CN102409806B - Method for determining relation between neutral layers of reinforced concrete beams with usable bending moment and reinforcement ratios - Google Patents

Abstract

The invention discloses a method for determining the relationship between the neutral layer and reinforcement ratio of a reinforced concrete beam under the use of bending moment. Make a group of rectangular cross-section reinforced concrete beams, the concrete strength, steel bar grade, beam length, section size and protective layer thickness of each beam are consistent, and the reinforcement ratio ρ _i is distributed from 0.2% to 2% from small to large Within the range, then simply support two-point symmetrical loading test for each beam, and use the data obtained from the test to obtain the height h ₁ of the neutral floor distance from the beam and the height h of the beam under the applied bending moment of the steel concrete beam The analytical expression H(ρ) of the ratio H changing with the reinforcement ratio ρ, the obtained H(ρ) is of great significance to the design and analysis of reinforced concrete structures.

Description

A kind of method of steel concrete beam neutral line and reinforcement ratio relation under definite use moment of flexure

Technical field

The present invention relates to reinforced concrete beam in definite method of using neutral line position and reinforcement ratio relation under 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 of some dispersions that just existed, under load or External Force Acting, these microcracks will increase gradually and expand, 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), usually all in the work with cracking state.Cracking is subject 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 (only in the tensile region of beam, configuring the beam of longitudinal reinforcement), carry out freely-supported two point symmetry load bendings tests, the impact of research cracking on the reinforced concrete beam mechanical property, as shown in Figure 1.For the bend test of under-reinforced beam (being the beam that the arrangement of reinforcement ratio is suitable), at first the concrete strain of tensile region reaches its tension failure strain value, and strain stress ftractures _cr, corresponding moment of flexure is cracking moment M _cr; Macroscopic crack appears in the concrete that continues loading ，La district, and along with the carrying out loaded, tensile reinforcement stress reaches yield strength f _y, corresponding moment of flexure is yield moment M _y, and now compressive concrete strain not yet reaches the Compressive failure strain stress _cu, therefore claim again that 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 _u.The destruction of under-reinforced beam is generally a kind of ductile failure pattern,, destroys front reinforcing bar strain large, so beam has very large distortion before destroying, also referred to as tensile failure that is.If the experimental test result of the amount of deflection w of moment M and beam is depicted as to 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 applied reaches breaking bending moment M _uafter, moment M will downward trend occur along with the increase of amount of deflection w, on M (w) figure, having a Maximum bending moment (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, moment of flexure increment Delta M=M like this _y-M _crcan try one's best large, and wish that using moment of flexure (being the maximum service moment of flexure) is M _k≈ (0.5～0.6) M _u.Therefore, we can use the relation of reinforced concrete beam neutral line position and reinforcement ratio under moment of flexure by research, hold better the macro-mechanical property of reinforced concrete structure (abbreviation Reinforced Concrete Structure), improve the reasonability of Reinforced Concrete Structure design.

Usually, the designer of reinforced concrete structure wishes according to certain design parameters, directly to determine reinforced concrete beam in the position of using neutral line under 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 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 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 meets 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, a kind of method of steel concrete beam neutral line and reinforcement ratio relation under definite use moment of flexure has been proposed: prefabricated one group of REINFORCED CONCRETE BEAM WITH SINGLE REINFORCEMENT, and it is carried out to freely-supported two point symmetry load tests, as shown in Figure 1, obtain the coordinate graph of a relation M (w) of each root reinforced concrete beam moment M with amount of deflection w variation in bottom in girder span, M (w) figure that those moment M and amount of deflection w is presented to the tri linear Variation Regularity of Morphological Characteristics is used as result of the test, as shown in Figure 2, get the breaking bending moment M that in these figure, Maximum bending moment is corresponding beam _u, and get (0.5～0.6) M (i) _u(i) be the use moment M of corresponding beam _k(i), corresponding M _k(i) in girder span, the bottom deflection value is w _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 _kand w (i) _k(i) can try to achieve the tension elastic mould value in every Gen Liangla district then, utilize the reinforcement ratio ρ of each root beam _iand correspondence

calculated value, and concrete elastic modulus E _c, try to achieve reinforced concrete beam and using under moment of flexure, the height h that neutral line is low apart from beam ₁the analytical expression H (ρ) changed with reinforcement ratio ρ with the ratio H of deck-molding.Like this, the designer only need to, according to design parameters reinforcement ratio ρ, just can determine designed reinforced concrete beam and use the neutral line height h low apart from beam under moment of flexure easily ₁=H (ρ) h, and for the reinforced concrete beam under same condition, the empirical formula H that disposable test obtains (ρ), can be used as permanent use.

The technical solution adopted for the present invention to solve the technical problems is:

Make the REINFORCED CONCRETE BEAM WITH SINGLE REINFORCEMENT 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, and allow the reinforcement ratio ρ of each root beam _ibe distributed in from small to large in 0.2% to 2% scope.All beams are carried out to freely-supported two point symmetry load bending tests, as shown in Figure 1, and record the coordinate graph of a relation M (w) of each root beam moment M with amount of deflection w variation in bottom in 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, get the breaking bending moment M that the Maximum bending moment in these figure is corresponding beam _u, and get (0.5～0.6) M (i) _u(i) be the use moment M of corresponding beam _k(i), corresponding M _k(i) in girder span, the bottom deflection value is w _k(i).According to the small deflection plain bending theory of shallow beam, every simply supported beam is under load action, and beam can deflection, and 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 _c.Every 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 Mubaerchu meter Yang 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 longitudinal fiber of neutral line is pressurized (h ₂and the following longitudinal fiber of neutral line is tension (0<y≤h≤y<0), ₁).According to generalized elastic laws, the normal stress σ of tensile region ⁺normal stress σ with pressure zone ^-should be respectively

$\left\{\begin{array}{cc}{\mathrm{\&sigma;}}^{+}=E^{+}\frac{y}{R},& 0<y\&le;{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081506760000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1\\ {\mathrm{\&sigma;}}^{-}=E^{-}\frac{y}{R},& -h_2\&le;y<0\end{array}\right.$

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

${\&Integral;}_{-h_2}^0{\mathrm{\&sigma;}}_x^{-}\mathrm{bdy}+{\&Integral;}_0^{{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081506760000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1}{\mathrm{\&sigma;}}_x^{+}\mathrm{bdy}=0$

And

${\&Integral;}_{-h_2}^0{\mathrm{\&sigma;}}_x^{-}\mathrm{ybdy}+{\&Integral;}_0^{{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081506760000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1}{\mathrm{\&sigma;}}_x^{+}\mathrm{ybdy}=M$

By above expression formula, we can push away

simultaneous equation h=h so ₁+ h ₂after, can obtain

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

In addition, by above, can also be pushed away

$\frac{1}{R}[\frac{E^{-}b{h}_2^3}{3}+\frac{E^{+}b{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="HDA0000081506760000012.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^3}{3}]=M$

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

$[\frac{E_i^{-}b{h}_2^3\left(i\right)}{3}+\frac{E_i^{+}\mathrm{<figure-callout\; id="1"\; label="b\; h"\; filenames="HDA0000081506760000012.png"\; state="\{\{state\}\}">b</figure-callout>}{h}_{}^{}{}_1^3\left(i\right)}{3}]=R\left(i\right)M_k\left(i\right)$

Geometrical relationship during according to the beam deflection can obtain

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

Consider we finally can obtain so

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

By M _kand w (i) _k(i) the above equation of substitution, can try to achieve the tension elastic mould value in every Gen Liangla district

utilize

The reinforcement ratio ρ of each root beam _iand correspondence calculated value, can try to achieve regression equation E ⁺factor alpha in=α ρ+β and β,

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

The tensile region height h of beam for let us ₁with the H of the relative position than value representation neutral line of deck-molding h, i.e. H=h ₁/ h, and consider E ^-=E _cand E ⁺=α ρ+β, we finally have so

$H\left(\mathrm{\&rho;}\right)=\frac{h_1}{h}=\frac{\sqrt{E_c}}{\sqrt{\mathrm{\&alpha;\&rho;}+\mathrm{\&beta;}}+\sqrt{E_c}}$

Again by α, β, E _cthe above expression formula of substitution, just can obtain using the analytical expression H (ρ) of the relative position H of reinforced concrete beam neutral line under moment of flexure with reinforcement ratio ρ variation, and the unit of all physical quantitys all adopts the International System of Units.

The invention has the beneficial effects as follows: only need to, according to design parameters reinforcement ratio ρ, just can determine easily designed reinforced concrete beam and use the neutral line height h low apart from beam under moment of flexure ₁=H (ρ) h, and for the reinforced concrete beam under the same terms, the empirical formula H that disposable test obtains (ρ), can be used as permanent use, thus reached the purpose that improves the experimental study business efficiency.

The accompanying drawing explanation

The mechanical model of the reinforced concrete rectangular beam that Fig. 1 is the both sides freely-supported that adopts of the present invention under two point symmetries load.In figure, x, y, z is that rectangular co-ordinate, l are that shear span length, the b that simply supported beam span length, a are beam 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 while 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.In figure, " 1 " is first obvious turning point that load on M (w) figure-deflection curve occurs, 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, 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, the top concrete strain of signal beam reaches failure strain (being concrete compressive strain limiting value); M _cr(i) mean the cracking moment of each root beam, in corresponding girder span, the bottom deflection value is w _cr(i); M _k(i) mean the use moment of flexure of each root beam, in corresponding girder span, the bottom deflection value is w _k(i); M _y(i) mean the yield moment of each root beam, in corresponding girder span, the bottom deflection value is w _y(i); M _u(i) mean the breaking bending moment of each root beam, in corresponding girder span, the bottom deflection value is w _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 to freely-supported two point symmetry load tests, as shown in Figure 1, the shear span that l is the span length of all reinforced concrete beams while carrying out freely-supported two point symmetry load test, a is beam is long, b is that deck-siding, h are deck-molding, the point load applied when in figure, P is two point symmetries loadings, therefore between two Concentrated load points that apply, be the pure bending test section of beam, its moment M=ap.Each root is tested reinforced concrete beam 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% scope, and, in the shear span district, each root is tested reinforced concrete beam used and is all disposed enough stirrup amounts, so that shear failure does not occur the guarantee test beam in loading procedure.By load test, record the coordinate graph of a relation M (w) of each root beam moment M with amount of deflection w variation in bottom in 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, get the breaking bending moment M that the Maximum bending moment in these figure is corresponding beam _u, and get (0.5～0.6) M (i) _u(i) be the use moment M of corresponding beam _k(i), corresponding M _k(i) in girder span, the bottom deflection value is w _k(i).Get the pressurized elastic mould value

be all concrete elastic modulus E _c, by l, b, h, E _cand M _kand w (i) _k(i) the following equation of substitution

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

Try to achieve respectively the tension elastic mould value in each Gen Liangla district

reinforcement ratio ρ by 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, then by the factor alpha and the β that try to achieve, and E _cthe following formula of substitution

$H\left(\mathrm{\&rho;}\right)=\frac{\sqrt{E_c}}{\sqrt{\mathrm{\&alpha;\&rho;}+\mathrm{\&beta;}}+\sqrt{E_c}}$

Try to achieve the analytical expression H (ρ) of the relative position H of reinforced concrete beam neutral line under the use moment of flexure with reinforcement ratio ρ variation, the unit of all physical quantitys all adopts the International System of Units.

Claims (1)

1. A method for determining the relationship between the neutral layer and the reinforcement ratio of steel-concrete beams under bending moments, characterized in that: make n reinforced concrete beams with rectangular cross-sections, wherein n≥12, and each beam is only configured on the tension side Longitudinal reinforcement, so that the concrete strength, steel grade, beam length, beam width, beam height and cover thickness of all beams are basically consistent, and the reinforcement ratio ρ _i of each beam is distributed from 0.2% to 2% from small to large Within the range of , carry out the simply supported two-point symmetrical loading test on all the beams, and measure the coordinate relationship diagram M(w) of the bending moment M of each beam changing with the deflection w at the bottom of the beam mid-span, and select those bending moment M and deflection w The M(w) diagrams showing the shape change law of the tri-fold line are used as the test results, and the maximum bending moment in these diagrams is taken as the breaking moment M _u (i) of the corresponding beam, and (0.5～0.6)M _u ( i) is the service bending moment M _k (i) of the corresponding beam, the deflection value of the bottom of the beam mid-span corresponding to M _k (i) is w _k (i), and the compressive elastic modulus of all beams

Both are taken as the elastic modulus E _c of concrete, and E _c , M _k (i) and w _k (i) are substituted into the equation $\mathrm{<span\; class="notranslate">\; b</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}\frac{{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; +</span>}{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}}{{<span\; class="notranslate">\; (</span>\sqrt{{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; +</span>}}<span\; class="notranslate">\; +</span>\sqrt{{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; h</span>}{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>\frac{\sqrt{{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}}}{\sqrt{{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; +</span>}}<span\; class="notranslate">\; +</span>\sqrt{{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 3</span>}{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>\frac{\mathrm{<span\; class="notranslate">\; 4</span>}{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 8</span>}{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ,</span>$ Obtain the tensile modulus of elasticity of the corresponding beam Among them, l is the span length of all reinforced concrete beams in the simply supported two-point symmetrical loading test, b is the beam width, h is the beam height, and the reinforcement ratio ρ _i of each beam and the corresponding

The calculated value of is substituted into the following formula, $\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; no</span>}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; +</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; +</span>}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; no</span>}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; =</span>\frac{\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; +</span>}<span\; class="notranslate">\; -</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; +</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; no</span>}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}$ Obtain the coefficients α and β, and then substitute α, β, and E _c into the following expressions $\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&rho;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\sqrt{{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}}}{\sqrt{\mathrm{<span\; class="notranslate">\; \&alpha;\&rho;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}}<span\; class="notranslate">\; +</span>\sqrt{{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}}}<span\; class="notranslate">\; ,</span>$ Obtain the analytical expression H(ρ) of the ratio H of the height h ₁ of the neutral floor distance from the beam to the height h of the beam under the applied bending moment of the reinforced concrete beam with the variation of the reinforcement ratio ρ, and the units of all physical quantities are international unit system.

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