAt respectively left and right arch springing section simultaneously function unit torque when it is caused left and right
Arch springing sectional twisting angle and horizontal displacement；The v1LFor at left and right arch springing section simultaneously function unit torque when left arch springing section at
Vertical displacement；The β2L、β2R、u2LAnd u2RIt is at respectively left and right arch springing section while caused when function unit horizontal force
Left and right arch springing sectional twisting angle and horizontal displacement；The v2LFor at left and right arch springing section simultaneously function unit horizontal force when left arch springing
Horizontal displacement at section；The β3L、β3R、u3LAnd u3RAt respectively left and right arch springing section simultaneously function unit vertical force when institute
Caused left and right arch springing sectional twisting angle and horizontal displacement；The v3LFor at left and right arch springing section simultaneously function unit vertical force when
Vertical displacement at left arch springing section；The βpL、βpR、upLAnd upRRespectively left and right arch springing section caused by country rock load turns
Angle and horizontal displacement；The vpLFor the vertical displacement at arch springing section left under country rock load action；The q0It is erected for tunnel is total
To country rock load, unit N/m can be calculated public by the pressure from surrounding rock of shallow tunnel and deep tunnel in associated tunnel design specification
Formula is respectively calculated；η is the country rock load combination ratio acted on steelframe；γ is country rock bulk density, unit N/m3；λ is side pressure
Force coefficient；L is that top bar excavates span, unit m；F is the tunnel excavation height above interim inverted arch (stull), unit m；Angle for interim inverted arch (stull) end relative to tunnel middle line, unit are °.
According to the constituent parts displacement at steelframe arch springing and displacement (formula (40)~(51)) is carried, and steelframe can be acquired using force method
Redundant force at vault and arch springing (see Fig. 7) are as follows:
Wherein
In formula
Have for shallow embedding no-bias tunnel
Have for deep tunnel
The redundant force X1For the moment of flexure of steelframe crown section, unit Nm；The redundant force X2For steelframe
The axle power of crown section, unit N；The redundant force X3Ground for steelframe arch springing under interim inverted arch (stull) supporting is anti-
Power, unit N；The AfFor the contact area of steelframe arch springing and substrate, unit m2；The R is tunnel arch radius, unit
For m；EsIsFor the bending stiffness of steelframe, unit Nm2；The δ11、δ12、δ21And δ22When being rigidly fixed for steelframe arch springing, X1、
X2It acts at lower crown section respectively along X1、X2The unit displacement that direction generates；The Δ1p、Δ2pRespectively country rock load action
Respectively along X at crown section when lower steelframe arch springing rigidly fixes1、X2The displacement that direction generates.
In the redundant force X for acquiring steelframe vault and arch springing1、X2And X3Afterwards, that is, it can determine that arch steelframe acts on to face
When inverted arch (stull) both ends shearing, moment of flexure and axle power value be respectively as follows:
Wherein have for shallow tunnel
Have for deep tunnel
The shear value and moment (formula (63)) at interim inverted arch (stull) both ends acquired using the above process and interim
The amount of deflection, corner, moment of flexure, shearing and subgrade reaction expression formula (formula (22)~(26)) of inverted arch (stull) arbitrary section, obtain
The amount of deflection, corner, moment of flexure, shearing and subgrade reaction of interim inverted arch (stull) arbitrary section under the load action of arch steelframe transmitting
Value, is finally completed the stress and deformation analysis of interim inverted arch (stull).
Claims (2)
1. a kind of interim inverted arch or the stress of stull and deformation analysis method, it is characterised in that: the method and step are as follows:
Step 1): the elastic foundation Curved Beams model of interim inverted arch or stull is established；
Step 2: the elastic foundation Curved Beams model of the interim inverted arch or stull established according to step 1) establishes interim inverted arch
Or the curve differential equation of stull, interim inverted arch or stull amount of deflection, moment of flexure, shearing, axle power and sectional twisting angle are obtained through solving
General expression；
Step 3): interim inverted arch or stull amount of deflection, moment of flexure, shearing, axle power and the sectional twisting angle obtained according to step 2 it is general
Expression formula acts on identical shearing, moment of flexure and arbitrary section when axle power at both ends using interim inverted arch or stull is further acquired
Internal force and deformation expression formula；
Step 4): any when identical shearing, moment of flexure and axle power are born in both ends according to interim inverted arch obtained by step 3) or stull
The internal force and deformation expression formula in section, further obtain steelframe arch springing needed for interim inverted arch or stull and steelframe integrated solution
The constituent parts at place conjugate and carry displacement；
Step 5): displacement is conjugated and carried according to the constituent parts at steelframe arch springing obtained by step 4), solves to obtain steelframe using force method
Redundant force at vault and arch springing；
Step 6): the redundant force at steelframe vault and arch springing obtained according to step 5), determination act on interim inverted arch
Or shearing, moment of flexure and the axle power value at stull both ends；
Step 7): shearing, moment of flexure and the axle power value and step 3) of the interim inverted arch or stull both ends that are obtained using step 6) are obtained
To interim inverted arch or stull both ends be respectively subjected to identical shearing, moment of flexure and when axle power arbitrary section internal force and deformation table
Up to formula, obtains the amount of deflection of interim inverted arch or stull arbitrary section under the load action that arch steelframe transmits, corner, moment of flexure, cuts
Power and subgrade reaction value are finally completed the stress and deformation analysis of interim inverted arch or stull.
2. a kind of interim inverted arch according to claim 1 or the stress of stull and deformation analysis method, it is characterised in that: institute
The interim inverted arch or stull elastic foundation curved Beam Model for stating step 1) foundation should meet the following conditions:
A: temporarily inverted arch or stull is the Curved Beams on elastic foundation, and suffered radial direction subgrade reaction obeys Winkler's assumption；
B: temporarily inverted arch or stull are elastomer, obey Hooke's law；
C: interim inverted arch or stull thickness compare arc radius very little, belong to thin beam.
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Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN103093050A (en) * | 2013-01-14 | 2013-05-08 | 北京特泽热力工程设计有限责任公司 | Data processing method for heat engineering shallow tunnel structure and Computer-Aided Design (CAD) system |
| CN103324811A (en) * | 2013-03-07 | 2013-09-25 | 河北工业大学 | Long-span bridge bending deformation handling method |
| CN105735375A (en) * | 2016-03-21 | 2016-07-06 | 长安大学 | Method and system for monitoring stability of loess tunnel bottom and construction method |
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN103093050A (en) * | 2013-01-14 | 2013-05-08 | 北京特泽热力工程设计有限责任公司 | Data processing method for heat engineering shallow tunnel structure and Computer-Aided Design (CAD) system |
| CN103324811A (en) * | 2013-03-07 | 2013-09-25 | 河北工业大学 | Long-span bridge bending deformation handling method |
| CN105735375A (en) * | 2016-03-21 | 2016-07-06 | 长安大学 | Method and system for monitoring stability of loess tunnel bottom and construction method |
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