CN109446715A - A kind of Longspan Bridge seismic response Time-History Analysis Method - Google Patents
A kind of Longspan Bridge seismic response Time-History Analysis Method Download PDFInfo
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Abstract
The present invention relates to a kind of Longspan Bridge seismic response Time-History Analysis Methods, steps are as follows: the first step, it is discrete that Space finite element is carried out to bridge structure, establish the finite element model of bridge structure, the components such as girder, leaning tower, suspension cable, main push-towing rope are all made of Euler's beam element, and Rayleigh damping matrix is used, and by the movement difference equations of Hamilton principle export discrete system, equivalent inertia force is calculated by input acceleration；Second step, access time step-length, time step are taken as n times of accelerogram time interval；Third step calculates by time step, calculates displacement, the velocity and acceleration of each time step；Dynamic time history analysis method of the present invention has many advantages, such as that higher order accuracy, computational efficiency are high, can filter the false higher-order of oscillation due to caused by spatial spreading completely, be highly suitable for the solution of this stiff problem of Loads of Long-span Bridges.
Description
Technical field
The present invention relates to a kind of Longspan Bridge seismic response Time-History Analysis Methods, and in particular to a kind of calculating long-span bridge
Unconditional stability that beam responds under geological process, higher order accuracy, the Dynamic time history analysis method with optimal values damping.
Background technique
Bridge is the multi-purpose project of transportation system, is the important component of lifeline engineering, guarantees bridge on ground
The safety and normal use in earthquake centre, it is important to all having in the Seismic Resistance And Disaster Reduction work and post earthquake recovery reconstruction in city and area
Meaning.In the design phase, numerical simulation means are usually used, the analysis under geological process are carried out to bridge, to guarantee bridge
Safety of the structure under the severe earthquake action that happens suddenly.For convenient for calculating, there has been proposed some such as equivalent base shear method, response spectrum method
Method.Continuous development however as civil engineering structure form and to kinematic analysis reliability increasingly higher demands, we
The behavior to structure under dynamic load is needed to carry out more accurate prediction.
Direct dynamic analysis method is grown up with the increasing the extensive use with computer technology of STRONG MOTION DATA,
It is generally acknowledged explication de texte method.It is all pointed out in domestic and international most of earthquake resistant engineering design specifications, for the ground of Longspan Bridge
Shake response analysis, Yao Caiyong Dynamic Response History Analysis Method.That is, the structural finite element model of Yao Caiyong multinode freedom degree,
The one seismic acceleration time-histories of excitation of earthquake forced vibration is directly inputted, SEISMIC RESPONSE is carried out to structure.Power
Time history analysis method can be accurate consideration ground and structure interaction, seismic wave phase difference and differently seismic wave multi -components are more
The problems such as point input, various complex nonlinear factors and the piecemeal damping of structure.
Currently available technology be directed to geological process under long-span bridges carry out Dynamic time history analysis mainly using
Newmark method, Wilson method and HHT method etc., but these current methods there is a problem of more or less.For Longspan Bridge knot
Structure, due to structure the longest period and most short cycle differs greatly, which belongs to stiff problem, for stiff problem
Dynamic time history analysis is easy to produce unwanted oscillation, this concussion is due to finite element especially for the calculating of acceleration responsive
Caused by discrete, the influence to the real response of structure be it is negative, need to filter out by numerical damping.Common time-histories point
Analysis method can not filter out this false concussion due to not sufficiently large numerical damping, or even cause method unstable, cause
Calculating does not restrain.
To solve this problem, Bathe has developed a kind of effective method, mesh highly effective for the solution of stiff problem
Before have been embedded in Adina software.However the computational efficiency of Bathe method is lower, needs to carry out two on each time step
Submatrix decomposes, and method only has 2 rank precision.There is the analysis method of 2 rank precision to carry out Dynamic time history using these
When analysis, to guarantee precision, it is necessary to use shorter time step.Because of Loads of Long-span Bridges model freedom degree substantial amounts, time
Step-length is selected too short, and Seismic input duration is constant, and total time step quantity is excessive, and it is huge also to have resulted in analytic process time consumption
Greatly.
Summary of the invention
For defect of the traditional power Time-History Analysis Method in terms of solving Longspan Bridge Dynamic time history analysis, the present invention
Be designed to provide it is a kind of calculating Longspan Bridge responded under geological process unconditional stability, higher order accuracy, have most
The Dynamic time history analysis method of good numerical damping.
Technical scheme is as follows:
A kind of Longspan Bridge seismic response Time-History Analysis Method, its step are as follows:
The first step, discrete to long-span bridge girder construction progress Space finite element, the finite element model for establishing the structure is discrete
System, the components such as girder, leaning tower, suspension cable, main push-towing rope are all made of space Bernoulli Jacob's Euler's beam element, and are damped using Rayleigh
Unit damping matrix is established, Bulk stiffness matrix, whole is integrated by element stiffness matrix, element mass matrix and unit damping matrix
Constitution moment matrix and integral damping matrix, and by the movement difference equations of Hamilton principle export discrete system:
Second step, access time step-length, time step are taken as n times of accelerogram time interval, and wherein n is integer,
Since method has higher order accuracy, n can be chosen for biggish value；
Third step is calculated by time step, for i-th of time step, it is known that ti-1The displacement components u at momenti-1With speed vi-1, by
Following sub-step calculates tiThe displacement components u at momenti, speed vi；
A. by equation D11ui=R1Solve displacement components ui, wherein
R1=P1+G11ui-1+ΔtG12vi-1
B. by equation D11vi=R2Solve displacement vi, wherein
R2=(P2+G21ui-1+ΔtG22vi-1-D21ui)/Δt
C. by equation D11ei=δ R1Solve the error e of displacementi, wherein
D. by equation D11εi=δ R2The error ε of solving speedi, wherein
E. u is corrected by following formulaiAnd vi
ui=ui+ei vi=vi+εi
F. acceleration a is calculated by following formulai
Coefficient matrix and vector involved in formula be respectively
Preferably, in the first step, element stiffness matrix are as follows:
WhereinThe element stiffness matrix for being space bar member in local coordinate system, is 12 × 12 symmetrical matrix, right
It is coordinately transformed, the element stiffness matrix stiffness matrix under global coordinate system can be obtained, i.e.,
Preferably, in the first step, element mass matrix are as follows:
Using HRZ method pairDiagonalization is carried out, obtains lumped mass matrix, as
It is rightIt is coordinately transformed, the element stiffness matrix stiffness matrix under global coordinate system can be obtained, i.e.,
Preferably, it in the first step, is damped using Rayleigh, establishes unit damping matrix
Ce=a0Me+a1Ke
Wherein
ωiAnd ωjThe 1st rank and the 3rd order frequency of structure are generally taken respectively, ζ is damping ratio, generally 0.05.
Preferably, in the first step, calculating for F, need to the seismic wave accelerogram to each moment adopt
ü is obtained with linear interpolationg(t), it is calculated by following formula
F=-M üg(t)
Preferably, in the second step, n is taken as 4-10.
Preferably, in the third step, unit level parallel method is can be used in the item on the right of equation, with R1Calculating
For be illustrated, calculating each unit firstIt is parallel that each unit can be used herein
Method, then byIntegrate R1, the calculating of other equation right-hand vectors is using the same manner processing；
Compared with prior art, the present invention advantage is:
1) Dynamic time history analysis method of the invention has 4 rank precision, with traditional second order such as Newmark method, Bathe method
Dynamic time history analysis method is compared, high 2 rank of precision.
2) in Dynamic time history analysis method of the invention, each time step is only needed to dimension Neq (the total number of degrees of freedom, of particle
Mesh) matrix D11It asks primary inverse, the band-like sparse property of Space finite element discrete matrix is able to maintain in calculating, can also be counted parallel
It calculates, the tradition second-order dynamic Time-History Analysis Method such as calculation amount and Newmark method is suitable, lower than the calculation amount of Bathe method, because
This, for long-span bridge girder construction, Dynamic time history analysis method of the invention can be using longer time step-length namely less
Time step, obtain comparable with traditional power Time-History Analysis Method precision as a result, solution efficiency can be substantially improved.
3) Dynamic time history analysis method of the invention and Bathe method all have progressive cancellation characteristic, are L- antihunt means,
Performance much surmounts Newmark method, can filter the oscillation of the falseness due to caused by finite element discretization, guarantees to calculate and stablize.
Detailed description of the invention
Fig. 1 oblique pull composite bridge structural schematic diagram across suspension cable greatly；
Fig. 2 FEM model schematic diagram；
Fig. 3 spatial beam schematic diagram；
Fig. 4 El-Centro wave schematic diagram；
The tower top displacement diagram that Bathe method and Dynamic time history analysis method of the present invention calculate under Fig. 5 difference step-length；
Bathe method and Dynamic time history analysis method of the present invention calculate tower top speed schematic diagram under Fig. 6 difference step-length；
Bathe method and Dynamic time history analysis method of the present invention calculate tower top acceleration schematic diagram under Fig. 7 difference step-length.
Specific embodiment
Next combined with specific embodiments below invention is further explained, but does not limit the invention to these tools
Body embodiment.One skilled in the art would recognize that present invention encompasses may include in Claims scope
All alternatives, improvement project and equivalent scheme.
Structural principle and working principle of the invention are described in detail with reference to the accompanying drawing:
Using a long-span bridges as example, it is specifically described Dynamic time history analysis method of the present invention, certain is oblique across suspension cable greatly
Composite bridge is drawn as shown in Figure 1, this bridge is self-anchored type oblique pull-suspention co-operative system bridge.Access bridge part, the across footpath of the bridge are not considered
Are as follows: specific structure construction are as follows: (1) girder: 132m+400m+132m=664m uses six Room armored concrete of integral cast-in-situ list case
Box beam section.The high 2.6m of girder, be main span 1/155, top plate thickness 18-26cm, web 40cm, bottom plate 25-26cm, pavement and
Bar Anchorage area is arranged on the outside of box beam.End of main beam overstriking thickeies, and inside is arranged from anchor anchorage.120m is steel case in the middle part of girder
Beam arrangement.(2) Sarasota: for A font concrete structure, the above tower height 70m of bridge floor, Sarasota gradually overstriking from the top down, tower top face
Product 30m2, it is 56.5m at tower root2, section linearly becomes larger.Tower top is made into rounded nose, Pylon pier consolidation, the separation of tower beam.(3) suspension cable:
Using fan-shaped double rope faces on Sarasota, each rope face is made of 17 pairs of drag-lines, asymmetric arrangement.Oblique pull between abutment pier and auxiliary pier
Rope spacing 6m, rest part spacing 8m, 4 rope faces of full-bridge, in addition No. 0 rope at king-tower, is altogether 140 ropes.All oblique pulls
Suo Jun is anchored in girder.Diaphragm is arranged in girder at suspension cable.(4) main push-towing rope: because main push-towing rope instead of side rope position,
Simultaneously Non-completety symmetry is arranged to suspension cable along bridge.Main push-towing rope is anchored at the girder overstriking on abutment pier.(5) pier footing.Tentatively
Material shows that bridge site physical state is relatively good herein, can use end-bearing pile, dimension of platform 22.75,16.5m, and cushion cap is one whole
Body, linkage section is having a size of 22.75 × 16.5m, the bored concrete pile of use, every 24 pile of cushion cap.A kind of calculating large span of the application
Unconditional stability that bridge responds under geological process, higher order accuracy, the Dynamic time history analysis method with optimal values damping
It comprises the following steps:
The first step, it is discrete to Loads of Long-span Bridges progress Space finite element, mould is distinguished using space Bernoulli Jacob's Euler's beam element
Intend the components such as girder, Sarasota, suspension cable, the main push-towing rope of the bridge, obtain the finite element model of the bridge, as shown in Fig. 2, arrow in figure
Head direction is seismic direction.Spatial beam is most typical component in bridge structure, as shown in Figure 3.The Space Beam
The element displacement vector of unit is
The element stiffness matrix of the spatial beam is
WhereinThe element stiffness matrix for being space bar member in local coordinate system, is 12 × 12 symmetrical matrix.
It is rightIt is coordinately transformed, the element stiffness matrix under global coordinate system can be obtained, i.e.,
Mass matrix uses diagonal matrix, is
It is rightIt is coordinately transformed, the element stiffness matrix under global coordinate system can be obtained, i.e.,
It is damped using Rayleigh, establishes unit damping matrix
Ce=a0Me+a1Ke
Wherein
For general structure.Here ωiAnd ωjThe 1st rank and the 3rd order frequency of structure are generally taken respectively.ζ is damping ratio, is
0.05.After derived above element stiffness matrix, mass matrix and damping matrix, Bulk stiffness matrix, mass matrix are integrated
And it is as follows that movement difference equations can be obtained by Hamilton principle in damping matrix
Second step, access time step-length, time step are taken as n times of accelerogram time interval, and n takes 10；
Third step is calculated by time step, for i-th of time step, it is known that ti-1The displacement components u at momenti-1With speed vi-1, by
Following sub-step calculates tiThe displacement components u at momenti, speed vi；
A. by equation D11ui=R1Solve displacement components ui, wherein
R1=P1+G11ui-1+ΔtG12vi-1
B. by equation D11vi=R2Solve displacement vi, wherein
R2=(P2+G21ui-1+ΔtG22vi-1-D21ui)/Δt
C. by equation D11ei=δ R1Solve the error e of displacementi, wherein
D. by equation D11εi=δ R2The error ε of solving speedi, wherein
E. u is corrected by following formulaiAnd vi
ui=ui+ei vi=vi+εi
F. acceleration a is calculated by following formulai
Coefficient matrix and vector involved in formula be respectively
El Centro seismic wave as shown in Figure 4 is inputted in substrate, is divided into 0.01s between the accelerogram of the seismic wave,
Action direction is perpendicular to bridge plane.In Dynamic time history analysis method of the invention, Analysis on Selecting step-length is accelerogram
10 times, i.e. Δ t=0.1s, calculate tower top dynamic respond.Damping ratio chooses 0.05, the initial displacement of each particle and initially speed
Degree is 0.
In order to show Dynamic time history analysis method high efficiency and accuracy of the invention, first using most common second order essence
Degree Newmark method is analyzed, and the parameter of Newark method is selected as α=1/2, β=1/4, and step-length uses Δ t=0.01s, and analysis is lost
Surely, calculated result tends to be infinitely great.Further analyzed using Bathe method, analysis result as illustrated in figs. 5-7, when consuming
Between be 65s.Still Bathe method is used, attempts step-length amplifying original 10 times, even Δ t=0.1s, result have deviated from Δ
It is when t=0.01s as a result, the result illustrate, using Bathe method analyze, if amplification time step-length, will lead to analysis result it is serious
Deviate from.And Dynamic time history analysis method of the invention is used, enabling step-length is respectively Δ t=0.01s and Δ t=0.1s, result
With using Bathe method, the result that step-length is Δ t=0.01s is very close to (3 response curves are almost overlapped), such as Fig. 5-7 institute
Show.The comparative example intuitively shows that as Δ t=0.1s, Dynamic time history analysis method that is, of the invention is when step-length very much
At 10 times of Bathe method, as a result still very accurate, expending the time is only 20.4s, about the 1/3 of Newmark method.Sufficiently
Demonstrate the accuracy and high efficiency of the Dynamic time history analysis method of long-span bridges of the present invention.
It should be understood that the present invention describe method the step of be only exemplary description, it is successively carried out
Time sequencing does not have special requirement, unless itself there is inevitable sequencing relationship.
As it appears from the above, although the present invention is illustrated with reference to limited embodiment and attached drawing, belonging to the present invention
Have can carrying out various modifications and deform from this record per capita for usual knowledge in field.Other embodiments and power as a result,
Sharp claim and equivalent belong to scope of protection of the claims.
Claims (7)
1. a kind of Longspan Bridge seismic response Time-History Analysis Method, which is characterized in that steps are as follows:
The first step, it is discrete to long-span bridge girder construction progress Space finite element, the finite element model discrete system of the structure is established,
The components such as girder, leaning tower, suspension cable, main push-towing rope are all made of space Bernoulli Jacob's Euler's beam element, and are established using Rayleigh damping
Unit damping matrix integrates Bulk stiffness matrix, whole matter by element stiffness matrix, element mass matrix and unit damping matrix
Moment matrix and integral damping matrix, and by the movement difference equations of Hamilton principle export discrete system:
Second step, access time step-length, time step are taken as n times of accelerogram time interval, and wherein n is integer, and n is optional
It is taken as biggish value；
Third step is calculated by time step, for i-th of time step, it is known that ti-1The displacement components u at momenti-1With speed vi-1, by following
Several sub-steps calculate tiThe displacement components u at momenti, speed vi；
A. by equation D11ui=R1Solve displacement components ui, wherein
R1=P1+G11ui-1+ΔtG12vi-1
B. by equation D11vi=R2Solve displacement vi, wherein
R2=(P2+G21ui-1+ΔtG22vi-1-D21ui)/Δt
C. by equation D11ei=δ R1Solve the error e of displacementi, wherein
D. by equation D11εi=δ R2The error ε of solving speedi, wherein
E. u is corrected by following formulaiAnd vi
ui=ui+ei vi=vi+εi
F. acceleration a is calculated by following formulai
Coefficient matrix and vector involved in formula be respectively
2. Longspan Bridge seismic response Time-History Analysis Method according to claim 1, which is characterized in that described first
In step, element stiffness matrix are as follows:
WhereinThe element stiffness matrix for being space bar member in local coordinate system, is 12 × 12 symmetrical matrix, rightIt carries out
The element stiffness matrix stiffness matrix under global coordinate system can be obtained, i.e., in coordinate transform
3. Longspan Bridge seismic response Time-History Analysis Method according to claim 1, which is characterized in that described first
In step, element mass matrix are as follows:
Using HRZ method pairDiagonalization is carried out, obtains lumped mass matrix, as
It is rightIt is coordinately transformed, the element stiffness matrix stiffness matrix under global coordinate system can be obtained, i.e.,
4. Longspan Bridge seismic response Time-History Analysis Method according to claim 1, which is characterized in that described first
It in step, is damped using Rayleigh, establishes unit damping matrix
Ce=a0Me+a1Ke
Wherein
ωiAnd ωjThe 1st rank and the 3rd order frequency of structure are generally taken respectively, and ζ is damping ratio, is 0.05.
5. Longspan Bridge seismic response Time-History Analysis Method according to claim 1, which is characterized in that described first
In step, calculating for F, need to the seismic wave accelerogram to each moment obtained using linear interpolationBy following formula
It calculates
6. Longspan Bridge seismic response Time-History Analysis Method according to claim 1, which is characterized in that n is taken as 4-10.
7. Longspan Bridge seismic response Time-History Analysis Method according to claim 1, which is characterized in that the third
In step, unit level parallel method is can be used in the item on the right of equation, with R1Calculating for be illustrated, calculate first each
UnitEach unit parallel method can be used herein, then byIntegrate R1,
The calculating of other equation right-hand vectors is handled using the same manner.
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| CN110909500A (en) * | 2019-11-19 | 2020-03-24 | 东南大学 | Unconditionally stable multi-point excitation collapse explicit analysis method for oversized bridge span |
| WO2021082802A1 (en) * | 2019-10-31 | 2021-05-06 | 李鲁 | IMPROVED GENERALIZED-α METHOD FOR NONLINEAR DYNAMIC ANALYSIS OF COMPLEX STRUCTURE |
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Cited By (7)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN110008635A (en) * | 2019-04-19 | 2019-07-12 | 陕西新西商工程科技有限公司 | Using Newmark precise integration combined techniques to the method for Elasto-Plastic Structures seismic response analysis |
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| WO2021082802A1 (en) * | 2019-10-31 | 2021-05-06 | 李鲁 | IMPROVED GENERALIZED-α METHOD FOR NONLINEAR DYNAMIC ANALYSIS OF COMPLEX STRUCTURE |
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| CN113392450A (en) * | 2021-06-07 | 2021-09-14 | 国能朔黄铁路发展有限责任公司 | Method, device and system for evaluating health state of railway pier |
| CN116306177A (en) * | 2023-05-18 | 2023-06-23 | 石家庄铁道大学 | Method and device for analyzing earthquake response of river-valley-crossing terrain bridge and terminal equipment |
| CN116306177B (en) * | 2023-05-18 | 2023-08-04 | 石家庄铁道大学 | Method and device for analyzing earthquake response of river-valley-crossing terrain bridge and terminal equipment |
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