CN110929321A - Method for calculating height of relative boundary compression zone of bonded steel reinforced prestressed concrete beam - Google Patents

Method for calculating height of relative boundary compression zone of bonded steel reinforced prestressed concrete beam Download PDF

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CN110929321A
CN110929321A CN201911128632.5A CN201911128632A CN110929321A CN 110929321 A CN110929321 A CN 110929321A CN 201911128632 A CN201911128632 A CN 201911128632A CN 110929321 A CN110929321 A CN 110929321A
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concrete beam
height
steel
prestressed
prestressed concrete
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CN110929321B (en
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张能伟
韩子健
吴畏
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Architecture Design and Research Institute of Tongji University Group Co Ltd
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Abstract

The invention relates to a method for calculating the height of a relative limit compression area of a bonded steel reinforced prestressed concrete beam, which comprises the following steps: 1. acquiring basic parameters; 2. obtaining the equivalent effective height and the height of a section compression area; 3. calculating the relative residual bending-resistant bearing capacity provided by the common tensioned steel bar, the relative residual bending-resistant bearing capacity provided by the prestressed tendon and the relative residual bending-resistant bearing capacity provided by the adhered steel plate; 4. judging the relative residual bending resistance bearing capacity of the three materials, and obtaining structural control conditions for the bonded steel reinforced prestressed concrete beam to be damaged; 5. and according to the structural control conditions, calculating the strain generated at the edge of the tension zone when the bonded steel reinforced prestressed concrete beam is damaged, and further calculating the height of the relative limit compression zone of the bonded steel reinforced prestressed concrete beam under the control of the bonded steel reinforced prestressed concrete beam. Compared with the prior art, the method has the advantages of accurate calculation, strong practicability and the like.

Description

Method for calculating height of relative boundary compression zone of bonded steel reinforced prestressed concrete beam
Technical Field
The invention relates to the technical field of component design, in particular to a method for calculating the height of a relative limit compression area of a bonded steel reinforced prestressed concrete beam.
Background
With the continuous development of social economy, the existing buildingsThe concrete structure is designed to ensure that the structure can absorb and dissipate most of energy under the action of earthquake, and the height ξ of the relative compression zone of the member is not more than the height ξ of the compression zone of the relative limitbFor the conventional prestressed concrete beam or bonded steel reinforced concrete beam, the design specification of prestressed concrete structure and the design specification of concrete structure reinforcement have given the relative limit compression zone height ξbHowever, in the engineering design of the bond steel reinforced prestressed concrete beam, there is no accurate and practical calculation method for the height of the relative limit compression zone, and the calculation formula has a great influence on the engineering design of the bond steel reinforced prestressed concrete beam.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a method for calculating the height of the relative limit compression zone of the bonded steel reinforced prestressed concrete beam.
The purpose of the invention can be realized by the following technical scheme:
the method for calculating the height of the relative boundary compression zone of the bond steel reinforced prestressed concrete beam follows the following three basic assumptions:
1. the cross section strain after reinforcement conforms to the assumption of a flat cross section;
2. the stress state of the components is in an elastic stage;
3. before the limit state of the flexural bearing capacity is reached, the bonding stripping damage between the bonded steel plate and the prestressed concrete beam does not occur.
The method specifically comprises the following steps:
step one, determining basic parameters.
Height ξ of relative limit compression zone of bond steel reinforced prestressed concrete beambThe basic parameters specifically include:
1. component geometry: the width b of the prestressed concrete beam; beam height (viscous steel)The height of the section of the reinforced prestressed concrete beam) h; width b of reinforced steel platea(ii) a Thickness ta(ii) a Area A of ordinary tension steel bars0(ii) a Area of ordinary pressed reinforcing steel bar A's0(ii) a Area A of prestressed tendonp0(ii) a Area A of the reinforced steel platea0。
2. Member material information: ultimate compressive strain epsilon of concretecuDesign value f of yield strength of ordinary steel barsyYield strain epsilon of ordinary steel barsyModulus of elasticity E of ordinary steel barsDesign value f of yield strength of prestressed tendonpyYield tensile strain epsilon of prestressed reinforcementpyModulus of elasticity E of prestressed reinforcing steelpDesign value f of yield strength of reinforced steel plateayYield tensile strain epsilon of bonded steel plateayModulus of elasticity E of reinforced Steel platea。
3. Member arrangement position: distance a between common tensioned steel bar and tensioned edge of concretes(ii) a Distance a 'between common compressed steel bars and tension edge of concrete's(ii) a Distance a between prestressed tendon and tensile edge of concretep。
And step two, acquiring the equivalent effective height of the section and the height of the pressed area of the section. Specifically, the method comprises the following steps:
31) and acquiring the stretching control stress and the prestress loss of the prestressed concrete, and acquiring the prestressed tendon stress when the concrete normal stress at the prestressed tendon resultant force point is equal to zero according to the stretching control stress and the prestress loss of the prestressed concrete.
32) Obtaining the effective height h of the section of the bonded steel reinforced prestressed concrete beam according to the torque equivalence principle0；
The method comprises the following steps of (1) obtaining according to a moment equivalent principle of a section of a bonded steel reinforced prestressed concrete beam:
fayAayha+fsyAs0hs+fpyAp0hp＝(fayAay+fsyAs0+fpyAp0)h0
in the formula ha、hs、hpRespectively as the resultant action point of the adhered steel plates and the common tensionDistance between the resultant action point of the steel bars and the resultant action point of the prestressed tendons and the pressed edge, ha＝h，hs＝h-as，hp＝h-ap。
33) And obtaining the height x of the compression area of the bonded steel reinforced prestressed concrete beam according to the tension-compression balance relation of the section of the bonded steel reinforced prestressed concrete beam.
The expression of the tension-compression balance relation of the section of the bond steel reinforced prestressed concrete beam is as follows:
fayAa0+fsyAs0+fpyAp0＝α1fcbx+f’syA’s0
the calculation formula of the height x of the compression zone of the bonded steel reinforced prestressed concrete beam is as follows:
Figure BDA0002277644950000021
in the formula, α1The ratio of the stress value of the concrete rectangular stress diagram of the compression area to the designed value of the concrete axial compressive strength is shown. The values are given in the following table:
TABLE 1 α1Value taking
Figure BDA0002277644950000031
And step three, obtaining the relative residual bending resistance bearing capacity provided by the common tensioned steel bar, the prestressed tendons and the adhered steel plate. And judging the damage sequence of the common tensioned steel bar, the prestressed tendon and the adhered steel plate according to the values of the three.
The damage sequence of the bond steel reinforced prestressed concrete beam is controlled by the relative residual bending bearing capacity of the common tension steel bar, the prestressed tendon and the bonded steel plate, and the larger the relative residual bending bearing capacity is, the later the damage is. The relative residual bending-resistant bearing capacity is composed of the residual bending-resistant bearing capacity and the weight occupied by the residual bending-resistant bearing capacity.
The assumption of a flat section shows that the strain increment of a common tension steel bar, a prestressed tendon and a reinforced steel plate is linearly related, namely:
Figure BDA0002277644950000032
the above equation can be approximated as:
Figure BDA0002277644950000033
in the formula,. DELTA.epsilona、Δεs、ΔεpRespectively the strain increment of a common tensioned steel bar, a prestressed tendon and a bonded steel plate after the steel plate is bonded.
Assuming that the weight of the residual bending resistance bearing capacity provided by the adhered steel plate is 1, the weight of the residual bending resistance bearing capacity provided by the common tension steel bar is as follows:
Figure BDA0002277644950000034
the weight of the residual bending resistance bearing capacity provided by the prestressed tendons is as follows:
Figure BDA0002277644950000035
the above derivation can obtain the relative residual bending resistance bearing capacity M provided by the common tension steel barsComprises the following steps:
Figure BDA0002277644950000036
relative residual bending resistance bearing capacity M provided by prestressed tendonspComprises the following steps:
Figure BDA0002277644950000041
relative residual bending resistance bearing capacity M provided by pasting steel plateaComprises the following steps:
Figure BDA0002277644950000042
for the acquired Ms、Mp、MaThe values of the tension bars and the prestressed bars are judged to obtain the damage sequence of the common tensioned steel bars, the prestressed bars and the adhered steel plates, Ms、Mp、MaThe larger the value of the three is, the later the material damage of the corresponding bond steel reinforced prestressed concrete beam is, which indicates that the damage of the bond steel reinforced prestressed concrete beam is controlled by the condition.
Step four, calculating the height ξ of the relative limit compression zone of the bonded steel reinforced prestressed concrete beam according to the judgment result of the step threeb。
When the common tension steel bar, the prestressed reinforcing bar and the adhesive steel plate reach the yield strength of the common tension steel bar, the prestressed reinforcing bar and the adhesive steel plate at the same time, the height ξ of the relative compression area of the adhesive steel reinforced prestressed concrete beambThe calculation of the method is related to the damage sequence of common tension steel bars, prestressed tendons and adhered steel plates, and for the adhered steel reinforced prestressed concrete beam with different damage sequences, the calculation of the height of a relative limit compression area is divided into the following three conditions:
(1) when the concrete in the compression zone reaches its ultimate compressive strain epsiloncuWhen the steel plate is used, the prestressed reinforcing steel bars and the adhered steel plates are firstly subjected to yielding, and the common tensioned reinforcing steel bars are finally subjected to yielding. At which point the strain value epsilon of the edge in tensionsEqual to the yield strain epsilon of common tensioned steel barssyAnd obtaining the height of the relative limit compression area of the bonded steel reinforced prestressed concrete beam according to the plane geometric relation as follows:
Figure BDA0002277644950000043
(2) when the concrete in the compression zone reaches its ultimate compressive strain epsiloncuWhen the steel plate is used, the common tensioned steel bar and the bonded steel plate yield first, and the prestressed reinforcing steel bar yields finally. At which point the strain value epsilon of the edge in tensionsThe stress increment of the prestressed tendon from the moment when the concrete at the position where the combined force of the prestressed tendons acts is decompressed to the moment when the prestressed tendon yields, and for the prestressed tendon with an obvious yield step, the value is epsilonpy-εp0For a tendon without a significant yield step, a residual strain of 0.2% is considered, and the value is εpy-εp0+0.002, obtained according to a planar geometry:
under the condition of the prestressed reinforcing steel with the obvious yield step, the height of the relative boundary compression area of the bonded steel reinforced prestressed concrete beam is as follows:
Figure BDA0002277644950000044
under the condition of no obvious yield step prestressed tendon, the height of the relative boundary compression zone of the bonded steel reinforced prestressed concrete beam is as follows:
Figure BDA0002277644950000051
in the formula sigmap0、εp0Stress and strain of the prestressed tendon when the concrete section stress is zero at the resultant force action position of the prestressed tendon, respectivelyp0＝σp0/Ep。
(3) When the concrete in the compression zone reaches its ultimate compressive strain epsiloncuWhen the steel plate is attached to the steel plate, the common tensioned steel bars and the prestressed steel bars yield first and the attached steel plate yields last. At which point the strain value epsilon of the edge in tensionsEqual to the strain value epsilon of the tensile edge of the beam when the bonded steel is reinforceds0With steel plate yield tensile strain epsilonayAnd (3) obtaining the height of the relative limit compression area of the bonded steel reinforced prestressed concrete beam according to the plane geometric relation:
Figure BDA0002277644950000052
in the formula, σs0、εs0Respectively the initial stress and strain, epsilon, of a common tensioned steel bar when the steel plate is adhereds0＝σs0/Es。
Wherein, the tensile edge tensile strain epsilon of the beam is generated when the bond steel is reinforceds0The acquisition process comprises the following steps:
before the prestressed beam is bonded with steel and reinforced, under the action of original use load, the tensile steel bar at the bottom of the beam is stressed to generate stress, and the following equilibrium equation is obtained according to concrete structure design specifications:
M0k＝0.87(σs0As0hs+σpAphp)
wherein
σs0＝εs0Es
σp＝εpAp
εp＝εp0+εp,p0
In the formula M0kThe standard value of the bending moment stressed by the prestressed concrete beam during the reinforcement of the bonded steel; epsilonpThe strain value of the prestressed tendon when the bonded steel is reinforced; epsilonp,p0The strain increment of the prestressed tendon from the concrete pressure relief of the prestressed tendon resultant force action position to the bonding steel reinforcementp0The strain of the prestressed tendon when the concrete section stress is zero at the position of the resultant force action of the prestressed tendon.
The comprehensive formula can be obtained:
Figure BDA0002277644950000053
then
Figure BDA0002277644950000061
In the formula, αEpsα being the ratio of the modulus of elasticity of the tendon to that of the ordinary tendon in tensionApsα is the ratio of the cross-sectional area of the prestressed bar to that of the ordinary tension barEps＝Ep/Es；αAps＝Ap0/As0。
Step five, according to the obtained height ξ of the relative limit compression zone of the bonded steel reinforced prestressed concrete beambAnd judging whether the construction requirements are met, and if so, carrying out the flexural design of the bonded steel reinforced prestressed concrete beam according to the data.
The limitations of the construction requirements include:
the height x of the compression zone of the bond steel reinforced prestressed concrete beam should satisfy x being less than or equal to ξbh0。
And if the structural requirements are not met, returning to the step one, and reselecting the material parameters.
Compared with the prior art, the invention has the beneficial effects that:
according to the method, according to the basic principle of the concrete structure and the plane geometric relation, the influence of different damage sequences on the structural stress is fully considered, and the effective calculation method of the height of the relative boundary limited compression area of the bond steel reinforced prestressed concrete is provided;
the invention provides a concept of relative residual bending resistance bearing capacity according to the basic principle of the concrete structure and the plane geometric relation, judges the material failure sequence of the prestressed concrete beam reinforced by the bonded steel according to the relative residual bending resistance bearing capacity, is more suitable for the actual situation, and can obtain more accurate height of the relative boundary limited pressure zone of the bonded steel reinforced prestressed concrete under three conditions aiming at different failure sequences.
Drawings
FIG. 1 is a schematic flow diagram of the process of the present invention;
FIG. 2 is a cross-sectional strain distribution diagram of a prestressed concrete beam in a boundary failure state;
FIG. 3 is a cross-sectional strain distribution diagram of a bonded steel reinforced prestressed concrete beam when a common tensioned steel bar is finally yielding;
FIG. 4 is a cross-sectional strain distribution diagram of a bonded steel reinforced prestressed concrete beam when the prestressed tendons are finally yielding;
FIG. 5 is a cross-sectional strain distribution diagram of the bonded steel reinforced prestressed concrete beam under the condition that the bonded steel plate is finally yielding.
Detailed Description
The invention is described in detail below with reference to the figures and specific embodiments. It is to be understood that the embodiments described are only a few embodiments of the present invention, and not all embodiments. All other embodiments, which can be obtained by a person skilled in the art without any inventive step based on the embodiments of the present invention, shall fall within the scope of protection of the present invention.
The invention relates to a method for calculating the height of a relative limit compression area of a bond steel reinforced prestressed concrete beam, which is applied to the engineering design of the bond steel reinforced prestressed concrete beam. When the height of the relative boundary compression zone of the bonded steel reinforced prestressed concrete beam is obtained, the following three basic assumptions are followed:
1. the cross section strain after reinforcement conforms to the assumption of a flat cross section;
2. the stress state of the components is in an elastic stage;
3. before the limit state of the flexural bearing capacity is reached, the bonding stripping damage between the bonded steel plate and the prestressed concrete beam does not occur.
When the common tension steel bar, the prestressed tendon and the pasting steel plate are damaged at the same time, the section strain distribution diagram under the critical state is shown in figure 4, and the stress-strain:
Figure BDA0002277644950000071
the calculation formula of the height of the relative limit compression zone of the bonded steel reinforced prestressed concrete beam is as follows:
Figure BDA0002277644950000072
in the formula, epsiloncuIs the ultimate compressive strain of the concrete; epsilonsThe strain generated by the edge of the tension area when the bonded steel reinforced prestressed concrete beam is damaged. h is0The equivalent effective height of the section of the prestressed concrete beam is reinforced by the bond steel, and β is the height x of the compression zone of the rectangular stress diagrambAnd a neutral axis height xcbThe ratio of (a) to (b), which is related to the concrete strength rating, is given in the following table.
TABLE 2 rectangular stress map compression zone height xbAnd a neutral axis height xcbRatio β of
Figure BDA0002277644950000073
Example 1
Based on the above assumptions, the method of the present embodiment actually calculates the height of the compression zone of the relative limit of the bond steel reinforced prestressed concrete beam, and specifically includes the following steps:
basic information of the component:
the prestressed concrete beam has the cross-sectional dimension of 500mm multiplied by 800mm and the strength grade of C40. The common longitudinal bars of the tension area are 4 HRB 335-grade steel bars with the diameter of 25mm, and the distance a from the action position of the resultant force point to the edge of the tension areas50 mm; common longitudinal bars of the compression area are 4 HRB 335-grade steel bars with the diameter of 25mm, and the action position of the resultant force point is a 'away from the edge of the tension area's50 mm; the prestressed tendon adopts 3 holes and 6 strands of 1860 steel strands (1x7), and the acting position of the resultant point of the prestressed tendon is a distance a from the edge of a tension areap200 mm. The reinforced steel plate is Q345 and adopts two wide plates bspA steel plate of 150mm thickness tsp5mm in total. The bending effect on the prestressed concrete beam during the bonding steel reinforcement is M0k＝1500kN·m。
From the above, it can be obtained: designed value f of concrete compressive strengthc19.1MPa, concrete ultimate compressive strain epsiloncu0.0033, design value f of tensile strength of common steel barsy300MPa, and the design value f 'of the compressive strength of common steel bars'sy300MPa, modulus of elasticity E of plain reinforcing steels＝2.0×105MPa, design value f of tensile strength of prestressed tendonpy1320MPa, modulus of elasticity E of prestressed tendonp＝1.95×105MPa, design value f of tensile strength of reinforced steel plateay305MPa, modulus of elasticity Ea＝2.06×105MPa。
At the same time, it can be calculated that: area of normal tension bar
Figure BDA0002277644950000081
Area of normal stressed steel bar
Figure BDA0002277644950000082
Area A of prestressed tendonp0＝3×6×140＝2520mm2Area A of the reinforced steel platea0＝1×2×bsp×tsp＝2×150×5＝1500mm2. Under normal tensionDistance h between combined force action point of reinforcing steel bars and pressed edge of concrete beam0＝h-as800-50-750 mm; distance h between the resultant action point of the prestressed reinforcement and the edge of the compression area of the concrete beamp＝h-ap＝800-200＝600mm。
Step one, calculating equivalent effective height h0And a cross-sectional compression zone height x.
Firstly, obtaining the prestress loss, and adopting phi to prestress the steel bars15.2 pre-stressed steel strands, so that the tensioning control stress of the pre-stressed concrete is as follows:
σcon＝0.75fptk＝0.75×1860＝1395MPa
the prestress loss is taken as:
σl＝0.2σcon＝0.2×1395＝279MPa
the prestressed beam adopts a pretensioning method construction process, and the prestressed tendon stress at the prestressed tendon resultant force point is equal to zero:
σp0＝σcon-σl＝1395-279＝1116MPa
the effective height of the section of the bonded steel reinforced prestressed concrete beam can be obtained according to a moment equivalent principle, and a moment equivalent formula is as follows:
fayAa0ha+fsyAs0hs+fpyAp0hp＝(fayAay+fsyAs0+fpyAp0)h0
then:
Figure BDA0002277644950000091
the tension-compression balance relation of the section of the prestressed concrete beam reinforced by the bonded steel is obtained by the following formula:
fayAa0+fsyAs0+fpyAp0＝α1fcbx+f’syA’s0
then:
Figure BDA0002277644950000092
step two, calculating a strain value epsilon generated at the edge of a tension area when the beam is damageds0。
Under the original use load action of the bonded steel reinforced prestressed concrete beam, the stress of the common tension steel bar is as follows:
Figure BDA0002277644950000093
the steel bar is obtained by the constitutive relation of common tension steel bars:
Figure BDA0002277644950000094
calculating the strain increment epsilon of the prestressed tendon from the concrete pressure relief of the prestressed tendon resultant force action position to the bonded steel reinforcement by the geometric relationp,p0Comprises the following steps:
Figure BDA0002277644950000095
the strain of the prestressed tendon when the concrete section stress is zero at the resultant force action position of the prestressed tendon is as follows:
Figure BDA0002277644950000096
the strain value of the prestressed tendon during the steel bonding reinforcement is as follows:
εp＝εp0+εp,p0＝5.7×10-3+5.5×10-5＝575.5×10-5
step three, calculating the relative residual bending resistance bearing capacity M provided by the common tension steel bar, the prestressed tendon and the adhered steel plates、Mp、Ma。
The yield tensile strain of the bonded steel plate is as follows:
Figure BDA0002277644950000101
the yield tensile strain of the tendon is
Figure BDA0002277644950000102
The yield strain of a common tensioned steel bar is:
Figure BDA0002277644950000103
the residual bending resistance bearing capacity provided by the pasting steel plate is as follows:
Figure BDA0002277644950000104
the residual bending bearing capacity provided by the longitudinal tension steel bar is as follows:
Figure BDA0002277644950000105
the residual bending resistance bearing capacity provided by the prestressed tendons is as follows:
Figure BDA0002277644950000106
step four, providing relative residual bending resistance bearing capacity M for common tensioned steel bars, prestressed bars and adhered steel platess、Mp、MaThe larger the value is, the later the damage sequence is, the height of the relative limit compression zone is controlled by the last damaged material, and the height ξ of the relative limit compression zone of the bond steel reinforced prestressed concrete beam is calculatedb。
The residual bending resistance bearing capacity M provided by the longitudinal tension steel bar can be obtained by the third stepsThe maximum value of (A) indicates that the damage of the bond steel reinforced prestressed concrete beam is controlled by the common tension steel bar, and the height ξ of the relative limit compression zone of the bond steel reinforced prestressed concrete beambComprises the following steps:
Figure BDA0002277644950000107
step five, according to the obtained height ξ of the relative limit compression zone of the bonded steel reinforced prestressed concrete beambAnd judging whether the construction requirements are met.
x＝369mm≤0.7001×641＝448.8mm
And meeting the requirements, and carrying out bending design on the bonded steel reinforced prestressed concrete beam according to the data.
Example 2
Based on the above assumptions, the actual calculation of the height of the relative limit compression zone of the bond steel reinforced prestressed concrete beam in the embodiment specifically includes the following contents:
basic information of the component:
the prestressed concrete beam has the cross-sectional dimension of 600mm multiplied by 800mm and the strength grade of C40. The common longitudinal bars of the tension area are 4 HRB 335-grade steel bars with the diameter of 22mm, and the distance a from the action position of the resultant force point to the edge of the tension areas100 mm; common longitudinal bars of the compression area are 4 HRB 335-grade steel bars with the diameter of 22mm, and the action position of the resultant force point is a 'away from the edge of the tension area's100 mm; the prestressed tendon adopts 3 holes and 6 strands of 1860 steel strands (1x7), and the acting position of the resultant point of the prestressed tendon is a distance a from the edge of a tension areap100 mm. The reinforced steel plate is Q345 and adopts two wide plates bspA steel plate of 150mm thickness tsp4mm, one layer. The bending effect on the prestressed concrete beam during the bonding steel reinforcement is M0k＝1800kN·m。
From the above, it can be obtained: designed value f of concrete compressive strengthc19.1MPa, concrete ultimate compressive strain epsiloncu0.0033, design value f of tensile strength of common steel barsy300MPa, and the design value f 'of the compressive strength of common steel bars'sy300MPa, modulus of elasticity E of plain reinforcing steels＝2.0×105MPa, design value f of tensile strength of prestressed tendonpy1320MPa, modulus of elasticity E of prestressed tendonp＝1.95×105MPa, design value f of tensile strength of reinforced steel plateay305MPa, modulus of elasticity Ea＝2.06×105MPa。
At the same time, it can be calculated that: area of normal tension bar
Figure BDA0002277644950000111
Area of normal stressed steel bar
Figure BDA0002277644950000112
Area A of prestressed tendonp0＝3×6×140＝2520mm2Area A of the reinforced steel platea0＝1×2×bsp×tsp＝2×150×4＝1200mm2. Distance h between joint action point of common tension steel bars and pressed edge of concrete beams＝h-as800-; distance h between the resultant action point of the prestressed reinforcement and the edge of the compression area of the concrete beamp＝h-ap＝800-100＝700mm。
Step one, calculating equivalent effective height h0And a cross-sectional compression zone height x.
Firstly, obtaining the prestress loss, and adopting phi to prestress the steel bars15.2 pre-stressed steel strands, so that the tensioning control stress of the pre-stressed concrete is as follows:
σcon＝0.75fptk＝0.75×1860＝1395MPa
the prestress loss is taken as:
σl＝0.2σcon＝0.2×1395＝279MPa
the prestressed beam adopts a pretensioning method construction process, and the prestressed tendon stress at the prestressed tendon resultant force point is equal to zero:
σp0＝σcon-σl＝1395-279＝1116MPa
the effective height of the section of the bonded steel reinforced prestressed concrete beam can be obtained according to a moment equivalent principle, and a moment equivalent formula is as follows:
fayAa0ha+fsyAs0hs+fpyAp0hp＝(fayAay+fsyAs0+fpyAp0)h0
then:
Figure BDA0002277644950000121
the tension-compression balance relation of the section of the prestressed concrete beam reinforced by the bonded steel is obtained by the following formula:
fayAa0+fsyAs0+fpyAp0＝α1fcbx+f’syA’s0
then:
Figure BDA0002277644950000122
step two, calculating a strain value epsilon generated at the edge of a tension area when the beam is damageds0。
Under the original use load action of the bonded steel reinforced prestressed concrete beam, the stress of the common tension steel bar is as follows:
Figure BDA0002277644950000123
the steel bar is obtained by the constitutive relation of common tension steel bars:
Figure BDA0002277644950000124
calculating the strain increment epsilon of the prestressed tendon from the concrete pressure relief of the prestressed tendon resultant force action position to the bonded steel reinforcement by the geometric relationp,p0Comprises the following steps:
Figure BDA0002277644950000125
the strain of the prestressed tendon when the concrete section stress is zero at the resultant force action position of the prestressed tendon is as follows:
Figure BDA0002277644950000131
the strain value of the prestressed tendon during the steel bonding reinforcement is as follows:
εp＝εp0+εp,p0＝5.7×10-3+16.8×10-5＝586.8×10-5
step three, calculating the relative residual bending resistance bearing capacity M provided by the common tension steel bar, the prestressed tendon and the adhered steel plates、Mp、Ma。
The yield tensile strain of the bonded steel plate is as follows:
Figure BDA0002277644950000132
the yield tensile strain of the prestressed tendon is as follows:
Figure BDA0002277644950000133
the yield strain of a common tensioned steel bar is:
Figure BDA0002277644950000134
the residual bending resistance bearing capacity provided by the pasting steel plate is as follows:
Figure BDA0002277644950000135
the residual bending bearing capacity provided by the longitudinal tension steel bar is as follows:
Figure BDA0002277644950000136
the residual bending resistance bearing capacity provided by the prestressed tendons is as follows:
Figure BDA0002277644950000137
step four, providing relative residual bending resistance bearing capacity M for common tensioned steel bars, prestressed bars and adhered steel platess、Mp、MaIs judged to beThe larger the value, the later the damage sequence, the height of the relative limit compression zone is controlled by the last damaged material, and the height ξ of the relative limit compression zone of the bond steel reinforced prestressed concrete beam is calculatedb。
The residual bending resistance bearing capacity M provided by the prestressed tendons can be obtained by the third steppThe maximum value of (A) indicates that the damage of the bond steel reinforced prestressed concrete beam is controlled by the prestressed tendons, and the height ξ of the relative limit compression zone of the bond steel reinforced prestressed concrete beambComprises the following steps:
Figure BDA0002277644950000141
step five, according to the obtained height ξ of the relative limit compression zone of the bonded steel reinforced prestressed concrete beambAnd judging whether the construction requirements are met.
x＝322mm≤0.4994×709＝354.1mm
And if the requirements are met, performing flexural design on the bonded steel reinforced prestressed concrete beam according to the data.
Example 3
Based on the above assumptions, the actual calculation of the height of the relative limit compression zone of the bond steel reinforced prestressed concrete beam in the embodiment specifically includes the following contents:
basic information of the component:
the prestressed concrete beam has the cross-sectional dimension of 600mm multiplied by 800mm and the strength grade of C40. The common longitudinal bars of the tension area are 4 HRB 335-grade steel bars with the diameter of 25mm, and the distance a from the action position of the resultant force point to the edge of the tension areas50 mm; common longitudinal bars of the compression area are 4 HRB 335-grade steel bars with the diameter of 25mm, and the action position of the resultant force point is a 'away from the edge of the tension area's50 mm; the prestressed tendon adopts 3 holes and 6 strands of 1860 steel strands (1x7), and the acting position of the resultant point of the prestressed tendon is a distance a from the edge of a tension areap200 mm. The reinforced steel plate is Q345 and adopts two wide plates bspA steel plate of 150mm thickness tsp5mm in total. The bending effect on the prestressed concrete beam during the bonding steel reinforcement is M0k＝1500kN·m。
From the above, it can be obtained: designed value f of concrete compressive strengthc19.1MPa, concrete ultimate compressive strain epsiloncu0.0033, design value f of tensile strength of common steel barsy300MPa, and the design value f 'of the compressive strength of common steel bars'sy300MPa, modulus of elasticity E of plain reinforcing steels＝2.0×105MPa, design value f of tensile strength of prestressed tendonpy1320MPa, modulus of elasticity E of prestressed tendonp＝1.95×105MPa, design value f of tensile strength of reinforced steel plateay305MPa, modulus of elasticity Ea＝2.06×105MPa。
At the same time, it can be calculated that: area of normal tension bar
Figure BDA0002277644950000142
Area of normal stressed steel bar
Figure BDA0002277644950000143
Area A of prestressed tendonp0＝3×6×140＝2520mm2Area A of the reinforced steel platea0＝1×2×bsp×tsp＝2×150×5＝1500mm2. Distance h between joint action point of common tension steel bars and pressed edge of concrete beam0＝h-as800-50-750 mm; distance h between the resultant action point of the prestressed reinforcement and the edge of the compression area of the concrete beamp＝h-ap＝800-200＝600mm。
Step one, calculating equivalent effective height h0And a cross-sectional compression zone height x.
Firstly, obtaining the prestress loss, and adopting phi to prestress the steel bars15.2 pre-stressed steel strands, so that the tensioning control stress of the pre-stressed concrete is as follows:
σcon＝0.75fptk＝0.75×1860＝1395MPa
the prestress loss is taken as:
σl＝0.2σcon＝0.2×1395＝279MPa
the prestressed beam adopts a pretensioning method construction process, and the prestressed tendon stress at the prestressed tendon resultant force point is equal to zero:
σp0＝σcon-σl＝1395-279＝1116MPa
the effective height of the section of the bonded steel reinforced prestressed concrete beam can be obtained according to a moment equivalent principle, and a moment equivalent formula is as follows:
fayAa0ha+fsyAs0hs+fpyAp0hp＝(fayAay+fsyAs0+fpyAp0)h0
then:
Figure BDA0002277644950000151
the tension-compression balance relation of the section of the prestressed concrete beam reinforced by the bonded steel is obtained by the following formula:
fayAa0+fsyAs0+fpyAp0＝α1fcbx+f’syA’s0
then:
Figure BDA0002277644950000152
step two, calculating a strain value epsilon generated at the edge of a tension area when the beam is damageds0。
Under the original use load action of the bonded steel reinforced prestressed concrete beam, the stress of the common tension steel bar is as follows:
Figure BDA0002277644950000153
the steel bar is obtained by the constitutive relation of common tension steel bars:
Figure BDA0002277644950000161
calculating the strain of the prestressed tendon from the concrete pressure relief of the prestressed tendon resultant force action position to the bonded steel reinforcement by the geometric relationIncrement of epsilonp,p0Comprises the following steps:
Figure BDA0002277644950000162
the strain of the prestressed tendon when the concrete section stress is zero at the resultant force action position of the prestressed tendon is as follows:
Figure BDA0002277644950000163
the strain value of the prestressed tendon during the steel bonding reinforcement is as follows:
εp＝εp0+εp,p0＝5.7×10-3+6.3×10-5＝576.3×10-5
step three, calculating the relative residual bending resistance bearing capacity M provided by the common tension steel bar, the prestressed tendon and the adhered steel plates、Mp、Ma。
The yield tensile strain of the bonded steel plate is as follows:
Figure BDA0002277644950000164
the yield tensile strain of the prestressed tendon is as follows:
Figure BDA0002277644950000165
the yield strain of a common tensioned steel bar is:
Figure BDA0002277644950000166
the residual bending resistance bearing capacity provided by the pasting steel plate is as follows:
Figure BDA0002277644950000167
the residual bending bearing capacity provided by the longitudinal tension steel bar is as follows:
Figure BDA0002277644950000168
the residual bending resistance bearing capacity provided by the prestressed tendons is as follows:
Figure BDA0002277644950000169
Figure BDA0002277644950000171
step four, providing relative residual bending resistance bearing capacity M for common tensioned steel bars, prestressed bars and adhered steel platess、Mp、MaThe larger the value is, the later the damage sequence is, the height of the relative limit compression zone is controlled by the last damaged material, and the height ξ of the relative limit compression zone of the bond steel reinforced prestressed concrete beam is calculatedb。
The residual bending resistance bearing capacity M provided by the adhered steel plate can be obtained by the third stepaThe maximum value of (A) indicates that the damage of the bond steel reinforced prestressed concrete beam is controlled by the bonded steel plate, and the height ξ of the relative limit compression zone of the bond steel reinforced prestressed concrete beambComprises the following steps:
Figure BDA0002277644950000172
step five, according to the obtained height ξ of the relative limit compression zone of the bonded steel reinforced prestressed concrete beambAnd judging whether the construction requirements are met.
x＝330mm≤0.6812×638＝434.6mm
And if the requirements are met, performing flexural design on the bonded steel reinforced prestressed concrete beam according to the data.
According to the basic principle and the plane geometric relation of the concrete structure, the influence of different damage sequences on the structural stress is fully considered, the concept of relative residual bending resistance bearing capacity is provided, the material damage sequence of the prestressed concrete beam after bonded steel reinforcement is judged according to the relative residual bending resistance bearing capacity, the actual situation is better fitted, and the more accurate height of the relative boundary limited pressure zone of the bonded steel reinforced prestressed concrete under three conditions can be obtained according to different damage sequences.
While the invention has been described with reference to specific embodiments, the invention is not limited thereto, and those skilled in the art can easily conceive of various equivalent modifications or substitutions within the technical scope of the invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims (10)

1. The method for calculating the height of the relative limit compression area of the bonded steel reinforced prestressed concrete beam is characterized by comprising the following steps of:
s1, acquiring the geometric dimension, material information and arrangement position of the component;
s2, acquiring the equivalent effective height and the height of the section compression zone according to the basic parameters of the step S1;
s3, obtaining the relative residual bending-resistant bearing capacity provided by the common tension steel bar, the relative residual bending-resistant bearing capacity provided by the prestressed tendons and the relative residual bending-resistant bearing capacity provided by the adhered steel plate according to the equivalent effective height and the height of the section compression zone;
s4, judging the sizes of the three relative residual bending resistance bearing capacities obtained in the step S3, and obtaining structural control conditions for the final damage of the bonded steel reinforced prestressed concrete beam;
s5, according to the structural control conditions obtained in the step S4, obtaining strain generated at the edge of a tension area when the bond steel reinforced prestressed concrete beam is damaged, and further calculating the height of a relative limit compression area of the bond steel reinforced prestressed concrete beam under the control of the bond steel reinforced prestressed concrete beam;
and S6, judging whether the construction requirements are met according to the obtained height of the relative limit compression area of the bonded steel reinforced prestressed concrete beam, and if so, performing bending design on the bonded steel reinforced prestressed concrete beam according to the data.
2. The method for calculating the height of the relative limit compression area of the viscous steel reinforced prestressed concrete beam according to claim 1, wherein in step S4, the larger a certain value among the relative residual bending bearing forces provided by the normal tension bar, the prestressed bar and the bonded steel plate is, the later the material corresponding to the certain value is damaged, and if the certain value is the maximum, the material corresponding to the certain value is the structural control condition that the viscous steel reinforced prestressed concrete beam is finally damaged, that is, the material corresponding to the certain value is finally yielded.
3. The method for calculating the height of the relative limit compression zone of the viscous steel reinforced prestressed concrete beam according to claim 2, characterized in that when the concrete in the compression zone reaches its limit compression strain epsiloncuWhen the prestressed reinforcing steel bar and the bonded steel plate are firstly yielding, the common tensioned reinforcing steel bar is finally yielding, and according to the plane geometric relationship, the height of the relative limit compression area of the bonded steel reinforced prestressed concrete beam is obtained as follows:
Figure FDA0002277644940000011
wherein h is the height of the cross section of the bonded steel reinforced prestressed concrete beam, h0Reinforcing the effective equivalent height h of the prestressed concrete beam section for the bond steelsThe distance f between the resultant force action point of the common tension steel bar and the edge of the concrete compression zonesyDesigned value for yield strength of common steel bar, EsIs the elastic modulus of common steel bars, β is the height x of the compression zone of a rectangular stress diagrambAnd a neutral axis height xcbRatio of (e ∈)syIs the yield tensile strain of ordinary tensioned steel bars.
4. The method for calculating the height of the relative limit compression zone of the viscous steel reinforced prestressed concrete beam according to claim 2, characterized in that when the concrete in the compression zone reaches its limit compression strain epsiloncuWhen the prestressed reinforcing steel bar is used, the common tensile reinforcing steel bar and the adhered steel plate firstly yield, the prestressed reinforcing steel bar finally yields, and the adhered steel reinforces the prestress under the condition that the prestressed reinforcing steel bar with an obvious yield step exists at the momentThe height of the concrete beam relative limit compression zone is as follows:
Figure FDA0002277644940000021
under the condition of no obvious yield step prestressed tendon, the height of the relative boundary compression zone of the bonded steel reinforced prestressed concrete beam is as follows:
Figure FDA0002277644940000022
in the formula, σp0、εp0Stress and strain of the prestressed tendon when the concrete section stress is zero at the resultant force action position of the prestressed tendon, respectivelyp0＝σp0/Ep，EpThe modulus of elasticity of the prestressed reinforcement; epsilonpyThe yield tensile strain of the prestressed reinforcement; h is0Reinforcing the effective equivalent height h of the prestressed concrete beam section for the bond steelpThe distance between the acting point of the prestressed reinforcement and the edge of the concrete compression area, epsilonpyβ is the height x of compression zone of rectangular stress diagram for the yield tensile strain of prestressed reinforcementbAnd a neutral axis height xcbRatio of (a) to (b), fpyThe yield strength of the prestressed reinforcement.
5. The method for calculating the height of the relative limit compression zone of the viscous steel reinforced prestressed concrete beam according to claim 2, characterized in that when the concrete in the compression zone reaches its limit compression strain epsiloncuWhen the prestressed concrete beam is in a plane geometric relationship, the height of a relative limit compression area of the bonded steel reinforced prestressed concrete beam is obtained by the following steps:
Figure FDA0002277644940000023
in the formula, σs0、εs0Respectively the initial stress and strain, epsilon, of a common tensioned steel bar when the steel plate is adhereds0＝σs0/Es，EsIs the elastic modulus of common steel bars; h is0Reinforcing the effective equivalent height of the prestressed concrete beam section, fayThe design value of the yield strength of the adhered steel plate is shown, β is the height x of the compression zone of the rectangular stress diagrambAnd a neutral axis height xcbRatio of (e ∈)ayThe yield tensile strain of the bonded steel plate.
6. The method for calculating the height of the relative limit compression area of the bonded steel reinforced prestressed concrete beam as claimed in claim 1, wherein the relative residual bending resistance bearing capacity M provided by the ordinary tension steel barsThe expression of (a) is:
Figure FDA0002277644940000031
in the formula, EsIs the modulus of elasticity of ordinary steel bar, EaModulus of elasticity, ε, for sticking steel platess0Strain of reinforcing bars in original concrete beam h when steel plates are adheredsThe distance between the resultant force action point of common steel bars and the edge of the concrete compression zone, x is the height of the compression zone of the bonded steel reinforced prestressed concrete beam, As0The area of the section of the original common longitudinal tension steel bar; a. thea0The cross section area of the adhered steel plate;
relative residual bending resistance bearing capacity M provided by prestressed tendonspThe expression of (a) is:
Figure FDA0002277644940000032
in the formula, αEpa＝Ep/Ea，αAPa＝Ap0/Aay，EpIs the modulus of elasticity, h, of the prestressed reinforcementpThe distance between the acting point of the prestressed reinforcement and the edge of the concrete compression area, epsilonpyFor the yield tensile strain of the prestressed reinforcement, Ap0The area of the original prestressed tendon section is obtained;
relative residual bending resistance bearing capacity M provided by pasting steel plateaExpression (2)Comprises the following steps:
Figure FDA0002277644940000033
in the formula, epsilonayThe yield tensile strain of the bonded steel plate.
7. The method for calculating the height of the compression zone of the relative limit of the bonded steel reinforced prestressed concrete beam as claimed in claim 3, wherein in step S5, when the ordinary tension steel bar is finally yielded, the strain epsilon generated at the edge of the tension zone when the bonded steel reinforced prestressed concrete beam is damagedsThe expression of (a) is:
εs＝εsy。
8. the method for calculating the height of the relative limit compression zone of the bonded steel reinforced prestressed concrete beam as claimed in claim 4, in step S5, when the prestressed reinforcement is finally yielding, for the prestressed reinforcement with obvious yielding step, the strain epsilon produced by the edge of the tension zone when the bonded steel reinforced prestressed concrete beam is damagedsThe expression of (a) is:
Figure FDA0002277644940000034
for the prestressed reinforcing steel without obvious yield step, the strain epsilon generated at the edge of the tension area when the bonded steel reinforced prestressed concrete beam is damagedsThe expression of (a) is:
Figure FDA0002277644940000035
9. the method for calculating the height of the compression zone of the opposite boundary of the bonded steel reinforced prestressed concrete beam as claimed in claim 5, wherein in step S5, when the bonded steel plate is finally yielding, the strain epsilon generated at the edge of the tension zone when the bonded steel reinforced prestressed concrete beam is damagedsThe expression of (a) is:
εs＝εs0+εay。
10. the method for calculating the height of the relative limit compression area of the bonded steel reinforced prestressed concrete beam according to claim 1, wherein the step S6 specifically comprises the following steps:
height ξ of relative limit compression zone of obtained bonded steel reinforced prestressed concrete beambJudging whether the height of the compression area of the bond steel reinforced prestressed concrete beam meets the structural requirement, if so, performing bending design on the bond steel reinforced prestressed concrete beam according to the data, if not, returning to the step S1, and reselecting material parameters, wherein the structural requirement is limited by the condition that the height x of the compression area of the bond steel reinforced prestressed concrete beam meets the condition that x is not more than ξbh0。
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