CN102359229B - Suitable method for determining ideal ratio of reinforcement of reinforced concrete beam under usage of bending moment - Google Patents

Abstract

The invention discloses a suitable method for determining an ideal ratio of reinforcement of a reinforced concrete beam under usage of a bending moment. A group of rectangular cross section reinforced concrete beams is prepared; concrete strength, steel bar grade, beam length, cross section dimension and protective layer thickness of each beam are kept the same, and ratio of reinforcement rhoi of each beam is between 0.2% to 2% from small to large; then a simply supported two point symmetrical loading test is carried on each beam; the obtained data is utilized to calculate an experience formula of the ideal ratio of reinforcement rho' of the reinforced concrete beam under usage of the bending moment. The obtained experience formula of rho' has great significance to design of a reinforced concrete structure.

Description

The usability methods of the desirable reinforcement ratio of steel concrete beam under a kind of definite use moment of flexure

Technical field

The present invention relates to reinforced concrete beam in definite method of using desirable reinforcement ratio under the moment of flexure.

Background technology

Because concrete (abbreviation concrete) contraction of coarse aggregate and cement mortar in process of setting is poor, and the microstress field action that produces of inhomogeneous temperature, humidity place, structural concrete is before bearing load or external force, the inner microcrack that has just had some dispersions, under load or External Force Acting, these microcracks will increase and expand gradually, develop into gradually macrocrack by micro crack, until final member is destroyed, therefore, in commission reinforced concrete beam (being called for short steel concrete beam) all is in the work with cracking state usually.Cracking is subjected to change the bending rigidity of the position of bar bending concrete beam neutral line, the macro-mechanical property that affects reinforced concrete beam, weakening reinforced concrete beam.Usually, adopt prefabricated rectangle cross section REINFORCED CONCRETE BEAM WITH SINGLE REINFORCEMENT (namely only configuring the beam of longitudinal reinforcement in the tensile region of beam), carry out the test of freely-supported two point symmetry load bendings, the research cracking is on the impact of reinforced concrete beam mechanical property, as shown in Figure 1.For the bend test of under-reinforced beam (being the suitable beam of arrangement of reinforcement ratio), the concrete strain of tensile region at first reaches its tension failure strain value, and strain stress namely ftractures _Cr, corresponding moment of flexure is cracking moment M _CrContinue to load, draw the concrete in district macroscopic crack to occur, along with the carrying out that loads, tensile reinforcement stress reaches yield strength f _y, corresponding moment of flexure is yield moment M _y, and this moment, the compressive concrete strain not yet reached the Compressive failure strain stress _Cu, claim again that therefore under-reinforced beam is the low amount of reinforcing bar; Along with proceeding of loading, the compressive concrete strain will reach the Compressive failure strain stress _Cu, corresponding moment of flexure is breaking bending moment M _uThe destruction of under-reinforced beam is generally a kind of ductile failure pattern,, destroys front reinforcing bar strain large that is, so beam has very large distortion before destroying, is also referred to as tensile failure.If the experimental test result of the amount of deflection w of moment M and beam is depicted as coordinate diagram M (w), the M of under-reinforced beam (w) figure presents the Changing Pattern of tri linear form usually, and the moment M that applies reaches breaking bending moment M _uAfter, moment M will downward trend occur along with the increase of amount of deflection w, and namely having a Maximum bending moment on M (w) figure (is breaking bending moment M _u), as shown in Figure 2.For the design of under-reinforced beam, usually wish M _Cr≈ (0.2～0.3) M _u, M _y≈ (0.9～0.95) M _u, such moment of flexure increment Delta M=M _y-M _CrCan try one's best large, and wish to use moment of flexure (being the maximum service moment of flexure) to be M _k≈ (0.5～0.6) M _uTherefore, we can use by research the desirable reinforcement ratio of reinforced concrete beam under the moment of flexure, and the design of holding better the actual reinforcement ratio of steel concrete under-reinforced beam improves the reasonability of Reinforced Concrete Structure design.

Usually, the designer of reinforced concrete structure wishes and can according to certain design parameters, directly determine the steel concrete under-reinforced beam in the actual reinforcement ratio of using under the moment of flexure very much.Yet, the most Test And Research Work all is based on the modulus elasticity theory such as classical, in the basic unit of result of the test, qualitative or cracking is discussed quantitatively to using the influence degree of reinforced concrete beam bending rigidity under the moment of flexure, and these achievements in research are very inconvenient to the design and analysis that instructs reinforced concrete structure.As everyone knows, experimental study will consume a large amount of expense inputs.For reaching the purpose that improves the experimental study business efficiency, can be so that disposable economic input, obtain the permanent experimental study achievement that conveniently instructs design of reinforced concrete structure and analysis, this field is in the urgent need to new experimental study method, with accuracy and the convenience demand that satisfies design and analysis work.

Summary of the invention

The problem and shortage part of bringing in order to overcome modulus elasticity theory such as having Test And Research Work employing classics now, the present invention is based on the tension and compression different modulus elasticity theory, the usability methods of the desirable reinforcement ratio of steel concrete beam under a kind of definite use moment of flexure has been proposed: prefabricated one group of rectangular reinforced concrete beam with single reinforcement, and it is carried out freely-supported two point symmetry load tests, as shown in Figure 1, obtain the coordinate graph of a relation M (w) that each root reinforced concrete beam moment M changes with bottom amount of deflection w in the girder span, M (w) figure that those moment M and amount of deflection w is presented the tri linear Variation Regularity of Morphological Characteristics uses as result of the test, as shown in Figure 2, get that Maximum bending moment is the breaking bending moment M of corresponding beam among these figure _uAnd get (0.5～0.6) M (i), _u(i) be the use moment M of corresponding beam _k(i), corresponding M _k(i) the bottom deflection value is w in the girder span _k(i).Because the nip of each beam only has concrete, so the pressurized elastic mould value

Can be taken as concrete elastic modulus E _c, so by M _k(i) and w _k(i) can try to achieve the tension elastic mould value in every Liang La district

Recycle the reinforcement ratio ρ of each root beam _iAnd correspondence

Calculated value, try to achieve regression equation E ⁺Factor alpha among=α ρ+β and β.Then, according to the compressive stress of reinforced concrete beam top concrete and the tensile stress of lower curtate tensile reinforcement, the principle that should keep balanced growth in the deflection of beam loading procedure is tried to achieve reinforced concrete beam in the empirical formula of using the desirable reinforcement ratio ρ ' under the moment of flexure.Like this, the designer only need to be according to design parameters, just can determine easily designed reinforced concrete beam at the desirable reinforcement ratio ρ ' that uses under the moment of flexure, then just can reasonably determine the actual reinforcement ratio ρ of a under-reinforced beam according to ρ ', thereby avoid because designing the unreasonable waste of material that causes.And for the reinforced concrete beam under the condition, the empirical formula of the desirable reinforcement ratio ρ ' that disposable test obtains can be used as permanent use.

The technical solution adopted for the present invention to solve the technical problems is: the REINFORCED CONCRETE BEAM WITH SINGLE REINFORCEMENT of making n root square-section; n 〉=12 wherein; allow concrete strength, reinforcing bar grade, beam length, deck-siding, deck-molding and the protective layer thickness of all beams substantially be consistent, and allow the reinforcement ratio ρ of each root beam _iBe distributed in from small to large in 0.2% to 2% the scope.All beams are carried out the test of freely-supported two point symmetry load bendings, as shown in Figure 1, and record the coordinate graph of a relation M (w) that each root beam moment M changes with bottom amount of deflection w in the girder span.Choose M (w) figure that those moment M and amount of deflection w present the tri linear Variation Regularity of Morphological Characteristics and use as result of the test, as shown in Figure 2, the Maximum bending moment of getting among these figure is the breaking bending moment M of corresponding beam _uAnd get (0.5～0.6) M (i), _u(i) be the use moment M of corresponding beam _k(i), corresponding M _k(i) the bottom deflection value is w in the girder span _k(i).Theoretical according to the small deflection plain bending of shallow beam, every simply supported beam is under load action, and beam can deflection, and is in the bottom tension and the stress of upper portion pressurized, thereby forms neither the also neutral line of pressurized not of tension.Suppose that the tension and compression modulus of elasticity is designated as

With

Because the nip of every beam only has concrete, so the pressurized elastic mould value

Can be taken as concrete elastic modulus E _cEvery simply supported beam span length is designated as l, deck-siding and is designated as the tensile region height that b, deck-molding be designated as h, beam and is designated as h ₁(i), the depth of compression zone of beam is designated as h ₂(i), using moment M _k(i) radius of curvature of lower neutral line is designated as R (i), is using moment M _k(i) underbeam span centre bottom deflection value is designated as w _k(i), therefore h=h is arranged ₁(i)+h ₂(i).

According to tension and compression different modulus pure bending beam theoretical (C.A. A Mubaerchumiyang work. Wu Ruifeng, Zhang Yunzhen etc. translate. different modulus elasticity theory [M]. and Beijing: China Railway Press, 1986.), investigation point for distance neutral line y place, its strain can be expressed as e=y/R, therefore, the above (h of neutral line ₂≤ y＜0) longitudinal fiber is pressurized, and the following (0＜y≤h of neutral line ₁) longitudinal fiber be tension.According to generalized elastic laws, the normal stress σ of tensile region ⁺Normal stress σ with pressure zone ^-Should be respectively

$\left\{\begin{array}{cc}{\mathrm{\&sigma;}}^{+}=E^{+}\frac{y}{R},& 0<y\&le;{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1\\ {\mathrm{\&sigma;}}^{-}=E^{-}\frac{y}{R},& -{\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_2\&le;y<0\end{array}\right.$

The projection of all normal force on the x axle equals zero, and the moment M that their moment equals to act on can obtain following equilibrium equation like this

${\&Integral;}_{-{\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_2}^0{\mathrm{\&sigma;}}_x^{-}\mathrm{bdy}+{\&Integral;}_0^{{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1}{\mathrm{\&sigma;}}_x^{+}\mathrm{bdy}=0$

And

${\&Integral;}_{-{\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_2}^0{\mathrm{\&sigma;}}_x^{-}\mathrm{ybdy}+{\&Integral;}_0^{{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1}{\mathrm{\&sigma;}}_x^{+}\mathrm{ybdy}=M$

By above expression formula, we can push away

Simultaneous equation h=h so ₁+ h ₂After, can get

$h_1=\frac{\sqrt{E^{-}}}{\sqrt{E^{+}}+\sqrt{E^{-}}}h,$ $h_2=\frac{\sqrt{E^{+}}}{\sqrt{E^{+}}+\sqrt{E^{-}}}h$

In addition, can also be pushed away by above

$\frac{1}{R}[\frac{E^{-}{\mathrm{<figure-callout\; id="2"\; label="bh"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">bh</figure-callout>}}_2^3}{3}+\frac{E^{+}{\mathrm{<figure-callout\; id="1"\; label="bh"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">bh</figure-callout>}}_1^3}{3}]=M$

If tension and compression different modulus pure bending beam is introduced the bending stiffness concept Then following formula can be changed into the deformation formula D=RM of well-known pure bending beam.Therefore, we have

$\left\{\begin{array}{cc}{\mathrm{\&sigma;}}^{+}=\frac{E^{+}}{D}\mathrm{My}& 0<y\&le;{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1\\ {\mathrm{\&sigma;}}^{-}=\frac{E^{-}}{D}\mathrm{My}& -{\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_2\&le;y<0\end{array}\right.$

Because the nip of every beam only has concrete, so pressurized elastic mould value E ^-Can be taken as concrete elastic modulus E _cThereby the compressive concrete maximum crushing stress of beam is

${\mathrm{\&sigma;}}_c^{-}=\frac{\sqrt{E^{+}}+\sqrt{E_c}}{\sqrt{E^{+}}}\frac{3\mathrm{<figure-callout\; id="2"\; label="M\; bh"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">M</figure-callout>}}{{\mathrm{bh}}^{}}$

Obviously, maximum crushing stress

Appear at the longitudinal fiber layer of back section.And at La Qu, maximum stretching strain Appear at the longitudinal fiber layer of beam lower curtate

${\mathrm{\&epsiv;}}_{\max }^{+}=\frac{\sqrt{E^{+}}+\sqrt{E_c}}{\sqrt{E_c}{E}^{+}}\frac{3\mathrm{<figure-callout\; id="2"\; label="M\; bh"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">M</figure-callout>}}{{\mathrm{bh}}^{}}$

For simplicity, ignore the topping of reinforcing bar, so the tensile stress of reinforcing bar

Can be approximated to be

${\mathrm{\&sigma;}}_s^{+}=E_s{\mathrm{\&epsiv;}}_{\max }^{+}=\frac{\sqrt{E^{+}}+\sqrt{E_c}}{\sqrt{E_c}{E}^{+}}\frac{{3E}_s\mathrm{<figure-callout\; id="2"\; label="M\; bh"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">M</figure-callout>}}{{\mathrm{bh}}^{}}$

Wherein, E _sModulus of elasticity for tensile reinforcement.So, for above-mentioned test beam, we have

$h_1\left(i\right)=\frac{\sqrt{E_i^{-}}}{\sqrt{E_i^{+}}+\sqrt{E_i^{-}}}h,$ $h_2\left(i\right)=\frac{\sqrt{E_i^{+}}}{\sqrt{E_i^{+}}+\sqrt{E_i^{-}}}h$

And

$\frac{E_i^{-}{\mathrm{<figure-callout\; id="2"\; label="bh"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">bh</figure-callout>}}_2^3\left(i\right)}{3}+\frac{E_i^{+}{\mathrm{<figure-callout\; id="1"\; label="bh"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">bh</figure-callout>}}_1^3\left(i\right)}{3}=R\left(i\right)M_k\left(i\right)$

Geometrical relationship during according to the beam deflection can get

$R\left(i\right)=\frac{w_k^2\left(i\right)+\frac{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">l</figure-callout>}}^2}{4}}{{2w}_k\left(i\right)-h_1\left(i\right)}$

Consider We finally can obtain so

${\mathrm{bh}}^3\frac{E_i^{+}{E}_c}{{(\sqrt{E_i^{+}}+\sqrt{E_c})}^2}+3h{M}_k\left(i\right)\frac{\sqrt{E_c}}{\sqrt{E_i^{+}}+\sqrt{E_c}}=3M_k\left(i\right)\frac{4w_k{\left(i\right)}^2+{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">l</figure-callout>}}^2}{{8w}_k\left(i\right)}$

With M _k(i) and w _k(i) the above equation of substitution then can be tried to achieve the tension elastic mould value in every Liang La district

Utilize the reinforcement ratio ρ of each root beam _iAnd correspondence

Calculated value, can try to achieve regression equation E ⁺Factor alpha among=α ρ+β and β,

$\mathrm{\&alpha;}=\frac{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i^{+}\&CenterDot;{\mathrm{\&rho;}}_i-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i^{+}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i}{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i^2-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i},$ $\mathrm{\&beta;}=\frac{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i^2\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i^{+}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i^{+}\&CenterDot;{\mathrm{\&rho;}}_i\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i}{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i^2-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i}$

Suppose that concrete compressive strength is designated as f _c, reinforcing bar yield strength be designated as f _y, so, with regard to the design of reinforced concrete beam, usually wish the concrete compressive stress of back section

Tensile stress with the lower curtate tensile reinforcement

In the deflection of beam loading procedure, can keep balanced growth, namely have

Therefore set up, can be pushed away by above

$E^{+}=\frac{E_s^2{f}_c^2}{E_c{f}_y^2}$

Consider the above regression equation E that is tried to achieve by experimental data ⁺=α ρ+β, and take into account existing design specifications to the consideration of under-reinforced beam design, we will satisfy

The reinforcement ratio of condition is designated as ρ ', and is called desirable reinforcement ratio.Obviously, for under-reinforced beam, actual reinforcement ratio ρ should be slightly less than this desirable reinforcement ratio ρ ', and in other words, this desirable reinforcement ratio ρ ' can be used as the design considerations of determining the actual reinforcement ratio ρ of under-reinforced beam.Like this, we finally have

${\mathrm{\&rho;}}^{\&prime;}=\frac{E_s^2{\mathrm{<figure-callout\; id="2"\; label="f\; c"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">f</figure-callout>}}_c^{}}{\mathrm{\&alpha;}{E}_c{f}_y^2}-\frac{\mathrm{\&beta;}}{\mathrm{\&alpha;}}$

As long as with α, β, E _c, E _s, f _c, f _yThe above expression formula of substitution just can obtain using the desirable reinforcement ratio ρ ' of reinforced concrete beam under the moment of flexure.The experimental data used under the moment of flexure tries to achieve because parameter alpha and β are based on, therefore claim ρ ' for reinforced concrete beam in the desirable reinforcement ratio of using under the moment of flexure.The unit of all physical quantitys all adopts the International System of Units.

The invention has the beneficial effects as follows: only need to be according to design parameters, just can determine easily designed reinforced concrete beam at the desirable reinforcement ratio ρ ' that uses under the moment of flexure, then just can reasonably determine the actual reinforcement ratio ρ of a under-reinforced beam according to ρ ', thereby avoid because designing the unreasonable waste of material that causes.And for the reinforced concrete beam under the same terms, the empirical formula of the desirable reinforcement ratio ρ ' that disposable test obtains can be used as permanent use, thereby has reached the purpose that improves the experimental study business efficiency.

Description of drawings

Fig. 1 is the mechanical model of reinforced concrete rectangular beam under two point symmetries load of the both sides freely-supported that adopts of the present invention.Among the figure, x, y, z are that rectangular co-ordinate, l are that simply supported beam span length, a are that the shear span of beam is long, b is that deck-siding, h are deck-molding, h ₁Tensile region height, h for beam ₂For depth of compression zone, the P of beam is two point loads that two point symmetries apply when loading.

Fig. 2 is M (w) schematic diagram that moment M and amount of deflection w present the tri linear Variation Regularity of Morphological Characteristics.Among the figure, " 1 " is first obvious turning point that load on M (w) figure-deflection curve occurs, and the bottom concrete strain of signal beam reaches cracking strain (being concrete pulling strain limit value); " 2 " are second obvious turning point that load on M (w) figure-deflection curve occurs, and the strain of signal longitudinal tensile reinforcing bar reaches yield strain; " 3 " are the 3rd the obvious turning point that load on M (w) figure-deflection curve occurs, and the top concrete strain of signal beam reaches failure strain (being concrete compressive strain limiting value); M _Cr(i) represent the cracking moment of each root beam, the bottom deflection value is w in the corresponding girder span _Cr(i); M _k(i) represent the use moment of flexure of each root beam, the bottom deflection value is w in the corresponding girder span _k(i); M _y(i) represent the yield moment of each root beam, the bottom deflection value is w in the corresponding girder span _y(i); M _u(i) represent the breaking bending moment of each root beam, the bottom deflection value is w in the corresponding girder span _u(i).

The specific embodiment

Make the reinforced concrete beam of n root square-section, wherein n 〉=12 allow concrete strength, reinforcing bar grade, beam length, deck-siding, deck-molding and the protective layer thickness of all beams substantially be consistent.All beams are carried out freely-supported two point symmetry load tests, as shown in Figure 1, the shear span that span length when l carries out freely-supported two point symmetry load test for all reinforced concrete beams, a are beam is long, b is that deck-siding, h are deck-molding, the point load that applies when P is the loading of two point symmetries among the figure, therefore between two Concentrated load points that apply, be pure bending of beam test section, its moment M=aP.The reinforced concrete beam that the test of each root is used only at tension side configuration longitudinal reinforcement, and allows the reinforcement ratio ρ of each root beam _iBe distributed in from small to large in 0.2% to 2% the scope, and in the shear span district, the used reinforced concrete beam of each root test all disposes enough stirrup amounts, so that shear failure does not occur in loading procedure the guarantee test beam.By load test, record the coordinate graph of a relation M (w) that each root beam moment M changes with bottom amount of deflection w in the girder span.Choose M (w) figure that those moment M and amount of deflection w present the tri linear Variation Regularity of Morphological Characteristics and use as result of the test, as shown in Figure 2, the Maximum bending moment of getting among these figure is the breaking bending moment M of corresponding beam _uAnd get (0.5～0.6) M (i), _u(i) be the use moment M of corresponding beam _k(i), corresponding M _k(i) the bottom deflection value is w in the girder span _k(i).Get the pressurized elastic mould value

Be all concrete elastic modulus E _c, with l, b, h, E _cAnd M _k(i) and w _k(i) the following equation of substitution

${\mathrm{bh}}^3\frac{E_i^{+}{E}_c}{{(\sqrt{E_i^{+}}+\sqrt{E_c})}^2}+3h{M}_k\left(i\right)\frac{\sqrt{E_c}}{\sqrt{E_i^{+}}+\sqrt{E_c}}=3M_k\left(i\right)\frac{4w_k{\left(i\right)}^2+{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">l</figure-callout>}}^2}{{8w}_k\left(i\right)},$

Try to achieve respectively the tension elastic mould value in each root Liang La district

Reinforcement ratio ρ with each root beam _iAnd correspondence

The following formula of calculated value substitution

$\mathrm{\&alpha;}=\frac{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i^{+}\&CenterDot;{\mathrm{\&rho;}}_i-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i^{+}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i}{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i^2-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i},$ $\mathrm{\&beta;}=\frac{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i^2\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i^{+}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i^{+}\&CenterDot;{\mathrm{\&rho;}}_i\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i}{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i^2-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i}$

Try to achieve factor alpha and β, again with α, β, E _c, E _s, f _c, f _yThe substitution following formula

${\mathrm{\&rho;}}^{\&prime;}=\frac{E_s^2{\mathrm{<figure-callout\; id="2"\; label="f\; c"\; filenames="HDA0000081503960000012.png"\; state="\{\{state\}\}">f</figure-callout>}}_c^{}}{\mathrm{\&alpha;}{E}_c{f}_y^2}-\frac{\mathrm{\&beta;}}{\mathrm{\&alpha;}}$

Try to achieve the desirable reinforcement ratio ρ ' that uses reinforced concrete beam under the moment of flexure, wherein, E _sBe the modulus of elasticity of tensile reinforcement, f _cBe concrete compressive strength, f _yYield strength for tensile reinforcement.The unit of all physical quantitys all adopts the International System of Units.

Claims (1)

1. usability methods that determine to use the desirable reinforcement ratio of steel concrete beam under the moment of flexure; it is characterized in that: make n root rectangular reinforced concrete beam; n 〉=12 wherein; every beam is only at tension side configuration longitudinal reinforcement; allow concrete strength, reinforcing bar grade, beam length, deck-siding, deck-molding and the protective layer thickness of all beams substantially be consistent, and allow the reinforcement ratio ρ of each root beam _iBe distributed in from small to large in 0.2% to 2% the scope, all beams are carried out freely-supported two point symmetry load tests, record the coordinate graph of a relation M (w) that each root beam moment M changes with bottom amount of deflection w in the girder span, choose M (w) figure that those moment M and amount of deflection w present the tri linear Variation Regularity of Morphological Characteristics and use as result of the test, the Maximum bending moment of getting among these figure is the breaking bending moment M of corresponding beam _uAnd get (0.5～0.6) M (i), _u(i) be the use moment M of corresponding beam _k(i), corresponding M _k(i) the bottom deflection value is w in the girder span _k(i), the pressurized elastic mould value of all beams

All be taken as concrete elastic mould value E _c, with E _cAnd M _k(i) and w _k(i) substitution equation

${\mathrm{bh}}^3\frac{E_i^{+}{E}_c}{{(\sqrt{E_i^{+}}+\sqrt{E_c})}^2}+3h{M}_k\left(i\right)\frac{\sqrt{E_c}}{\sqrt{E_i^{+}}+\sqrt{E_c}}=3M_k\left(i\right)\frac{4w_k{\left(i\right)}^2+l^2}{{8w}_k\left(i\right)},$

Try to achieve the tension elastic mould value of corresponding beam

Wherein, span length, the b when l carries out freely-supported two point symmetry load test for all reinforced concrete beams is that deck-siding, h are deck-molding, with the reinforcement ratio ρ of each root beam _iAnd correspondence

The following formula of calculated value substitution,

$\mathrm{\&alpha;}=\frac{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i^{+}\&CenterDot;{\mathrm{\&rho;}}_i-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i^{+}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i}{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i^2-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i},$ $\mathrm{\&beta;}=\frac{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i^2\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i^{+}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i^{+}\&CenterDot;{\mathrm{\&rho;}}_i\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i}{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i^2-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&rho;}}_i}$

Try to achieve factor alpha and β, again with α, β, E _c, E _s, f _c, f _yThe substitution following formula,

${\mathrm{\&rho;}}^{\&prime;}=\frac{E_s^2{f}_c^2}{\mathrm{\&alpha;}{E}_c{f}_y^2}-\frac{\mathrm{\&beta;}}{\mathrm{\&alpha;}},$

Try to achieve reinforced concrete beam at the desirable reinforcement ratio ρ ' that uses under the moment of flexure, wherein, E _sBe the modulus of elasticity of tensile reinforcement, f _cBe concrete compressive strength, f _yBe the yield strength of tensile reinforcement, the unit of all physical quantitys all adopts the International System of Units.

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