CN112861213A - Calculation method for lateral support rigidity of longitudinal bar wrapped with FRP (fiber reinforced Plastic) - Google Patents

Calculation method for lateral support rigidity of longitudinal bar wrapped with FRP (fiber reinforced Plastic) Download PDF

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CN112861213A
CN112861213A CN202110024647.8A CN202110024647A CN112861213A CN 112861213 A CN112861213 A CN 112861213A CN 202110024647 A CN202110024647 A CN 202110024647A CN 112861213 A CN112861213 A CN 112861213A
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白玉磊
孙鹏轩
贾俊峰
张玉峰
杨洪龙
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Abstract

The invention discloses a calculation method for lateral support rigidity of longitudinal bars wrapped with FRP (fiber reinforced plastic), which is characterized in that equal section division is carried out according to the number of longitudinal bars of the section of a column and the symmetry of the section, and a curve beam model is determined; adopting an analysis guiding stress-strain model to obtain axial strain, axial stress, lateral strain, lateral stress and increment forms thereof; determining the axial shear modulus of the FRP constraint concrete and the annular elastic modulus of the FRP constraint concrete protective layer; the combined section beam is equivalent to a single material section, and the section characteristics of the combined section beam are calculated; the method for solving the statically indeterminate structure by using structural mechanics solves the displacement generated at the action point under the action of any given load; and (5) calculating the equivalent stiffness. The method has the advantages of high accuracy, renewability, modification and the like, and the mature calculation method for solving the statically indeterminate structure by using structural mechanics is simple in calculation, high in accuracy and high in research and application value.

Description

Calculation method for lateral support rigidity of longitudinal bar wrapped with FRP (fiber reinforced Plastic)
Technical Field
The invention relates to a calculation method for solving the lateral support rigidity of FRP (fiber reinforced plastic) confined concrete to longitudinal bars in an externally-wrapped FRP confined circular reinforced concrete column, in particular to a calculation method which is suitable for the circular reinforced concrete column and has high updatable and modifiable accuracy for the lateral support rigidity of the FRP confined concrete to the longitudinal bars.
Background
It has been found that buckling of the longitudinal ribs is common, both from column testing and post-seismic structural investigations, and that, particularly for structures built to specifications before the 70's of the 20 th century, the relatively large inter-stirrup spacing does not provide sufficient lateral support for the longitudinal ribs. In recent years, fiber composite materials (FRP) are increasingly used in seismic reinforcement of reinforced concrete columns, the first function being to increase the strength and ductility of the concrete; the second function is to prevent or delay buckling of the reinforcing bars. Buckling of the longitudinal bars is usually accompanied by the shedding of the concrete of the protective layer, which may lead to sudden failure of conventional reinforced concrete columns when the compressive deformation reaches a certain critical value. In order to ensure the safety of an old building structure under the action of strong earthquake, earthquake-resistant reinforcement is necessary, the reinforcement technology of wrapping FRP provides powerful technical support for earthquake-resistant reinforcement of the reinforced concrete column, but due to the complexity of buckling of longitudinal ribs, most of the researches on buckling of the longitudinal ribs in the FRP constraint reinforced concrete column are stopped on perceptual knowledge at present. Therefore, based on the elastic foundation beam model (BOEF model), the analysis and calculation of the lateral support stiffness of the longitudinal bar of the FRP constraint circular reinforced concrete column with uniform constraint are carried out, and the basis for analyzing the buckling deformation of the longitudinal bar is formed.
The calculation method of the lateral support rigidity of the longitudinal bar applies a model of confined concrete, divides the section of the column into equal sections, takes part of the equal sections for analysis, enables the combined section formed by the FRP and the protective layer concrete to be equivalent to the section of a single material, converts the problem into the hyperstatic structure in the basic structural mechanics to solve the internal force, and solves the rigidity of the lateral support according to the definition of the rigidity. At present, the existing research aims at the calculation method for calculating the lateral support stiffness of the longitudinal bar of the FRP constraint concrete in the FRP constraint reinforced concrete column, and mainly comprises a Vincenzo Giamundo (2014) model and the like, wherein the longitudinal bar is equivalent to a horizontal Euler beam in the model, and the lateral support stiffness of the longitudinal bar is defined as the constraint stiffness of the FRP multiplied by the length of the FRP action.
In order to solve the accuracy problem, the interaction relation among the FRP, the concrete and the steel bars needs to be established, a combined section formed by the FRP and the concrete of the protective layer can be equivalent to a single concrete section by an equivalent rigidity method, the simplification of the section is carried out according to the symmetry, the combined section is converted into mature structural mechanics, the problem of statically indeterminate structure is solved, and the supporting rigidity of the FRP constraint concrete to the longitudinal bars is obtained according to the definition of the rigidity. At present, the existing calculation method is few, interaction relation among the three is not established, and certain accuracy and theoretical support are lacked, so that the method is not substantially popularized in actual research.
Disclosure of Invention
Aiming at the defects of the existing calculation method for calculating the lateral support rigidity of the FRP restrained concrete to the longitudinal bar in the aspect of solving accuracy, the invention aims to provide the calculation method for the lateral support rigidity of the FRP restrained concrete to the longitudinal bar, which is universally applicable to the circular reinforced concrete column and has updatable and modifiable high accuracy.
The technical scheme of the invention is as follows:
a calculation method for lateral support rigidity of longitudinal bars wrapped with FRP (fiber reinforced Plastic) comprises the following steps:
the method comprises the following steps of firstly, according to the number n of longitudinal ribs of the column section and the symmetry of the section, performing equal-section division on the section of the FRP-constrained circular reinforced concrete column. Based on the elastic foundation beam model (BOEF), when evaluating the rigidity of the horizontal slice of the FRP-constrained concrete protective layer for restraining buckling of the longitudinal bars, the FRP-constrained protective layer concrete is idealized into a curved beam, two ends of the curved beam are fixed and radially loaded, that is, an outward concentrated load P is radially loaded by the reinforcing steel bars during spanning, and then the equivalent rigidity of the spring can be expressed as: and K is P/delta, wherein delta is the curvilinear beam mid-span displacement. Of curved beamsThe angle phi is expressed as phi 2 pi/n, wherein n is the number of longitudinal ribs of the column section. The radius r of the curved beam is: r ═ R-c + hcWherein R is the radius of the cylindrical section; c is the thickness of the concrete protective layer; h iscThe height of the neutral axis of the curved beam.
And secondly, analyzing the section of the curve beam model, and paying more attention to the lateral (circumferential) stress-strain relation of the section compared with the axial stress-strain relation. Given that the concrete itself is an isotropic material, given a certain lateral strain, axial stress and lateral stress can be determined according to the (Dai2011) analysis-oriented model.
The general relationship between lateral strain and confining pressure for encased FRPs is:
Figure BDA0002889973610000021
wherein: sigmal: the confining pressure (or lateral stress) of the FRP; efrp: secant modulus of FRP. A constant for a conventional FRP with a wire elastic stress-strain relationship; for a high strain FRP with a nonlinear stress-strain relationship, the value depends on the hoop strain (equal to the lateral strain ε)l)；tfrp: nominal thickness of the FRP; r: radius of the core concrete.
Jiang and Teng (2007) proposes a steel bar constraint concrete model for predicting the stress-strain relationship of actively constrained concrete:
Figure BDA0002889973610000022
wherein the content of the first and second substances,
Figure BDA0002889973610000023
and
Figure BDA0002889973610000024
respectively under the action of a specific constant constraint pressure, the peak axial compressive stress of the concrete and the corresponding axial compressive strain; sigmacAnd εcRespectively the compressive stress and the compressive strain of the FRP constraint concrete; constant m is defined as:
Figure BDA0002889973610000025
wherein E iscThe modulus of elasticity of the unconstrained concrete; the peak stress of the stress-strain curve of the actively-constrained concrete is as follows:
Figure BDA0002889973610000026
the axial strain corresponding to the peak compressive stress is:
Figure BDA0002889973610000027
wherein, f'co: the compressive strength of the unconstrained concrete; epsilonco: and the compressive strain corresponding to the compressive strength of the unconstrained concrete.
Dai (2011) et al, by performing least squares regression on the experimental data, have better described the expression of equivalent normalized axial strain versus normalized lateral strain:
Figure BDA0002889973610000031
where, from regression analysis, a ═ 1.024, b ═ 0.350, and c ═ 0.089. Thus, the modified lateral strain equation is:
Figure BDA0002889973610000032
given the initial value, maximum value and increment of the lateral strain, the relationship between the axial stress and the strain and the relationship between the lateral stress and the strain can be obtained by substituting the model.
Thirdly, according to the principle of equal difference, the axial direction of the total quantity form obtained in the second stepThe stress to strain and lateral stress to strain relationships are converted to incremental forms. For example, assuming an increment of p, the new axial strain is Np+1-N1，Np+2-N2And by analogy, the axial stress, the lateral stress and the lateral strain are treated in the same way to obtain the relationship between the axial stress strain and the lateral stress strain in an incremental manner. FRP-constrained concrete is assumed to be an isotropic material, and therefore, it is possible to obtain:
Figure BDA0002889973610000033
wherein, E: elastic modulus of FRP constraint concrete; v: the Poisson's ratio of the FRP constraint concrete; sigmaa: the FRP restrains the axial stress of the concrete; sigmaθ: the FRP restrains the lateral stress of the concrete; epsilonθ: the FRP constrains the lateral strain of the concrete.
By solving the above equation system, the values of the elastic modulus E and poisson ratio ν based on the analysis-oriented model can be obtained as follows:
Figure BDA0002889973610000034
determining the value of the elastic modulus and the Poisson ratio according to the axial stress, the axial strain, the lateral stress and the lateral strain, wherein the elastic modulus E changes along with the change of the axial strain, but when the axial strain value is larger, the elastic modulus tends to a determined value, and the determined value is taken as the shear modulus E of the FRP constraint concrete in the axial direction2。
Can roughly estimate the elastic modulus E of the FRP constraint concrete protective layer in the circumferential directionconThe tangential modulus of the FRP constraint concrete in the axial direction (namely the slope of the FRP constraint concrete axial stress-strain curve) is assumed to be equal. Namely:
Econ＝E2
fourthly, the section of the curved beam is a combined section formed by FRP and FRP constrained protective layer concrete, and the position of a neutralization shaft in the combined section is as follows:
Figure BDA0002889973610000041
wherein, w: the width of the concrete protective layer; w is aequ: the equivalent width of the FRP; t is tfrpAnd tcon: the thickness of the FRP and the thickness of the concrete protective layer are respectively set; a. theconAnd Aequ: respectively the cross section area of the concrete protective layer and the equivalent cross section area of the FRP; efrp: elastic modulus of FRP; h is1And h2: the centroid height of the concrete protective layer cross section area and the equivalent FRP cross section area is respectively.
Fifthly, calculating the bending rigidity (EI) of the section of the composite beam according to the determined position of the neutral axisCBAnd axial stiffness (EA)CBComprises the following steps:
Figure BDA0002889973610000042
(EA)CB＝Econwtcon+Efrpwtfrp
determining the section radius r of the curve beam as follows: r ═ R-c + hc。
Sixthly, the curve beam is of a cubic statically indeterminate structure, and three redundant forces on the right end are restrained by three redundant forces1Shear force X2And bending moment X3Instead, in this case, the structure is transformed into a statically determinate structure without redundant constraints, and X is determined1、X2And X3The value of (2) can be used for solving the internal force of the section according to a structural mechanics method.
First, when the redundant force X is applied1When applied to a curved beam, the forces in the section along the length of the beam are obtained by balancing:
M1＝r(1-cosθ)
N1＝cosθ
V1＝sinθ
similarly, sequentially adding the redundant force X21 and X31-section internal force along the length of a curved beam when applied to the beamBy balancing it is possible to obtain:
M2＝r sinθ
N2＝-sinθ
V2＝cosθ
and
M3＝1
N3＝0
V3＝0
in the above formula, Mi、NiAnd Vi(i ═ 1, 2, 3) are the bending moment, axial force and shear force of the beam section, respectively.
Thus, when in X1When a single force is applied in one direction, the displacement of the curved beam in three redundant force directions is:
Figure BDA0002889973610000051
Figure BDA0002889973610000052
Figure BDA0002889973610000053
wherein, deltaijIndicating the displacement in the ith direction of the beam load point when a unit load is applied in the jth direction. (Note: delta)ij＝δji)。
Similarly, the time at X can be found2And X3Unit load is applied to the directions respectively, and the displacement of the curve beam in the directions of three redundant forces is respectively as follows:
Figure BDA0002889973610000054
Figure BDA0002889973610000061
Figure BDA0002889973610000062
Figure BDA0002889973610000063
when an external load P is applied to the curved beam, the cross-sectional internal forces along the length of the curved beam are:
Figure BDA0002889973610000064
Figure BDA0002889973610000065
Figure BDA0002889973610000066
Mp＝0；Np＝0；Vp＝0 θ＜φ/2
therefore, the displacement of the external load in three directions can be expressed as:
Figure BDA0002889973610000067
Figure BDA0002889973610000068
Figure BDA0002889973610000071
Figure BDA0002889973610000072
wherein, DeltaiPWhich represents the displacement in the i-th direction that occurs when an external load P is applied to the curved beam.
The standard formula of the third statically indeterminate beam in the flexibility method is as follows:
δ11X1+δ12X2+δ13X3+Δ1P＝0
δ21X1+δ22X2+δ23X3+Δ2P＝0
δ31X1+δ32X2+δ33X3+Δ3P＝0
can be simplified into:
Figure BDA0002889973610000073
combining the above formulas to obtain:
Figure BDA0002889973610000074
when all the three redundant forces are determined, the internal force of any section of the hyperstatic beam under the action of the external force P can be calculated. For ease of discussion, the curved beam is divided into two portions, one on the left side (abbreviated "L") and one on the right side (abbreviated "R") of the concentrated load P. Therefore, the internal force of the cross section of the right side beam of the concentrated load P can be obtained as follows:
MPR＝M1X1+M2X2+M3X3
＝r(1-cosθ)X1+r sinθX2+X3
NPR＝N1X1+N2X2+N3X3
＝X1cosθ-X2sinθ
VPR＝V1X1+V2X2+V3X3
＝X1sinθ+X2cosθ
similarly, the beam section load on the left side of the concentrated load P is:
Figure BDA0002889973610000081
Figure BDA0002889973610000082
Figure BDA0002889973610000083
the virtual work principle can be used for calculating the deformation of the hyperstatic curve beam under the action of the concentrated load P, and the flexibility method provides a solving method of an internal force diagram of the beam section under the action of the external load P. Similarly, applying a virtual force at the location of the applied load P can also obtain the beam cross-sectional internal force as follows:
M'PR＝r(1-cosθ)X'1+r sinθX'2+X'3
N'PR＝X'1cosθ-X'2sinθ
V'PR＝X'1sinθ+X'2cosθ
Figure BDA0002889973610000084
Figure BDA0002889973610000085
Figure BDA0002889973610000086
wherein, X'1，X'2And X'3The internal force value under the action of the deficiency force.
According to the virtual work principle, the deformation of a beam with specific section characteristics under the action of concentrated load can be recorded as:
Figure BDA0002889973610000087
wherein M, N and V are cross-sectional internal forces under the action of a load P; m ', N ' and N ' are cross-sectional internal forces under the action of the virtual force. For convenience, the curved beam is considered to be an euler-bernoulli beam, without considering the effect of shear deformation. Therefore, the displacement expression of the point of the curve beam under the action of the load P can be obtained as follows:
Figure BDA0002889973610000091
and seventhly, according to the definition of the rigidity, the equivalent section rigidity K is as follows:
Figure BDA0002889973610000092
the specific calculation sequence is as follows:
a) according to the number of longitudinal ribs of the section of the column and the symmetry of the section, performing equal section division to determine a curve beam model;
b) adopting a Dai (2011) oriented analysis stress-strain model to obtain axial strain, axial stress, lateral strain and lateral stress and incremental forms thereof;
c) the method comprises the steps of taking the confined concrete as an isotropic material, solving the elastic modulus E and Poisson's ratio v of the FRP (fiber reinforced Plastic) confined concrete according to an elastoplasticity mechanical relationship, wherein the elastic modulus changes along with the magnitude of axial strain and tends to be constant when the axial strain is larger, namely the value is E2. Let Econ＝E2Then E can be obtainedcon。
d) The curve beam is a combined beam, equivalently converted into a section of a single material, and the section characteristic of the equivalent section is solved;
e) the curve beam is regarded as a statically indeterminate structure, structural mechanics methods such as a force method are used for solving, and displacement delta generated at an action point under the action of a given load P is solved.
f) From the definition of the stiffness of the structure mechanics, K is known as P/Δ.
Preferably, in the first step, when the curved beam is determined, the column section is divided into equal sections according to the symmetry of the column section and the number of the longitudinal ribs, and the method is generally applicable to reinforced concrete columns constrained by the FRP.
Preferably, in the first step, the curved beam approximately ignores the simultaneous lateral movement of the rebar and concrete cap caused by core concrete expansion, which is believed to have a minor effect on the spring rate that needs to be evaluated.
Preferably, in the second step, when the analysis of the section of the curved beam is performed, attention should not be paid to only the axial stress and strain of the whole column, and the problem is transformed to the analysis of the section, and more attention should be paid to the lateral stress and strain of the section of the curved beam.
Preferably, in the second step, an incremental Dai (2011) model is used for determining the axial stress strain and the lateral stress strain, and the predicted value of the model is more accurate when the axial strain is larger. As the research progresses, the constraint model can be modified and updated.
Preferably, in said third step, when the axial stress exceeds the strength of the unconstrained concrete under the action of the axial pressure, the FRP constraining effect is obviously caused by the expansion of the concrete, and the second part of the linear rise controls the FRP constraining performance of the concrete. Buckling of the rebar is unlikely to occur at lower stress levels, so it can be assumed that FRP-constrained concrete is controlled by the linear second stage of the stress-strain curve in the rebar buckling analysis. Therefore, it can be roughly estimated that the FRP restrains the elastic modulus E of the concrete protective layer in the circumferential directionconThe tangential modulus of the FRP constraint concrete in the axial direction (namely the slope of the FRP constraint concrete axial stress-strain curve) is assumed to be equal.
Preferably, in the third step, the obtained axial stress strain and the lateral stress strain in the full quantity form are converted into the incremental form, and the conversion is performed according to the principle of equal difference.
Preferably, in the fourth step, the original section is converted into an equivalent inverted-T section of the concrete of the protective layer, and the height of the centroid of the section of the single material of the curved beam is determined.
Preferably, in the fifth step, the characteristics of the section of the curved beam and the radius of the section are determined according to the height of the centroid of the equivalent rear section.
Preferably, in the sixth step, the curved beam is a composite section beam, and for convenience, the curved beam is considered to be an euler-bernoulli beam, irrespective of the effect of shear deformation.
Compared with the prior art, the invention has the advantages that:
1) according to the calculation method for the lateral support rigidity of the longitudinal bar by wrapping the FRP, the applied constraint model can be modified and deformed along with the in-depth research of students, the calculation can be defined according to the actual characteristics of the column, and the calculation method applied in the calculation is a mature and simple method for solving the statically indeterminate structure through structural mechanics, so that the calculation can be accurately performed according to the requirements, the calculation is more complete and accurate along with the updating of the in-depth constraint model, and the solution efficiency can be greatly improved by combining the applied structural mechanics method with solution software.
2) According to the calculation method for the lateral support rigidity of the longitudinal bar wrapped by the FRP, the solved longitudinal bar theoretical model is compared with the test result, certain accuracy is achieved, and a powerful theoretical calculation basis is provided for the quantitative analysis of the longitudinal bar in the FRP constraint reinforced concrete column.
3) The calculation method for the lateral support rigidity of the longitudinal bar wrapped with the FRP is simple in steps, high in calculation efficiency and easy to implement.
Drawings
FIG. 1 is a schematic diagram of the interaction between concrete, steel bars and FRP in an FRP restrained reinforced concrete column;
FIG. 2 is a schematic view of a model of an elastic foundation beam;
FIG. 3 is a schematic cross-sectional view of a cylinder;
FIG. 4 is a schematic view of a curved beam model;
FIG. 5 is a graph illustrating axial stress-strain versus lateral stress-strain;
FIG. 6 is a graph of axial strain versus modulus of elasticity;
FIG. 7 is a schematic cross-sectional view of a curved beam;
FIG. 8 is a schematic diagram of a basic system of a cubic hyperstatic curved beam based on a flexibility method;
FIG. 9 redundancy X based on compliance method11 acts in a basic system schematic;
FIG. 10 redundancy X based on compliance method21 acts in a basic system schematic;
FIG. 11 redundancy X based on compliance method31 acts in a basic system schematic;
FIG. 12 is a schematic diagram of a basic system based on the effect of a compliance method load P;
detailed description of the invention
The present invention is further illustrated by the following examples, which are not intended to limit the invention to these embodiments. It will be appreciated by those skilled in the art that the present invention covers all FRP constraints that may be included within the scope of the claims as the support stiffness K of the longitudinal bars in a circular reinforced concrete column.
The calculation method of the present invention is described in detail below with reference to the accompanying drawings:
taking a single-layer PEN-constrained circular reinforced concrete column as an example, the lateral side support of longitudinal bars by FRP-constrained concrete in the FRP-constrained circular reinforced concrete column based on curve beam model evaluation is specifically explained, the section of the column is shown in figure 1, the size of the section of the column is 200 x 200mm, the longitudinal bars are 4 phi 20, the strength of the concrete is 35MPa, the fracture strain of the FRP is 0.07, and the elastic modulus of the FRP is 12 GPa. The method for calculating the lateral supporting rigidity of the FRP constraint concrete to the longitudinal bar in the FRP constraint circular reinforced concrete column comprises the following steps:
firstly, according to the symmetry of the column section and the number of longitudinal bars, uniform section division is carried out, and a curve beam model of FRP constraint protective layer concrete is determined, as shown in figure 2. The two ends of the curved beam are fixed, and outward concentrated load P is radially loaded by the reinforcing steel bars in the midspan, namely the C point. The angle phi of the curved beam is: phi is 2 pi/4 pi/2.
And secondly, assuming that the initial strain is 0.0026, the strain increment is 0.0001, the fracture strain of the FRP is 0.7, and given each lateral strain, the corresponding axial strain, axial stress and lateral stress can be obtained according to a constraint model of Dai (2011). Fig. 3 shows an axial stress-strain curve and a lateral stress-strain curve.
In the third step, the analysis-oriented model of Dai (2011) is in an incremental form, and the full-scale form obtained in the second step needs to be converted into the incremental form. Assuming an increment of 2, the new axial strain is N3-N1、N4-N2、N5-N3And analogizing, and similarly, performing the same treatment on the axial stress, the lateral stress and the lateral strain to obtain the axial stress strain and the lateral stress strain in an incremental form. Assuming that FRP constraint concrete is an isotropic material, the stress-strain relationship is simplified as follows:
Figure BDA0002889973610000111
wherein, E: elastic modulus of FRP constraint concrete; v: the Poisson's ratio of the FRP constraint concrete; sigmaa: the FRP restrains the axial stress of the concrete; sigmaθ: the FRP restrains the lateral stress of the concrete; epsilonθ: the FRP constrains the lateral strain of the concrete.
By solving the above formula, the values of the elastic modulus E and the Poisson ratio v of the FRP constraint concrete can be obtained, which are:
Figure BDA0002889973610000112
from this equation, each set of values of axial stress, axial strain, lateral stress and lateral strain can be solved for a unique elastic modulus and poisson's ratio. FIG. 4 is a graph of modulus of elasticity versus axial strain, where the modulus of elasticity tends to be constant when the axial strain is relatively large, and this value is taken as the shear modulus E2The average value of the stationary latter segment value can also be approximated. In this example, the approximate average of the elastic moduli corresponding to 0.051797-0.055326 is:
(348.23+348.78+349.32+349.87+350.41+350.95+351.49+352.03+352.57+353.10+353.64+354.17+354.70+355.23+355.76+356.29+356.82+357.34+357.86+358.39+358.91)/21 ≈ 354MPa, namely the FRP constraint concrete has the axial shear modulus: e2354MPa, and roughly estimated, the modulus of elasticity E of the FRP-constrained concrete protective layer in the hoop directionconThe tangential modulus of the FRP constraint concrete in the axial direction (namely the slope of the FRP constraint concrete axial stress-strain curve) is assumed to be equal. I.e. Econ＝E2＝354MPa。
Fourthly, the section of the curved beam is a combined section (as shown in fig. 5) composed of the FRP and the FRP constrained concrete protective layer, and the positions of the neutralization axes in the combined section are as follows:
Figure BDA0002889973610000121
in this example, taking the width of the concrete protective layer as 1mm, the thickness of the concrete protective layer as 20mm, and the top surface as a zero point, the position of the neutral axis in the combined cross section of the curved beam is:
Figure BDA0002889973610000122
the position of a neutral axis of the equivalent rear section is determined, and the composite beam can be converted into an inverted T-shaped section beam made of a single material.
Fifthly, calculating the bending rigidity (EI) of the section of the composite beam by using the height of the neutral axis of the equivalent rear section obtained in the fourth stepCBAxial stiffness (EA)CBAnd the radius r of the section of the curved beam are respectively:
(EI)CB＝354×[1×203/12+1×20×(17.2658-20/2)2]
+12000×[1×1.2723/12+1×1.272×(20+1.272/2-17.2658)2]＝785200MPa
(EA)CB＝354×1×20+12000×1×1.272＝22344MPa
r＝100-20+17.2658＝97.2658mm
sixthly, the curve beam is of a cubic statically indeterminate structure, and three redundant beams at the right end are combinedAxial force X of three redundant forces for bundling1Shear force X2And bending moment X3Instead (as in fig. 6), the structure is now transformed into a statically determinate structure without unnecessary constraints. Determine X1、X2And X3The value of (2) can be used for solving the internal force of the section according to a structural mechanics method.
First, as shown in FIG. 7, when the redundant force X is applied1When applied to a curved beam, the forces in the section along the length of the beam are obtained by balancing:
M1＝97.2658×(1-cosθ)
N1＝cosθ
V1＝sinθ
similarly, as shown in FIGS. 8 and 9, the redundant force X is sequentially applied21 and X3When applied to a curved beam, the forces in the section along the length of the beam are obtained by balancing:
M2＝97.2658sinθ
N2＝-sinθ
V2＝cosθ
and
M3＝1
N3＝0
V3＝0
in the above formula, Mi、NiAnd Vi(i ═ 1, 2, 3) are the bending moment, axial force and shear force of the beam section, respectively.
For convenience, the curved beam is considered to be an euler-bernoulli beam, without considering the effect of shear deformation. Thus, when in X1When a single force is applied in one direction, the displacement of the curved beam in three redundant force directions is:
Figure BDA0002889973610000131
Figure BDA0002889973610000132
Figure BDA0002889973610000133
wherein, deltaijIndicating the displacement in the ith direction of the beam load point when a unit load is applied in the jth direction. (Note: delta)ij＝δji)。
Similarly, the time at X can be found2And X3Unit load is applied to the directions respectively, and the displacement of the curve beam in the directions of three redundant forces is respectively as follows:
Figure BDA0002889973610000141
Figure BDA0002889973610000142
Figure BDA0002889973610000143
when an external load P is applied to a curved beam (as shown in fig. 10), the cross-sectional internal forces along the length of the curved beam are:
Figure BDA0002889973610000144
Figure BDA0002889973610000145
Figure BDA0002889973610000151
Mp＝0；Np＝0；Vp＝0 θ＜φ/2
therefore, the displacement of the external load in three directions can be expressed as:
Figure BDA0002889973610000152
Figure BDA0002889973610000153
Figure BDA0002889973610000154
wherein, DeltaiPWhich represents the displacement in the i-th direction that occurs when an external load P is applied to the curved beam. In particular, when P is 1, Δ1P＝-0.2540；Δ2P＝-0.3266；Δ3P＝-0.0035。
The standard formula of the third statically indeterminate beam in the flexibility method is as follows:
Figure BDA0002889973610000161
can be simplified into:
Figure BDA0002889973610000162
combining the above formulas to obtain:
Figure BDA0002889973610000163
when all the three redundant forces are determined, the internal force of any section of the hyperstatic beam under the action of the external force P can be calculated. For ease of discussion, the curved beam is divided into two portions, one on the left side (abbreviated "L") and one on the right side (abbreviated "R") of the concentrated load P.
Therefore, the internal force of the cross section of the right side beam of the concentrated load P can be obtained as follows:
MPR＝M1X1+M2X2+M3X3＝r(1-cosθ)X1+r sinθX2+X3
＝97.2658×(1-cosθ)×0.8867+97.2658×sinθ×(-0.1796)+(-2.0823)
＝86.2456×(1-cosθ)-17.4689×sinθ-2.0823
NPR＝N1X1+N2X2+N3X3＝X1cosθ-X2sinθ＝0.8867cosθ+0.1796sinθ
VPR＝V1X1+V2X2+V3X3＝X1sinθ+X2cosθ＝0.8867sinθ-0.1796cosθ
similarly, the beam section load on the left side of the concentrated load P is:
Figure BDA0002889973610000164
Figure BDA0002889973610000165
Figure BDA0002889973610000166
Figure BDA0002889973610000171
the virtual work principle can be used for calculating the deformation of the hyperstatic curve beam under the action of the concentrated load P, and the flexibility method provides a solving method of an internal force diagram of the beam section under the action of the external load P. Similarly, applying a virtual unit force at the location of the applied load P can also yield the beam cross-sectional internal force as follows:
M'PR＝r(1-cosθ)X'1+rsinθX'2+X'3
＝97.2658×(1-cosθ)×0.8867+97.2658×sinθ×(-0.1796)+(-2.0823)
＝86.2456×(1-cosθ)-17.4689×sinθ-2.0823
N'PR＝X'1cosθ-X'2sinθ＝0.8867cosθ+0.1796sinθ
V'PR＝X'1sinθ+X'2cosθ＝0.8867sinθ-0.1796cosθ
Figure BDA0002889973610000172
Figure BDA0002889973610000173
Figure BDA0002889973610000174
wherein, X1'，X'2And X3' is the internal force value under the action of the virtual unit force.
According to the virtual work principle, the deformation of a beam with specific section characteristics under the action of concentrated load can be recorded as:
Figure BDA0002889973610000175
wherein M, N and V are cross-sectional internal forces under the action of a load P; m ', N ' and N ' are cross-sectional internal forces under the action of the virtual force. For convenience, the curved beam is considered to be an euler-bernoulli beam, without considering the effect of shear deformation. Therefore, the displacement of the point of the curved beam under the action of the load P can be obtained as follows:
Figure BDA0002889973610000176
and seventhly, according to the relation between the rigidity and the flexibility, the section rigidity K of the curved beam is known as follows:
Figure BDA0002889973610000181
it should be understood that the steps of the method described in the present invention are not merely exemplary, and are applicable to all FRP-restrained reinforced concrete columns, and lay the foundation for the deep study of longitudinal bar buckling analysis.
While the present invention has been described with reference to a limited number of embodiments and drawings, as described above, various modifications and changes will become apparent to those skilled in the art to which the present invention pertains. Accordingly, other embodiments are within the scope and spirit of the following claims and equivalents thereto.

Claims (6)

1. A calculation method for lateral support rigidity of longitudinal bars wrapped with FRP is characterized by comprising the following steps:
the method comprises the following steps that firstly, according to the number n of longitudinal ribs of the column section and the symmetry of the section, the section of the FRP-restrained circular reinforced concrete column is divided into equal sections; based on the elastic foundation beam model BOEF, when the rigidity of the horizontal slice of the FRP restrained concrete protective layer for inhibiting buckling of the longitudinal ribs is evaluated, the FRP restrained concrete protective layer is idealized into a curved beam, two ends of the curved beam are fixed and radially loaded, namely outward concentrated load P is radially loaded by the reinforcing steel bars in the midspan; the equivalent stiffness of the spring can be expressed as: k is P/delta, wherein delta is the span-middle displacement of the curved beam; the angle phi of the curve beam is expressed as phi being 2 pi/n, wherein n is the number of longitudinal ribs of the column section;
secondly, analyzing the section of the curve beam model, and paying more attention to the lateral (circumferential) stress-strain relation of the section compared with the axial stress-strain relation; assuming that concrete is an isotropic material, according to a model which is obtained by subjecting experimental data to least square regression through Dai (2011) and describes equivalent normalized axial strain and lateral strain and is oriented to analysis, and a steel bar constraint concrete model proposed by Jiang and Teng (2007), a determined lateral strain can be given, and axial strain, axial stress and lateral stress can be determined; therefore, given the initial value, the maximum value and the increment of the lateral strain, and substituting the initial value, the maximum value and the increment into the model, the corresponding axial stress, the axial strain and the lateral stress can be obtained;
thirdly, converting the relationship between the axial stress and the strain and the relationship between the lateral stress and the strain in the full-scale form obtained in the second step into an incremental form according to an arithmetic principle by taking the Dai (2011) analysis-oriented model as an incremental model; for example, assume the increment isp, new axial strain Np+1-N1，Np+2-N2By analogy, the axial stress, the lateral stress and the lateral strain are treated in the same way to obtain the relationship between the axial stress strain and the lateral stress strain in an incremental form; FRP-constrained concrete is assumed to be an isotropic material, and therefore, it is possible to obtain:
Figure FDA0002889973600000011
wherein, E: elastic modulus of FRP constraint concrete; v: the Poisson's ratio of the FRP constraint concrete; sigmaa: the FRP restrains the axial stress of the concrete; sigmaθ: the FRP restrains the lateral stress of the concrete; epsilonθ: the FRP restrains the lateral strain of the concrete;
by solving the above equation system, the values of the elastic modulus E and the poisson ratio ν based on the analysis-oriented model can be obtained as follows:
Figure FDA0002889973600000012
determining the value of the elastic modulus and the Poisson ratio according to the axial stress, the axial strain, the lateral stress and the lateral strain, wherein the elastic modulus E changes along with the change of the axial strain, but when the axial strain value is larger, the elastic modulus tends to a determined value, and the determined value is taken as the shear modulus E of the FRP constraint concrete in the axial direction2；
Can roughly estimate the elastic modulus E of FRP constraint concrete in the circumferential directionconAssuming that the tangential modulus of the FRP constraint concrete in the axial direction is equal to the slope of the FRP constraint concrete axial stress-strain curve, namely: econ＝E2；
Fourthly, the section of the curved beam is a combined section formed by FRP and FRP constrained concrete protective layers, and the position of a neutralization shaft in the combined section is as follows:
Figure FDA0002889973600000021
wherein, w: the width of the concrete protective layer; w is aequ: the equivalent width of the FRP; t is tfrpAnd tcon: the thickness of the FRP and the thickness of the concrete protective layer are respectively set; a. theconAnd Aequ: respectively the cross section area of the concrete protective layer and the equivalent cross section area of the FRP; efrp: elastic modulus of FRP; h is1And h2: the height of the centroid of the cross section area of the concrete protective layer and the equivalent FRP cross section area are respectively obtained;
fifthly, calculating the bending rigidity (EI) of the section of the composite beam according to the determined position of the neutral axisCBAnd axial stiffness (EA)CBComprises the following steps:
Figure FDA0002889973600000022
(EA)CB＝Econwtcon+Efrpwtfrp
determining the section radius r of the curve beam as follows: r ═ R-c + hcWherein R is the radius of the cylindrical section; c is the thickness of the concrete protective layer; h iscIs the height of the neutral axis of the curved beam;
sixthly, the curve beam is of a cubic statically indeterminate structure, and three redundant forces on the right end are restrained by three redundant forces1Shear force X2And bending moment X3Instead, in this case, the structure is transformed into a statically determinate structure without redundant constraints, and X is determined1、X2And X3The value of (2) can be used for solving the internal force of the section according to a structural mechanics method; for convenience, the curved beam is considered to be an euler-bernoulli beam, without considering the effect of shear deformation;
will redundant force X1＝1、X21 and X3When the force is respectively applied to a simplified basic system of the curve beam, the internal force of the section along the length of the beam can be determined by a balance equation; according to the knowledge of the compliance method in the structural mechanics, three components are calculated and applied respectivelyDisplacement delta generated in the direction of three redundant forces in the event of redundant forcesij(ii) a Similarly, an external load P is independently applied to a curve beam basic system, and the cross-section internal force along the length of the curve beam and the displacement delta generated in the three redundant force directions are calculatediP；
According to the standard formula of the third statically indeterminate beam in the flexibility method and the obtained displacement, the displacement is comprehensively obtained:
Figure FDA0002889973600000031
when all the three redundant forces are determined, the internal force of any section of the hyperstatic beam under the action of the external force P can be calculated; for the convenience of discussion, the curved beam is divided into two parts, which are respectively positioned at the left side (abbreviated as "L") and the right side (abbreviated as "R") of the concentrated load P; therefore, the internal force of the cross section of the right side beam of the concentrated load P can be obtained as follows:
MPR＝r(1-cosθ)X1+rsinθX2+X3
NPR＝X1cosθ-X2sinθ
similarly, the beam section load on the left side of the concentrated load P is:
Figure FDA0002889973600000032
Figure FDA0002889973600000033
the virtual work principle is used for calculating the deformation of the hyperstatic curve beam under the action of concentrated load P, and the flexibility method provides a solving method of an internal force diagram of the beam section under the action of external load P; similarly, applying a virtual force at the location of the applied load P can also obtain the beam cross-sectional internal force as follows:
M′PR＝r(1-cosθ)X′1+rsinθX′2+X′3
N′PR＝X′1cosθ-X′2sinθ
Figure FDA0002889973600000034
Figure FDA0002889973600000035
wherein, X'1，X′2And X'3The internal force value under the action of the deficiency force;
according to the virtual work principle, the deformation of a beam with specific section characteristics under the action of concentrated load can be recorded as:
Figure FDA0002889973600000036
wherein M, N and V are cross-sectional internal forces under the action of a load P; m ', N ' and N ' are cross-sectional internal forces under the action of the virtual force; therefore, the displacement expression of the point of the curve beam under the action of the load P can be obtained as follows:
Figure FDA0002889973600000037
and seventhly, according to the definition of the rigidity, the equivalent section rigidity K is as follows:
Figure FDA0002889973600000041
the specific calculation sequence is as follows:
a) dividing according to the number of longitudinal ribs of the section of the column and the symmetry of the section to determine a curve beam model;
b) adopting a Dai (2011) oriented analysis stress-strain model to obtain axial strain, axial stress, lateral strain and lateral stress and incremental forms thereof;
c) the method comprises the steps of taking the confined concrete as an isotropic material, solving the elastic modulus E and Poisson's ratio v of the FRP (fiber reinforced Plastic) confined concrete according to an elastoplasticity mechanical relationship, wherein the elastic modulus changes along with the magnitude of axial strain and tends to be constant when the axial strain is larger, namely the value is E2(ii) a Let Econ＝E2Then E can be obtainedcon；
d) The curve beam is a combined beam, equivalently converted into a section of a single material, and the section characteristic of the equivalent section is solved;
e) the curve beam is regarded as a statically indeterminate structure, structural mechanics methods such as a force method are used for solving, and displacement delta generated at an action point under the action of a given load P is solved;
f) from the definition of stiffness in structural mechanics, K is known as P/Δ.
2. The method for calculating the lateral supporting rigidity of the outsourcing FRP to the longitudinal bar according to claim 1, is characterized in that: in the first step, the FRP constraint reinforced concrete column is a circular column, and equal sections are divided according to the number of longitudinal ribs on the section of the column; the curved beam approximately ignores simultaneous lateral movement of the rebar and the concrete cap caused by core concrete expansion.
3. The method for calculating the lateral supporting rigidity of the outsourcing FRP to the longitudinal bar according to claim 1, is characterized in that: in the second step, as long as the relationship between the constraint pressure provided by the outer layer constraint and the circumferential constraint can be established, the model of the surface phase analysis is theoretically suitable for concrete under the constraint of various materials.
4. The method for calculating the lateral supporting rigidity of the outsourcing FRP to the longitudinal bar according to claim 1, is characterized in that: in the third step, under the action of axial pressure, when the axial stress exceeds the strength of the unconstrained concrete, the FRP constraint effect is obviously caused by the expansion of the concrete, and the linearly rising second part controls the performance of the FRP constraint concrete; less buckling of the reinforcing barsMay occur at a lower stress level, so it may be assumed that in the rebar buckling analysis, the FRP-constrained concrete is controlled by the linear second stage of the stress-strain curve; therefore, it can be roughly estimated that the elastic modulus E of FRP constraint concrete in the hoop directionconThe tangential modulus of the FRP constraint concrete in the axial direction (namely the slope of the FRP constraint concrete axial stress-strain curve) is assumed to be equal.
5. The method for calculating the lateral supporting rigidity of the outsourcing FRP to the longitudinal bar according to claim 1, is characterized in that: in the first step, delta is the midspan displacement of the bending beam, if the section characteristics of the bending beam are known, the solution can be carried out by using a mature hyperstatic bending beam method in the structural analysis, namely the hyperstatic bending beam under the action of concentrated load P in the midspan is analyzed, and the displacement delta of the point is solved; and fourthly, converting the original section into an equivalent inverted T-shaped section of the concrete protective layer, determining the section centroid height of the single material of the curved beam, and determining the section characteristics in the fifth step.
6. The method for calculating the lateral supporting rigidity of the outsourcing FRP to the longitudinal bar according to claim 1, is characterized in that: in the sixth step, the curved beam is a combined section beam, and the curved beam is regarded as an euler-bernoulli beam, without considering the influence of shear deformation.
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Publication number Priority date Publication date Assignee Title
US20090125282A1 (en) * 2005-11-07 2009-05-14 Keio University Numerical structural analysis system based on the load-transfer-path method
CN104166792A (en) * 2014-08-06 2014-11-26 中国科学院工程热物理研究所 Finite element analysis method for temperature action on prestressed reinforced concrete continuous rigid frame bridge
CN110111423A (en) * 2019-04-11 2019-08-09 河海大学 A kind of constrained concrete free form surface form creation method for considering to creep

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20090125282A1 (en) * 2005-11-07 2009-05-14 Keio University Numerical structural analysis system based on the load-transfer-path method
CN104166792A (en) * 2014-08-06 2014-11-26 中国科学院工程热物理研究所 Finite element analysis method for temperature action on prestressed reinforced concrete continuous rigid frame bridge
CN110111423A (en) * 2019-04-11 2019-08-09 河海大学 A kind of constrained concrete free form surface form creation method for considering to creep

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