CN113642087B - Method for predicting shearing performance of square-section reinforced concrete beam - Google Patents
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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 an improved internal force arch truss model; s2, solving a crack angle; s3, arch inclination angle; s4, solving the tension of the stirrup; s5, solving the pressure of the concrete pressing rod; s6, solving the shearing force of the inner force arch; s7, solving the tension of the lower tensioned steel bar; s8, solving the upper concrete pressure and the microcrack cohesive force; and S9, comprehensively solving through all the sub items to obtain the shearing performance of the reinforced concrete beam with the square cross 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
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 the shear performance of a square-section reinforced concrete beam, which considers the bonding slippage between special concrete and steel bars in a square-section reinforced concrete structure and the 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 the structure safety evaluation and the 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, a beam shear balance formula:
in the formula: v sum Is the total pressure; t is a pulling force; z is a radical of r The 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:
in the formula: c 1 Concrete pressure provided for the upper concrete of the fracture toughness zone; c 2 Providing pressure for concrete between cracks, namely the compression web members in the truss model; t is a unit of 1 Tension provided for the stirrup; t is 2 Tension provided to the lower tensioned reinforcement; t is 3 Tensile forces generated by microcracks near the tip of the crack;
is the crack angle; theta is an arch inclination angle; v t The shear force is to be calculated;
s2, calculating a crack angle
And fitting a critical fracture expression determined by the normalized fracture initiation position on the basis of the critical fracture:
in the formula: alpha (alpha) ("alpha") 0 Relative 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:
in the formula: alpha is alpha 1 Relative abscissa of critical crack tip; h is the beam height;
s3, calculating the 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:
in the formula: x is the number of n Is the equivalent width of the arch;
obtaining an expression of the camber angle theta through a geometrical relation:
s4, calculating the tension T of the stirrup 1
Assuming that the stress of each stirrup is equal, the number N of equivalent stirrups cut by the crack in the separator I sv Comprises the following steps:
get the tension T of the stirrup 1 The expression of (a) is:
T 1 ＝N sv T′ (8)
in the formula: t' is the stress of a single stirrup;
s5, calculating the tension C of the compression bar 2
Taking the spacer III, the length of AM is
The expression of the compression bar tension is:
s6, calculating the shearing force V of the inner force arch arch
According to the hypothesis, 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:
the relational expression of the concrete compressive stress in the internal force arch and the concrete compressive stress in the truss is as follows:
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:
from formulas (11) and (12):
in the formula:
from the geometric relationship how to obtain the arch shear force V arch The expression of (c) is:
V arch ＝(1-β 0 )f c bx n sinθ (14)
in the formula: a. The s Is the section area of the steel bar; n is a radical of hydrogen st The number of longitudinal ribs at the bottom in the cross section of the beam is;
s7, calculating the tension T of the lower tensioned steel bar 2
Taking stress sigma of insulator IV and steel bar elongation arch model s ' stress plus truss model σ s Providing steel in which the stresses and the bonding forces to the steel bars are balanced in accordance with the balance of forcesThe expression for the amount of rib elongation is:
in the formula: tau (S) is a relational expression of the bonding slippage behavior of the steel bar and the concrete;
according to the geometrical compatibility condition, the slippage is equal to the elongation of the reinforcing steel bars plus the compression of the concrete, and for the sake of simplicity, the compression of the concrete is assumed to be approximately equal to
Multiplying the elongation of the steel bar to obtain an equation:
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:
in the formula: omega c (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 bar 2 The expression of (a) is:
s8, solving the upper concrete pressure C 1 And microcrack cohesion T 3
The softening constitutive model of the tensile concrete is calculated by adopting a two-section model:
in the formula: f. of t Is the tensile strength of concrete, omega * For the virtual crack width, calculated from equation (23), ω 0 The width of the crack at the starting position of the virtual crack is taken as 0.1mm;
computational expression of cohesion near the fracture tip:
in the formula: d virt Is the virtual crack segment distance; d pl The length of a plastic zone of a flexible zone at the front part of the tip of the concrete crack; wherein d is s And s is the differential of the distance from the crack tip to the plastic or virtual crack zone and the distance,
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
The tensile force provided by the tensile plastic zone;
to account for size effects, T is obtained 3 The calculation expression of (1):
in the formula (I), the compound is shown in the specification,
is the Hillerberg brittleness constant;
is a characteristic length;
is a constant;
assuming pressure C provided by concrete above the fracture-tip ductile band 1 Provided as concrete from the crack tip plastic zone to the top of the beam;
the constitutive model expression of the concrete is as follows:
in the formula: epsilon u Representing the peak stress corresponding strain;
assuming that the pressure distribution shape of the concrete is similar to the constitutive relation sigma of the concrete cm A rising section of (2), then C 1 The expression of (a) is:
in the formula:
represents the upper concrete reference coefficient,
d s represents a distance differential;
s9 shear performance prediction
Alpha content was determined by the steps (1) to (8) osv And V t Two parameters and with respect to alpha 1 C of (A) 1 (α 1 )、 C 2 (α 1 ,α osv )、T 1 (α 1 ,α osv )、T 2 (α 1 ,α osv ,V t )、T 3 (α 1 ,V t )、V arch (α 1 ,α osv )、θ(α 1 ) And
expression, substituting expressions (4), (6), (8), (9), (14), (19), (22) and (24) into expression (2), and eliminating parameter alpha osv To obtain V t With respect to alpha 1 And 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, the form factor of the ribbed square steel in S2 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:
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 the single stirrup is obtained:
in the formula: a. The sv The area of the cross section of the stirrup; n is the number of hooped limbs; f. of yv The yield strength of the stirrup; b is the cross-sectional width; s is the stirrup spacing; alpha is alpha osv The participation coefficient of the stirrup is shown; rho sv The 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:
in the formula: tau is 0 For each set of experimental initial bond strengths, the arithmetic mean, τ u Is the arithmetic mean of the experimental ultimate bond strengths, tau, of the groups r Residual bonding for each set of experimentsArithmetic mean of intensity, S u The arithmetic mean value of the corresponding slip values of the experimental ultimate bond strengths of each group, S r The arithmetic mean of the slip values corresponding to the initial residual bond strengths of each set of experiments,
for the rise function shape coefficients, regression is given by:
rewrite equation (27) to unity using the Heaviside function:
(H(S-S u )-H(S-S r ))+τ r H(S-S r )
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 steel 0 The value is 0.34-0.41, tau u The value is 9.33 to 11.41, tau r The value is 3.36-4.10 u The value is 1.73 to 2.11 r The value is 2.81-3.43,
the value is 0.50-0.62; for flat section steels,. Tau 0 The value is 0.25-0.30, tau u The value is 8.85 to 10.82, tau r The value is 1.84-2.25 u The value is 3.29 to 4.02S r The value is 5.69 to 6.95,
value of 1.03-1.25; for helical steels,. Tau 0 The value is 0.29 to 0.36, tau u The value is 8.43 to 10.31, tau r The value of S is 1.40-1.71 u The value of S is 4.14-5.07 r The value is 5.30 to 6.47,
the value is 0.23-0.29.
Further, the width omega of the crack in the S7 c (z) the solving process specifically comprises the following steps:
the crack width expression of the beam under the concentrated force is as follows:
wherein psi is the correction coefficient of the steel bar body shape, h * The height of the crack at the desired width;
is a polynomial regression equation:
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 section distance d in S8 virt Plastic zone length d of the flexible zone in front of the concrete crack tip pl The specific process comprises the following steps:
s81, assuming the coordinate of the virtual crack starting position as (x) 0 ,y 0 ) And satisfies the crack coordinate equation (3) and brings it into equation (23) to make the crack width equal to 0.1, find x 0 And y 0 ；
Since the virtual crack segment is short relative to the full length of the crack, its length is expressed as:
in the formula: | a | non-woven phosphor 2 Represents the 2 norm of vector a;
s82, the length of the plastic zone of the flexible belt at the front part of the concrete crack tip is as follows:
in the formula: σ is the distal stress; l crack Is the crack length;
the distal stress σ is reduced using the following equation:
crack length l crack Is calculated by the expression is as follows:
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 the insulator I truss 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 diagram of deformation coordination analysis of insulator 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, a beam shear balance formula:
in the formula: v sum Is the total pressure; t is aForce; z is a radical of r The 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:
in the formula: c 1 Concrete pressure provided for the upper concrete (zone 3, zone 4 concrete) of the fracture toughness zone; c 2 The pressure provided for the concrete between the cracks, i.e. the compression web members in the truss model (zone 1, zone 2 concrete); t is 1 Tension provided for the stirrup; t is 2 Tension provided for the lower tension bars; t is a unit of 3 Tensile forces generated by microcracks near the tip of the crack;
is the crack angle; theta is the camber angle; v t To request the shearing force.
S2, calculating the crack angle
Based on the critical fracture, the normalized fracture initiation position (alpha) is fitted 0 ) The determined critical crack expression:
in the formula: alpha is alpha 0 Relative 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:
in the formula: alpha is alpha 1 Relative abscissa of critical crack tip; h is the beam height.
S3, calculating the 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:
in the formula: x is the number of n Is the equivalent width of the arch.
The expression for camber angle θ can be obtained from the geometric relationship:
s4, calculating the tension T of the stirrup 1
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:
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:
in the formula: a. The sv The area of the cross section of the stirrup; n is the number of hooped limbs; f. of yv The yield strength of the stirrup; b is the cross-sectional width; s is the stirrup spacing; alpha is alpha osv The participation coefficient of the stirrup is shown; rho sv The hoop ratio is used.
If the stress of each stirrup is equal, the stirrup will crack in the insulator INumber N of equivalent stirrups obtained by sewing sv Comprises the following steps:
the tension T of the stirrup obtained by the formulas (8) and (9) 1 The expression of (c) is:
s5, calculating the tension C of the compression bar 2
Taking the spacer III, the length of AM is
The expression of the compression bar tension is:
s6, calculating the shearing force V of the inner force arch arch
According to the hypothesis, 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:
the relational expression of the concrete compressive stress in the internal force arch and the concrete compressive stress in the truss is as follows:
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:
from formulas (13) and (14):
in the formula:
from the geometric relationship how to obtain the arch shear force V arch The expression of (a) is:
V arch ＝(1-β 0 )f c bx n sinθ (16)
in the formula: a. The s Is the section area of the steel bar; n is a radical of st The number of the longitudinal ribs at the bottom in the cross section of the beam.
S7, calculating the tension T of the lower tensioned steel bar 2
The relation expression of the bonding and slipping behavior of the steel bars and the concrete adopts a three-section constitutive model:
in the formula: tau is 0 For each set of experimental initial bond strengths, the arithmetic mean, τ u Is the arithmetic mean of the experimental ultimate bond strengths, tau, of the groups r Is the arithmetic mean of the residual bond strengths, S, of each set of experiments u The arithmetic mean value of the corresponding slip values of the experimental ultimate bond strengths of each group, S r The arithmetic mean of the slip values corresponding to the initial residual bond strengths of each set of experiments,
for shape coefficient of ascending segment functionRegression can be obtained from the following formula:
rewrite equation (18) to unity using the Heaviside function:
(H(S-S u )-H(S-S r ))+τ r H(S-S r )
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' s Stress sigma plus truss model s Providing that, in terms of the balance of forces, the stresses and the adhesion to the bars are balanced, the elongation of the bars is expressed by:
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
The elongation of the steel bar is multiplied, and then the equation can be obtained:
the crack width expression of the beam under the concentrated force is as follows:
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;
is a polynomial regression equation:
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:
the tensile force T provided by the tension bar can be obtained by the formulas (17) to (25) 2 The expression of (a) is:
s8, solving the upper concrete pressure C 1 And microcrack cohesion T 3
The softening constitutive model of the tensile concrete is calculated by adopting a two-section model:
in the formula: f. of t Is the tensile strength of concrete, omega * For the virtual crack width, ω can be calculated from equation (23) 0 Is a virtualThe crack width at the initiation of the crack may be 0.1mm.
Assume a virtual crack initiation coordinate of (x) 0 ,y 0 ) And satisfies the crack coordinate equation (3) and brings it into equation (23) to make the crack width equal to 0.1, find x 0 And y 0 。
Since the virtual crack segment is short relative to the full length of the crack, its length can be approximated as:
in the formula: | a | non-woven phosphor 2 Representing 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:
in the formula: σ is the distal stress; l crack Is the crack length.
The distal stress σ can be reduced using the following equation:
crack length l crack The calculation expression of (a) is:
the calculation expression of the cohesive force in the vicinity of the crack tip can be obtained from the expressions (27) to (31):
in the formula: wherein d is s And s is the differential of the distance from the crack tip to the plastic or virtual crack zone and the distance,
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
Is stretched by the tension provided by the plastic zone.
To take into account the size effect, T is obtained 3 The calculation expression of (1):
in the formula (I), the compound is shown in the specification,
is the Hillerberg brittleness constant;
is the characteristic length;
is a constant.
Assuming the pressure C provided by the upper concrete (3-zone, 4-zone concrete) of the fracture tip ductile zone 1 Can 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:
in the formula: epsilon u Representing peak stressCorresponding to the strain.
Assuming that the pressure distribution shape of the concrete is similar to the constitutive relation sigma of the concrete cm A rising section of (2), then C 1 The expression of (a) is:
in the formula:
represents the upper concrete reference coefficient,
d s representing the distance differential.
Shear performance prediction
Thus, the content of α has been determined through the steps (1) to (8) osv And V t Two parameters and with respect to alpha 1 C of (A) 1 (α 1 )、 C 2 (α 1 ,α osv )、T 1 (α 1 ,α osv )、T 2 (α 1 ,α osv ,V t )、T 3 (α 1 ,V t )、V arch (α 1 ,α osv )、θ(α 1 ) And
expression substituting expressions (4), (6), (10), (11), (26), (33) and (35) into expression (2) can eliminate the parameter α osv To obtain V t With respect to alpha 1 And 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, tau 0 The suggested value is 0.34-0.41, tau u The suggested value is 9.33-11.41, tau r The suggested value is 3.36-4.10, S u The suggested value is 1.73-2.11, S r The suggested value is 2.81-3.43,
the suggested value is 0.50-0.62; for flat section steels,. Tau 0 The suggested value is 0.25-0.30, tau u The suggested value is 8.85-10.82, tau r The suggested value is 1.84-2.25, S u The suggested value is 3.29-4.02 r The suggested value is 5.69-6.95,
the suggested value is 1.03-1.25; for helical steels,. Tau 0 The suggested value is 0.29-0.36, tau u The suggested value is 8.43-10.31, tau r The suggested value is 1.40-1.71, S u The suggested value is 4.14-5.07 r The suggested value is 5.30-6.47,
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
Note: the shear span ratio, the hoop ratio and the reinforcement ratio are all divided by the effective height h in the calculation formula 0 Which can be represented by the formula h 0 And h-c, wherein h is the section height and c is the thickness of the protective layer.
Table 2: comparison of the prediction formula with other normative calculation results
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 stirrup allocation rate; and the shear failure of the beam is supposed to occur in the ACI formula and the GB50010-2010 specification, so 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 theoretically greater than an experimental value, but the influence of the concrete and the bottom longitudinal bar on the shear bearing is excessively estimated due to the conservative value in the specification. The formula provided by the invention considers the bonding slippage between the concrete and the steel bars, the constitutive models of the steel bars and the concrete, and simultaneously considers the influences of the size effect and the critical cracks in the calculation formula, so that the difference between the calculation result and the test value is smaller, and the shearing resistance of the square-section reinforced concrete beam 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, a beam shear balance formula:
in the formula: v sum Is the total pressure; t is a pulling force; z is a radical of r The 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:
in the formula: c 1 Concrete pressure provided for the upper concrete of the fracture toughness zone; c 2 Providing pressure for concrete between cracks, namely the compression web members in the truss model; t is 1 Tension provided to the stirrup; t is a unit of 2 Tension provided to the lower tensioned reinforcement; t is 3 Tensile forces generated by microcracks near the tip of the crack;
is the crack angle; theta is an arch inclination angle; v t The shear force is to be calculated;
s2, calculating the crack angle
And fitting a critical fracture expression determined by the normalized fracture initiation position on the basis of the critical fracture:
in the formula: alpha is alpha 0 Relative 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:
in the formula: alpha (alpha) ("alpha") 1 Relative abscissa of critical crack tip; h is the beam height;
s3, calculating the inclination angle theta of the arch
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:
in the formula: x is the number of n Is the equivalent width of the arch;
the expression of the camber angle theta is obtained through the geometrical relation:
s4, calculating the tension T of the stirrup 1
Assuming that the stress of each stirrup is equal, the number N of equivalent stirrups cut by the crack in the separator I sv Comprises the following steps:
get the tension T of the stirrup 1 The expression of (a) is:
T 1 ＝N sv T′ (8)
in the formula: t' is the stress of a single stirrup;
s5, calculating the tension C of the compression bar 2
Taking the spacer III, the length of AM is
The expression of the compression bar tension is:
s6, calculating the shearing force V of the inner force arch arch
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:
the relational expression of the concrete compressive stress in the internal force arch and the concrete compressive stress in the truss is as follows:
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:
from formulas (11) and (12):
in the formula:
from the geometric relationship how to obtain the arch shear force V arch The expression of (a) is:
V arch ＝(1-β 0 )f c bx n sinθ (14)
in the formula: a. The s Is the section area of the steel bar; n is a radical of hydrogen st The number of longitudinal ribs at the bottom in the cross section of the beam is;
s7, calculating the tension T of the lower tensioned steel bar 2
Taking the isolated body IV and the stress sigma of the arch model of the elongation of the steel bar s ' stress plus truss model σ s Providing that, in terms of the balance of forces, the stresses and the adhesion to the bars are balanced, the elongation of the bars is expressed by:
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
Multiplying the elongation of the steel bar to obtain an equation:
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:
in the formula: omega c (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 bar 2 The expression of (c) is:
s8, solving the upper concrete pressure C 1 And microcrack cohesion T 3
The softening constitutive model of the tensile concrete is calculated by adopting a two-section model:
in the formula: f. of t Is the tensile strength of concrete, omega * Calculated for the virtual crack width from equation (23) (. Omega.) 0 The width of the crack at the starting position of the virtual crack is taken as 0.1mm;
computational expression of cohesion near the fracture tip:
in the formula: d virt Is the virtual crack segment distance; d pl The length of a plastic zone of a flexible zone in front of the tip of the concrete crack is determined; wherein d is s And s is the differential of the distance from the crack tip to the plastic or virtual crack zone and the distance,
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
The tensile force provided by the tensile plastic zone;
to account for size effects, T is obtained 3 The calculation expression of (1):
in the formula (I), the compound is shown in the specification,
is the Hillerberg brittleness constant;
is the characteristic length;
assuming pressure C provided by concrete above the fracture-tip ductile band 1 Provided as concrete from the crack tip plastic zone to the top of the beam;
the constitutive model expression of the concrete is as follows:
in the formula: epsilon u Representing the peak stress corresponding strain;
assuming that the pressure distribution shape of the concrete is similar to the constitutive relation sigma of the concrete cm A rising section of (2), then C 1 The expression of (a) is:
in the formula:
represents the upper concrete reference coefficient,
d s represents a distance differential;
s9 shear performance prediction
Alpha content was determined by the steps (1) to (8) osv And V t Two parameters and with respect to alpha 1 C of (A) 1 (α 1 )、C 2 (α 1 ,α osv )、T 1 (α 1 ,α osv )、T 2 (α 1 ,α osv ,V t )、T 3 (α 1 ,V t )、V arch (α 1 ,α osv )、θ(α 1 ) And
expression, substituting expressions (4), (6), (8), (9), (14), (19), (22) and (24) into expression (2), and eliminating parameter alpha osv To obtain V t With respect to alpha 1 And finally obtaining the shear force calculation result of the beam by using the expression of (1).
2. The method for predicting the shear stress of the square-section reinforced concrete beam according to claim 1, wherein the square-section reinforced concrete beam adopts a reinforced steel 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 ribbed square steel in S2 is 1.131, the form factor of flat-section steel is 0.899, and the form factor of spiral bars is 0.738.
4. The method for predicting the shear performance of the square-section reinforced concrete beam according to 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:
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:
in the formula: a. The sv The area of the cross section of the stirrup; n is the number of hooped limbs; f. of yv The yield strength of the stirrup; b is the cross-sectional width; s is the stirrup spacing; alpha is alpha osv The participation coefficient of the stirrup is shown; rho sv The hoop ratio is used.
5. The method for predicting the shear performance of the square-section reinforced concrete beam according to claim 2, wherein in the step S7, a three-section constitutive model is adopted for the relational expression of the bonding slip behavior of the steel bar and the concrete:
in the formula: tau is 0 For each set of experimental initial bond strengths, the arithmetic mean, τ u Is the arithmetic mean of the experimental ultimate bond strengths, τ, of each group r Is the arithmetic mean of the residual bond strengths, S, of each set of experiments u The arithmetic mean value of the corresponding slip values of the experimental ultimate bond strengths of each group, S r The arithmetic mean of the slip values corresponding to the initial residual bond strengths of each set of experiments,
for the rise function shape coefficients, regression is given by:
rewrite equation (27) to unity using the Heaviside function:
(H(S-S u )-H(S-S r ))+τ r H(S-S r )
in the formula: h (x) represents the Heaviside function.
6. The method for predicting the shear performance of the square-section reinforced concrete beam according to claim 5, wherein the three-section constitutive model of the bonding and sliding of the steel bars and the concrete is a ribbed square steel τ 0 The value is 0.34-0.41, tau u The value is 9.33 to 11.41, tau r The value is 3.36-4.10 u The value is 1.73 to 2.11 r The value is 2.81-3.43,
the value is 0.50-0.62; for flat section steels,. Tau 0 The value is 0.25-0.30, tau u The value is 8.85-10.82, tau is r The value is 1.84-2.25 u The value of S is 3.29 to 4.02 r The value is 5.69-6.95,
the value is 1.03-1.25; for spiral steels,. Tau. 0 The value is 0.29-0.36, tau u The value is 8.43 to 10.31, tau r The value of S is 1.40-1.71 u The value of S is 4.14-5.07 r The value is 5.30 to 6.47,
the value is 0.23-0.29.
7. The method for predicting the shearing performance of the square-section reinforced concrete beam according to claim 1, wherein the width ω of the crack in the S7 is c (z) the solving process specifically comprises the following steps:
the crack width expression of the beam under the concentrated force is as follows:
wherein psi is the correction coefficient of the steel bar body shape, h * The height of the crack at the desired width;
is a polynomial regression equation:
in 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 in the step S8 isd virt Plastic zone length d of the flexible zone in front of the concrete crack tip pl The specific process comprises the following steps:
s81, assuming the coordinate of the virtual crack starting position as (x) 0 ,y 0 ) And satisfies the crack coordinate equation (3) and brings it into equation (23) to make the crack width equal to 0.1, find x 0 And y 0 ；
Since the virtual crack segment is short relative to the full length of the crack, its length is expressed as:
in the formula: | a | non-woven phosphor 2 Represents the 2 norm of vector a;
s82, the length of the plastic zone of the flexible belt at the front part of the concrete crack tip is as follows:
in the formula: σ is the distal stress; l crack Is the length of the crack;
the distal stress σ is converted using the following equation:
crack length l crack The calculation expression of (a) is:
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| CN111220467A (en) * | 2020-03-11 | 2020-06-02 | 中铁科学研究院有限公司 | Method for measuring normal section bearing capacity and reinforcing bar of steel bar-steel fiber concrete shield segment |
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