CN107341317B - Calculation method for rigidity of top support of large-gradient rigid frame column and calculated length of rigid frame column - Google Patents

Calculation method for rigidity of top support of large-gradient rigid frame column and calculated length of rigid frame column Download PDF

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CN107341317B
CN107341317B CN201710571093.7A CN201710571093A CN107341317B CN 107341317 B CN107341317 B CN 107341317B CN 201710571093 A CN201710571093 A CN 201710571093A CN 107341317 B CN107341317 B CN 107341317B
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段熙宾
张贵海
蔡玉军
孙建龙
樊轶江
高志宏
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China Railway First Survey and Design Institute Group Ltd
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Abstract

The invention relates to a method for calculating the rigidity of a rigid frame column top support with large gradient and the calculated length of a rigid frame column, which determines the conditions of the rigid frame column top support and respectively makes a calculation diagram for two spring support forms; performing hyperstatic structure solution, and calculating the rigidity of the horizontal spring support and the rotating spring support at the top of the column; establishing a rigid frame column buckling differential equation, and obtaining an transcendental equation of two parameters of the calculated length coefficient about the horizontal rigidity and the rotational rigidity of the support end; fitting and solving the equation to determine the calculated length coefficient of the rigid frame columnμTheoretical formulas for the stiffness of the horizontal spring support and the rotational spring support. The method considers the semi-rigid characteristic of the beam-column connection node of the large-gradient rigid frame, is suitable for the actual engineering design of the calculation length coefficient of the large-gradient rigid frame column, can also be suitable for the structural design of the rigid frame with the gradient less than 1:5, has the obvious advantages of strong structural characteristic pertinence, simple and convenient calculation process and the like, and fills the blank in the field in China.

Description

Calculation method for rigidity of top support of large-gradient rigid frame column and calculated length of rigid frame column
Technical Field
The invention belongs to the field of steel structure engineering calculation, and particularly relates to a calculation method for the rigidity of a top support of a rigid frame column with large gradient and the calculated length of the rigid frame column.
Background
The calculation length coefficient value of the structural column specified by the current steel structure design specification in China is to determine the calculation length of the frame column according to the constraint condition of the beam-column line stiffness ratio in the frame structure. In the calculation model, the beams are vertically intersected with the columns, and the method has good application significance in general building structures. For a portal rigid frame structure, the technical code of portal rigid frame light house steel structure (CECS 102:2002) provides a calculation formula for the calculated length of a beam column in a structure form with side movement and without side movement respectively, but the specification stipulates a limit value for the gradient of an oblique beam of the portal rigid frame so as to meet the basic assumed requirement of the calculation formula. For the mountain-shaped portal rigid frame, when the oblique beam is less than 23 degrees, a calculation method of flattening the beam is generally adopted, and then the calculated length of the column is determined according to the rigid frame with the horizontal beam.
In modern buildings, a rigid frame structure with large gradient is gradually generated due to the requirements of building appearance, space use, roof drainage and the like, the gradient of an oblique beam of the structure is larger, the gradient of the oblique beam exceeds the technical code of steel structures of light buildings with portal rigid frames in 4.1.5, and the gradient of the roof of the portal rigid frame is required to be limited to 1/8-1/20. The calculation of the large-gradient rigid frame structure cannot meet the basic assumption of a related calculation length formula in steel structure design specifications and steel structure technical regulations of portal rigid frame light houses, so theoretical basis and guidance are lacked in actual engineering design. At present, domestic and foreign researches are directed at traditional frame structures or portal rigid frame structures, and few researches are conducted on structural forms of large-gradient rigid frames. The increase of the slope of the oblique beam directly influences the internal force distribution and the rigidity supporting effect of the structure, so that the special derivation and research work is carried out on the calculated length coefficient value of the structural column, and the method has very important significance in providing scientific and reliable design basis for the design work.
Disclosure of Invention
The invention aims to provide a method for calculating the rigidity of a top support of a rigid frame column with large gradient and the calculated length of the rigid frame column, which is suitable for the actual engineering design of the calculated length coefficient of the rigid frame column with large gradient and is also suitable for the structural design of the rigid frame with the gradient less than 1: 5.
The technical scheme adopted by the invention is as follows:
the method for calculating the rigidity of the top support of the large-gradient rigid frame column and the calculated length of the rigid frame column is characterized by comprising the following steps of:
the method is realized by the following steps:
the method comprises the following steps: determining the conditions of the top support of the rigid frame column, and respectively making a calculation diagram for two spring support forms of a horizontal spring support and a rotary spring support;
step two: performing hyperstatic structure solution, and calculating the rigidity of the horizontal spring support and the rotating spring support at the top of the column;
step three: establishing a rigid frame column buckling differential equation, and obtaining an transcendental equation of two parameters of the calculated length coefficient about the horizontal rigidity and the rotational rigidity of the support end;
step four: and fitting and solving the equation, and determining a theoretical formula of the calculated length coefficient mu of the rigid frame column on the rigidity of the horizontal spring support and the rotating spring support.
In the first step, when a calculation diagram is established, the bottom support of a column in a large-gradient rigid frame structure form with a column base hinged is hinged, and a horizontal spring support and a rotating spring support are arranged at the top of the column; for the structural form of the large-gradient rigid frame with the rigid connection of the column foot, the support condition at the bottom of the column is rigid connection, and the horizontal spring support and the rotating spring support are arranged at the top of the column.
In the second step, for the structural form of the large-gradient rigid frame with the hinged column foot, the horizontal and rotating spring support displacement calculation sketch diagrams are both in a one-time statically indeterminate structure; for the structural form of the large-gradient rigid frame with the rigid connection of the column foot, the horizontal and rotating spring support displacement calculation sketch is a secondary statically indeterminate structure.
The invention has the following advantages:
the calculation method comprises the steps of respectively solving the rigidity of a horizontal support and the rigidity of a rotary support at the top of a large-gradient rigid frame structure column in two column foot forms, then establishing a balance differential equation of the rigid frame column, determining the relation between the rigidity of the horizontal support and the rigidity of the rotary support at the top of the column and the calculated length coefficient of the rigid frame column, carrying out numerical fitting by using MATLAB, and determining a fitting formula of the calculated length coefficient of the rigid frame column. The semi-rigid characteristic of the beam-column connection node of the large-gradient rigid frame is considered by the formula, the method has the obvious advantages of strong structural characteristic pertinence, simple and convenient calculation process and the like, and fills the blank in the field in China.
Drawings
Fig. 1 is a calculation diagram of a rigid frame of a hinged column base.
Fig. 2 is a diagram for calculating the displacement of the horizontal spring support of the hinge pedestal.
Fig. 3 is a diagram of the calculation of the displacement of the pivoting spring support of the hinge stub.
Fig. 4 shows a basic system for calculating the displacement of the horizontal spring support of the hinged column base.
Fig. 5 shows a basic system for calculating the displacement of the pivoting pedestal swivel spring mount.
Fig. 6 is a solution for the spring rate of the hinge shoe.
Fig. 7 is a modified view of the hinged pedestal column.
Figure 8 is a hinged stub unit insulation.
Fig. 9 is a calculation coefficient change curved surface of the column base hinged rigid frame column.
Fig. 10 is a calculation diagram of a rigid frame of a rigid connection column foot.
Fig. 11 is a schematic diagram of calculation of displacement of the horizontal spring support of the rigid connection column foot.
Fig. 12 is a schematic diagram of calculation of displacement of the rigid connection column foot rotating spring support.
FIG. 13 is a basic system for calculating the displacement of the horizontal spring support of the rigid connection column base.
FIG. 14 is a basic system for calculating the displacement of the rigid connection column base rotating spring support.
FIG. 15 is a rigid connection column base spring stiffness solution idea.
Fig. 16 is a deformed view of the rigid connection column foot rigid frame column.
Fig. 17 is a rigid stub unit spacer.
Fig. 18 is a rigid joint column foot rigid frame column calculation coefficient change curved surface.
Fig. 19 is a value of the calculated length coefficient of the column base hinged rigid frame column.
Fig. 20 is a value of the calculated length coefficient of the rigid frame column connected to the column base.
Detailed Description
The present invention will be described in detail with reference to specific embodiments.
The invention relates to a calculation method for the rigidity of a top support of a large-gradient rigid frame column and the calculated length of the rigid frame column. In establishing the buckling equilibrium equation, the following basic assumptions are made:
a. the component is an ideal straight rod with equal section;
b. the pressure acts along the original axis of the component;
c. the material conforms to Hooke's law, namely the stress and the strain are in a linear relationship;
d. the flat section before the deformation of the component is still a plane after the bending deformation;
e. the bending deformation of the member is slight, and the curvature can be approximately expressed by the second derivative of the deformation, i.e., -y ".
The invention is realized by the following steps:
the method comprises the following steps: determining the conditions of the top support of the rigid frame column, and respectively making a calculation diagram for two spring support forms of a horizontal spring support and a rotary spring support;
the bottom support of the column in a large-gradient rigid frame structure form with a column base hinged is hinged, and the top is provided with a horizontal spring support and a rotating spring support; for the structural form of the large-gradient rigid frame with the rigid connection of the column foot, the support condition at the bottom of the column is rigid connection, and the horizontal spring support and the rotating spring support are arranged at the top of the column.
Step two: performing hyperstatic structure solution, and calculating the rigidity of the horizontal spring support and the rotating spring support at the top of the column;
for the structural form of the large-gradient rigid frame with the hinged column foot, the calculation sketch of the displacement of the horizontal spring support and the rotation spring support is a statically indeterminate structure; for the structural form of the large-gradient rigid frame with the rigid connection of the column foot, the horizontal and rotating spring support displacement calculation sketch is a secondary statically indeterminate structure.
Step three: and establishing a rigid frame column buckling differential equation to obtain an transcendental equation of the calculated length coefficient relative to two parameters of the horizontal rigidity and the rotational rigidity of the support end.
Step four: and fitting and solving the equation, and determining a theoretical formula of the calculated length coefficient mu of the rigid frame column on the rigidity of the horizontal spring support and the rotating spring support.
(1) Hinged column base rigid frame
For the structural form of the large-gradient rigid frame with the hinged column base, when a buckling differential equation is established, the bottom support of the column is hinged, and the top is provided with the horizontal spring support and the rotating spring support, so that the boundary condition of the bottom end of the column is considered to be that the column displacement and the bending moment value are 0, and the top of the column is provided with the elastic restraint of translation and the elastic restraint of rotation displacement. Therefore, a calculation diagram can be drawn according to the constraint conditions, as shown in FIGS. 1 to 3.
For the structural form of the large-gradient rigid frame with hinged column feet, the calculation sketch of the displacement of the horizontal spring support and the rotation spring support are both statically indeterminate structures, a redundant constraint force is respectively removed in the calculation formula, and the constraint force is replaced by an unknown force X1As shown in fig. 4 and 5.
The horizontal stiffness and the rotational stiffness of the column top are obtained according to the solving process of fig. 6:
Figure BDA0001349688390000061
Figure BDA0001349688390000062
in the formula, K1For linear stiffness of the oblique beams of the rigid frame, K2The rigidity of the rigid frame column line is shown, and xi is a length proportionality coefficient which is respectively calculated according to the following formula.
Figure BDA0001349688390000063
Figure BDA0001349688390000064
Figure BDA0001349688390000065
The column foot articulated rigid frame column calculation can be simplified as the bottom support condition is articulated and the top is in the form of a horizontal spring support and a rotating spring support, as shown in fig. 7. Under the action of a load P, the rigid frame column deforms under the support condition. The clockwise rotation angle is positive, the rightward translation is positive, the moment and the horizontal force of the column end are positive when the moment and the horizontal force are in the same direction with the displacement, and negative when the moment and the horizontal force are in different directions.
Taking the separator shown in fig. 8, an equilibrium equation is established:
Figure BDA0001349688390000071
Figure BDA0001349688390000072
the connection type (6) and the formula (7) can obtain a fourth-order differential equation of the stressed axis of the rigid frame column, as shown in the formula (8),
EI2y(4)+Py″＝0 (8)
let k2＝P/EI2The general solution of the differential equation can be obtained,
y＝C1sinkx+C2coskx+C3x+C4(9)
y′＝C1kcoskx-C2ksinkx+C3(10)
y″＝-C1k2sinkx-C2k2coskx (11)
from the boundary conditions y (0) ═ 0 and y' (0) ═ 0, C can be obtained2＝0，C4＝0。
From Q (l) ═ PC3＝-kBy (l) and M (l) ═ EI2y″(l)＝rBy' (l) is available:
kBC1sinkl+(kBl-P)C3＝0 (12)
(EI2k2sin kl-rBkcoskl)C1-rBC3＝0 (13)
C1、C3the condition that a non-zero solution is to be present is that the determinant of the coefficients in equations (12) and (13) is zero, i.e. that
Figure BDA0001349688390000073
The determinant expansion can obtain:
kBlEI2k2sinkl-kBrBlkcoskl-PEI2k2sinkl+PrBkcoskl+kBrBsinkl＝0
(15)
since the parameters in the formula have the following relationship:
Figure BDA0001349688390000081
Figure BDA0001349688390000082
Figure BDA0001349688390000083
at the same time, let the support parameter RB、KBAre respectively as
Figure BDA0001349688390000084
Figure BDA0001349688390000085
And (3) substituting the formula (16) into the formula (15) to obtain a transcendental equation of the calculated length coefficient of the hinged column base rigid frame relative to the column:
Figure BDA0001349688390000086
when the section and the size of the rigid frame structure are known, the value of the spring stiffness can be determined according to the formula (1) and the formula (2) in a calculation mode, and then the calculated length coefficient of the column is obtained.
Iterative solution of the real root of equation (18) using secant methodThe solution, secant method is a point-by-point linearization method, and its basic idea is to give an initial value x in advance-1And x0Using the interval [ xk-1,xk]The upper secant approximation replaces the derivative function curve of the objective function, and the abscissa of the intersection of the secant and the abscissa is used as the approximate solution of the equation.
The iterative formula of the secant method is:
Figure BDA0001349688390000091
when | f (x)k)|<At 0.001, the iteration is ended, and the iteration result is taken as an approximate solution for calculating the length coefficient, and part of the result is shown in fig. 19.
Will KB、RBThe results of the calculations from 0 to 100 are plotted as a curved surface, as shown in fig. 9. Using MATLAB to carry out numerical fitting on the calculated data, wherein the fitting formula of the calculated length coefficient of the column base hinged rigid frame column is as follows:
Figure BDA0001349688390000092
the determination coefficient of the fitting formula (20) is 0.9879, the standard deviation is 0.1115, the calculation result is slightly larger than the calculation result of the transcendental equation, and the fitting formula calculation is accurate enough and can be applied to actual engineering calculation values.
(2) Rigid connection column foot rigid frame:
for the structural form of the large-gradient rigid frame with the rigid connection of the column foot, when a buckling differential equation is established, the conditions of the bottom support of the column are rigid connection, and the top is provided with the horizontal spring support and the rotating spring support, so that the boundary conditions of the bottom end of the column are considered to be that the displacement and the rotation angle of the column are 0, and the top of the column is provided with translational elastic constraint and rotational elastic constraint. A calculation diagram can be drawn according to the constraint conditions, as shown in fig. 10 to 12.
For the structural form of the large-gradient rigid frame with the rigid connection of the column foot, the calculation sketch of the displacement of the horizontal spring support and the displacement of the rotary spring support are both in a secondary statically indeterminate structure, two redundant constraint forces are respectively removed in the calculation formula, and the two redundant constraint forces are replaced by unknown forces X1、X2As shown in FIG. 13 and fig. 14.
The horizontal stiffness and the rotational stiffness of the column top are obtained according to the solving process of fig. 15:
Figure BDA0001349688390000101
Figure BDA0001349688390000102
the calculation of the column foot rigid connection rigid frame column can be simplified into that the bottom support condition is rigid connection, and the top is in the form of a horizontal spring support and a rotating spring support, as shown in fig. 16. Under the action of a load P, the rigid frame column deforms under the support condition. The clockwise rotation angle is positive, the rightward translation is positive, the moment and the horizontal force of the column end are positive when the moment and the horizontal force are in the same direction with the displacement, and negative when the moment and the horizontal force are in different directions.
The same method as that for establishing the buckling equation of the column base hinged rigid frame column is adopted, the isolated body shown in fig. 17 is taken, and the buckling equation of the column base rigid frame column is established:
EI2y(4)+Py″＝0 (23)
formula (23) is a fourth order differential equation of the axis compression of the rigid frame column, let k2＝P/EI2A general solution of the differential equation can be obtained:
y＝C1sinkx+C2coskx+C3x+C4(24)
y′＝C1kcoskx-C2ksinkx+C3(25)
y″＝-C1k2sinkx-C2k2coskx (26)
from the boundary conditions y (0) to 0 and y' (0) to 0, we obtain:
C2+C4＝0 (27)
C1k+C3＝0 (28)
from Q (l) ═ PC3＝-kBy (l) and M (l) ═ EI2y″(l)＝rBy' (l) is available:
kBC1sinkl+(kBl-P)C3＝0 (29)
(EI2k2sinkl-rBkcoskl)C1-rBC3＝0 (30)
C1、C3the condition of having a non-zero solution is that the determinant of the coefficient in the expressions (27) to (30) is zero, i.e., that
Figure BDA0001349688390000111
The determinant expansion can obtain:
kBEI2k2sinkl-kBrBkcoskl+2kBrBk-kBrBkcoskl-kBlEI2k3coskl
-kBrBlk2sinkl+PEI2k3coskl+PrBk2sinkl＝0 (32)
similarly, equations (16) and (17) are substituted into equation (30) together to obtain the transcendental equation of the calculated length coefficient of the articulated mast foundation relative to the column:
Figure BDA0001349688390000112
when the section size of the rigid frame and the size of the rigid frame are determined, the spring stiffness value can be determined by calculation according to the formula (21) and the formula (22), and then the calculated length coefficient of the column can be obtained.
Partial solutions of equation (33) were also numerically solved and the results are shown in fig. 20.
Will KB、RBThe results of the calculations from 0 to 100 are plotted as a curved surface, as shown in fig. 18. And (3) performing numerical fitting on the calculated data by using MATLAB, wherein a fitting formula of the calculation coefficient of the rigid frame column with the column base is as follows:
Figure BDA0001349688390000113
the determination coefficient of the fitting formula (34) is 0.9775, the standard deviation is 0.04818, the calculation result is slightly larger than the calculation result of the transcendental equation, and the fitting formula calculation is accurate enough and can be applied to actual engineering calculation values.
Solving the calculated length coefficient of the large-gradient rigid frame column, and if the large-gradient rigid frame column is hinged with a column base, calculating and determining the spring stiffness value according to the formula (1) and the formula (2); in the case of a rigid joint column base, the spring rigidity value can be determined by calculation according to equations (19) and (20). And calculating the length coefficient from the corresponding column in table 1 or table 2, or calculating the length coefficient according to the fitting formula (20) or (34).
The invention is not limited to the examples, and any equivalent changes to the technical solution of the invention by a person skilled in the art after reading the description of the invention are covered by the claims of the invention.

Claims (3)

1. The method for calculating the rigidity of the top support of the large-gradient rigid frame column and the calculated length of the rigid frame column is characterized by comprising the following steps of:
the method is realized by the following steps:
the method comprises the following steps: determining the conditions of the top support of the rigid frame column, and respectively making a calculation diagram for two spring support forms of a horizontal spring support and a rotary spring support;
step two: performing hyperstatic structure solution, and calculating the rigidity of the horizontal spring support and the rotating spring support at the top of the column;
step three: establishing a rigid frame column buckling differential equation, and obtaining an transcendental equation of two parameters of the calculated length coefficient about the horizontal rigidity and the rotational rigidity of the support end;
for the structural form of the large-gradient rigid frame with hinged column feet, the calculation sketch of the displacement of the horizontal spring support and the rotation spring support are both statically indeterminate structures, a redundant constraint force is respectively removed in the calculation formula, and the constraint force is replaced by an unknown force X1And obtaining the horizontal rigidity and the rotational rigidity of the column top:
Figure FDA0002560944040000011
Figure FDA0002560944040000012
in the formula, K1For linear stiffness of the oblique beams of the rigid frame, K2The rigidity of the rigid frame column line is shown, xi is a length proportionality coefficient, and the length proportionality coefficient is calculated according to the following formula respectively;
Figure FDA0002560944040000013
Figure FDA0002560944040000014
Figure FDA0002560944040000015
the calculation of the column foot hinged rigid frame column is simplified into that the bottom support is hinged, and the top is in the form of a horizontal spring support and a rotating spring support; under the action of a load P, the rigid frame column deforms under the support condition; wherein, clockwise rotation angle is positive, right translation is positive, moment and horizontal force of the column end are positive when the same direction as displacement and negative when different directions are opposite;
taking an isolated body, and establishing an equilibrium equation:
Figure FDA0002560944040000016
Figure FDA0002560944040000017
the four-order differential equation of the stressed rigid frame column axis is obtained by the joint vertical type (6) and the formula (7), as shown in the formula (8),
EI2y(4)+Py″＝0 (8)
let k2＝P/EI2The general solution of the differential equation can be obtained,
y＝C1sinkx+C2coskx+C3x+C4(9)
y′＝C1kcoskx-C2ksinkx+C3(10)
y″＝-C1k2sinkx-C2k2coskx (11)
from the boundary conditions y (0) ═ 0 and y' (0) ═ 0, C can be obtained2＝0，C4＝0；
From Q (l) ═ PC3＝-kBy (l) and M (l) ═ EI2y″(l)＝rBy' (l) is available:
kBC1sin kl+(kBl-P)C3＝0 (12)
(EI2k2sin kl-rBk cos kl)C1-rBC3＝0 (13)
C1、C3the condition that a non-zero solution is to be present is that the determinant of the coefficients in equations (12) and (13) is zero, i.e. that
Figure FDA0002560944040000021
The determinant expansion can obtain:
kBlEI2k2sin kl-kBrBlk cos kl-PEI2k2sin kl+PrBk cos kl+kBrBsin kl＝0(15)
since the parameters in the formula have the following relationship:
Figure FDA0002560944040000022
Figure FDA0002560944040000023
Figure FDA0002560944040000024
at the same time, let the support parameter RB、KBAre respectively as
Figure FDA0002560944040000025
Figure FDA0002560944040000026
And (3) substituting the formula (16) into the formula (15) to obtain a transcendental equation of the calculated length coefficient of the hinged column base rigid frame relative to the column:
Figure FDA0002560944040000031
step four: fitting and solving the equation, and determining a theoretical formula of the calculated length coefficient mu of the rigid frame column about the rigidity of the horizontal spring support and the rotating spring support;
the real root of equation (18) is solved iteratively by a secant method, which is a point-by-point linearization method that presets an initial value x-1And x0Using the interval [ xk-1,xk]The secant approximation replaces a derivative function curve of the target function, and the abscissa of the intersection point of the secant and the horizontal axis is used as the approximate solution of the equation;
the iterative formula of the secant method is:
Figure FDA0002560944040000032
when | f (x)k)|<When the length coefficient is 0.001, finishing the iteration, and taking an iteration result as an approximate solution of the calculated length coefficient;
will KB、RBDrawing a curved surface from the calculation results of 0 to 100; using MATLAB to carry out numerical fitting on the calculated data, wherein the fitting formula of the calculated length coefficient of the column base hinged rigid frame column is as follows:
Figure FDA0002560944040000033
2. the method for calculating the rigidity of the top support of the high-gradient rigid frame column and the calculated length of the rigid frame column according to claim 1, is characterized in that:
in the first step, when a calculation diagram is established, the bottom support of a column in a large-gradient rigid frame structure form with a column base hinged is hinged, and a horizontal spring support and a rotating spring support are arranged at the top of the column; for the structural form of the large-gradient rigid frame with the rigid connection of the column foot, the support condition at the bottom of the column is rigid connection, and the horizontal spring support and the rotating spring support are arranged at the top of the column.
3. The method for calculating the rigidity of the top support of the high-gradient rigid frame column and the calculated length of the rigid frame column according to claim 1, is characterized in that:
in the second step, for the structural form of the large-gradient rigid frame with the hinged column foot, the horizontal and rotating spring support displacement calculation sketch diagrams are both in a one-time statically indeterminate structure; for the structural form of the large-gradient rigid frame with the rigid connection of the column foot, the horizontal and rotating spring support displacement calculation sketch is a secondary statically indeterminate structure.
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