CN117150610A - General checking method for arbitrary section stress of bridge concrete structure - Google Patents
General checking method for arbitrary section stress of bridge concrete structure Download PDFInfo
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Abstract
The invention discloses a general calculation method for arbitrary section stress of a bridge concrete structure, which comprises the following steps: inputting material information, section information, reinforcement information and internal force information of a bridge concrete structure; translating an axial force action point in the internal force information to the centroid of the section, and synthesizing a translated bending moment; preliminarily setting a neutralization shaft angle, and rotating the cross section to enable the preliminarily set neutralization shaft to be horizontal; decomposing resultant bending moment of the rotated section, and iteratively calculating the height of the neutralization shaft; and calculating and judging whether the internal transverse bending moment and the external transverse bending moment are balanced or not, and detecting a stress result. The invention constructs a general section stress analysis and calculation method, and deduces a general stress calculation method applicable to concrete structures with arbitrary section and arbitrary reinforcement form through a neutralization shaft angle ladder encryption algorithm and a neutralization shaft height rapid iteration algorithm, thereby establishing a bridge concrete structure stress detection and calculation method. The invention solves the problem of the design accuracy of the complex concrete structure of the bridge.
Description
Technical Field
The invention belongs to the technical field of bridge engineering in the transportation industry, and particularly relates to a general calculation method for arbitrary section stress of a bridge concrete structure.
Background
The high-speed rail is a business card built by the infrastructure of China, and the high-speed rail network throughout the country has great effect in facilitating the travel of people and promoting the regional economic development, and is a locomotive for pulling economic growth. In the construction of the high-speed railway, in order to save land and ensure engineering quality, most road sections adopt bridge structures, and most of the high-speed railway bridge structures adopt concrete structures, because the concrete structures have lower manufacturing cost, are easy to maintain and simple and convenient to construct, and form a standard structure system, the high-speed railway bridge structure is widely applied to the railway engineering construction of China.
Besides high-speed rails, china also has national expressways and common highway networks, and is connected with national cities and villages. Bridge structures remain an option when crossing valleys, rivers and interchange requirements. Similar to railway engineering, highway bridges are mainly concrete structures and are widely applied due to low manufacturing cost and convenient construction and maintenance.
The bridge concrete structure is divided into a reinforced concrete structure and a prestressed concrete structure according to different configuration steel bars, wherein common steel bars are configured in the reinforced concrete structure, and prestressed steel bundles are configured in the prestressed concrete structure. When the structure inspection is carried out, the reinforced concrete structure generally needs to inspect and calculate stress and crack indexes, and the prestressed concrete structure needs to inspect and calculate stress and crack indexes. Stress, crack or crack detection at any time is an important basic parameter, so that accurate calculation of section stress is the basis of bridge concrete structure design calculation.
In the existing railway and highway bridge design specifications, only the stress calculation formulas of simple sections such as rectangle, T shape, I shape, round shape and the like are given, and only the simple conditions of single-layer reinforcement and unidirectional stress are aimed at. The structural reinforcement form faced by the standard regulations is single, the use condition is harsh, and the structural reinforcement form is far away from the complex structural forms and reinforcement types of box girders, round-end bridge piers and the like commonly used in actual engineering, and cannot be directly used for guiding design. In practical calculation, the simple section and the simple reinforcement form are used for replacing calculation, and the design accuracy of the concrete structure and the requirement of high-quality development of structural design calculation cannot be guaranteed.
Aiming at the problems, it is necessary to develop a general calculation method for the stress of any section of the bridge concrete structure, a general calculation method for the stress of the bridge concrete structure is established on the basis of the general calculation method by deducing a general section analysis theory and using a neutral axis angle step encryption algorithm and a neutral axis height rapid iteration algorithm, so that the problem of the accuracy of calculating the stress of the complex concrete structure of the bridge is solved.
Disclosure of Invention
The invention provides a general calculation method for arbitrary section stress of a bridge concrete structure, which aims to solve the problems existing in the prior art.
The technical scheme of the invention is as follows: the general checking and calculating method for the arbitrary section stress of the bridge concrete structure comprises the following steps:
A. inputting material information, section information, reinforcement information and internal force information of a bridge concrete structure;
B. translating an axial force action point in the internal force information to the centroid of the section, and synthesizing a translated bending moment;
C. preliminarily setting a neutralization shaft angle, and rotating the cross section to enable the preliminarily set neutralization shaft to be horizontal;
D. decomposing resultant bending moment of the rotated section, and iteratively calculating the height of the neutralization shaft;
E. f, calculating and judging whether the internal transverse bending moment and the external transverse bending moment are balanced, and if the internal transverse bending moment and the external transverse bending moment are balanced, executing the step F; c, if the internal force transverse bending moment and the external force transverse bending moment are unbalanced, returning to the step C;
F. and (5) checking a stress result.
Furthermore, the material information, the section information, the reinforcement information and the internal force information of the bridge concrete structure are input in the step A, and the concrete process is as follows:
firstly, dividing material information of a bridge concrete structure into concrete material parameters and reinforcing steel bar material parameters for input;
then, uniformly inputting the section information;
then, inputting reinforcement information consisting of coordinates of each beam of reinforcement and the number of beams of reinforcement;
and finally, inputting internal force information consisting of the magnitude of the axial force, the action point and bending moments in two directions.
Furthermore, the section information is input uniformly, and the specific process is as follows:
firstly, performing discrete treatment on a simple section and a complex section;
then, connecting a plurality of contour lines obtained after the discretization end to end;
finally, the contour array is used for input.
Further, the step B translates the axial force action point in the internal force information to the centroid of the section, and synthesizes the translated bending moment, and the specific process is as follows:
firstly, translating an axial force acting point in internal force information to a section centroid position in section information;
then, calculating an additional bending moment caused by the translation axis force;
then, the additional bending moment is overlapped with the original bending moment in the internal force information;
and finally, synthesizing the bending moment by using a vector method.
Further, the step C is to preliminarily set the neutralization axis angle, and the rotation section is to level the preliminarily set neutralization axis, and the specific process is as follows:
firstly, preliminarily setting a neutralization axis, and assuming a neutralization axis angle theta;
then, the section is rotated until the neutralization shaft is in a horizontal state;
and finally, in the process of section rotation, simultaneously rotating reinforcement information and internal force information.
Further, the step D is to decompose the resultant bending moment of the section after rotation, and the height of the neutralization shaft is calculated iteratively, and the concrete process is as follows:
firstly, a local coordinate system, namely a Y '-Z' coordinate system is newly arranged on a section after rotation;
then decomposing the resultant bending moment to the Y ' axis and the Z ' axis of the Y ' -Z ' coordinate system to obtain resultant bending moment M ' of the Y ' axis ' y And a resultant bending moment M of the Z' axis z ′；
Finally, the height y of the neutralization shaft is calculated and obtained by using the principle of material mechanics according to different stress forms.
Further, the Y '-Z' axis in the Y '-Z' coordinate system is aligned with the preliminarily set neutralization axis n 0 -n 0 Parallel.
Further, step E calculates and judges whether the internal force transverse bending moment and the external force transverse bending moment are balanced, if so, the step F is executed; if the internal force transverse bending moment and the external force transverse bending moment are unbalanced, returning to the step C, wherein the specific process is as follows:
firstly, dividing a section into a left part and a right part along a Z' axis;
then, calculating the transverse bending moment of the pressed concrete to the Z' axis;
then, calculating the transverse bending moment of the steel bar to the Z' axis;
then, summing the transverse bending moment calculated by the steel bar and the pressed concrete to obtain M zn ；
Finally, the calculated internal force transverse bending moment is M zn With calculated external transverse bending moment M z ' compare and determine if balanced.
Further, if the internal force transverse bending moment and the external force transverse bending moment are unbalanced, the angle theta of the neutralization shaft is preliminarily set to be wrong in the step C, the angle theta of the neutralization shaft is re-assumed, and calculation is continued until the balance condition of the internal force transverse bending moment and the external force transverse bending moment is met.
Further, under the condition that the internal transverse bending moment and the external transverse bending moment meet balance conditions, a section stress distribution state is obtained.
The beneficial effects of the invention are as follows:
the invention aims at the problem of calculating the stress of the bridge concrete structure, and a general stress calculation method suitable for the concrete structure with any cross section and any reinforcement form is constructed by deducing a general cross section analysis theory, a neutralization shaft angle step encryption algorithm and a neutralization shaft height rapid iteration algorithm, so that the method for detecting the stress of the bridge concrete structure is established, and the problem of accuracy of calculating the stress of the complex concrete structure of the bridge is solved.
The method can accurately calculate the stress of the bridge concrete structure in the field of transportation, and solves the problem of accuracy in calculating the stress of the complex bridge concrete structure.
Drawings
FIG. 1 is a schematic flow chart of the steps of the present invention;
FIG. 2 is a schematic diagram of a cross-sectional information input according to the present invention;
FIG. 3 is a schematic illustration of axial force translation of the present invention;
FIG. 4 is a schematic view of a rotating section and exploded resultant force bending moment of the present invention;
FIG. 5 is a schematic diagram of a stress calculation of a flexural member of this invention;
FIG. 6 is a schematic diagram of stress calculation of a small eccentric stressed member of the present invention;
FIG. 7 is a schematic diagram of stress calculation of a large eccentric stressed member of the present invention;
FIG. 8 is a schematic diagram of the transverse bending moment calculation of the present invention;
fig. 9 is a rectangular cross-sectional view of an embodiment of the present invention.
FIG. 10 is a graph showing the calculated result of the center axis position in a rectangular cross section according to an embodiment of the present invention;
FIG. 11 is a cross-sectional view of a two-round end of an embodiment of the present invention;
FIG. 12 is a graph showing the result of calculating the center axis position of a cross-section with two round ends according to the embodiment of the present invention.
Detailed Description
The present invention will be described in detail below with reference to the drawings and examples:
as shown in fig. 1 to 12, a general method for checking the stress of any section of a bridge concrete structure comprises the following steps:
A. inputting material information, section information, reinforcement information and internal force information of a bridge concrete structure;
B. translating an axial force action point in the internal force information to the centroid of the section, and synthesizing a translated bending moment;
C. preliminarily setting a neutralization shaft angle, and rotating the cross section to enable the preliminarily set neutralization shaft to be horizontal;
D. decomposing resultant bending moment of the rotated section, and iteratively calculating the height of the neutralization shaft;
E. f, calculating and judging whether the internal transverse bending moment and the external transverse bending moment are balanced, and if the internal transverse bending moment and the external transverse bending moment are balanced, executing the step F; c, if the internal force transverse bending moment and the external force transverse bending moment are unbalanced, returning to the step C;
F. and (5) checking a stress result.
Step A, inputting material information, section information, reinforcement information and internal force information of a bridge concrete structure, wherein the concrete process comprises the following steps:
firstly, dividing material information of a bridge concrete structure into concrete material parameters and reinforcing steel bar material parameters for input;
then, uniformly inputting the section information;
then, inputting reinforcement information consisting of coordinates of each beam of reinforcement and the number of beams of reinforcement;
and finally, inputting internal force information consisting of the magnitude of the axial force, the action point and bending moments in two directions.
The section information is input uniformly, and the specific process is as follows:
firstly, performing discrete treatment on a simple section and a complex section;
then, connecting a plurality of contour lines obtained after the discretization end to end;
finally, the contour array is used for input.
And B, translating an axial force action point in the internal force information to the centroid of the section, and synthesizing a translated bending moment, wherein the specific process is as follows:
firstly, translating an axial force acting point in internal force information to a section centroid position in section information;
then, calculating an additional bending moment caused by the translation axis force;
then, the additional bending moment is overlapped with the original bending moment in the internal force information;
and finally, synthesizing the bending moment by using a vector method.
C, preliminarily setting a neutralization shaft angle, and rotating the cross section to enable the preliminarily set neutralization shaft to be horizontal, wherein the specific process is as follows:
firstly, preliminarily setting a neutralization axis, and assuming a neutralization axis angle theta;
then, the section is rotated until the neutralization shaft is in a horizontal state;
and finally, in the process of section rotation, simultaneously rotating reinforcement information and internal force information.
And D, decomposing resultant force bending moment of the rotated section, and iteratively calculating the height of the neutralization shaft, wherein the concrete process is as follows:
firstly, a local coordinate system, namely a Y '-Z' coordinate system is newly arranged on a section after rotation;
then decomposing the resultant bending moment to the Y ' axis and the Z ' axis of the Y ' -Z ' coordinate system to obtain resultant bending moment M ' of the Y ' axis ' y And a resultant bending moment M of the Z' axis z ′；
Finally, the height y of the neutralization shaft is calculated and obtained by using the principle of material mechanics according to different stress forms.
The Y ' axis in the Y ' -Z ' coordinate system and the preliminarily set neutralization axis n 0 -n 0 Parallel.
E, calculating and judging whether the internal transverse bending moment and the external transverse bending moment are balanced, and if the internal transverse bending moment and the external transverse bending moment are balanced, executing the step F; if the internal force transverse bending moment and the external force transverse bending moment are unbalanced, returning to the step C, wherein the specific process is as follows:
firstly, dividing a section into a left part and a right part along a Z' axis;
then, calculating the transverse bending moment of the pressed concrete to the Z' axis;
then, calculating the transverse bending moment of the steel bar to the Z' axis;
then, summing the transverse bending moment calculated by the steel bar and the pressed concrete to obtain M zn ；
Finally, the calculated internal force transverse bending moment is M zn With calculated external transverse bending moment M z ' compare and determine if balanced.
If the internal force transverse bending moment and the external force transverse bending moment are unbalanced, the step C is to preliminarily set the angle theta of the neutralization shaft to be wrong, and the angle theta of the neutralization shaft is re-assumed, and the calculation is continued until the balance condition of the internal force transverse bending moment and the external force transverse bending moment is met.
And under the condition that the internal force transverse bending moment and the external force transverse bending moment meet balance conditions, obtaining the section stress distribution state.
Specifically, the material information in the step A refers to concrete material parameters and reinforcing steel bar material parameters, including labels, elastic modulus, reinforcing steel bar conversion area ratio and the like.
Specifically, in the step D, the neutralization axis height is unknown and cannot be obtained through qualitative analysis, so that a dichotomy can be used for quick solution.
Specifically, the step F is used for checking the stress result, and the specific process is as follows:
firstly, aiming at concrete structures of railways and highway bridges, obtaining allowable stress values;
then, the concrete compressive stress sigma is obtained by using the step D c And steel bar tensile stress sigma s ；
And finally, judging whether the allowable stress is exceeded, if so, checking that the allowable stress is not passed, and otherwise, checking that the allowable stress is passed.
More specifically, the invention aims at the concrete structures of the railway and highway bridges, the corresponding specifications are the railway bridge and culvert concrete structure design specification TB10092-2017, the railway bridge and culvert design specification (limit state method) Q/CR9300-2018 and the highway reinforced concrete and prestressed concrete bridge and culvert design specification JTG3362-2018, and for the three specifications, the allowable stress values are obtained by respectively searching tables 3.1.4, 7.5.21-2 and chapter 7;
specifically, the contour lines obtained after the section information in the step A is discretized are connected end to end, and the contour lines can only be straight lines or circular arcs. For one contour line, the rule of the input data is as follows: the contour line includes:
(a) Coordinates of starting point (y) 1 ,z 1 )；
(b) End point coordinates (y) 2 ,z 2 )；
(c) Radius r, greater than 0 when cycled clockwise; when the rotation is anticlockwise, the rotation is smaller than 0, and the straight line is 0;
(d) Arc marking, wherein the inferior arc or semicircle is 1, and the major arc is-1;
(e) The outline is marked, the outer outline is larger than 0, and the inner outline is smaller than 0.
As an illustration, a rectangular cross section with width 2 and height 1 as shown in fig. 2, the contour line data is:
table 1 rectangular section contour line data
And A, inputting the coordinates (y, z) of each beam of reinforcing steel bars and the number num of the beams of reinforcing steel bars for reinforcing steel bar arrangement information, wherein the coordinates of the reinforcing steel bars are consistent with the cross-section coordinate system.
Step A for the internal force information, the stress calculation needs to consider the section axial force N and the axial force acting point coordinates (y n ,z n ) And bending moment M y 、M z The positive direction defines the axial force to be positive when pulled and the bending moment to be positive when parallel to the positive direction of the local coordinate system.
The rectangular cross section shown in fig. 3 is subjected to both axial force and bending moment, wherein the axial force is pressure and thus negative, and the point of action coordinates (y n ,z n ) The method comprises the steps of carrying out a first treatment on the surface of the Bending moment M y The direction is parallel to the negative Y-axis direction and therefore negative; bending moment M z The direction is forward parallel to the Z axis and therefore positive.
Specifically, the axial force action point in the internal force information is translated to the centroid of the section in the step B, and the bending moment after the translation is synthesized. Preprocessing the internal force information input in the step A to generate standard internal force data, wherein the internal force data comprises the following specific steps:
first, the axial force is translated to the position of the cross-sectional centroid, as shown in FIG. 3, where the original axial force application point N is at (y n ,z n ) To be translated to the cross-section centroid N 1 The coordinates are (0, 0).
Then, the translation axial force can cause additional bending moment, and the additional bending moment needs to be overlapped with the original bending moment, and the bending moment overlapped calculation formula is as follows:
M yt ＝M y +N(z t -z c ) (1)
M zt ＝M z +N(y t -y c ) (2)
in (y) c ,z c ) For the point coordinates after axial force translation, (y) t ,z t ) For the point coordinates before axial force translation, a rectangular cross section is shown in FIG. 3For example, the total bending moment M after translation yt 、M zt The calculation formula is as follows:
M yt ＝M y +N(z n -0) (3)
M zt ＝M z +N(y n -0) (4)
and finally, synthesizing a bending moment by using a vector method, wherein the calculation formula of the synthesized bending moment M is as follows:
bending moment and local coordinate system Y-axis clamping angle theta M The calculation formula is as follows:
θ M ＝arctan(M zt /M yt ) (6)
through the above process, the cross-sectional internal force N, M after the treatment is obtained.
Specifically, the neutralization axis angle is preliminarily set in the step C, and the rotation section enables the preliminarily set neutralization axis to be horizontal. In general, it can be demonstrated from the hydrostatic balance that the neutralization axis is not necessarily perpendicular to the plane of action of the resultant bending moment. The angle of the neutralization shaft is related to the section, the reinforcement form and the stress condition, the influence factors are numerous, the mechanism is complex, the neutralization shaft cannot be directly determined by a deduction formula, and the neutralization shaft is determined by trial calculation. The specific calculation process is as follows:
first, a neutralization axis angle θ is assumed, and the angle refers to the rotation angle from the local coordinate axis Y axis to the assumed neutralization axis, and the clockwise rotation is positive and the counterclockwise rotation is negative.
With a round end section as shown in FIG. 4, the local coordinate system is Y-Z, the resultant bending moment M is parallel to the Y axis, and the neutralization axis is assumed to be n 0 -n 0 The included angle between the Y-axis and the Y-axis is theta.
Then, the rotation section enables the initial neutral axis to be horizontal, and the rotation process needs to simultaneously rotate the section, the reinforcing bars and the internal force.
Taking the round end section shown in FIG. 4 as an example, the section needs to be rotated counterclockwise by an angle θ, for any point P on the section i (y i ,z i ) After rotation ofCoordinates (y' i ,z′ i ) The calculation formula is as follows:
y′ i ＝y i cosθ+z i sinθ (7)
z′ i ＝-y i sinθ+z i cosθ (8)
similarly, all the section profile information, the reinforcement information and the internal force information are rotated to initially set the neutralization axis n 0 -n 0 The horizontal, rotated cross section is shown on the right side of fig. 4.
Specifically, the step D is to decompose the resultant bending moment of the section after rotation, and the height of the neutralization shaft is calculated in an iterative manner, wherein the specific process is as follows:
first, the local coordinate system is reset to Y ' -Z ' for the cross section after the rotation in the step C, so that the Y ' axis and the neutral axis n 0 -n 0 Parallel as shown in fig. 4.
Then decomposing the resultant bending moment to M 'in the Y' axis and Z 'axis' y And M z ' as shown in fig. 4.
Finally, an internal force bending moment M 'is used' y And the axial force N calculates the neutral axis height y.
More specifically, the force is divided into the following force-bearing forms according to the difference of internal forces:
the first and the bent state are as follows:
when the axial force N is zero, the bending member is used, the stress calculation schematic diagram of the bending member is shown in fig. 5, and the concrete compressive stress sigma is shown c And steel bar tensile stress sigma s The calculation formulas are respectively as follows:
wherein M' y To calculate the bending moment; n is the ratio of the converted areas of the reinforced steel bars and the concrete, and the value is obtained according to the standard requirement; w (W) 0 、W s The calculated cross section resisting moment is the calculated position, and the calculated cross section resisting moment is the ratio of the calculated cross section centering and shaft inertia moment to the calculated position to the centering and shaft distance.
The position of the neutral axis is determined according to the calculation stress, and the stress at the neutral axis is zero according to the principle of material mechanics, so that the position y of the neutral axis meets the following requirements:
S y ＝S hy +S gy ＝0 (11)
wherein S is hy For the compression zone and the steel bar conversion area to the static moment of the neutralization axis y, S gy And converting the area pair and the static moment of the neutral axis y for the steel bars in the tension zone.
Due to S hy 、S gy All are functions of y, and change when the neutralization axis moves. Again the flexural member must have compression and tension zones so that the neutral axis is located within the cross section so that the neutral axis position can be quickly resolved using a dichotomy.
When the neutral axis position is solved by the dichotomy, one neutral axis position y can be selected at will 0 S in the calculation formula (11) hy 、S gy And S is y If S y <0, which means that the area of the part of the pressed area is too small, the height of the pressed area should be increased, and the neutralization shaft is lowered; on the contrary if S y >0 indicates that the area of the nip portion is too large, the nip height should be reduced, and the neutralization shaft should be raised.
The iteration is repeated until equation (11) is satisfied. The convergence speed of the dichotomy is high, and the neutral axis position y can be found by calculating for several times.
Secondly, eccentric stress state, specifically as follows:
when the axial force N is not zero, the eccentric force-bearing member is an eccentric force-bearing member, and the eccentric force-bearing member can be divided into small eccentric force-bearing and large eccentric force-bearing according to whether a tension zone and a compression zone exist on the section at the same time, wherein the small eccentric force-bearing refers to the condition of full-section compression or full-section tension; large eccentric forces refer to sections where both tension and compression zones are present. The two cases are different in calculation method, and are discussed separately below.
(a) Small eccentric stress
The small eccentric stress member refers to the condition of full section compression or full section tension. For the former, the concrete and the steel bars are pressed, all materials on the section participate in work, the neutralization shaft is positioned at the full-section converted section-shaped mandrel (heavy mandrel), and the stress calculation schematic diagram is shown in fig. 6.
Distance y of small eccentric stress member inner distance conversion section shape mandrel (heavy mandrel) 1 The stress calculation formula at the position is:
wherein N is the axial force; a is that 0 、J 0 The reduced cross-sectional area and the reduced cross-sectional area are respectively the moment of inertia of the neutralization shaft, wherein the small eccentric compression member considers the reduced cross-section of the full-section concrete and the steel bar, and the small eccentric tension member only considers the reduced cross-section of the steel bar; η is an increase coefficient of the influence of deflection on the eccentricity, and is calculated according to specification; y is 1 To calculate the stress point position, the stress point position x is calculated from the upper edge of the cross section as shown in FIG. 6 1 At times x 1 ＝y-x；To neutralize the bending moment of the shaft, the calculation formula is:
wherein M' y Bending moment is generated at the pure concrete section mandrel; r is the distance from the pure concrete section-shaped mandrel to the upper edge of the section; y is the scaled cross-sectional mandrel to cross-sectional upper edge distance as shown in fig. 6.
Substituting formula (13) into formula (12) to obtain:
in the method, in the process of the invention,
(b) Large eccentric force
When the concrete in the small eccentric compression member has tensile stress and the steel bar in the small eccentric tension member has compressive stress, the calculation is needed according to the large eccentric stress. Unlike small eccentricity, the cross-sectional stress of the large eccentric stress member at the neutralization axis y is equal to zero, the concrete at the part below the neutralization axis is not considered to be pulled, all the tensile stress is born by the steel bars in the pulling area, and the stress calculation schematic diagram is shown in fig. 7.
The moment of inertia and the static moment of the reinforced concrete converted section (without considering the tension of the concrete) to the neutral axis y are respectively J y 、S y External forces N and M' y Bending moment to the neutralization axis y is N (y-R) -M' y Then, there are:
the micro section with the height h from the neutralization axis y in the figure 7 is taken, the width of the micro section is b (h), the stress is sigma h/y, and the area of the tension steel bar is A s Stress of n sigma x 2 And/y, the area of the pressed reinforcing steel bar is A' s Stress of n sigma x 1 And/y, according to the force balance equation:
the simultaneous expression (15) and the expression (16) can be obtained:
wherein J is y 、S y The moment of inertia and the static moment of the converted section to the y-axis, respectively. This is the calculation formula for finding the center and shaft positions of the large eccentric force-bearing member.
After the neutral axis is found, the distance x from the section of the large eccentric stress component to the neutral axis 1 The calculation formula of the stress is as follows:
wherein S is i ＝0，
As can be seen from the formula (17), the neutral axis position is related to the external force, the axial force and the bending moment, and the formula is shown at the left side J y 、S y All are functions of y, so the solution of the neutral axis is a process of iterative calculation, which can be calculated using the concept of dichotomy, similar to a flexural member. Because the neutralization shaft is positioned in the section, one neutralization shaft position y can be selected at will 0 And (3) calculating the terms on the two sides of the medium number in the formula (17), continuously adjusting the position of the medium axis according to the relative relation between the terms, and repeatedly iterating until the terms are met. The convergence speed of the dichotomy is high, and the neutral axis position y can be found by calculating for several times.
More specifically, whether the eccentric stress is large eccentric stress or small eccentric stress cannot be determined before the eccentric stress calculation, the small eccentric stress is calculated firstly, then the large eccentric stress and the small eccentric stress are judged, and if the small eccentric stress is confirmed, the stress is calculated by using a formula (14); if a large eccentric force is confirmed, the neutral axis position is calculated iteratively using a dichotomy according to equation (17), and then the stress is calculated using equation (18).
Specifically, the step of calculating and judging whether the internal transverse bending moment and the external transverse bending moment are balanced or not, and if the internal transverse bending moment and the external transverse bending moment are balanced, executing the step F; and C, if the internal force transverse bending moment and the external force transverse bending moment are unbalanced, returning to the step C. The process of obtaining the section stress distribution state by iteratively calculating the middle and shaft heights is deduced based on the principle of material mechanics, so that the balance condition of the shaft force and the Y 'shaft direction, namely the internal and external forces of the vertical bending moment, is met, but the balance condition of the Z' shaft direction, namely the internal and external forces of the transverse bending moment, is not necessarily met. The specific judging process is as follows:
first, the cross section is divided into left and right parts along the Z 'axis, because the transverse bending moment caused by the material stress on the left and right sides of the Z' axis has opposite signs. As shown in fig. 8, the position of the neutral axis calculated in the step D is n-n, and the axis is more compressed and less tensioned, and the concrete stress and all the steel bar stress are calculated only above the n-n axis because the concrete material is not considered to be tensioned.
Then, the transverse bending moment of the pressed concrete to the Z' axis is calculated.
Because the concrete stress is distributed in a triangle, the concrete stress cannot be directly integrated, the concrete stress can be divided into a plurality of small sections along the height direction, the stress at the center of each small section is taken as the stress of the whole section, and the transverse bending moment of the section to the Z' axis is calculated by summing the sections.
The left-hand shadow of the Z' axis in FIG. 8 is shown as A i The stress at the center of the segment is sigma i The distance from the center of the segment to the Z' axis is y i The section is transversely bent to Z' axis by a bending moment M zni The calculation formula is as follows:
M zni ＝σ i A i y i (19)
by analogy, the transverse bending moment of all the pressed concrete materials on the Z' axis can be calculated.
Then, the bending moment of the reinforcement bar transverse to the Z' axis is calculated and summed with the concrete portion.
The calculation principle of the transverse bending moment of the steel bar is the same as that of a concrete small section, and all effective materials of the section are used for calculating the Z' -axis transverse bending moment and M zn The calculation formula is as follows:
M zn ＝∑σ i A i y i +∑σ si A si y si (20)
finally, compare M z ' and M zn And judging whether the transverse bending moment of the internal and external forces is balanced or not. And (3) when the balance of the transverse bending moment of the internal and external forces is not met, the preset neutral axis direction theta of the step C is indicated to be wrong, the neutral axis direction is re-assumed, and the steps C-E are repeated until the balance condition of the transverse bending moment of the internal and external forces is met. The cross-sectional stress distribution state was obtained on the basis of this.
In actual calculation, the possible interval of the neutral axis direction cannot be determined through qualitative analysis, so that the calculated amount is large and the calculation speed is unstable due to blind trial calculation of the neutral axis angle, and as shown in fig. 4, the included angle between the neutral axis and the resultant bending moment direction is positioned in the interval (-pi/2, pi/2), so that the calculated amount can be reduced and the calculation speed can be improved by adopting a step encryption algorithm.
The specific calculation flow of the ladder encryption algorithm is as follows:
firstly, 5 DEG is selected as a first step length, the interval (-pi/2, pi/2) is divided into [ -85, -80, -75, …, -10, -5, 10, …, 75, 80, 85] angle arrays, the angles are calculated and judged one by using the steps B-E, the requirement is met when the angles are met, and the next step calculation is continued when all the angles are not met.
Then, 2 DEG is selected as a second step length, the interval (-pi/2, pi/2) similar to the 1 st step is divided into a calculated angle array and calculated one by one, and the calculated angles are not calculated any more. And when the calculated angles are not satisfied, continuing the next step calculation.
Finally, 1 degree is selected as the third step length, and the calculation process is the same as the above. And the same is done when the calculated angle is not satisfied … until satisfied.
Through the calculation process of gradual encryption of multiple steps, the calculation amount can be effectively reduced, and the calculation speed can be improved.
Specifically, step F provides for checking the stress results. The invention aims at concrete structures of railways and highway bridges, and the corresponding specifications are a railway bridge and culvert concrete structure design specification TB10092-2017, a railway bridge and culvert design specification (limit state method) Q/CR9300-2018 and a highway reinforced concrete and prestressed concrete bridge and culvert design specification JTG3362-2018, and the concrete processes are as follows:
firstly, for the 'concrete structural design Specification for railway bridge and culvert TB 10092-2017', firstly, the allowable stress value is inquired by using the specification table 3.1.4, and then the concrete compressive stress sigma is calculated by using the step D c And steel bar tensile stress sigma s And judging whether the allowable stress is exceeded, if so, checking that the allowable stress is not passed, and otherwise, checking that the allowable stress is passed.
Then, for the "railway bridge and culvert design Specification (Limit State method) Q/CR 9300-2018", firstly, the allowable stress value is queried by using the Specification Table 7.5.21-2, and then the concrete compressive stress sigma is obtained by using the calculation of the step D c And the tensile stress of the steel barσ s And judging whether the allowable stress is exceeded, if so, checking that the allowable stress is not passed, and otherwise, checking that the allowable stress is passed.
Finally, for the design specification JTG3362-2018 of reinforced concrete and prestressed concrete bridge culvert of highway, firstly, calculating the allowable stress value according to chapter 7 of the specification, and then using the step D to calculate the concrete compressive stress sigma c And steel bar tensile stress sigma s And judging whether the allowable stress is exceeded, if so, checking that the allowable stress is not passed, and otherwise, checking that the allowable stress is passed.
Example 1
The invention is used for calculating the stress of a rectangular section member which is subjected to the bending moment in two directions at the same time. The rectangular section is 0.6m high and 0.6m wide, 8 steel bars are uniformly distributed along the section, and the diameter is 20mm. The concrete material is C30, and the reinforcing steel bar material is HRB400. The cross section is shown in fig. 9. The cross section is bent, M y ＝90kN*m，M z ＝45kN*m。
The general calculation method provided by the invention is used for calculating the distribution state of the section stress, and the neutral axis position calculation result is shown in fig. 10. To verify the correctness of the results, the stresses of the effective materials such as the section compressed concrete and the steel bars are summed up, the vertical bending moment and the transverse bending moment are calculated, and the vertical bending moment and the transverse bending moment are compared with the internal force bending moment, and the results are shown in table 2.
Table 2 comparison of rectangular section bending moment results
As shown in the table above, the difference between the effective material bending moment and the internal force bending moment obtained by calculating the stress distribution state by using the method is small, the calculation accuracy is high, the method meets the static equilibrium condition, and the method provided by the invention is accurate in calculation.
Example two
The stress calculation is carried out on a round end-shaped section component which is subjected to the action of bending moments in two directions at the same time by using the invention. The round end section has a longitudinal width of 1m and a transverse width of 2m, 6 reinforcing steel bars are uniformly distributed along the section, and the diameter is 20mm. The concrete material is C40, and the reinforcing steel bar material is HRB400. The cross section is shown in fig. 11. The cross section is subjected to shaft pressure and two simultaneouslyMoment action, N= -1500kN (pressure), M y ＝200kN*m，M z ＝80kN*m。
The general calculation method provided by the invention is used for calculating the distribution state of the section stress, and the neutral axis position calculation result is shown in fig. 12. To verify the correctness of the results, the stresses of the effective materials such as the section compressed concrete and the steel bars are summed up, and the axial force, the vertical bending moment and the transverse bending moment are calculated and compared with the internal force, and the results are shown in table 3.
Table 3 comparison of bending moment results for round end sections
As shown in the table above, the difference between the effective material bending moment and the internal force bending moment obtained by using the stress distribution state calculated by the method is small, the calculation accuracy is high, the method meets the static equilibrium condition, and the method provided by the invention is accurate in calculation.
In summary, the general checking method for the arbitrary section stress of the bridge concrete structure provided by the invention has clear algorithm and high calculation precision, and meets the calculation requirements of actual engineering.
The invention aims at the problem of calculating the stress of the bridge concrete structure, and a general stress calculation method suitable for the concrete structure with any cross section and any reinforcement form is constructed by deducing a general cross section analysis theory, a neutralization shaft angle step encryption algorithm and a neutralization shaft height rapid iteration algorithm, so that the method for detecting the stress of the bridge concrete structure is established, and the problem of accuracy of calculating the stress of the complex concrete structure of the bridge is solved.
The method can accurately calculate the stress of the bridge concrete structure in the field of transportation, and solves the problem of accuracy in calculating the stress of the complex bridge concrete structure.
Claims (10)
1. A general calculation method for arbitrary section stress of a bridge concrete structure is characterized by comprising the following steps: the method comprises the following steps:
A. inputting material information, section information, reinforcement information and internal force information of a bridge concrete structure;
B. translating an axial force action point in the internal force information to the centroid of the section, and synthesizing a translated bending moment;
C. preliminarily setting a neutralization shaft angle, and rotating the cross section to enable the preliminarily set neutralization shaft to be horizontal;
D. decomposing resultant bending moment of the rotated section, and iteratively calculating the height of the neutralization shaft;
E. f, calculating and judging whether the internal transverse bending moment and the external transverse bending moment are balanced, and if the internal transverse bending moment and the external transverse bending moment are balanced, executing the step F; c, if the internal force transverse bending moment and the external force transverse bending moment are unbalanced, returning to the step C;
F. and (5) checking a stress result.
2. The general calculation method for arbitrary section stress of a bridge concrete structure according to claim 1, wherein the general calculation method is characterized by comprising the following steps: step A, inputting material information, section information, reinforcement information and internal force information of a bridge concrete structure, wherein the concrete process comprises the following steps:
firstly, dividing material information of a bridge concrete structure into concrete material parameters and reinforcing steel bar material parameters for input;
then, uniformly inputting the section information;
then, inputting reinforcement information consisting of coordinates of each beam of reinforcement and the number of beams of reinforcement;
and finally, inputting internal force information consisting of the magnitude of the axial force, the action point and the bending moment in two directions.
3. The general calculation method for arbitrary section stress of a bridge concrete structure according to claim 2, wherein the general calculation method is characterized by comprising the following steps: the section information is input uniformly, and the specific process is as follows:
firstly, performing discrete treatment on a simple section and a complex section;
then, connecting a plurality of contour lines obtained after the discretization end to end;
finally, the contour array is used for input.
4. The general calculation method for arbitrary section stress of a bridge concrete structure according to claim 1, wherein the general calculation method is characterized by comprising the following steps: and B, translating an axial force action point in the internal force information to the centroid of the section, and synthesizing a translated bending moment, wherein the specific process is as follows:
firstly, translating an axial force acting point in internal force information to a section centroid position in section information;
then, calculating an additional bending moment caused by the translation axis force;
then, the additional bending moment is overlapped with the original bending moment in the internal force information;
and finally, synthesizing the bending moment by using a vector method.
5. The general calculation method for arbitrary section stress of a bridge concrete structure according to claim 1, wherein the general calculation method is characterized by comprising the following steps: c, preliminarily setting a neutralization shaft angle, and rotating the cross section to enable the preliminarily set neutralization shaft to be horizontal, wherein the specific process is as follows:
firstly, preliminarily setting a neutralization axis, and assuming a neutralization axis angle theta;
then, the section is rotated until the neutralization shaft is in a horizontal state;
and finally, in the process of section rotation, simultaneously rotating reinforcement information and internal force information.
6. The general calculation method for arbitrary section stress of a bridge concrete structure according to claim 1, wherein the general calculation method is characterized by comprising the following steps: and D, decomposing resultant force bending moment of the rotated section, and iteratively calculating the height of the neutralization shaft, wherein the concrete process is as follows:
firstly, a local coordinate system, namely a Y '-Z' coordinate system is newly arranged on a section after rotation;
then decomposing the resultant bending moment to the Y ' axis and the Z ' axis of the Y ' -Z ' coordinate system to obtain resultant bending moment M ' of the Y ' axis ' y And a resultant bending moment M of the Z' axis z ′；
Finally, the height y of the neutralization shaft is calculated and obtained by using the principle of material mechanics according to different stress forms.
7. The general calculation method for arbitrary section stress of the bridge concrete structure according to claim 6, wherein the general calculation method is characterized by comprising the following steps: the Y ' axis in the Y ' -Z ' coordinate system and the preliminarily set neutralization axis n 0 -n 0 Parallel.
8. The general calculation method for arbitrary section stress of the bridge concrete structure according to claim 7, wherein the general calculation method is characterized by comprising the following steps: e, calculating and judging whether the internal transverse bending moment and the external transverse bending moment are balanced, and if the internal transverse bending moment and the external transverse bending moment are balanced, executing the step F; if the internal force transverse bending moment and the external force transverse bending moment are unbalanced, returning to the step C, wherein the specific process is as follows:
firstly, dividing a section into a left part and a right part along a Z' axis;
then, calculating the transverse bending moment of the pressed concrete to the Z' axis;
then, calculating the transverse bending moment of the steel bar to the Z' axis;
then, summing the transverse bending moment calculated by the steel bar and the pressed concrete to obtain M zn ；
Finally, the calculated internal force transverse bending moment is M zn With calculated external transverse bending moment M z ' compare and determine if balanced.
9. The general calculation method for arbitrary section stress of the bridge concrete structure according to claim 8, wherein the general calculation method is characterized by comprising the following steps: if the internal force transverse bending moment and the external force transverse bending moment are unbalanced, the step C is to preliminarily set the angle theta of the neutralization shaft to be wrong, and the angle theta of the neutralization shaft is re-assumed, and the calculation is continued until the balance condition of the internal force transverse bending moment and the external force transverse bending moment is met.
10. The general calculation method for arbitrary section stress of a bridge concrete structure according to claim 9, wherein the general calculation method is characterized by comprising the following steps: and under the condition that the internal force transverse bending moment and the external force transverse bending moment meet balance conditions, obtaining the section stress distribution state.
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