CN113642087A - Method for predicting shearing performance of square-section reinforced concrete beam - Google Patents

Method for predicting shearing performance of square-section reinforced concrete beam Download PDF

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CN113642087A
CN113642087A CN202111066539.3A CN202111066539A CN113642087A CN 113642087 A CN113642087 A CN 113642087A CN 202111066539 A CN202111066539 A CN 202111066539A CN 113642087 A CN113642087 A CN 113642087A
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淳庆
张承文
李盼
林怡婕
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Abstract

The invention discloses a method for predicting shearing performance of a square-section reinforced concrete beam, which comprises the following steps of: s1, determining a beam shear balance expression through the improved internal force arch truss model; s2, solving a crack angle; s3, camber angle; s4, solving the tension of the stirrup; s5, solving the pressure of the concrete pressure rod; s6, solving the shearing force of the inner force arch; s7, solving the tension of the lower tension steel bar; s8, solving the upper concrete pressure and the microcrack cohesive force; and S9, comprehensively solving through the sub items to obtain the shearing performance of the reinforced concrete beam with the square section. The method considers the bonding slippage between the concrete and the steel bars and the constitutive model of the steel bars and the concrete, and simultaneously considers the size effect and the influence of critical cracks in a calculation formula, so the calculation precision is higher; the invention can evaluate the shearing performance of the square-section reinforced concrete beam and provide a theoretical basis for scientifically evaluating the stress performance of the square-section reinforced concrete structural member.

Description

Method for predicting shearing performance of square-section reinforced concrete beam
Technical Field
The invention relates to the field of civil engineering, in particular to a method for predicting shearing performance of a square-section reinforced concrete beam.
Background
Historical buildings are precious material cultural heritages in China, wherein reinforced concrete buildings with square sections occupy a very important position, the buildings are used for more than seven and eighty years and far exceed the reasonable service life of reinforced concrete, and the buildings have damage and diseases of different degrees due to the deterioration of material performance and the change of external factors. The material performance and the construction method of the square-section reinforced concrete beam are obviously different from those of the modern reinforced concrete structure, so that a plurality of existing calculation methods are not suitable for calculation and evaluation of the stress performance of the square-section reinforced concrete beam, and therefore a scientific and accurate calculation model is quite necessary to be established for evaluating the bearing capacity of the square-section reinforced concrete beam.
The invention establishes the shear resistance calculation expression of the square section reinforced concrete beam by adopting an improved internal force arch-truss model, and obtains the shear resistance of the square section reinforced concrete beam by the subentry solution. The method considers the bonding slippage between the concrete and the steel bars and the constitutive model of the steel bars and the concrete, simultaneously considers the influence of the size effect and the critical cracks in the calculation formula, and can more accurately evaluate the shearing performance of the reinforced concrete beam with the square section. Meanwhile, the method can solve the resistance reserve of the concrete beam according to the cracking process of the crack, and the calculation of the solving process is simple, convenient and quick, and the result is accurate. The method can provide a theoretical basis for scientific evaluation of the stress performance of the square-section reinforced concrete structure member and provide a theoretical basis for structural safety evaluation and reinforcement and repair design of the square-section reinforced concrete building.
Disclosure of Invention
The invention aims to provide a method for predicting shear performance of a square-section reinforced concrete beam, which considers bonding slippage between special concrete and steel bars in a square-section reinforced concrete structure and constitutive models of the steel bars and the concrete, considers the influences of a size effect and a critical crack in a calculation formula, can accurately evaluate the shear performance of the square-section reinforced concrete beam, and can provide a theoretical basis for structural safety evaluation and reinforcement and repair design of a square-section reinforced concrete building.
The purpose of the invention can be realized by the following technical scheme:
a method for predicting shearing performance of a square-section reinforced concrete beam comprises the following steps:
s1 formula of beam shear balance:
Figure RE-GDA0003306088910000021
in the formula: vsumIs the total pressure; t is a pulling force; z is a radical ofrThe distance between the resultant force of the pressure and the tension;
and (3) carrying out stress analysis on the isolated body I, and establishing a balance equation:
Figure RE-GDA0003306088910000022
in the formula: c1Concrete pressure provided for the upper concrete of the fracture toughness zone; c2Providing pressure for concrete between cracks, namely the compression web members in the truss model; t is1Tension provided to the stirrup; t is2Tension provided to the lower tensioned reinforcement; t is3Tensile forces generated by microcracks near the tip of the crack;
Figure RE-GDA0003306088910000023
is the crack angle; theta is an arch inclination angle; vtThe shear force is to be calculated;
s2 calculating crack angle
Figure RE-GDA0003306088910000024
And fitting a critical fracture expression determined by the normalized fracture initiation position on the basis of the critical fracture:
Figure RE-GDA0003306088910000025
in the formula: alpha is alpha0Relative abscissa of critical crack; z is the relative height from the bottom of the beam to the stressed steel bar; kappa is the shape coefficient of different steel bars;
obtaining a calculation formula of a vertical coordinate of the crack tip and the crack angle at the moment:
Figure RE-GDA0003306088910000031
in the formula: alpha is alpha1Relative abscissa of critical crack tip; h is the beam height;
s3 calculating camber angle theta
In the internal force arch model, when the load is a concentrated load, the internal force arch is linear, and the calculation expression of the equivalent width of the arch is as follows:
Figure RE-GDA0003306088910000032
in the formula: x is the number ofnIs the equivalent width of the arch;
obtaining an expression of the camber angle theta through a geometrical relation:
Figure RE-GDA0003306088910000033
s4 calculating the tension T of the stirrup1
Assuming that the stress of each stirrup is equal, the number N of equivalent stirrups cut by the crack in the separator IsvComprises the following steps:
Figure RE-GDA0003306088910000034
get the tension T of the stirrup1The expression of (a) is:
T1＝NsvT′ (8)
in the formula: t' is the stress of a single stirrup;
s5 calculating the tension C of the compression bar2
Taking the spacer III, the length of AM is
Figure RE-GDA0003306088910000041
The expression of the compression bar tension is:
Figure RE-GDA0003306088910000042
s6 calculating the shearing force V of the inner force archarch
According to the assumption, when the concrete of the truss model reaches the ultimate compressive strength, the compressive stress expression of the concrete in the truss model is as follows:
Figure RE-GDA0003306088910000043
the relational expression of the concrete compressive stress in the internal force arch and the concrete compressive stress in the truss is as follows:
Figure RE-GDA0003306088910000044
in the formula: ν is a softening coefficient to be considered by compressive stress when the concrete is shear resistant, and is calculated by the following formula:
Figure RE-GDA0003306088910000045
from formulas (11) and (12):
Figure RE-GDA0003306088910000046
in the formula:
Figure RE-GDA0003306088910000047
from the geometric relationship how to obtain the arch shear force VarchThe expression of (a) is:
Varch＝(1-β0)fcbxnsinθ (14)
Figure RE-GDA0003306088910000048
in the formula: a. thesIs the section area of the steel bar; n is a radical ofstThe number of longitudinal ribs at the bottom in the cross section of the beam is;
s7 calculating the tension T of the lower tension bar2
Taking stress sigma of insulator IV and steel bar elongation arch models' stress plus truss model σsProviding that, according to the balance of forces, the stress and the bonding force applied to the steel bar are balanced, and the elongation of the steel bar is expressed as:
Figure RE-GDA0003306088910000051
in the formula: tau (S) is a relational expression of the bonding slippage behavior of the steel bar and the concrete;
according to the geometric compatibility condition, the slippage is equal to the elongation of the reinforcing steel bars plus the compression of the concrete, and for simplifying the calculation, the compression of the concrete is assumed to be approximately equal to
Figure RE-GDA0003306088910000052
Multiplying the elongation of the steel bar to obtain an equation:
Figure RE-GDA0003306088910000053
according to the geometrical relationship, it is assumed that the width of the crack is generally equal to the elongation of the bar plus the amount of slip:
Figure RE-GDA0003306088910000054
in the formula: omegac(z) the width of the crack is the height z of the section where the tension steel bar is located;
the equations (15) - (18) show that the tension T provided by the tension bar2The expression of (a) is:
Figure RE-GDA0003306088910000055
s8 solving for upper concrete pressure C1And microcrack cohesion T3
The softening constitutive model of the tensile concrete is calculated by adopting a two-section model:
Figure RE-GDA0003306088910000056
in the formula: f. oftIs the tensile strength of concrete, omega*For the virtual crack width, calculated from equation (23), ω0The width of the crack at the starting position of the virtual crack is taken as 0.1 mm;
computational expression of cohesion near the fracture tip:
Figure RE-GDA0003306088910000061
in the formula: dvirtIs the virtual crack segment distance; dplThe length of a plastic zone of a flexible zone at the front part of the tip of the concrete crack; wherein d issAnd s is the differential of the distance from the crack tip to the plastic or virtual crack zone and the distance,
Figure RE-GDA0003306088910000062
represents ω in the formula (20)*(s) is expressed in distance-dependent form using equation (3) and the fracture width expression, the second term on the right of the integral being assumed to be before the concrete fracture tip
Figure RE-GDA0003306088910000063
The tensile force provided by the tensile plastic zone;
to take into account the size effect, T is obtained3The calculation expression of (1):
Figure RE-GDA0003306088910000064
in the formula (I), the compound is shown in the specification,
Figure RE-GDA0003306088910000065
is the Hillerberg brittleness constant;
Figure RE-GDA0003306088910000066
is the characteristic length;
Figure RE-GDA0003306088910000067
is a constant;
assuming pressure C provided by concrete above the fracture-tip ductile band1Provided as concrete from the crack tip plastic zone to the top of the beam;
the constitutive model expression of the concrete is as follows:
Figure RE-GDA0003306088910000068
in the formula: epsilonuRepresenting the peak stress corresponding strain;
assuming that the pressure distribution shape of the concrete is similar to the constitutive relation sigma of the concretecmA rising section of (2), then C1The expression of (a) is:
Figure RE-GDA0003306088910000071
in the formula:
Figure RE-GDA0003306088910000072
indicating upper concrete participationThe number of the first and second groups is,
Figure RE-GDA0003306088910000073
dsrepresents a distance differential;
s9 shear performance prediction
Alpha content was determined by the steps (1) to (8)osvAnd VtTwo parameters and with respect to alpha1C of (A)1(α1)、 C2(α1,αosv)、T1(α1,αosv)、T2(α1,αosv,Vt)、T3(α1,Vt)、Varch(α1,αosv)、θ(α1) And
Figure RE-GDA0003306088910000076
expression formula (4), (6), (8), (9), (14), (19), (22) and (24) are substituted into formula (2), and parameter alpha is eliminatedosvTo obtain VtWith respect to alpha1And finally obtaining the shear force calculation result of the beam by using the expression of (1).
Furthermore, the reinforced concrete beam with the square cross section adopts the reinforced cross section in the form of ribbed square steel, flat-section steel and spiral steel.
Further, in S2, the form factor of the ribbed square steel is 1.131, the form factor of the flat-section steel is 0.899, and the form factor of the spiral rib is 0.738.
Further, the solving process of the stress T' of the single stirrup in S4 specifically includes the following steps:
taking the stirrups cut by the cracks and the nearby concrete as an isolator II, and obtaining the following expression according to the geometrical relationship:
Figure RE-GDA0003306088910000074
in the formula: AB is a fracture toughness zone extension line; b is a crack tip; m is the projection of the point B at the bottom of the beam;
then an expression of the stress of a single stirrup is obtained:
Figure RE-GDA0003306088910000075
in the formula: a. thesvThe area of the cross section of the stirrup; n is the number of hooped limbs; f. ofyvThe yield strength of the stirrup; b is the cross-sectional width; s is the stirrup spacing; alpha is alphaosvThe participation coefficient of the stirrup is shown; rhosvThe hoop ratio is used.
Further, in S7, the relational expression of the adhesion slip behavior of the steel bar and the concrete adopts a three-stage constitutive model:
Figure RE-GDA0003306088910000081
in the formula: tau is0For each set of experimental initial bond strengths, the arithmetic mean, τuIs the arithmetic mean of the experimental ultimate bond strengths, τ, of each grouprIs the arithmetic mean of the residual bond strengths, S, of each set of experimentsuThe arithmetic mean value of the corresponding slip values of the experimental ultimate bond strengths of each group, SrThe arithmetic mean of the slip values corresponding to the initial residual bond strengths of each set of experiments,
Figure RE-GDA0003306088910000082
for the rise function shape coefficients, regression is given by:
Figure RE-GDA0003306088910000083
rewrite equation (27) to unity using the Heaviside function:
Figure RE-GDA0003306088910000084
(H(S-Su)-H(S-Sr))+τrH(S-Sr)
Figure RE-GDA0003306088910000085
in the formula: h (x) represents the Heaviside function.
Furthermore, the reinforcing steel bar and concrete bonding slip three-section type constitutive model is tau for ribbed square steel0The value is 0.34-0.41, tauuThe value is 9.33-11.41, taurThe value is 3.36-4.10, SuThe value is 1.73-2.11, SrThe value is 2.81 to 3.43,
Figure RE-GDA0003306088910000086
the value is 0.50-0.62; for flat section steels,. tau0The value is 0.25-0.30, tauuThe value is 8.85-10.82, taurThe value is 1.84-2.25, SuThe value is 3.29-4.02SrThe value is 5.69 to 6.95,
Figure RE-GDA0003306088910000087
the value is 1.03-1.25; for helical steels,. tau0The value is 0.29-0.36, tauuThe value is 8.43-10.31, taurThe value of S is 1.40-1.71uThe value is 4.14-5.07SrThe value is 5.30 to 6.47,
Figure RE-GDA0003306088910000088
the value is 0.23-0.29.
Further, the width ω of the crack in S7c(z) the solving process specifically comprises the following steps:
the crack width expression of the beam under the concentrated force is as follows:
Figure RE-GDA0003306088910000091
wherein psi is the correction coefficient of the steel bar body shape, h*The height of the crack at the desired width;
Figure RE-GDA0003306088910000092
is a polynomial regression equation:
Figure RE-GDA0003306088910000093
in the insulator IV, h*Taken as the height z of the section where the tensioned steel bar is located.
Further, the steel bar body shape correction coefficient is determined by secant rigidity of the bonding slippage ascending section limit bonding value and the initial bonding value, the square steel is 1, the flat section steel is 2.5, and the spiral rib is 3.4.
Further, the virtual crack segment distance d in S8virtPlastic zone length d of the flexible zone in front of the concrete crack tipplThe specific process comprises the following steps:
s81, assuming the coordinate of the virtual crack starting point as (x)0,y0) And satisfies the crack coordinate equation (3) and brings it into equation (23) to make the crack width equal to 0.1, find x0And y0；
Since the virtual crack segment is short relative to the full length of the crack, its length is expressed as:
Figure RE-GDA0003306088910000096
in the formula: | a | non-woven phosphor2Represents the 2 norm of vector a;
s82, the plastic zone length of the flexible belt at the front part of the concrete crack tip is as follows:
Figure RE-GDA0003306088910000094
in the formula: σ is the distal stress; lcrackIs the crack length;
the distal stress σ is reduced using the following equation:
Figure RE-GDA0003306088910000095
crack length lcrackThe calculation expression of (a) is:
Figure RE-GDA0003306088910000101
Figure RE-GDA0003306088910000102
the invention has the beneficial effects that:
1. the prediction method considers the bonding slippage between special concrete and steel bars in the square-section reinforced concrete structure and the constitutive model of the steel bars and the concrete, simultaneously considers the influence of the size effect and the critical cracks in a calculation formula, and has higher calculation precision;
2. the method can accurately evaluate the shearing resistance of the square-section reinforced concrete beam, and can provide a theoretical basis for the structural safety evaluation and the reinforcement and repair design of the square-section reinforced concrete building.
Drawings
The invention will be further described with reference to the accompanying drawings.
FIG. 1 is a flow chart of a prediction method of the present invention;
FIG. 2 is a schematic diagram of an internal force arch in the truss model of the present invention;
FIG. 3 is a schematic view of a truss in the truss model of the present invention;
FIG. 4 is an analysis of the internal force arch of spacer I of the present invention;
FIG. 5 is an analysis of a truss of insulator I of the present invention;
FIG. 6 is a sectional view of a concrete of the present invention;
FIG. 7 is an analysis diagram of separator II of the present invention;
FIG. 8 is an analysis diagram of the separator III of the present invention;
FIG. 9 is a graph of deformation coordination analysis of the spacer IV of the present invention;
FIG. 10 is a graph of stress versus deformation for a spacer IV of the present invention;
FIG. 11 is a diagram of a virtual microcrack analysis in accordance with the invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The embodiment discloses a method for predicting shearing performance of a square-section reinforced concrete beam, which comprises the following steps of:
s1 formula of beam shear balance:
Figure RE-GDA0003306088910000111
in the formula: vsumIs the total pressure; t is a pulling force; z is a radical ofrThe distance between the resultant of the pressure and the tension.
And (3) carrying out stress analysis on the isolated body I (the concrete on the upper surface of the critical crack), and establishing an equilibrium equation:
Figure RE-GDA0003306088910000112
in the formula: c1Concrete pressure provided for the upper concrete (zone 3, zone 4 concrete) of the fracture toughness zone; c2The pressure provided for the concrete between the cracks, i.e. the compression web members in the truss model (zone 1, zone 2 concrete); t is1Tension provided to the stirrup; t is2Tension provided to the lower tensioned reinforcement; t is3Tensile forces generated by microcracks near the tip of the crack;
Figure RE-GDA0003306088910000113
is the crack angle; theta is an arch inclination angle; vtTo request the shearing force.
S2 calculating crack angle
Figure RE-GDA0003306088910000114
Based on the critical fracture, the normalized fracture initiation position (alpha) is fitted0) The determined critical crack expression:
Figure RE-GDA0003306088910000115
in the formula: alpha is alpha0Relative abscissa of critical crack; z is the relative height from the bottom of the beam to the stressed steel bar; kappa is the form factor of different steel bars, the ribbed square steel is 1.131, the flat section steel is 0.899, and the spiral rib is 0.738.
From this, the formula for the ordinate of the crack tip and the crack angle at that time can be derived:
Figure RE-GDA0003306088910000121
in the formula: alpha is alpha1Relative abscissa of critical crack tip; h is the beam height.
S3 calculating camber angle theta
In the internal force arch model, when the load is a concentrated load, the internal force arch is linear, and the calculation expression of the equivalent width of the arch is as follows:
Figure RE-GDA0003306088910000122
in the formula: x is the number ofnIs the equivalent width of the arch.
The expression for camber angle θ can be obtained from the geometric relationship:
Figure RE-GDA0003306088910000123
s4 calculating the tension T of the stirrup1
Taking the stirrups cut by the cracks and the nearby concrete as an isolator II, and obtaining the following expression according to the geometrical relationship:
Figure RE-GDA0003306088910000124
in the formula: AB is a fracture toughness zone extension line; b is a crack tip; m is the projection of the point B on the bottom of the beam.
Then an expression for the force of a single stirrup can be obtained:
Figure RE-GDA0003306088910000125
in the formula: a. thesvThe area of the cross section of the stirrup; n is the number of hooped limbs; f. ofyvThe yield strength of the stirrup; b is the cross-sectional width; s is the stirrup spacing; alpha is alphaosvThe participation coefficient of the stirrup is shown; rhosvThe hoop ratio is used.
Assuming that the stress of each stirrup is equal, the number N of equivalent stirrups cut by the crack in the separator IsvComprises the following steps:
Figure RE-GDA0003306088910000131
the tension T of the stirrup obtained by the formulas (8) and (9)1The expression of (a) is:
Figure RE-GDA0003306088910000132
s5 calculating the tension C of the compression bar2
Taking the spacer III, the length of AM is
Figure RE-GDA0003306088910000133
The expression of the compression bar tension is:
Figure RE-GDA0003306088910000134
s6 calculating the shearing force V of the inner force archarch
According to the assumption, when the concrete of the truss model reaches the ultimate compressive strength, the compressive stress expression of the concrete in the truss model is as follows:
Figure RE-GDA0003306088910000135
the relational expression of the concrete compressive stress in the internal force arch and the concrete compressive stress in the truss is as follows:
Figure RE-GDA0003306088910000136
in the formula: ν is the softening coefficient to be taken into account by the compressive stress when the concrete is shear resistant, and can be calculated by the following formula:
Figure RE-GDA0003306088910000137
from formulas (13) and (14):
Figure RE-GDA0003306088910000138
in the formula:
Figure RE-GDA0003306088910000141
from the geometric relationship how to obtain the arch shear force VarchThe expression of (a) is:
Varch＝(1-β0)fcbxnsinθ (16)
Figure RE-GDA0003306088910000142
in the formula: a. thesIs the section area of the steel bar; n is a radical ofstIn the beam cross sectionThe number of the bottom longitudinal ribs.
S7 calculating the tension T of the lower tension bar2
The relation expression of the bonding and slipping behavior of the steel bars and the concrete adopts a three-section constitutive model:
Figure RE-GDA0003306088910000143
in the formula: tau is0For each set of experimental initial bond strengths, the arithmetic mean, τuIs the arithmetic mean of the experimental ultimate bond strengths, τ, of each grouprIs the arithmetic mean of the residual bond strengths, S, of each set of experimentsuThe arithmetic mean value of the corresponding slip values of the experimental ultimate bond strengths of each group, SrThe arithmetic mean of the slip values corresponding to the initial residual bond strengths of each set of experiments,
Figure RE-GDA0003306088910000144
for the rise function shape coefficients, regression can be made from:
Figure RE-GDA0003306088910000145
rewrite equation (18) to unity using the Heaviside function:
Figure RE-GDA0003306088910000146
(H(S-Su)-H(S-Sr))+τrH(S-Sr)
Figure RE-GDA0003306088910000147
in the formula: h (x) represents the Heaviside function.
Taking an insulator IV, and enabling the elongation of the steel bar to be equal to the stress sigma 'of the arch model'sStress sigma plus truss modelsProviding, according to the balance of forces, the sum of the stresses and the steelThe bonding force that the muscle receives balances each other, then the expression of reinforcing bar elongation is:
Figure RE-GDA0003306088910000151
according to the geometric compatibility condition, the slippage is equal to the elongation of the reinforcing steel bars plus the compression of the concrete, and for simplifying the calculation, the compression of the concrete is assumed to be approximately equal to
Figure RE-GDA0003306088910000152
The elongation of the steel bar is multiplied, and then the equation can be obtained:
Figure RE-GDA0003306088910000153
the crack width expression of the beam under the concentrated force is as follows:
Figure RE-GDA0003306088910000154
in the formula, psi is a steel bar body shape correction coefficient and can be determined by secant rigidity of a bonding slippage ascending section limit bonding value and an initial bonding value, the square steel is 1, the flat section steel can be 2.5, and the spiral rib is 3.4; h is*The height of the crack at the desired width;
Figure RE-GDA0003306088910000155
is a polynomial regression equation:
Figure RE-GDA0003306088910000156
in the insulator IV, h*It may be taken as the height z of the section where the tension bar is located.
According to the geometrical relationship, it is assumed that the width of the crack is generally equal to the elongation of the bar plus the amount of slip:
Figure RE-GDA0003306088910000157
the tensile force T provided by the tension bar can be obtained from the formulas (17) to (25)2The expression of (a) is:
Figure RE-GDA0003306088910000158
s8 solving for upper concrete pressure C1And microcrack cohesion T3
The softening constitutive model of the tensile concrete is calculated by adopting a two-section model:
Figure RE-GDA0003306088910000161
in the formula: f. oftIs the tensile strength of concrete, omega*For the virtual crack width, ω can be calculated from equation (23)0The width of the crack at the start of the virtual crack may be 0.1 mm.
Assuming the virtual crack initiation coordinates are (x)0,y0) And satisfies the crack coordinate equation (3) and brings it into equation (23) to make the crack width equal to 0.1, find x0And y0。
Since the virtual crack segment is short relative to the full length of the crack, its length can be approximated as:
Figure RE-GDA0003306088910000162
in the formula: | a | non-woven phosphor2Representing the 2 norm of vector a.
The plastic zone length of the toughness zone at the front part of the concrete crack tip is as follows:
Figure RE-GDA0003306088910000163
in the formula: σ is the distal stress; lcrackIs the crack length.
The distal stress σ can be reduced using the following equation:
Figure RE-GDA0003306088910000164
crack length lcrackThe calculation expression of (a) is:
Figure RE-GDA0003306088910000165
Figure RE-GDA0003306088910000166
the calculation expression of the cohesive force in the vicinity of the crack tip can be obtained from the expressions (27) to (31):
Figure RE-GDA0003306088910000171
in the formula: wherein d issAnd s is the differential of the distance from the crack tip to the plastic or virtual crack zone and the distance,
Figure RE-GDA0003306088910000172
represents ω in the formula (27)*(s) is expressed in distance-dependent form using equations (3) and (23), and the second term on the right of the integral is assumed to be before the concrete crack tip
Figure RE-GDA0003306088910000173
Is stretched by the tension provided by the plastic zone.
To take into account the size effect, T is obtained3The calculation expression of (1):
Figure RE-GDA0003306088910000174
in the formula (I), the compound is shown in the specification,
Figure RE-GDA0003306088910000175
is the Hillerberg brittleness constant;
Figure RE-GDA0003306088910000176
is the characteristic length;
Figure RE-GDA0003306088910000177
is a constant.
Assuming the pressure C provided by the upper concrete (3-zone, 4-zone concrete) of the fracture tip ductile band1Can be viewed as a concrete supply from the crack tip plastic zone to the top of the beam.
The constitutive model expression of the concrete is as follows:
Figure RE-GDA0003306088910000178
in the formula: epsilonuRepresenting peak stress versus strain.
Assuming that the pressure distribution shape of the concrete is similar to the constitutive relation sigma of the concretecmA rising section of (2), then C1The expression of (a) is:
Figure RE-GDA0003306088910000179
in the formula:
Figure RE-GDA0003306088910000181
represents the upper concrete reference coefficient,
Figure RE-GDA0003306088910000182
dsrepresenting the distance differential.
Shear performance prediction
Thus, the content of α has been determined through the steps (1) to (8)osvAnd VtTwo parameters and with respect to alpha1C of (A)1(α1)、 C2(α1,αosv)、T1(α1,αosv)、T2(α1,αosv,Vt)、T3(α1,Vt)、Varch(α1,αosv)、θ(α1) And
Figure RE-GDA0003306088910000184
expression, substituting the expressions (4), (6), (10), (11), (26), (33) and (35) into the expression (2), the parameter α can be eliminatedosvTo obtain VtWith respect to alpha1And finally obtaining the shear force calculation result of the beam by using the expression of (1).
In the invention, the reinforced concrete beam with the square section adopts the reinforced section forms of ribbed square steel, flat steel and spiral steel. In the step (7), for the steel bar and concrete bonding slip three-section constitutive model, for the ribbed square steel, tau0The suggested value is 0.34-0.41, tauuThe suggested value is 9.33-11.41, taurThe suggested value is 3.36-4.10, SuThe suggested value is 1.73-2.11, SrThe suggested value is 2.81-3.43,
Figure RE-GDA0003306088910000185
the suggested value is 0.50-0.62; for flat section steels,. tau0The suggested value is 0.25-0.30, tauuThe suggested value is 8.85-10.82, taurThe suggested value is 1.84-2.25, SuThe suggested value is 3.29-4.02SrThe suggested value is 5.69-6.95,
Figure RE-GDA0003306088910000186
the suggested value is 1.03-1.25; for helical steels,. tau0The suggested value is 0.29-0.36, tauuThe suggested value is 8.43-10.31, taurThe suggested value is 1.40-1.71, SuThe suggested value is 4.14-5.07, SrThe suggested value is 5.30-6.47,
Figure RE-GDA0003306088910000187
the suggested value is 0.23-0.29.
In order to evaluate the accuracy of the method for predicting the shear resistance of the reinforced concrete beam, the shear test results of the reinforced concrete beam with the square cross section shown in the following table are compared and analyzed.
TABLE 1 test shear test piece parameters
Figure RE-GDA0003306088910000183
Figure RE-GDA0003306088910000191
Note: the shear span ratio, the hoop ratio and the reinforcement ratio are all divided by the effective height h in the calculation formula0Which can be represented by the formula h0H-c, where h is the section height and c is the protective layer thickness.
Table 2: comparison of the prediction formula with other normative calculation results
Figure RE-GDA0003306088910000192
As can be seen from table 2, the agreement between the experimental values of the three test beams and the prediction formula of the present invention is higher than that of the other two calculation methods. The stirrup of the square-section reinforced concrete beam cannot yield due to the high hoop ratio; and the shear damage of the beam is assumed in the ACI formula and the GB50010-2010 specification, so that the stirrup is considered to be yielding in the formula, and the shear bearing value calculated according to the ACI formula and the GB50010-2010 specification is reasonably larger than an experimental value, but the influence of the concrete and the bottom longitudinal bar on the shear bearing is estimated too low due to the conservative value in the specification. According to the formula provided by the invention, the bonding slippage between the concrete and the steel bars and the constitutive model of the steel bars and the concrete are considered, and meanwhile, the influence of the size effect and the critical crack is considered in the calculation formula, so that the difference between the calculation result and the test value is small, and the shearing resistance of the reinforced concrete beam with the square section can be accurately evaluated.
In the description herein, references to the description of "one embodiment," "an example," "a specific example" or the like are intended to mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
The foregoing shows and describes the general principles, essential features, and advantages of the invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed.

Claims (9)

1. A method for predicting shearing performance of a square-section reinforced concrete beam is characterized by comprising the following steps:
s1 formula of beam shear balance:
Figure RE-FDA0003306088900000011
in the formula: vsumIs the total pressure; t is a pulling force; z is a radical ofrThe distance between the resultant force of the pressure and the tension;
and (3) carrying out stress analysis on the isolated body I, and establishing a balance equation:
Figure RE-FDA0003306088900000012
in the formula: c1Concrete pressure provided for the upper concrete of the fracture toughness zone; c2Providing pressure for concrete between cracks, namely the compression web members in the truss model; t is1Tension provided to the stirrup; t is2Tension provided to the lower tensioned reinforcement; t is3Tensile forces generated by microcracks near the tip of the crack;
Figure RE-FDA0003306088900000013
is the crack angle; theta is an arch inclination angle; vtThe shear force is to be calculated;
s2 calculating crack angle
Figure RE-FDA0003306088900000014
And fitting a critical fracture expression determined by the normalized fracture initiation position on the basis of the critical fracture:
Figure RE-FDA0003306088900000015
in the formula: alpha is alpha0Relative abscissa of critical crack; z is the relative height from the bottom of the beam to the stressed steel bar; kappa is the shape coefficient of different steel bars;
obtaining a calculation formula of a vertical coordinate of the crack tip and the crack angle at the moment:
Figure RE-FDA0003306088900000021
in the formula: alpha is alpha1Relative abscissa of critical crack tip; h is the beam height;
s3 calculating camber angle theta
In the internal force arch model, when the load is a concentrated load, the internal force arch is linear, and the calculation expression of the equivalent width of the arch is as follows:
Figure RE-FDA0003306088900000022
in the formula: x is the number ofnIs the equivalent width of the arch;
obtaining an expression of the camber angle theta through a geometrical relation:
Figure RE-FDA0003306088900000023
s4 calculating the tension T of the stirrup1
Assuming that the stress of each stirrup is equal, the number N of equivalent stirrups cut by the crack in the separator IsvComprises the following steps:
Figure RE-FDA0003306088900000024
get the tension T of the stirrup1The expression of (a) is:
T1＝NsvT′ (8)
in the formula: t' is the stress of a single stirrup;
s5 calculating the tension C of the compression bar2
Taking the spacer III, the length of AM is
Figure RE-FDA0003306088900000025
The expression of the compression bar tension is:
Figure RE-FDA0003306088900000031
s6 calculating the shearing force V of the inner force archarch
According to the assumption, when the concrete of the truss model reaches the ultimate compressive strength, the compressive stress expression of the concrete in the truss model is as follows:
Figure RE-FDA0003306088900000032
the relational expression of the concrete compressive stress in the internal force arch and the concrete compressive stress in the truss is as follows:
Figure RE-FDA0003306088900000033
in the formula: ν is a softening coefficient to be considered by compressive stress when the concrete is shear resistant, and is calculated by the following formula:
Figure RE-FDA0003306088900000034
from formulas (11) and (12):
Figure RE-FDA0003306088900000035
in the formula:
Figure RE-FDA0003306088900000036
from the geometric relationship how to obtain the arch shear force VarchThe expression of (a) is:
Varch＝(1-β0)fcbxnsinθ (14)
Figure RE-FDA0003306088900000037
in the formula: a. thesIs the section area of the steel bar; n is a radical ofstThe number of longitudinal ribs at the bottom in the cross section of the beam is;
s7 calculating the tension T of the lower tension bar2
Taking stress sigma of insulator IV and steel bar elongation arch models' stress plus truss model σsProviding that, according to the balance of forces, the stress and the bonding force applied to the steel bar are balanced, and the elongation of the steel bar is expressed as:
Figure RE-FDA0003306088900000041
in the formula: tau (S) is a relational expression of the bonding slippage behavior of the steel bar and the concrete;
according to the geometric compatibility condition, the slippage is equal to the elongation of the reinforcing steel bars plus the compression of the concrete, and for simplifying the calculation, the compression of the concrete is assumed to be approximately equal to
Figure RE-FDA0003306088900000042
Multiplying the elongation of the steel bar to obtain an equation:
Figure RE-FDA0003306088900000043
according to the geometrical relationship, it is assumed that the width of the crack is generally equal to the elongation of the bar plus the amount of slip:
Figure RE-FDA0003306088900000044
in the formula: omegac(z) the width of the crack is the height z of the section where the tension steel bar is located;
the equations (15) - (18) show that the tension T provided by the tension bar2The expression of (a) is:
Figure RE-FDA0003306088900000045
s8 solving for upper concrete pressure C1And microcrack cohesion T3
The softening constitutive model of the tensile concrete is calculated by adopting a two-section model:
Figure RE-FDA0003306088900000046
in the formula: f. oftIs the tensile strength of concrete, omega*For the virtual crack width, calculated from equation (23), ω0The width of the crack at the virtual crack initiation was taken to be 0.1mm；
Computational expression of cohesion near the fracture tip:
Figure RE-FDA0003306088900000047
in the formula: dvirtIs the virtual crack segment distance; dplThe length of a plastic zone of a flexible zone at the front part of the tip of the concrete crack; wherein d issAnd s is the differential of the distance from the crack tip to the plastic or virtual crack zone and the distance,
Figure RE-FDA0003306088900000051
represents ω in the formula (20)*(s) is expressed in distance-dependent form using equation (3) and the fracture width expression, the second term on the right of the integral being assumed to be before the concrete fracture tip
Figure RE-FDA0003306088900000052
The tensile force provided by the tensile plastic zone;
to take into account the size effect, T is obtained3The calculation expression of (1):
Figure RE-FDA0003306088900000053
in the formula (I), the compound is shown in the specification,
Figure RE-FDA0003306088900000054
is the Hillerberg brittleness constant;
Figure RE-FDA0003306088900000055
is the characteristic length;
Figure RE-FDA0003306088900000056
is a constant;
upper concrete lift of ductile band assuming crack tipPressure of supply C1Provided as concrete from the crack tip plastic zone to the top of the beam;
the constitutive model expression of the concrete is as follows:
Figure RE-FDA0003306088900000057
in the formula: epsilonuRepresenting the peak stress corresponding strain;
assuming that the pressure distribution shape of the concrete is similar to the constitutive relation sigma of the concretecmA rising section of (2), then C1The expression of (a) is:
Figure RE-FDA0003306088900000058
in the formula:
Figure RE-FDA0003306088900000061
represents the upper concrete reference coefficient,
Figure RE-FDA0003306088900000062
dsrepresents a distance differential;
s9 shear performance prediction
Alpha content was determined by the steps (1) to (8)osvAnd VtTwo parameters and with respect to alpha1C of (A)1(α1)、C2(α1,αosv)、T1(α1,αosv)、T2(α1,αosv,Vt)、T3(α1,Vt)、Varch(α1,αosv)、θ(α1) And
Figure RE-FDA0003306088900000063
expression formula (4), (6), (8), (9), (14), (19), (22) and (24) are substituted into formula (2), and parameter alpha is eliminatedosvTo obtain VtWith respect to alpha1And finally obtaining the shear force calculation result of the beam by using the expression of (1).
2. The method for predicting the shear performance of the square-section reinforced concrete beam as recited in claim 1, wherein the square-section reinforced concrete beam adopts a reinforced section form of ribbed square steel, flat-section steel and spiral steel.
3. The method for predicting the shear stress of a square-section reinforced concrete beam according to claim 2, wherein the form factor of the ribbed square steel is 1.131, the form factor of the flat-section steel is 0.899, and the form factor of the spiral rib is 0.738, in S2.
4. The method for predicting the shearing performance of the square section reinforced concrete beam as recited in claim 1, wherein the process of solving the stress T' of the single stirrup in the S4 specifically comprises the following steps:
taking the stirrups cut by the cracks and the nearby concrete as an isolator II, and obtaining the following expression according to the geometrical relationship:
Figure RE-FDA0003306088900000064
in the formula: AB is a fracture toughness zone extension line; b is a crack tip; m is the projection of the point B at the bottom of the beam;
then an expression of the stress of a single stirrup is obtained:
Figure RE-FDA0003306088900000065
in the formula: a. thesvThe area of the cross section of the stirrup; n is the number of hooped limbs; f. ofyvThe yield strength of the stirrup; b is the cross-sectional width; s is the stirrup spacing; alpha is alphaosvThe participation coefficient of the stirrup is shown; rhosvThe hoop ratio is used.
5. The method for predicting the shear performance of the square section reinforced concrete beam as claimed in claim 2, wherein in the step S7, the relational expression of the adhesion slip behavior of the steel bar and the concrete adopts a three-section constitutive model:
Figure RE-FDA0003306088900000071
in the formula: tau is0For each set of experimental initial bond strengths, the arithmetic mean, τuIs the arithmetic mean of the experimental ultimate bond strengths, τ, of each grouprIs the arithmetic mean of the residual bond strengths, S, of each set of experimentsuThe arithmetic mean value of the corresponding slip values of the experimental ultimate bond strengths of each group, SrThe arithmetic mean of the slip values corresponding to the initial residual bond strengths of each set of experiments,
Figure RE-FDA0003306088900000072
for the rise function shape coefficients, regression is given by:
Figure RE-FDA0003306088900000073
rewrite equation (27) to unity using the Heaviside function:
Figure RE-FDA0003306088900000074
(H(S-Su)-H(S-Sr))+τrH(S-Sr)
Figure RE-FDA0003306088900000075
in the formula: h (x) represents the Heaviside function.
6. A square cross-section according to claim 5The method for predicting the shearing performance of the reinforced concrete beam is characterized in that the three-section constitutive model of the bonding and sliding of the reinforced steel and the concrete is a ribbed square steel tau0The value is 0.34-0.41, tauuThe value is 9.33-11.41, taurThe value is 3.36-4.10, SuThe value is 1.73-2.11, SrThe value is 2.81 to 3.43,
Figure RE-FDA0003306088900000076
the value is 0.50-0.62; for flat section steels,. tau0The value is 0.25-0.30, tauuThe value is 8.85-10.82, taurThe value is 1.84-2.25, SuThe value is 3.29-4.02SrThe value is 5.69 to 6.95,
Figure RE-FDA0003306088900000081
the value is 1.03-1.25; for helical steels,. tau0The value is 0.29-0.36, tauuThe value is 8.43-10.31, taurThe value of S is 1.40-1.71uThe value is 4.14-5.07SrThe value is 5.30 to 6.47,
Figure RE-FDA0003306088900000082
the value is 0.23-0.29.
7. The method for predicting the shear strength of a square section reinforced concrete beam according to claim 1, wherein the width ω of the crack in S7 is larger than the width ω of the crack in S7c(z) the solving process specifically comprises the following steps:
the crack width expression of the beam under the concentrated force is as follows:
Figure RE-FDA0003306088900000083
wherein psi is the correction coefficient of the steel bar body shape, h*The height of the crack at the desired width;
Figure RE-FDA0003306088900000084
is a polynomial regression equation:
Figure RE-FDA0003306088900000085
in the insulator IV, h*Taken as the height z of the section where the tensioned steel bar is located.
8. The method for predicting the shear strength of a square-section reinforced concrete beam as recited in claim 7, wherein the reinforcement shape correction coefficient is determined by the secant stiffness of the ultimate bond value and the initial bond value of the bond slip ascending section, the square steel is 1, the flat steel is 2.5, and the spiral rib is 3.4.
9. The method for predicting the shear performance of a square-section reinforced concrete beam according to claim 1, wherein the virtual crack segment distance d in the step S8 is the virtual crack segment distance dvirtPlastic zone length d of the flexible zone in front of the concrete crack tipplThe specific process comprises the following steps:
s81, assuming the coordinate of the virtual crack starting point as (x)0,y0) And satisfies the crack coordinate equation (3) and brings it into equation (23) to make the crack width equal to 0.1, find x0And y0；
Since the virtual crack segment is short relative to the full length of the crack, its length is expressed as:
Figure RE-FDA0003306088900000086
in the formula: | a | non-woven phosphor2Represents the 2 norm of vector a;
s82, the plastic zone length of the flexible belt at the front part of the concrete crack tip is as follows:
Figure RE-FDA0003306088900000091
in the formula: σ is the distal stress; lcrackIs the crack length;
the distal stress σ is reduced using the following equation:
Figure RE-FDA0003306088900000092
crack length lcrackThe calculation expression of (a) is:
Figure RE-FDA0003306088900000093
Figure RE-FDA0003306088900000094
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