CN102359228A - Method for determining relationship between cracking height and reinforcement ratio of steel-concrete beam under use bending moment - Google Patents

Abstract

The invention discloses a method for determining relationship between cracking height and reinforcement ratio of a steel-concrete beam under a use bending moment. The method comprises the following steps of: making a group of steel-concrete beams with rectangular sections, wherein concrete strengths, reinforcing steel bar grades, beam lengths, section sizes and protective layer thicknesses of the beams are kept consistent with each other, and reinforcement ratios pi of the beams are distributed within a range of 0.2-2% from small to large; and then carrying out symmetric loading test to two simply support points of each beam; and by using the data obtained through test, figuring out an analytical expression Hcr (p) of the variation of a ratio Hcr between the cracking height hcr and beam height h of the steel-concrete beam under the use bending moment along with the reinforcement ratio p, wherein the obtained Hcr (p) is significant to the design and analysis of the steel-concrete structure.

Description

The method of steel concrete beam cracking height and reinforcement ratio relation under a kind of definite use moment of flexure

Technical field

The present invention relates to reinforced concrete beam in definite method of using cracking height and reinforcement ratio relation 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 of uneven temperature, the generation of 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 effect, these microcracks will increase and expand gradually, develop into macrocrack gradually by micro crack; Be destroyed up to final member; Therefore, in commission reinforced concrete beam (being called for short steel concrete beam) all is in band crack duty usually.Cracking will influence the macro-mechanical property of reinforced concrete beam, the bending rigidity of weakening reinforced concrete beam.Usually, adopt prefabricated rectangle cross section REINFORCED CONCRETE BEAM WITH SINGLE REINFORCEMENT (promptly only disposing the beam of longitudinal reinforcement in the tensile region), carry out the test of freely-supported two point symmetry load bendings, the research cracking is to the influence of reinforced concrete beam mechanical property, and is as shown in Figure 1.As far as 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 promptly 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 nip concrete strain did not reach resistance to compression failure strain ε as yet _Cu, claim again that therefore under-reinforced beam is the low amount of reinforcing bar; Proceed along with what load, the nip concrete strain will reach resistance to compression failure strain ε _Cu, corresponding moment of flexure is breaking bending moment M _uThe destruction of under-reinforced beam is generally a kind of ductile failure pattern, that is, it is big to destroy preceding reinforcing bar strain, so beam has very big distortion before destroying, is also referred to as pulling force and destroys.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) schemes to demonstrate usually the Changing Pattern of tri linear form, and the moment M that is applied reaches breaking bending moment M _uAfter, moment M will downward trend occur along with the increase of amount of deflection w, and promptly having a maximal bending moment value on M (w) figure (is breaking bending moment M _u), as shown in Figure 2.As far as the design of under-reinforced beam, hope M usually _Cr≈ (0.2～0.3) M _u, M _y≈ (0.9～0.95) M _u, moment of flexure increment Delta M=M like this _y-M _CrIt is big to try one's best, and hopes to use moment of flexure (being the maximum service moment of flexure) to be M _k≈ (0.5～0.6) M _uTherefore, we can hold the macro-mechanical property of reinforced concrete structure (abbreviation Reinforced Concrete Structure) better through the relation of reinforced concrete beam cracking height under the research use moment of flexure with reinforcement ratio, improve the Reinforced Concrete Structure designed rationality.

Usually, the designer of reinforced concrete structure hopes and can directly determine reinforced concrete beam in the cracking degree of using under the moment of flexure according to certain design parameters very much.Yet; Present most of experimental study work all is based on 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 make disposable economy drop into; Obtain the permanent experimental study achievement that conveniently instructs the reinforced concrete structure design and analysis; This field presses for 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 experimental study work employing classics now; The present invention is based on the tension and compression different modulus elasticity theory; The method of steel concrete beam cracking height and reinforcement ratio relation under a kind of definite use moment of flexure has been proposed: prefabricated one group of 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, those moment M are schemed to use as result of the test with the M (w) that amount of deflection w demonstrates tri linear metamorphosis rule; As shown in Figure 2, get these figure in the maximal bending moment value be 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).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 elastic mould value that drawn in every Liang La district Then, utilize the reinforcement ratio ρ of each root beam _iAnd correspondence

Calculated value and concrete elastic modulus E _cElastic modulus E with tensile reinforcement _s, try to achieve cracking height h _CrRatio H with deck-molding h _CrAnalytical expression H with reinforcement ratio ρ variation _Cr(ρ).Like this; The designer only need just can determine the cracking degree of reinforced concrete beam under the use moment of flexure that is designed according to design parameters reinforcement ratio ρ easily, and; As far as with the reinforced concrete beam under the condition, the empirical formula H that disposable test obtained _Cr(ρ), 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 manufacturing n root square-section, wherein n>=12 let concrete strength, reinforcing bar grade, beam length, deck-siding, deck-molding and the protective layer thickness of all beams be consistent basically, and let 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 demonstrate tri linear metamorphosis rule and use as result of the test, as shown in Figure 2, the maximal bending moment value 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 and is drawn and the stress of upper portion pressurized, is neither drawn also the not neutral line of pressurized thereby form.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 neutral line is designated as R (i), is using moment M down _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.), for the investigation point apart from 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 a pressurized, and (0＜y≤h below the neutral line ₁) longitudinal fiber drawn.According to the broad sense law of elasticity, the normal stress σ of tensile region ⁺Normal stress σ with pressure zone ^-Should be respectively

$\left\{\begin{array}{cc}{\mathrm{\&sigma;}}^{+}=E^{+}\frac{y}{R},& 0<y\&le;{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081505510000011.png,HDA0000081505510000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1\\ {\mathrm{\&sigma;}}^{-}=E^{-}\frac{y}{R},& -h_2\&le;y<0\end{array}\right.$

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

${\&Integral;}_{-h_2}^0{\mathrm{\&sigma;}}_x^{-}\mathrm{bdy}+{\&Integral;}_0^{{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081505510000011.png,HDA0000081505510000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1}{\mathrm{\&sigma;}}_x^{+}\mathrm{bdy}=0$

And

${\&Integral;}_{-h_2}^0{\mathrm{\&sigma;}}_x^{-}\mathrm{ybdy}+{\&Integral;}_0^{{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081505510000011.png,HDA0000081505510000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1}{\mathrm{\&sigma;}}_x^{+}\mathrm{ybdy}=M$

By above expression formula, we can push away

Simultaneous equality 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 push away by above

$\frac{1}{R}[\frac{E^{-}{\mathrm{bh}}_2^3}{3}+\frac{E^{+}{\mathrm{<figure-callout\; id="1"\; label="bh"\; filenames="HDA0000081505510000011.png,HDA0000081505510000012.png"\; state="\{\{state\}\}">bh</figure-callout>}}_1^3}{3}]=M$

If to tension and compression different modulus pure bending beam introduce bending stiffness notion

then following formula can be changed into

The deformation formula D=RM of well-known pure bending beam.Therefore, we have

$\left\{\begin{array}{cc}{\mathrm{\&sigma;}}^{+}=\frac{E^{+}}{D}\mathrm{My}& 0<y\&le;{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081505510000011.png,HDA0000081505510000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1\\ {\mathrm{\&sigma;}}^{-}=\frac{E^{-}}{D}\mathrm{My}& -h_2\&le;y<0\end{array}\right.$

Because the nip of every beam only has concrete, so pressurized elastic mould value E ^-Can be taken as concrete elastic modulus E _cThe effective height of supposing tensile region concrete cracking part is designated as h _Cr, and note H _Cr=h _Cr/ h, so in the tensile region, the total pulling force N of all longitudinal fibers ⁺Should be,

$N^{+}={\&Integral;}_0^{{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081505510000011.png,HDA0000081505510000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1}{\mathrm{\&sigma;}}^{+}\mathrm{ds}={\&Integral;}_0^{h_1}{E}^{+}\frac{y}{R}\mathrm{bdy}=\frac{{\mathrm{bE}}^{+}}{2\mathrm{<figure-callout\; id="1"\; label="R\; h"\; filenames="HDA0000081505510000011.png,HDA0000081505510000012.png"\; state="\{\{state\}\}">R</figure-callout>}}{h}_{}^{}{}_1^2$

In fact, this total pulling force N ⁺It is pulling force by tensile reinforcement

With the tensile region do not ftracture the part concrete pulling force

Form, promptly Wherein

$N_s^{+}=E_s\frac{h_1}{R}{A}_S,$ $N_c^{+}=\frac{{\mathrm{bE}}_c}{2R}{\left(h_1{-h}_{\mathrm{cr}}\right)}^2$

In the formula, E _sBe the modulus of elasticity of tensile reinforcement, A _sThe gross area for tensile reinforcement.Therefore, we finally can push away

$\frac{h_{\mathrm{cr}}}{h}=\frac{\sqrt{E_c}}{\sqrt{E^{+}}+\sqrt{E_c}}(1-\sqrt{\frac{E^{+}}{E_c}-2\frac{E_s}{E_c}\frac{\sqrt{E^{+}}+\sqrt{E_c}}{\sqrt{E_c}}\frac{A_S}{\mathrm{bh}}})$

So, as far as 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

$\frac{E_i^{-}{\mathrm{bh}}_2^3\left(i\right)}{3}+\frac{E_i^{+}{\mathrm{<figure-callout\; id="1"\; label="bh"\; filenames="HDA0000081505510000011.png,HDA0000081505510000012.png"\; state="\{\{state\}\}">bh</figure-callout>}}_1^3\left(i\right)}{3}=R\left(i\right)M_k\left(i\right)$

Geometrical relationship during by the beam deflection can get

$R\left(i\right)=\frac{w_k^2\left(i\right)+\frac{l^2}{4}}{{2w}_k\left(i\right)-h_1\left(i\right)}$

We finally can obtain so to consider

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

With M _k(i) and w _k(i) the above equation of substitution then can be tried to achieve the elastic mould value that drawn in every Liang La district Because in most cases, the protective layer thickness of reinforcing bar remains 2.5cm basically, so we can define reinforcement ratio ρ _i=A _s(i)/bh, wherein A _s(i) be the tensile reinforcement gross area of each root beam, utilize the reinforcement ratio ρ of each root beam so _iAnd correspondence

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

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

Consider E ⁺=α ρ+β, we finally can obtain so

$H_{\mathrm{cr}}\left(\mathrm{\&rho;}\right)=\frac{\sqrt{E_c}}{\sqrt{\mathrm{\&alpha;\&rho;}+\mathrm{\&beta;}}+\sqrt{E_c}}(1-\sqrt{\frac{\mathrm{\&alpha;\&rho;}+\mathrm{\&beta;}}{E_c}-2\frac{E_s}{E_c}\frac{\sqrt{\mathrm{\&alpha;\&rho;}+\mathrm{\&beta;}}+\sqrt{E_c}}{\sqrt{E_c}}\mathrm{\&rho;}})$

At last, with α, β, E _c, E _sIn the above expression formula of substitution, just can be in the hope of cracking height h _CrRatio H with deck-molding h _CrAnalytical expression H with reinforcement ratio ρ variation _Cr(ρ), wherein, the unit of all physical quantitys all adopts the International System of Units.

The invention has the beneficial effects as follows: only need be according to design parameters reinforcement ratio ρ; Just can determine the cracking degree of reinforced concrete beam under the use moment of flexure that is designed easily; And, as far as the reinforced concrete beam under the same terms, the empirical formula H that disposable test obtained _Cr(ρ), can be used as permanent use, thereby 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 of the present invention's employing.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 are applied when loading.

Fig. 2 demonstrates M (w) sketch map of tri linear metamorphosis rule for moment M and amount of deflection w.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

The reinforced concrete beam of manufacturing n root square-section, wherein n >=12 let concrete strength, reinforcing bar grade, beam length, deck-siding, deck-molding and the protective layer thickness of all beams be consistent basically.All beams are carried out freely-supported two point symmetry load tests; As shown in Figure 1; Span length when l carries out freely-supported two point symmetry load tests for all reinforced concrete beams, the shear span that a is beam length, b are that deck-siding, h are deck-molding; Therefore the point load that is applied when P is the loading of two point symmetries among the figure is the pure bending test section of beam, its moment M=aP between two point load application points that applied.The reinforced concrete beam that the test of each root is used only at tension side configuration longitudinal reinforcement, and lets 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 take place in loading procedure the guarantee test beam.Through 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 demonstrate tri linear metamorphosis rule and use as result of the test, as shown in Figure 2, the maximal bending moment value 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

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

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

Reinforcement ratio ρ with each root beam _i=A _s(i)/bh and correspondence Calculated value, the following formula of 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 alpha and β, again with the alpha of trying to achieve and β and E _cAnd E _sThe following formula of substitution

$H_{\mathrm{cr}}\left(\mathrm{\&rho;}\right)=\frac{\sqrt{E_c}}{\sqrt{\mathrm{\&alpha;\&rho;}+\mathrm{\&beta;}}+\sqrt{E_c}}(1-\sqrt{\frac{\mathrm{\&alpha;\&rho;}+\mathrm{\&beta;}}{E_c}-2\frac{E_s}{E_c}\frac{\sqrt{\mathrm{\&alpha;\&rho;}+\mathrm{\&beta;}}+\sqrt{E_c}}{\sqrt{E_c}}\mathrm{\&rho;}})$

Try to achieve cracking height h _CrRatio H with deck-molding h _CrAnalytical expression H with reinforcement ratio ρ variation _Cr(ρ), wherein, the unit of all physical quantitys all adopts the International System of Units.

Claims (1)

1. method of confirm using steel concrete beam cracking height and reinforcement ratio relation under the moment of flexure; It is characterized in that: manufacturing n root rectangular reinforced concrete beam; N>=12 wherein; Every beam only at tension side configuration longitudinal reinforcement, lets concrete strength, reinforcing bar grade, beam length, deck-siding, deck-molding and the protective layer thickness of all beams be consistent basically, and lets 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 demonstrate tri linear metamorphosis rule and use as result of the test, the maximal bending moment value 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 elastic mould value that drawn of corresponding beam

Wherein, span length, the b when l carries out freely-supported two point symmetry load tests for all reinforced concrete beams is that deck-siding, h are deck-molding, with the reinforcement ratio ρ of each root beam _iAnd correspondence Calculated value, the following formula of 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 alpha and β, again with α, β, E _c, E _sThe following expression formula of substitution

$H_{\mathrm{cr}}\left(\mathrm{\&rho;}\right)=\frac{\sqrt{E_c}}{\sqrt{\mathrm{\&alpha;\&rho;}+\mathrm{\&beta;}}+\sqrt{E_c}}(1-\sqrt{\frac{\mathrm{\&alpha;\&rho;}+\mathrm{\&beta;}}{E_c}-2\frac{E_s}{E_c}\frac{\sqrt{\mathrm{\&alpha;\&rho;}+\mathrm{\&beta;}}+\sqrt{E_c}}{\sqrt{E_c}}\mathrm{\&rho;}}),$

Try to achieve cracking height h _CrRatio H with deck-molding h _CrAnalytical expression H with reinforcement ratio ρ variation _Cr(ρ), wherein, E _sBe the modulus of elasticity of tensile reinforcement, the unit of all physical quantitys all adopts the International System of Units.

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