CN112613103B - Method for calculating transfer length of pretensioned member under influence of concrete spalling - Google Patents
Method for calculating transfer length of pretensioned member under influence of concrete spalling Download PDFInfo
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Abstract
The invention discloses a method for calculating the transfer length of a pretensioned member under the influence of concrete spalling, which comprises the following steps: s1, considering the common influence of Hall effect and corrosion, and predicting the interfacial expansive force of the steel strand and the concrete under the influence of the concrete expansion crack; s2, calculating the bonding strength of the steel strand under the influence of the concrete spalling according to the expansion force and the friction coefficient of the steel strand-concrete interface; s3, dispersing the pretensioned concrete member into a plurality of micro units, establishing stress and strain expressions of the steel strands in each micro unit and the corresponding surrounding concrete, and calculating the transfer length of the pretensioned concrete member under the influence of the concrete expansion crack based on the interface strain coordination relationship. The method can comprehensively consider the common influence of the Hall effect and the corrosion, can accurately calculate the transmission length of the pretensioned member under the influence of the concrete expansion crack, and can be widely applied to engineering practice.
Description
Technical Field
The invention relates to a method for calculating the transfer length of a pretensioning prestressed concrete member, in particular to a method for calculating the transfer length of a pretensioning member under the influence of concrete spalling.
Background
The pre-tensioned prestressed concrete structure occupies a large proportion in small and medium span bridges due to the characteristics of simple construction, industrialized batch prefabrication and the like. After the pre-tensioned prestressed tendon is released, a retraction effect can be generated in a stress transfer area at the end part of the member, the diameter of the pretensioned prestressed tendon expands to form a wedge shape, the phenomenon is called Hall effect, the Hall effect enables the prestress of the steel strand to be gradually increased from zero to effective prestress along the transfer area, and the length of the area is the transfer length. The pre-tensioned prestressed concrete structure transfers the prestress of the steel strand to the concrete through the transfer length, and the effective bonding of the steel strand and the concrete interface is a key factor for ensuring the prestress transfer.
Under the corrosion of the external adverse environment, the corrosion can cause the concrete to swell and the interface bonding property to deteriorate, thereby causing the change of the transmission length. The change of the transmission length can cause the loss of prestress, cause the reduction of the bearing capacity of the structure, and even cause the structural collapse in serious cases. Some researchers have developed researches on the transfer length of the pre-tensioned prestressed concrete structure, and analyzed the influence of the diameter of the steel strand, the effective prestress, the concrete strength and the concrete protection layer on the transfer length, and in addition, some specifications also give a prediction formula of the transfer length. However, the research on the transmission length of the pretensioned concrete element under the influence of concrete spalling is not reported, and the reasonable evaluation of the transmission length prediction of the pretensioned prestressed concrete element under the influence of concrete spalling is a difficult point of the current research due to the problems of concrete spalling, bond strength degradation and the like.
Disclosure of Invention
The present invention is directed to solving at least one of the problems of the prior art. Therefore, the invention provides a method for calculating the transfer length of a pretensioned member under the influence of concrete spalling, which can comprehensively consider the common influence of Hall effect and corrosion, can calculate the transfer length of the pretensioned member under the influence of concrete spalling more accurately, and can be widely applied to engineering practice.
According to the embodiment of the invention, the method for calculating the transfer length of the pretensioned member under the influence of the concrete expansion crack comprises the following steps:
s1, predicting the interfacial expansive force of the rusty steel strand and concrete: on the basis of a thick-wall cylinder theory, according to a displacement coordination condition, the joint influence of Hall effect and corrosion is comprehensively considered, and the expansion force of the steel strand and the concrete interface under the influence of concrete expansion crack is predicted;
s2, analyzing the bonding strength of the corrosion steel strand: calculating the bonding strength of the steel strand under the influence of the concrete spalling according to the interfacial expansion force of the steel strand and the concrete and the friction coefficient between the steel strand and the concrete;
s3, calculating the transmission length of the corrosion pretensioned member: dispersing the pretensioned concrete member into a plurality of micro units, establishing stress and strain expressions of steel strands in each micro unit and corresponding surrounding concrete, and calculating the transfer length of the pretensioned concrete member under the influence of concrete expansion crack based on the interface strain coordination relationship.
According to the method for calculating the transmission length of the pretensioned member under the influence of the concrete spalling, at least the following technical effects are achieved:
according to the method for calculating the transmission length of the pretensioned member under the influence of the concrete spalling, the influence of Hall effect and corrosion can be comprehensively considered, the interfacial expansion force of the steel strand and the concrete can be predicted, the bonding strength of the corrosion steel strand can be calculated, the pretensioned member is dispersed into a plurality of micro units, the stress and strain expressions of the steel strand in each micro unit and the corresponding surrounding concrete are established, and the transmission length of the pretensioned member under the influence of the concrete spalling is calculated based on the interfacial strain coordination relationship. According to the embodiment of the invention, the common influence of the Hall effect and the corrosion can be comprehensively considered, and the transmission length of the pretensioned member under the influence of the concrete expansion crack can be accurately calculated, so that the pretensioning member can be widely applied to engineering practice.
According to some embodiments of the present invention, in step S1, when the concrete is not cracked, the hall effect and the rusting together affect the expansion force P of the steel strand and the concrete interface, which can be expressed as the following formula:
in the formula, R 0 Is the initial radius, R, of the steel strand t Is the radius, R, of the steel strand after the prestress is released c Is the distance from the center of the steel strand to the concrete surface, f pz Is the axial stress of the steel strand, E p Is a steel strandModulus of elasticity, v p Is the Poisson's ratio of the steel strand, E c Is the modulus of elasticity, v, of concrete c Is the Poisson's ratio of concrete, f' cz Is the axial stress, f 'of the concrete surrounding the steel strand after rusting' cz ＝(1-ρ)f cz Rho is the corrosion rate of the steel strand, f cz The axial stress of the concrete around the steel strand when the steel strand is not rusted;
concrete radial displacement u under combined influence of Hall effect and corrosion c Can be expressed as the formula:
according to some embodiments of the present invention, in step S1, when the concrete part is cracked, the hall effect and the rust together affect the expansion force P of the steel strand and the concrete interface, which can be expressed as the following formula:
in the formula, p c Constraint stress, R, for the location of the interface of spalled and unflatten concrete u Radius of spalled concrete, R is any position of the spalled concrete region, R t ≤r≤R u ，σ θ (r) is the hoop stress of the spalled concrete.
According to some embodiments of the present invention, in step S1, when the concrete is completely cracked, the hall effect and the rusting together affect the expansion force P of the steel strand and the concrete interface, which can be expressed as the following formula:
according to some embodiments of the invention, the radius R of the steel strand after prestressing relaxation t Can be expressed as the formula:
in the formula (f) pt Is the tensile strength of the steel strand.
According to some embodiments of the invention, the constraint stress p at the interface position of the cracked concrete and the non-cracked concrete is determined according to the stress distribution coordination principle c Can be expressed as the formula:
in the formula (f) t The tensile strength of concrete.
According to some embodiments of the present invention, based on the thick-walled cylinder theory, the radial displacement u (r) of the unexpanded concrete when the concrete is partially cracked can be expressed as the formula:
according to the condition of radial displacement compatibility of the steel strand and the concrete interface, u c ＝u(R t ) The compatibility equation can be expressed as the formula:
from which R can be solved u 。
According to some embodiments of the invention, the hoop stress σ of the concrete takes into account the softening behavior of the tensile strength of the spalled concrete θ (r) can be expressed as the formula:
in the formula, epsilon θ (r) is the tangential strain of the concrete, ε cr Is the strain, epsilon, corresponding to the tensile strength of the concrete cr ＝f t /E c ，ε 1 Is the strain, epsilon, corresponding to a concrete stress of 15% tensile strength 1 ＝0.0003，ε u Is the ultimate strain of concrete, epsilon u ＝0.002。
According to some embodiments of the invention, the tangential strain epsilon of the unexpanded concrete is such that, when the concrete is partially cracked, the unexpanded concrete is subjected to a thick-walled cylinder theory θ (r) can be expressed as the formula:
substituting into boundary condition R when concrete is completely cracked u ＝R c Of tangential strain epsilon θ (r) can be expressed as the formula:
in the formula, epsilon qc Is the concrete surface tangential strain;
its radial displacement u (r) can be expressed as the formula:
according to the condition that the steel strand is compatible with the radial displacement of the concrete interface, i.e. u c ＝u(R t ) The compatibility equation can be expressed as the formula:
from which e can be solved qc 。
According to some embodiments of the invention, in the step S2, the bonding strength of the rusted steel strand can be expressed as the following formula:
τ＝μ·P
wherein μ is the coefficient of friction at the interface of the steel strand and the concrete.
According to some embodiments of the invention, in the step S3, the step of calculating the transfer length is as follows:
firstly, the pretensioned concrete member of 1/2 is scattered into a plurality of microcells, each microcell is numbered from 1 to n, and the local stress increment delta f of the microcell i p，i Can be expressed as the formula:
in the formula (II), d' p Is the diameter, tau, of the rusted steel strand i Is the bonding stress, A 'of the steel strand at the ith microcell position' p Is the cross-sectional area of the rusted steel strand, and Δ l is the length of the microcell;
for pre-tensioned concrete elements, the stress of the steel strands at the end position is 0, i.e. f p，0 0, for microcell i, stress f of its internal steel strand p，i Can be expressed as the formula:
f p，i ＝f p，i-1 +Δf p，i
in the formula,. DELTA.f p，i I is more than or equal to 1 and less than or equal to n and is the local stress increment of the microcell i;
strain increment delta epsilon of steel strand at ith microcell position p，i Can be expressed as the formula:
in the formula, epsilon p，0 Is the initial strain after the steel strand is released, E p Is the modulus of elasticity of the steel strand;
strain epsilon of concrete around the ith microcell c，i Can be expressed as the formula:
wherein A is the cross-sectional area of the concrete, e p Is eccentricity of steel strandDistance, I c Is the moment of inertia, y, of the concrete section b Is the distance from the neutral axis of the pretensioned concrete element to the bottom end of the pretensioned concrete element;
the stress of the steel strand gradually increases from the relaxation end to the midspan direction until the effective prestress is reached, and according to the interface strain coordination relationship, the strain epsilon of the concrete around the micro-unit where the stress of the steel strand reaches the effective prestress c，i Should be equal to the strain increase delta epsilon of the steel strand at the microcell p，i The distance between the position where the steel strand stress reaches the effective prestress and the tension releasing end is the transfer length, and then the transfer length can be expressed as a formula:
l t ＝k·Δl
in the formula, k is the number of the micro unit where the stress of the steel strand reaches the effective prestress, and k is more than or equal to 1 and less than or equal to n.
Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
Drawings
The above and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
FIG. 1 is a schematic diagram of the axial stress distribution of a steel strand under the influence of Hall effect and corrosion in the present invention;
FIG. 2 is a graph showing the distribution of the radial expansion force of concrete under the influence of Hall effect and corrosion in the present invention;
FIG. 3 is a theoretical schematic of a concrete spalling thick-walled cylinder of the present invention;
FIG. 4 is a graph of the softening curve of the tensile concrete of the present invention;
FIG. 5 is a force diagram of a plurality of sequential microcells;
FIG. 6 is a force diagram of a single microcell;
FIG. 7 is a flow chart of the calculation of the transfer length of the present invention;
FIG. 8 is a schematic view of a test beam and a rust apparatus of the present invention;
FIG. 9 is a partial cross-sectional view of a test beam;
FIG. 10 is a schematic view of the arrangement of the separate distance measuring tab of the present invention;
FIG. 11 is another perspective view of FIG. 10;
FIG. 12 is a graph of the strain distribution of concrete according to the present invention;
reference numerals:
the device comprises a test beam 100, a test steel strand 200, a first deformed steel bar 300, a second deformed steel bar 400, a stirrup 500, a direct current power supply 600, a corrosion groove 700, a stainless steel plate 800, a PVC sleeve 900, a distance measuring sheet 1000 and a high-precision vernier caliper 1100.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.
In the description of the present invention, it is to be understood that the terms "center", "longitudinal", "lateral", "length", "width", "thickness", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", "axial", "radial", "circumferential", and the like, indicate orientations and positional relationships based on the orientations and positional relationships shown in the drawings, and are used merely for convenience in describing the present invention and for simplicity in description, and do not indicate or imply that the device or element so referred to must have a particular orientation, be constructed and operated in a particular orientation, and therefore, should not be construed as limiting the present invention. Furthermore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of the present invention, "a plurality" means two or more unless otherwise specified.
In the description of the present invention, it should be noted that unless otherwise specifically stated or limited, the terms "mounted" and "connected" are to be construed broadly, e.g., directly or indirectly through intervening media, as well as both elements in communication. The specific meanings of the above terms in the present invention can be understood in specific cases to those skilled in the art.
A method for calculating the transfer length of a pretensioned element under the influence of concrete spalling according to an embodiment of the present invention will be described with reference to fig. 1 to 12.
According to the embodiment of the invention, the method for calculating the transfer length of the pretensioned member under the influence of the concrete expansion crack comprises the following steps:
s1, predicting the interfacial expansive force of the rusty steel strand and concrete: on the basis of a thick-wall cylinder theory, according to a displacement coordination condition, the joint influence of Hall effect and corrosion is comprehensively considered, and the expansion force of the steel strand and the concrete interface under the influence of concrete expansion crack is predicted;
s2, analyzing the bonding strength of the corrosion steel strand: calculating the bonding strength of the steel strand under the influence of the concrete spalling according to the interfacial expansion force of the steel strand and the concrete and the friction coefficient between the steel strand and the concrete;
s3, calculating the transmission length of the corrosion pretensioned member: dispersing the pretensioned concrete member into a plurality of micro units, establishing stress and strain expressions of steel strands in each micro unit and corresponding surrounding concrete, and calculating the transfer length of the pretensioned concrete member under the influence of concrete expansion crack based on the interface strain coordination relationship.
According to the method for calculating the transmission length of the pretensioned member under the influence of the concrete spalling, the influence of Hall effect and corrosion can be comprehensively considered, the interfacial expansion force of the steel strand and the concrete can be predicted, the bonding strength of the corrosion steel strand can be calculated, the pretensioned member is dispersed into a plurality of micro units, the stress and strain expressions of the steel strand in each micro unit and the corresponding surrounding concrete are established, and the transmission length of the pretensioned member under the influence of the concrete spalling is calculated based on the interfacial strain coordination relationship. According to the embodiment of the invention, the common influence of the Hall effect and the corrosion can be comprehensively considered, the transmission length of the pretensioned member under the influence of the concrete expansion crack can be accurately calculated, and the calculation method can be widely applied to engineering practice.
Specifically, in some embodiments of the present invention, in step S1, the calculation step of the expansion force of the steel strand and the concrete interface under the combined effect of the hall effect and the corrosion is:
after the prestressed steel strand is released, the radial expansion of the prestressed steel strand is distributed in a wedge shape due to the retraction of the steel strand, which is called as the hall effect, as shown in fig. 1 and 2, and the radius R of the steel strand after the release is shown t Can be expressed as equation 1:
in the formula, R 0 Is the initial radius of the steel strand, f pt Is the tensile strength of the steel strand, E p And v p The elastic modulus and the poisson ratio of the steel strands are respectively.
According to the thick-wall cylinder theory, as shown in FIG. 3, when the concrete is not cracked, the Hall effect influences the expansion force P of the steel strand and the concrete interface h Can be expressed as equation 2:
in the formula, R t Is the radius, R, of the steel strand after the prestress is released c Is the distance from the center of the steel strand to the concrete surface, f pz Is the axial stress of the steel strand, f cz Is the axial stress, v, of the concrete surrounding the strand when it is not rusted c Is the Poisson's ratio of concrete, E c Is the modulus of elasticity of the concrete. In the process of transmitting the prestress of the steel strand, the axial stress relation of the concrete around the steel strand can be expressed as formula 3:
in the formula, A p Is the cross-sectional area of the steel strand, A is the cross-sectional area of the concrete, e p Is the eccentricity of the steel strand, I c Is a concrete sectionMoment of inertia of a surface, y b Is the distance from the neutral axis of the pretensioned concrete element to the bottom end of the pretensioned concrete element.
Concrete radial displacement u under the influence of Hall effect c Can be expressed as equation 4:
the rust corrosion can cause the section area of the steel strand to be reduced, and the rusted section area A' p Can be expressed as equation 5:
A′ p ＝A p (1-ρ)
where ρ is a corrosion ratio.
When the steel strand is rusted, the axial stress of the concrete around the interface can be expressed as formula 6:
f′ cz ＝(1-ρ)f cz
the corrosion can lead to the steel strand wires sectional area to reduce, causes the steel strand wires axial stress to change, and then influences the axial stress of concrete, and simultaneous formula 2-6, based on thick wall drum theory, when the concrete is not spalled, steel strand wires and concrete interfacial expansive force P can be expressed as formula 7 under hall effect and the corrosion common influence:
concrete radial displacement u c Can be expressed as equation 8:
in some embodiments of the present invention, the stress of the steel strand gradually develops from zero to an effective pre-stress within the range of the transmission length, the bonding strength of the steel strand is a key factor affecting the transmission length, the magnitude of the bonding strength of the steel strand is mainly determined by the expansion force of the steel strand and the concrete interface, the stress-strain relationship in the concrete spalling state is analyzed according to the deformation coordination relationship between the steel strand and the concrete interface, a concrete stress balance equation at the partial spalling and complete spalling stages of the concrete is established, and an expansion force expression of the steel strand and the concrete interface in any spalling state under the joint influence of the hall effect and corrosion in step S1 can be obtained.
When the concrete is partially cracked, the concrete consists of a cracked inner ring and a non-cracked outer ring, as shown in fig. 3. Based on thick-walled cylinder theory, the radial displacement u (r) of the unexpanded concrete can be expressed as formula 9:
tangential strain epsilon of non-spalled concrete θ (r) can be expressed as equation 10:
in equations 9 and 10, f t Is the tensile strength, R, of concrete u The distance from the center of the steel strand to the tip of the concrete crack, namely the radius of the spalling concrete, R is any position of a spalling concrete area, R t ≤r≤R u 。
According to the compatibility condition of the steel strand-concrete interface radial displacement, i.e. u c ＝u(R t ) Combining equations 8 and 9 solves for R u The compatibility equation can be expressed as equation 11:
taking into account the softening behavior of the tensile strength of the spalled concrete, a softening curve is introduced, as shown in FIG. 4, the hoop stress σ of the concrete θ (r) can be expressed as equation 12:
in the formula, epsilon θ (r) is the tangential strain of the concrete,. epsilon cr Is the strain corresponding to the tensile strength of the concrete, i.e. the cracking critical strain, epsilon, of the concrete cr ＝f t /E c ，ε 1 Is the strain, epsilon, corresponding to a concrete stress of 15% tensile strength 1 ＝0.0003，ε u Is the ultimate strain of concrete, epsilon u ＝0.002。
According to the principle of stress distribution coordination, the stress at the interface position of the cracked concrete and the un-cracked concrete is equal to the tensile strength of the concrete, namely sigma θ (R u )＝f t The confining stress p of the concrete at the interface c Can be expressed as equation 13:
according to the force balance condition, the expansion force P of the steel strand and the concrete interface is mainly determined by the constraint stress P of the concrete at the tip of the crack when the concrete is partially cracked c And the hoop stress (i.e. residual tensile stress) σ of the spalled concrete θ (r) is determined jointly and can be expressed as equation 14:
after the concrete is completely cracked, substituting into the boundary condition R u ＝R c The radial displacement u (r) can be expressed as equation 15:
its tangential strain epsilon θ (r) can be expressed as formula 16:
in equations 15 and 16,. epsilon qc Is the concrete surface tangential strain.
According to the condition that the steel strand is compatible with the radial displacement of the concrete interface, i.e. u c ＝u(R t ) Combining equations 8 and 15 solves for ε qc The compatibility equation can be expressed as equation 17:
at this time, the expansion force P of the steel strand and concrete interface is mainly determined by the hoop stress (i.e., residual tensile stress) σ of the spalled concrete according to the balance condition of the forces θ (r) to resist, which can be expressed as equation 18:
it should be noted that the expression of the expansion force P when the concrete is not cracked, formula 7, can be used to derive the expressions for r in formulas 14 and 18. In particular, the axial stress f of the steel strand at the end of the releasing end of the pretensioned concrete element pz When the concrete is not cracked, the expansion force P can be calculated, and the concrete radial displacement u can be calculated c And then R can be solved according to the condition of the radial displacement compatibility of the steel strand-concrete interface u And epsilon qc And thus the expressions for r in equations 14 and 18 can be finally derived.
In some embodiments of the present invention, in step S2, the calculation method of the bonding strength of the rusted steel strand under the influence of the concrete cracking includes:
the bonding strength of the rusted steel strand is mainly determined by the interfacial expansive force between the steel strand and the concrete and the friction coefficient between the steel strand and the concrete, and can be expressed as formula 19:
τ＝μ·P
where μ is the coefficient of friction at the interface between the steel strand and the concrete, μ is preferably 0.34.
It will be appreciated that the expansive force P in equation 19 is required to depend on whether the concrete is spalled or notAnd the degree of spalling, specifically, when the concrete is not spalled, the expansive force P in the formula 19 is calculated through a formula 7; when the concrete is partially cracked, the expansion force P in the formula 19 is calculated through a formula 14; when the concrete is completely cracked, the expansive force P in formula 19 is calculated by formula 18. As for how to judge the spalling condition of the concrete, specifically, the expression of the concrete tangential strain at the steel strand-concrete interface can be
Judging epsilon q (R t ) Whether or not greater than critical strain epsilon of concrete cracking cr If epsilon q (R t ) Less than epsilon cr Indicates that the concrete has not cracked if ε q (R t ) Greater than epsilon cr And then the concrete is cracked. Continuing to discuss the concrete cracking degree and judge R u Whether or not greater than R c If it is less than R c If it is greater than R, the concrete is still in partial cracking stage c Indicating that the concrete has fully swelled.
In some embodiments of the present invention, as shown in fig. 5 and 6, in step S3, the calculation method of the rust pretensioned member transmission length under the influence of concrete spalling is:
the pretensioned concrete element of 1/2 may be discretized into microcells numbered 1 to n, the local stress increment Δ f of microcell i p，i Can be expressed as equation 20:
of formula (II) to' p Is the diameter of the rusted steel strand, τ i Is the bonding stress at the ith microcell location, and Δ l is the microcell length.
For rusted pretensioned concrete elements, the stress of the steel strands at the location of the ends of the element is 0, i.e. f p，0 For microcell i, stress f of steel strand p，i Can be expressed as equation 21:
f p，i ＝f p，i-1 +Δf p，i
in the formula,. DELTA.f p，i I is more than or equal to 1 and less than or equal to n and is the local stress increment of the microcell i;
strain increment delta epsilon of steel strand at ith microcell position p，i Can be expressed as equation 22:
in the formula, epsilon p0 Is the initial strain of the steel strand after being released;
strain epsilon of concrete around steel strand at ith microcell position c，i Can be expressed as equation 23:
when the internal stress of the steel strand gradually increases from the relaxation end to the midspan direction to reach the effective prestress, in order to meet the interface strain coordination relationship, the stress of the steel strand reaches the strain epsilon of the concrete around the micro-unit where the effective prestress is located c，i Should be equal to the strain increment delta epsilon of the steel strand at the microcell p，i The relationship between the steel strand and the concrete strain can be expressed as formula 24:
Δε p，i ＝ε c，i
when the internal stress of the steel strand reaches the effective prestress, the distance between the position of the effective prestress and the releasing end is the transfer length, and at the moment, the transfer length l t I.e., can be expressed as equation 25:
l t ＝k·Δl
in the formula, k is the number of the micro unit where the stress of the steel strand reaches the effective prestress, and k is more than or equal to 1 and less than or equal to n.
In summary, the invention provides a method for predicting the transfer length of a pretensioned member under the influence of concrete spalling, and the influence under the combined action of the Hall effect and the concrete spalling can be comprehensively considered. Referring to the flow chart of FIG. 7, a pretensioned structure under the influence of concrete spallingThe main calculation flow of the piece transfer length is as follows: firstly, calculating the radius R of the steel strand after the steel strand is released t (ii) a Secondly, enabling the axial stress f of the steel strand at the end part of the first micro-unit p，0 Is equal to 0, i.e. f cz 0; thirdly, calculating the axial stress f 'of the concrete around the steel strand' cz (ii) a Fourthly, calculating the interfacial expansion force P of the steel strand and the concrete when the concrete is not cracked; fifthly, calculating the concrete radial displacement u c (ii) a A sixth step of passing the formula
Calculating the tangential strain epsilon of the concrete at the interface of the steel strand and the concrete q (R t ) (ii) a The seventh step, judge q (R t ) Whether greater than the critical cracking strain epsilon of concrete cr If it is greater than ε cr To indicate that the concrete has cracked, the degree of cracking will be discussed below cr The concrete is not cracked, and the bonding strength can be directly calculated through the expansion force P when the concrete is not cracked; eighthly, calculating the concrete cracking radius R according to the steel strand-concrete interface radial displacement compatible condition u And according to the cracking radius R u Confirming the expansion crack degree of the concrete; the ninth step, when R u Less than R c According to the cracking radius R u Calculating the concrete tangential strain epsilon at any position θ (r) and determining the concrete hoop stress sigma at different positions according to the softening curve θ (r) expression by solving
Obtaining the expansive force P when the concrete is partially cracked; the tenth step, when R is u Greater than or equal to R c When the crack is shown to have developed to the concrete surface (the concrete is completely cracked), let R u ＝R c And calculating the surface strain epsilon of the concrete qc (ii) a The tenth step is based on the surface strain epsilon of concrete qc Calculating the tangential strain epsilon at any position θ (r) and determining the concrete hoop stress sigma at different positions according to the softening curve θ (r) expression, solving for
The expansive force P when the concrete is completely cracked can be obtained; step ten, substituting the expansion force P calculated under the corresponding condition into a formula tau mu.P to obtain the bonding strength of the steel strand and the concrete interface according to the expansion crack condition of the concrete; the tenth step, calculating the ith micro-unit stress increment delta f p，i And stress f p，i And respectively calculating respective strain delta epsilon of the steel strand and the concrete at the corresponding positions p，i And ε c，i (ii) a The fourteenth step, determine Δ ε p，i Whether or not equal to epsilon c，i If the difference is not equal, the steel strand-concrete interface in the micro-unit slides relatively, is positioned in a stress transfer area, does not reach the transfer length end point, and needs to continue to carry out cyclic iteration; fifteenth step, overlapping the lengths of the micro units to obtain a predicted value l of the transmission length t ＝k·Δl。
To evaluate the transmission length of the rusted pre-tensioned prestressed concrete member, as shown in fig. 8 and 9, three test beams 100 (pre-tensioned prestressed concrete members) for testing were designed and manufactured, each of the test beams 100 having a cross-sectional size of 200mm × 350mm and a total length of 3800 mm. The test beam 100 is internally provided with a test steel strand 200 with the diameter of 15.2mm, and the test steel strand 200 is externally coated with concrete (a concrete protective layer), wherein the diameter of the concrete is 67.4 mm. The test strand 200 had a tensile stress of 1395MPa, which is 75% of its standard tensile strength. Two first deformed steel bars 300 with the diameter of 16mm are arranged on two sides of the inner bottom end of the test beam 100 and used as tension steel bars, and two second deformed steel bars 400 with the diameter of 10mm are arranged on two sides of the inner top end of the test beam 100 and used as compression steel bars. The frame construction overcoat that first deformed steel bar 300 and second deformed steel bar 400 are constituteed is equipped with a plurality of stirrups 500, and stirrup 500 adopts the diameter to be 8 mm's deformed steel bar, and the interval between two adjacent stirrups 500 strides well the position at experimental roof beam 100 and is 100mm, is 70mm at the tip of experimental roof beam 100. It should be noted that, in order to prevent the stress concentration at the end of the member at the moment of releasing the tension, a PVC sleeve 900 may be embedded in the end of the member to relieve the effect of releasing the tension of the steel strand 200.
The test beam 100 performs local corrosion on the test steel strand 200 by using an electrochemical acceleration method. The accelerated corrosion system is composed of a direct current power supply 600, a corrosion tank 700 and a stainless steel plate 800. The corrosion groove 700 is arranged 500mm away from the end of the test beam 100 and fixed at the bottom of the test beam 100 by sealant. The corrosion tank 700 is provided with a 5% NaCl solution therein, and the stainless steel plate 800 is immersed in the 5% NaCl solution. The anode of the dc power supply 600 is connected to the test steel strand 200, the cathode is connected to the stainless steel plate 800, and the current is controlled to 0.5A during the corrosion process. The corrosion time of the three test beams 100 is respectively 15, 25 and 35 days, the three test beams are sequentially numbered as S1, S2 and S3, the transmission length data of the test beams 100 before corrosion is taken as an un-corroded control group and is named as S0, and the specific results are shown in Table 1.
TABLE 1 summary of test measurements
The corrosion rate is reflected by adopting the quality loss rate of the steel strand, and the measuring process comprises the following steps: firstly, cutting off a line segment of the test steel strand 200 in a local corrosion area; then, cleaning a rusted product on the surface of the rusted test steel strand 200 by using a 12% hydrochloric acid solution, and neutralizing by using an alkaline solution; finally, the residual mass of the test strand 200 after rusting was measured. The mass loss rate of the steel strand refers to the difference between the initial mass and the residual mass of the steel strand divided by the initial mass of the steel strand. Table 1 shows the average mass loss rate (i.e., the rust rate) for each test beam 100.
After the test beam 100 is cured, as shown in fig. 10 and 11, the estimated transfer length l is measured t The concrete surface in the area (1000 mm from the releasing end) is stuck with the separated distance measuring pieces 1000 at the interval of 50mm, and the position of the distance measuring pieces is consistent with the position of the longitudinal steel strand in the concrete. In this experiment, three adjacent distance measuring pieces 1000 are defined as a measuring section, the length between the measuring sections before and after rusting is sequentially measured along the longitudinal direction by using a high-precision vernier caliper 1100, the average strain of the measuring section is calculated by dividing the initial length by the absolute value of the increment of the length of the measuring section,taking the average value of two adjacent groups of strains as the strain of the midpoint between the two groups, thereby obtaining the surface strain of the concrete in the estimated transfer length area and the strain epsilon at the point of the x measurement x Can be expressed as equation 26:
in the formula, L x-1 And L x+1 Are respectively the initial lengths before rusting of the x-1 section and the x +1 section, L' x-1 And L' x+1 The lengths of the sections x-1 and x +1 after corrosion respectively.
As shown in fig. 12, the concrete strain at the end of the member becomes zero (no strain), and the concrete strain gradually increases with the distance from the end of the member, which indicates that the internal prestress of the steel strand in the region is gradually transferred to the concrete; after the distance from the end part of the member reaches a certain length, the compressive strain of the concrete tends to be stable, and the strain distribution diagram is in a horizontal state, so that the transmission of the prestress of the steel strand is completed, and the prestress of the concrete becomes a constant value. The area where the concrete compressive strain increases from zero to tend to be stable is the transfer zone, and the length of the transfer zone is the transfer length. The test uses the 95% average maximum strain method (AMS) to evaluate the test transfer length, which uses the numerical average of all strains contained in the horizontal section of the concrete strain distribution curve as the Average Maximum Strain (AMS) of the test piece, and defines the length of the intersection point of the 95% AMS horizontal line and the strain distribution curve as the transfer length of the test piece, as shown in fig. 12. The transfer length of each test beam 100 is shown in table 1.
Based on the measured corrosion rate, the theoretical transfer length is predicted by using the calculation method provided by the invention. The average prediction error for the experimental and theoretical transfer lengths was 4%. The error may be due to simplification of the theoretical model or may be a measurement deviation of the experimental data. This prediction error is acceptable in view of the uncertainty of the rusting process. The analysis shows that the theoretical model provided by the invention can accurately predict the transmission length of the rusted pretensioning prestressed concrete member.
In the description herein, references to the description of the term "one embodiment," "some embodiments," "an illustrative embodiment," "an example," "a specific example," or "some examples" or the like 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.
While embodiments of the invention have been shown and described, it will be understood by those of ordinary skill in the art that: various changes, modifications, substitutions and alterations can be made to the embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.
Claims (1)
1. A method for calculating the transfer length of a pretensioned member under the influence of concrete spalling is characterized by comprising the following steps of:
s1, predicting the interfacial expansive force of the rusty steel strand and concrete: based on a thick-wall cylinder theory, according to a displacement coordination condition, considering the common influence of Hall effect and corrosion, and predicting the expansion force of the interface of the steel strand and the concrete under the influence of the concrete expansion crack;
when the concrete is not cracked, the expansion force P of the interface between the steel strand and the concrete under the joint influence of the Hall effect and the corrosion can be expressed as a formula:
in the formula, R 0 Is the initial radius, R, of the steel strand t Is the radius, R, of the steel strand after the prestress is released c Is the distance from the center of the steel strand to the concrete surface, f pz Is the axial stress of the steel strand, E p Is the modulus of elasticity, v, of the steel strand p Is the Poisson's ratio of the steel strand, E c Is coagulationModulus of elasticity, v, of soil c Is the Poisson's ratio of concrete, f' cz Is the axial stress f 'of the concrete around the steel strand after rusting' cz ＝(1-ρ)f cz ρ is the corrosion rate of the steel strand, f cz The axial stress of the concrete around the steel strand when the steel strand is not rusted;
when the concrete is partially cracked, the expansion force P of the interface between the steel strand and the concrete under the joint influence of the Hall effect and the corrosion can be expressed as a formula:
in the formula, p c Constraint stress, R, for the location of the interface of spalled and unflatten concrete u Radius of spalled concrete, R is any position of the spalled concrete region, R t ≤r≤R u ，σ θ (r) is the hoop stress of the spalling concrete;
when the concrete is completely cracked, the expansion force P of the interface between the steel strand and the concrete under the joint influence of the Hall effect and the corrosion can be expressed as a formula:
wherein R is an arbitrary position of a spalled concrete region, R t ≤r≤R c ，σ θ (r) is the hoop stress of the spalling concrete;
radius R of steel strand after prestress releasing and tensioning t Can be expressed as the formula:
in the formula, f pt Is the tensile strength of the steel strand;
concrete radial displacement u under combined influence of Hall effect and corrosion c Can be expressed as the formula:
in the formula, P is the interface expansion force of the steel strand and the concrete under the joint influence of Hall effect and corrosion when the concrete is not cracked;
based on the thick-wall cylinder theory, the radial displacement u (r) of the unexpanded concrete when the concrete is partially cracked can be expressed as the formula:
in the formula (f) t The tensile strength of concrete;
according to the condition of radial displacement compatibility of the steel strand and the concrete interface, u c ＝u(R t ) The compatibility equation can be expressed as the formula:
from which R can be solved u In the formula, P is the interface expansion force of the steel strand and the concrete under the joint influence of Hall effect and corrosion when the concrete is not cracked;
taking into account the softening behavior of the tensile strength of the spalled concrete, the hoop stress sigma of the concrete θ (r) can be expressed as the formula:
in the formula (f) t Is the tensile strength of concrete, epsilon θ (r) is the tangential strain of the concrete, ε cr Is the strain, epsilon, corresponding to the tensile strength of the concrete cr ＝f t /E c ，ε 1 Is the strain, epsilon, corresponding to a concrete stress of 15% tensile strength 1 ＝0.0003，ε u Is the ultimate strain of concrete, epsilon u ＝0.002；
Based on the thick-wall cylinder theory, when the concrete is partially cracked, the tangential strain epsilon of the un-cracked concrete θ (r) can be expressed as the formula:
substituting into boundary condition R when the concrete is completely cracked u ＝R c Of tangential strain epsilon θ (r) can be expressed as the formula:
in the formula, epsilon qc Is the concrete surface tangential strain;
at full concrete spalling, the radial displacement u (r) can be expressed as the formula:
according to the condition of radial displacement compatibility of the steel strand and the concrete interface, u c ＝u(R t ) The compatibility equation can be expressed as the formula:
from which e can be solved qc In the formula, P is the interface expansion force of the steel strand and the concrete under the joint influence of Hall effect and corrosion when the concrete is not cracked;
s2, analyzing the bonding strength of the rusted steel strand: calculating the bonding strength of the steel strand under the influence of the concrete spalling according to the interfacial expansion force of the steel strand and the concrete and the friction coefficient between the steel strand and the concrete;
the bonding strength of a rusted steel strand can be expressed as a formula:
τ＝μ·P
wherein mu is the friction coefficient of the steel strand and the concrete interface;
s3, calculating the transmission length of the corrosion pretensioned member: dispersing the pretensioned concrete member into a plurality of micro units, establishing stress and strain expressions of steel strands in each micro unit and corresponding surrounding concrete, and calculating the transfer length of the pretensioned concrete member under the influence of concrete expansion crack based on an interface strain coordination relationship;
the calculation steps of the transfer length are as follows:
firstly, the pretensioned concrete member of 1/2 is scattered into a plurality of microcells, each microcell is numbered from 1 to n, and the local stress increment delta f of the microcell i p,i Can be expressed as the formula:
of formula (II) to' p Is the diameter, tau, of the rusted steel strand i Is the bonding stress, A ', of the steel strand at the ith microcell location' p Is the cross-sectional area of the rusted steel strand, and Δ l is the length of the microcell;
for pre-tensioned concrete elements, the stress of the steel strands at the end positions is 0, i.e. f p,0 0, for microcell i, stress f of its internal steel strand p,i Can be expressed as the formula:
f p,i ＝f p,i-1 +Δf p,i
in the formula,. DELTA.f p,i I is more than or equal to 1 and less than or equal to n and is the local stress increment of the microcell i;
strain increment delta epsilon of steel strand at ith microcell position p,i Can be expressed as the formula:
in the formula, epsilon p,0 Is the initial strain of the steel strand after being released;
strain epsilon of concrete around ith microcell c,i Can be expressed as the formula:
wherein A is the cross-sectional area of the concrete, e p Is the eccentricity of the steel strand, I c Is the moment of inertia, y, of the concrete section b Is the distance from the neutral axis of the pretensioned concrete element to the bottom end of the pretensioned concrete element;
the stress of the steel strand gradually increases from the relaxation end to the midspan direction until the effective prestress is reached, and according to the interface strain coordination relationship, the strain epsilon of the concrete around the micro-unit where the stress of the steel strand reaches the effective prestress c,i Should be equal to the strain increase delta epsilon of the steel strand at the microcell p,i The distance between the position where the steel strand stress reaches the effective prestress and the tension releasing end is the transfer length, and then the transfer length can be expressed as a formula:
l t ＝k·Δl
in the formula, k is the number of the micro unit where the stress of the steel strand reaches the effective prestress, and k is more than or equal to 1 and less than or equal to n.
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