CN107341317A - The computational methods of heavy grade rigid-frame column topmast seat rigidity and rigid-frame column computational length - Google Patents
The computational methods of heavy grade rigid-frame column topmast seat rigidity and rigid-frame column computational length Download PDFInfo
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Abstract
The present invention relates to heavy grade rigid-frame column topmast seat rigidity and the computational methods of rigid-frame column computational length, determines rigid-frame column topmast seat condition, makes calculation diagram to two kinds of spring fastening forms respectively；Redundant structure solution is carried out, calculates capital horizontal spring bearing and rotation spring support stiffness；Rigid-frame column differential equation of buckling is established, obtains computational length coefficient on the horizontal rigidity at bearing end and the transcendental equation of two parameters of rotational stiffness；Solution is fitted to equation, determines rigid-frame column computational length coefficientμTheoretical formula on horizontal spring bearing and rotation spring support stiffness.This method considers the semi-rigid feature of heavy grade rigid frame beam-column connection, suitable for the actual engineering design of heavy grade rigid-frame column computational length coefficient, is equally applicable to the gradient less than 1:5 rigid-framed structure design, has the obvious advantages such as design feature is with strong points, calculating process is easy, has filled up the blank in the domestic field.
Description
Technical field
The present invention relates to belong to structural steelwork calculating field, and in particular to a kind of heavy grade rigid-frame column topmast seat rigidity and
The computational methods of rigid-frame column computational length.
Background technology
The structural column computational length coefficient value of the existing Code for design of steel structures defined in China, it is to be directed to frame structure
The restraint condition of beam-column line stiffness ratio determines the computational length of frame column.In computation model, beam intersects vertically with post,
In in general building structure, there is preferable significance of application.For portal frame structure,《Steel Portal Frames
Technical regulation》(CECS 102:2002) respectively to side moves and the beam column computational length without displaceable structure form provides calculating
Formula, but specification defines limit value to the portal-rigid frames cant beam gradient, to meet the basic assumption requirement of calculation formula.For chevron
Portal-rigid frames, when cant beam is less than 23 °, computational methods that generally use evens up crossbeam, further according to the rigid frame for having horizontal gird
Determine the computational length of post but can transmit horizontal thrust for portal frame structure, bean column node so that cant beam produces arch effect
Should.
In modern architecture, due to demands such as architectural appearance, space use, roof drainages, a kind of heavy grade is gradually produced
Rigid-framed structure, such a structure cant beam gradient is larger, and the cant beam gradient has exceeded《Steel Portal Frames technology is advised
Journey》4.1.5 in, the portal-rigid frames slope of roof preferably takes 1/8-1/20 limit value requirement.The calculating of heavy grade rigid-framed structure can not expire
Foot《Code for design of steel structures》With《Steel Portal Frames technical regulation》In about the basic of computational length formula
It is assumed that therefore it is theoretically unsound and instructs in actual engineering design.And research both domestic and external at present is directed to conventional frame knot more
Structure or portal frame structure, heavy grade rigid-framed structure form study few.The increase of the cant beam gradient, by direct shadow
Internal force distribution, the rigidity supporting role of structure are rung, thus special push away is carried out to the computational length coefficient value of this structural column
Lead and research work, providing the reliable design considerations tool of science for design work is of great significance.
The content of the invention
It is an object of the invention to provide the calculating side of a kind of heavy grade rigid-frame column topmast seat rigidity and rigid-frame column computational length
Method, suitable for the actual engineering design of heavy grade rigid-frame column computational length coefficient, the gradient is equally applicable to less than 1:5 rigid frame knot
Structure designs.
The technical solution adopted in the present invention is：
The computational methods of heavy grade rigid-frame column topmast seat rigidity and rigid-frame column computational length, it is characterised in that：
Realized by following steps：
Step 1：Rigid-frame column topmast seat condition is determined, respectively to two kinds of springs of horizontal spring bearing and rotation spring bearing
Support style makes calculation diagram；
Step 2：Redundant structure solution is carried out, calculates capital horizontal spring bearing and rotation spring support stiffness；
Step 3：Establish rigid-frame column differential equation of buckling, obtain computational length coefficient on bearing end horizontal rigidity with
The transcendental equation of two parameters of rotational stiffness；
Step 4：Solution is fitted to equation, rigid-frame column computational length coefficient μ is determined on horizontal spring bearing and turns
Move the theoretical formula of spring support stiffness.
In step 1, when establishing calculation diagram, for the heavy grade rigid-framed structure form bottom of the pillar branch of hinged column feet
Seat condition is be hinged, and top is horizontal spring bearing and rotation spring bearing；For the heavy grade rigid-framed structure shape of suspension column rigid connection
Formula, bottom of the pillar support condition are rigid connection, and top is horizontal spring bearing and rotation spring bearing.
In step 2, for the heavy grade rigid-framed structure form of hinged column feet, horizontal and rotation spring support displacement calculates
Sketch is a redundant structure；For the heavy grade rigid-framed structure form of suspension column rigid connection, horizontal and rotation spring branch seat
It is secondary redundant structure to move calculation diagram.
The present invention has advantages below：
Computational methods of the present invention, the heavy grade rigid-framed structure capital horizontal seat of two kinds of suspension column forms is solved respectively
With the rigidity of rotating stand, resettle the balance differential equation of rigid-frame column, determine capital horizontal seat and rotating stand rigidity with
The relation of rigid-frame column computational length coefficient, numerical fitting is carried out using MATLAB, determines the fitting of rigid-frame column computational length coefficient
Formula.Formula considers the semi-rigid feature of heavy grade rigid frame beam-column connection, with design feature is with strong points, calculated
The obvious advantages such as journey simplicity, the blank in the domestic field is filled up.
Brief description of the drawings
Fig. 1 is hinged column base firm frame computing sketch.
Fig. 2 is hinged column base horizontal spring support displacement calculation diagram.
Fig. 3 is hinged column base rotation spring support displacement calculation diagram.
Fig. 4 is that hinged column base horizontal spring support displacement calculates primary structure.
Fig. 5 is that hinged column base rotation spring support displacement calculates primary structure.
Fig. 6 is hinged column base spring rate solution throughway.
Fig. 7 is hinged column base rigid-frame column deformation pattern.
Fig. 8 is hinged column base cell isolation body.
Fig. 9 is that hinged column feet rigid-frame column design factor changes curved surface.
Figure 10 is rigid connection suspension column firm frame computing sketch.
Figure 11 is rigid connection suspension column horizontal spring support displacement calculation diagram.
Figure 12 is rigid connection suspension column rotation spring support displacement calculation diagram.
Figure 13 is that rigid connection suspension column horizontal spring support displacement calculates primary structure.
Figure 14 is that rigid connection suspension column rotation spring support displacement calculates primary structure.
Figure 15 is rigid connection suspension column spring rate solution throughway.
Figure 16 is rigid connection suspension column rigid-frame column deformation pattern.
Figure 17 is rigid connection suspension column cell isolation body.
Figure 18 is that rigid connection suspension column rigid-frame column design factor changes curved surface.
Figure 19 is hinged column feet rigid-frame column computational length coefficient value.
Figure 20 is suspension column rigid connection rigid-frame column computational length coefficient value.
Embodiment
With reference to embodiment, the present invention will be described in detail.
The present invention relates to heavy grade rigid-frame column topmast seat rigidity and the computational methods of rigid-frame column computational length, according to traditional power
Method, the respectively rigidity of the heavy grade rigid-framed structure capital horizontal seat of two kinds of suspension column forms of solution and rotating stand, then build
The balance differential equation of vertical rigid-frame column, determines capital horizontal seat and rotating stand rigidity and the pass of rigid-frame column computational length coefficient
System, numerical fitting is carried out using MATLAB, determines the fitting formula of rigid-frame column computational length coefficient.Establishing flexing equilibrium equation
When, make following basic assumption：
A. component is the straight and upright bar of preferable uiform section；
B. pressure acts on along the original axis of component；
C. material meets Hooke's law, i.e. stress and strain is linear；
D. the plane section before the deformation of member is still plane after flexural deformation；
E. the flexural deformation of component is small that curvature can be represented approx with the second differential of deformation, i.e. Φ=-
y″。
The present invention is specifically realized by following steps：
Step 1：Rigid-frame column topmast seat condition is determined, respectively to two kinds of springs of horizontal spring bearing and rotation spring bearing
Support style makes calculation diagram；
Heavy grade rigid-framed structure form bottom of the pillar support condition for hinged column feet is be hinged, and top is horizontal spring
Bearing and rotation spring bearing；For the heavy grade rigid-framed structure form of suspension column rigid connection, bottom of the pillar support condition is rigid connection, top
Portion is horizontal spring bearing and rotation spring bearing.
Step 2：Redundant structure solution is carried out, calculates capital horizontal spring bearing and rotation spring support stiffness；
For the heavy grade rigid-framed structure form of hinged column feet, horizontal and rotation spring support displacement calculation diagram is one
Secondary redundant structure；For the heavy grade rigid-framed structure form of suspension column rigid connection, horizontal and rotation spring support displacement calculation diagram
It is secondary redundant structure.
Step 3：Establish rigid-frame column differential equation of buckling, obtain computational length coefficient on bearing end horizontal rigidity with
The transcendental equation of two parameters of rotational stiffness.
Step 4：Solution is fitted to equation, rigid-frame column computational length coefficient μ is determined on horizontal spring bearing and turns
Move the theoretical formula of spring support stiffness.
(1) hinged column base rigid frame
For the heavy grade rigid-framed structure form of hinged column feet, when establishing differential equation of buckling, bottom of the pillar carrier strip
Part is be hinged, and top is horizontal spring bearing and rotation spring bearing, you can think pillar bottom boundary condition for post displacement and
Moment is 0, and has the elastic restraint of translation and the elastic restraint of rotation displacement at the top of pillar.So it can be drawn according to constraints
Go out calculation diagram, as shown in Figures 1 to 3.
For the heavy grade rigid-framed structure form of hinged column feet, horizontal and rotation spring support displacement calculation diagram is one
Secondary redundant structure, a superfluous constraint power is removed respectively in calculating formula, instead unknown force X1, as shown in Figure 4 and Figure 5.
Solution procedure according to Fig. 6 obtains the horizontal rigidity and rotational stiffness of capital：
In formula, K1For rigid frame cant beam Line stiffness, K2For rigid-frame column Line stiffness, ξ is length ratio coefficient, is counted as the following formula respectively
Calculate.
It is be hinged that hinged column feet rigid-frame column, which calculates and can be reduced to bottom support bracket condition, and top is horizontal spring bearing and rotation
Spring fastening form, as shown in Figure 7.Under load P effects, rigid-frame column deforms under its support condition.Wherein, with up time
The corner of pin for just, translation to the right for just, the torque of styletable and horizontal force using with displacement it is equidirectional when as just, be when incorgruous
It is negative.
Take the slider shown in Fig. 8, equilibrium establishment equation：
Simultaneous formula (6) and formula (7) can obtain the Fourth Order Differential Equations of rigid-frame column axial compression, as shown in formula (8),
EI2y(4)+ Py "=0 (8)
Make k2=P/EI2, the general solution of the differential equation can be obtained,
Y=C1sinkx+C2coskx+C3x+C4 (9)
Y '=C1kcoskx-C2ksinkx+C3 (10)
Y "=- C1k2sinkx-C2k2coskx (11)
By boundary condition y (0)=0 and y ' (0)=0, C can be obtained2=0, C4=0.
By Q (l)=- PC3=-kBY (l) and M (l)=- EI2Y " (l)=rBY ' (l) can be obtained：
kBC1sinkl+(kBl-P)C3=0 (12)
(EI2k2sin kl-rBkcoskl)C1-rBC3=0 (13)
C1、C3The condition for having untrivialo solution is that the determinant of coefficient in formula (12) and formula (13) is zero, i.e.,
Determinantal expansion can obtain：
kBlEI2k2sinkl-kBrBlkcoskl-PEI2k2sinkl+PrB kcoskl+kBrBSinkl=0
(15)
Because parameters have following relation in formula：
Meanwhile bearing parameter R can be madeB、KBRespectively
Substituted into together in formula (15) with (16) formula, obtain the side of surmounting of the relevant calculation length of column coefficient of hinged column base rigid frame
Journey：
When section and the rigid frame size of known rigid-framed structure, it can be calculated according to formula (1) and formula (2) and determine spring rate
Value, then try to achieve the computational length coefficient of pillar.
Solution is iterated to the real root of equation (18) using secant method, secant method is a kind of method of Linearization every node,
Its basic thought is previously given initial value x-1And x0, with section [xk-1,xk] on secant it is approximate replace object function lead letter
Number curve, the approximate solution of equation is used as by the use of the abscissa of secant and transverse axis intersection point.
The iterative formula of secant method is：
When | f (xk)|<When 0.001, terminate iteration, take approximate solution of the iteration result as computational length coefficient, part is tied
Fruit is as shown in figure 19.
By KB、RBResult of calculation from 0 to 100 is depicted as curved surface, as shown in Figure 9.Carried out using MATLAB to calculating data
Numerical fitting, hinged column feet rigid-frame column computational length Coefficient Fitting formula are：
The determination coefficient of fitting formula (20) is 0.9879, and standard deviation 0.1115, result of calculation is slightly larger than transcendental equation
Result of calculation, fitting formula calculating is accurate enough, goes for practical engineering calculation value.
(2) rigid connection suspension column rigid frame：
For the heavy grade rigid-framed structure form of suspension column rigid connection, when establishing differential equation of buckling, bottom of the pillar carrier strip
Part is rigid connection, and top is horizontal spring bearing and rotation spring bearing, you can think pillar bottom boundary condition for post displacement and
Corner is 0, and has the elastic restraint of translation and the elastic restraint of rotation at the top of pillar.So calculating can be drawn according to constraints
Sketch, as shown in Figure 10~Figure 12.
For the heavy grade rigid-framed structure form of suspension column rigid connection, horizontal and rotation spring support displacement calculation diagram is two
Secondary redundant structure, two superfluous constraint power is removed respectively in calculating formula, instead unknown force X1、X2, such as Figure 13 and Figure 14 institutes
Show.
Solution procedure according to Figure 15 obtains the horizontal rigidity and rotational stiffness of capital：
It is rigid connection that suspension column rigid connection rigid-frame column, which calculates and can be reduced to bottom support bracket condition, and top is horizontal spring bearing and rotation
Spring fastening form, as shown in figure 16.Under load P effects, rigid-frame column deforms under its support condition.Wherein, with suitable
The corner of hour hands for just, translation to the right for just, the torque of styletable and horizontal force using with displacement it is equidirectional when as just, be when incorgruous
It is negative.
With establishing, hinged column feet rigid-frame column Buckling Equation method is identical, takes the slider shown in Figure 17, establishes suspension column rigid connection
Rigid-frame column Buckling Equation：
EI2y(4)+ Py "=0 (23)
Formula (23) is the Fourth Order Differential Equations of rigid-frame column axial compression, makes k2=P/EI2, can obtain the general solution of the differential equation：
Y=C1sinkx+C2coskx+C3x+C4 (24)
Y '=C1kcoskx-C2ksinkx+C3 (25)
Y "=- C1k2sinkx-C2k2coskx (26)
By boundary condition y (0)=0 and y ' (0)=0, obtain：
C2+C4=0 (27)
C1k+C3=0 (28)
By Q (l)=- PC3=-kBY (l) and M (l)=- EI2Y " (l)=rBY ' (l) can be obtained：
kBC1sinkl+(kBl-P)C3=0 (29)
(EI2k2sinkl-rBkcoskl)C1-rBC3=0 (30)
C1、C3The condition for having untrivialo solution is that the determinant of coefficient in formula (27)~formula (30) is zero, i.e.,
Determinantal expansion can obtain：
kBEI2k2sinkl-kBrBkcoskl+2kBrBk-kBrBkcoskl-kBlEI2k3coskl
-kBrBlk2sinkl+PEI2k3coskl+PrBk2Sinkl=0 (32)
Equally, formula (16), (17) are substituted into formula (30) together, the relevant pillar of hinged column base rigid frame must be obtained and calculate length
Spend the transcendental equation of coefficient：
When determining the cross-sectional sizes and rigid frame size of rigid frame, it can be calculated according to formula (21) and formula (22) and determine spring rate
Value, then try to achieve the computational length coefficient of pillar.
The same part solution that equation (33) is tried to achieve using numerical method, as a result as shown in figure 20.
By KB、RBResult of calculation from 0 to 100 is depicted as curved surface, as shown in figure 18.Entered using MATLAB to calculating data
Row numerical fitting, suspension column rigid connection rigid-frame column design factor fitting formula are：
The determination coefficient of fitting formula (34) is 0.9775, and standard deviation 0.04818, result of calculation is slightly larger than transcendental equation
Result of calculation, fitting formula calculating is accurate enough, goes for practical engineering calculation value.
The computational length coefficient of heavy grade rigid frame pillar is solved, if hinged column base, can be calculated according to formula (1) and formula (2)
Determine spring stiffness values；If rigid connection suspension column, it can be calculated according to formula (19) and formula (20) and determine spring stiffness values.Again from table 1 or
The corresponding calculation length of column coefficient that table 2 is looked into, or calculated according to fitting formula (20) or (34) and try to achieve corresponding pillar calculating
Length factor.
Present disclosure is not limited to cited by embodiment, and those of ordinary skill in the art are by reading description of the invention
And any equivalent conversion taken technical solution of the present invention, it is that claim of the invention is covered.
Claims (3)
1. the computational methods of heavy grade rigid-frame column topmast seat rigidity and rigid-frame column computational length, it is characterised in that：
Realized by following steps：
Step 1：Rigid-frame column topmast seat condition is determined, respectively to two kinds of spring fastenings of horizontal spring bearing and rotation spring bearing
Form makes calculation diagram；
Step 2：Redundant structure solution is carried out, calculates capital horizontal spring bearing and rotation spring support stiffness；
Step 3：Rigid-frame column differential equation of buckling is established, obtains horizontal rigidity and rotation of the computational length coefficient on bearing end
The transcendental equation of two parameters of rigidity；
Step 4：Solution is fitted to equation, determines rigid-frame column computational length coefficientμOn horizontal spring bearing and rotate bullet
The theoretical formula of spring support stiffness.
2. the computational methods of heavy grade rigid-frame column topmast seat rigidity according to claim 1 and rigid-frame column computational length, its
It is characterised by：
In step 1, when establishing calculation diagram, for the heavy grade rigid-framed structure form bottom of the pillar carrier strip of hinged column feet
Part is be hinged, and top is horizontal spring bearing and rotation spring bearing；For the heavy grade rigid-framed structure form of suspension column rigid connection, post
Sub- bottom support bracket condition is rigid connection, and top is horizontal spring bearing and rotation spring bearing.
3. the computational methods of heavy grade rigid-frame column topmast seat rigidity according to claim 1 and rigid-frame column computational length, its
It is characterised by：
In step 2, for the heavy grade rigid-framed structure form of hinged column feet, horizontal and rotation spring support displacement calculation diagram
It is a redundant structure；For the heavy grade rigid-framed structure form of suspension column rigid connection, horizontal and rotation spring support displacement meter
It is secondary redundant structure to calculate sketch.
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| CN201710571093.7A CN107341317B (en) | 2017-07-13 | 2017-07-13 | Calculation method for rigidity of top support of large-gradient rigid frame column and calculated length of rigid frame column |
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| US20110232113A1 (en) * | 2010-10-14 | 2011-09-29 | Holland Carl A | Method and device for measuring the inclination of a roadway |
| CN203808260U (en) * | 2014-04-26 | 2014-09-03 | 中国能源建设集团山西省电力勘测设计院 | Transformer substation continuous rigid frame single-tube beam capable of releasing temperature stress |
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