CN105389432A - Method for calculating load distribution intensity and maximum bending moment of rectangular two-way slab system - Google Patents

Abstract

The present invention relates to a method for calculating load distribution intensity and a maximum bending moment of a rectangular two-way slab system. The method adopts a characteristic simulated strip method to calculate the load distribution intensity and the maximum bending moment and a relationship thereof of the rectangular two-way slab system, and specifically comprises the following steps: A, intercepting two quadrature slab strips with a unit width in the rectangular two-way slab; B, considering a width to length ratio and a four-side support condition of the rectangular two-way slab, and calculating the load distribution intensity of the slab strips; and C, calculating the maximum bending moment of the rectangular two-way slab and the relationship of the load distribution intensity and the maximum bending moment. Compared to the traditional and prior art, the method provided by the present invention discloses that under the action of a uniformly distributed load of an internal force field of the rectangular two-way slab with a conventional boundary condition, corresponding internal and logical relationships exist between a midspan of the two-way slab and job characteristics of the slab and between the maximum bending moment internal force of a support and the job characteristics of the slab, and the logical and internal relationships can be expressed by using a concise mathematical model; and the method has the advantages of improvement on the design security and convenience of the two-way slab, high calculation reliability and the like.

Description

A kind of rectangle two-way slab system lotus cloth intensity and maximal bending moment computing method

Technical field

The present invention relates to the new for etc. computing method of a kind of rectangle two-way slab system structural load and interior force data, especially relate to a kind of rectangle two-way slab system lotus cloth intensity and maximal bending moment computing method.

Background technology

In the design of various buildings, structures, " plate " is a kind of common structural unit.Such as, the floor system, storehouse plate body system, large-scale rectangular pond etc. of structure, be made up of armoured concrete slab, compoboard or steel plate usually.In this system, most the support by beam or wall three limit or four limits of each module unit plate forms an one-piece construction.Along with the development of technology, the Internal Forces Analysis of rectangle two-way slab system in the plank frames such as office building, multistoried storehouse, garage and pond can have been come by the software for calculation comprising finite element method or numerical analysis method computing function, simply, also can computation of table lookup.In order to adapt to the many-sided needs of engineering analysis, many researchers propose many practical approaches simplifying calculating, and strip coating method is exactly one of them.

The thinking of strip coating method is by the load-bearing two-way slab system of complexity, be decomposed into two orthogonal unidirectional plate-girder systems, the real load of plate is decomposed simultaneously, the plate-girder system in equity generation is become to calculate the analytic transformation of two-way operation plate, make it control force value in the internal force reality corresponding to plate close, thus reach practical and simplify the object calculated.

Existing bar (plate) band method mainly contains MARCUS strip coating method, broad sense strip coating method, Hillerborg strip coating method, distribute type strip coating method etc., specific as follows:

1) strip coating method of MARCUS strip coating method: MARCUS proposition, its basic thought is, the orthogonal strip midway deflection value that the Liang Tiao unit of plate span centre is wide is equal, and two strip evenly load sums equal plate actual distribution load, and boundary condition is consistent.Although its account form is very simple, comparison as calculated, there is no small difference in the strip coating method result of calculation of MARCUS and actual value.

2) broad sense strip coating method: J.S.FERNANDO proposes broad sense strip coating method.The basic thought of the method is, determines that load distributes according to each point on plate along the elastic deflection equal principle of X and Y-direction band.The computational accuracy obtained like this is high, but strip load is difficult to formation rule solution.

3) Hillerborg strip coating method: according to the lower bound theorem of limit analysis, the strip coating method that Hillerborg proposes comprises simple strip method and high strip coating method, and the former has some limitations, and the latter is very complicated, and practicality is limited.

4) distribute type strip coating method: calculating object is resolved into different plate territories, calculates respectively by different load allocation proportions, systematicness and counting yield limited.

Summary of the invention

Object of the present invention is exactly provide a kind of improve the high rectangle two-way slab system lotus cloth intensity of rectangle two-way slab design safety, convenience and calculating reliability and maximal bending moment computing method to overcome deficiency or inconvenience that above-mentioned prior art exists.

Object of the present invention can be achieved through the following technical solutions:

A kind of rectangle two-way slab system lotus cloth intensity and maximal bending moment computing method, the method adopts to be intended levying lotus cloth intensity and the maximal bending moment that strip coating method calculates rectangle two-way slab, specifically comprises the following steps:

The orthogonal strip that A, intercepting rectangle two-way slab Zhong Liangtiao unit are wide;

B, the breadth length ratio considering rectangle two-way slab and four limit support conditions, calculate the lotus cloth intensity of strip, described four limit support conditions comprise simply supported on four sides and four limit fixed ends;

C, calculating rectangle two-way slab maximal bending moment and relation thereof;

Described orthogonal strip meets following requirement: 1) maximum defluxion of orthogonal strip is equal with the actural deflection value of rectangle two-way slab, 2) rectangle two-way slab load carries out lotus regional partition with yield line morphogenesis characters position at the bottom of plate, 3) strip load form intends shape rectangle, triangle and trapezoidal and combination distribution substantially, 4) strip two ends boundary condition is consistent with the boundary condition of rectangle two-way slab, and meet the basis hypothesis of strip coating method, namely strip internal torque is zero.

For the rectangle two-way slab of simply supported on four sides, described step B is specially:

Definition rectangle two-way slab X-direction length is Lx, and Y-direction length is Ly, carries out load subregion to rectangle two-way slab, and load blueline is lotus territory subangle line;

The formula calculating the lotus cloth intensity of strip is specially:

$\left\{\begin{array}{c}{Q}_1=\{1920/\[(10/{\mathrm{\α}}^2-t{g}^2\mathrm{\θ})t{g}^2\mathrm{\θ}\]\}f{Q}_o={\mathrm{\ψ}}_1{Q}_o\\ Q_{21}=\[1920(1-\mathrm{\α}tg\mathrm{\θ})/(25-16\mathrm{\α}tg\mathrm{\θ})\]{\mathrm{fQ}}_o={\mathrm{\ψ}}_{21}{Q}_o\\ Q_{22}=\[1920\mathrm{\α}tg\mathrm{\θ}/(25-16\mathrm{\α}tg\mathrm{\θ})\]{\mathrm{fQ}}_o={\mathrm{\ψ}}_{22}{Q}_o\end{array}\right.$

In formula, Q ₁represent the lotus cloth intensity in direction, place, long limit in rectangle two-way slab, Q ₂₁, Q ₂₂represent two lotus cloth intensities in direction, minor face place in rectangle two-way slab, α=min (Lx, Ly)/max (Lx, Ly), θ represents lotus territory subangle, is the angle of load blueline and minor face, 30 °≤θ≤45 °, f represents the amount of deflection factor maximum immunity value of two-way slab (under the unit evenly load) of rectangle two-way slab, Q _orepresent the evenly load value of rectangle two-way slab, ψ ₁, ψ ₂₁, ψ ₂₂for lotus divides coefficient.

In described step C, rectangle two-way slab maximal bending moment comprises the strip span centre maximal bending moment of X, Y both direction, in X, Y both direction, and the strip span centre maximal bending moment M in direction, place, long limit ₁computing formula is:

M ₁＝[80/(10/α ²-tg ²θ)]ξfQ _oL ²＝m ₁Q _oL ²

The strip span centre maximal bending moment M in direction, minor face place ₂computing formula is:

M ₂＝[80(3-2αtgθ)/(25-16αtgθ)]fQ _oL ²＝m ₂Q _oL ²

In formula, L=min (Lx, Ly), ξ are correction factor, ξ=tg (θ+φ ₁), φ ₁for correction factor angle, m ₁, m ₂for bending moment coefficients.

After the strip span centre maximal bending moment obtaining X or Y-direction, obtained the strip span centre maximal bending moment of other direction by following relational expression:

M ₁:M ₂＝(25-16αtgθ)ξ/[(3-2αtgθ)(10/α ²-tg ²θ)]。

Described lotus territory subangle θ, correction factor angle φ ₁obtain according to following table with the value of α:

α

1.00

0.95

0.90

0.85

0.80

0.75

0.70

0.65

0.60

0.55

0.50

0.45

0.40

0.35

θ

45°

45°

45°

45°

45°

45°

45°

43°

41°

39°

37°

35°

33°

31°

φ 1

0°

0°

0°

0°

0°

0°

0°

1°

2°

3°

4°

5°

6°

7°

。

For the rectangle two-way slab of four limit fixed ends, described step B is specially:

Definition rectangle two-way slab X is Lx to length, and Y-direction length is Ly, carries out load subregion to rectangle two-way slab, and load blueline is lotus territory subangle line;

The formula calculating the lotus cloth intensity of strip is specially:

$\left\{\begin{array}{c}{Q}_1=\{3840/\[(5/\mathrm{\α}-2tg\mathrm{\θ})t{g}^3\mathrm{\θ}\]\}f{Q}_o={\mathrm{\ψ}}_1{Q}_o\\ Q_{21}=\[3840(1-\mathrm{\α}tg\mathrm{\θ})/(10-7\mathrm{\α}tg\mathrm{\θ})\]{\mathrm{fQ}}_o={\mathrm{\ψ}}_{21}{Q}_o\\ Q_{22}=\[3840\mathrm{\α}tg\mathrm{\θ}/(10-7\mathrm{\α}tg\mathrm{\θ})\]{\mathrm{fQ}}_o={\mathrm{\ψ}}_{22}{Q}_o\end{array}\right.$

In formula, Q ₁represent the lotus cloth intensity in direction, place, long limit in rectangle two-way slab, Q ₂₁, Q ₂₂represent two lotus cloth intensities in direction, minor face place in rectangle two-way slab, α=min (Lx, Ly)/max (Lx, Ly), θ represents lotus territory subangle, is the angle of load blueline and minor face, 30 °≤θ≤45 °, f represents the amount of deflection factor maximum immunity value of two-way slab (under the unit evenly load) of rectangle two-way slab, Q _orepresent the evenly load value of rectangle two-way slab, ψ ₁, ψ ₂₁, ψ ₂₂for lotus divides coefficient.

For the rectangle two-way slab of four limit fixed ends, in described step C, rectangle two-way slab maximal bending moment comprises the strip span centre maximal bending moment of X, Y both direction and the strip bearing maximal bending moment of X, Y both direction, in X, Y both direction, and the strip span centre maximal bending moment M in direction, place, long limit _{in 1}computing formula is:

M _{in 1}=[40 α ²/ (5-2 α tg θ)] ξ fQ _ol ²=m ₁q _ol ²

The strip span centre maximal bending moment M in direction, minor face place _{in 2}computing formula is:

M _{in 2}=40 [(4-3 α tg θ)/(10-7 α tg θ)] fQ _ol ²=m ₂q _ol ²

In X, Y both direction, the strip bearing maximal bending moment M in direction, place, long limit ₁computing formula is:

M ₁=-40{ α (4-α tg θ)/[(5-2 α tg θ) tg θ] } fQ _ol ²=m ₃q _ol ²

The strip span centre support moment M in direction, minor face place ₂computing formula is:

M ₂=-40 [(8-5 α tg θ)/(10-7 α tg θ)] fQ _ol ²=m ₄q _ol ²

In formula, L=min (Lx, Ly), ξ are correction factor, ξ=tg (θ+φ ₂), φ ₂for correction factor angle, m ₁, m ₂, m ₃, m ₄for bending moment coefficients.

After obtaining the arbitrary value in X, the strip span centre maximal bending moment of Y-direction and the strip bearing maximal bending moment of X, Y both direction, obtain other value by following relational expression:

M _{in 1}: M _{in 2}=α ²(10-7 α tg θ) ξ/[(5-2 α tg θ) (4-3 α tg θ)]

M ₁: M ₂=α (4-α tg θ) (10-7 α tg θ)/[(5-2 α tg θ) (8-5 α tg θ) tg θ]

M ₁: M _{in 1}=-(4-α tg θ)/(α ξ tg θ)

M ₂: M _{in 2}=-(8-5 α tg θ)/(4-3 α tg θ).

Described lotus territory subangle θ, correction factor angle φ ₂obtain according to following table with the value of α:

α

1.00

0.95

0.90

0.85

0.80

0.75

0.70

0.65

0.60

0.55

0.50

0.45

0.40

0.35

θ

45°

45°

45°

45°

45°

45°

45°

43°

41°

39°

37°

35°

33°

31°

φ 2

2°

2°

2°

2°

2°

1°

0°

0°

0°

-2°

-4°

-6°

-8°

-10°

。

Compared with prior art, the inventive method disclose there is conventional boundary condition rectangle Two-way Rc Slabs field under Uniform Load, two-way slab span centre and there is corresponding internal logical relationship between bearing maximal bending moment internal force with this plate operating characteristic, these logical relations and inner link can be expressed by simple and clear mathematical model.The present invention has following beneficial effect:

(1) the present invention adopts " plan levies strip coating method " to calculate lotus cloth intensity and the maximal bending moment of rectangle two-way slab, computing method are more convenient, result of calculation is more reliable, can not cause waste of material in the design of rectangle two-way slab, also improves the security of rectangle two-way slab design.

(2) the present invention adopts " plan levies strip coating method " considers the amount of deflection factor, the breadth length ratio of plate, the comprehensive relation factor such as plan shape lotus cloth intensity and lotus territory subangle etc. of two-way slab, and its computational accuracy obtained is apparently higher than " MARCUS strip coating method ".For armoured concrete slab, as long as accomplish in arrangement of reinforcement layout at the bottom of the plate of long side direction to meet bending resistance design and anchorage length, the safety operating calculating data would not be affected.

(3) when Calculation of Continuous Rectangular Plates, by calculating the lotus cloth intensity of strip respectively, multispan strip system can be set up, with the analytical approach computation structure internal force of continuous plate-girder, because precision is reliable, the utilization advantage of " plan levies strip coating method " can be given full play to.

(4) it is a variable with the change of two-way slab breadth length ratio that the present invention " plan levies strip coating method " discloses lotus territory subangle, but when the breadth length ratio of plate is not less than 0.7, basic maintenance 45 degree.For the two-way slab of simply supported on four sides, subangle change in lotus territory is less on the impact of calculated value, can omit and disregard on engineering uses.

(5) the present invention " plan levies strip coating method " is also applicable to the non-rectangular quadrilaterals that can carry out approximate envelope with rectangle quadrilateral.

Accompanying drawing explanation

Strip load decomposition schematic diagram when Fig. 1 is Lx >=Ly, θ=45 °;

Strip load decomposition schematic diagram when Fig. 2 is Lx≤Ly, θ=45 °;

Fig. 3 is that under simply supported on four sides condition, strip lotus cloth intensity calculates schematic diagram;

Fig. 4 is that under four limit fixed end conditions, strip lotus cloth intensity calculates schematic diagram;

Fig. 5 is plate long spring square finite element analysis cloud atlas when length breadth ratio is 1.5 under four limit fixed end conditions, wherein, and μ=0 in left figure, μ=1/6 in right figure;

Fig. 6 is plate long spring square finite element analysis cloud atlas when length breadth ratio is 2 under four limit fixed end conditions, wherein, and μ=0 in left figure, μ=1/6 in right figure.

Embodiment

Below in conjunction with the drawings and specific embodiments, the present invention is described in detail.The present embodiment is implemented premised on technical solution of the present invention, give detailed embodiment and concrete operating process, but protection scope of the present invention is not limited to following embodiment.

The invention provides a kind of rectangle two-way slab system lotus cloth intensity and maximal bending moment computing method, the method adopts to be intended levying lotus cloth intensity and the maximal bending moment that strip coating method calculates rectangle two-way slab, specifically comprises the following steps:

The orthogonal strip that A, intercepting rectangle two-way slab Zhong Liangtiao unit are wide;

Four side conditions of B, consideration rectangle two-way slab, calculate the lotus cloth intensity of strip;

C, calculating rectangle two-way slab maximal bending moment.

Wherein, orthogonal strip meets following requirement: 1) maximum defluxion of orthogonal strip is equal with the actural deflection value of rectangle two-way slab, 2) rectangle two-way slab load carries out lotus regional partition with yield line morphogenesis characters position at the bottom of plate, 3) strip load form intends shape rectangle, triangle and trapezoidal and combination distribution substantially, 4) strip two ends boundary condition is consistent with the boundary condition of rectangle two-way slab, and meet the basis hypothesis of strip coating method, namely plate internal torque is zero.

Principle and the mathematical model of the present invention's " plan levies strip coating method " are described as follows.

1, strip load form, lotus territory subangle and lotus cloth intensity definition

Strip load form, refers to the load assignment along a direction in strip region, and its each controlling value is called lotus cloth intensity.The rectangle two-way slab of simply supported on four sides and four limit consolidations, load blueline is that (in order to easy explanation, assuming that angle theta=45 °, note: regulation θ is lotus territory subangle to four angle bisection lines, is the angle of lotus territory angle separated time and plate minor face, as shown in Figure 1).The lotus cloth intensity of strip, generally has following several situation:

1) Lx >=Ly, Qx, Qy represent X and Y-direction strip lotus cloth intensity respectively, and Qo is two-way slab evenly load, α y=Ly/Lx (i.e. α), θ=45 °, definition Ly=2a, then Lx=Lx '+2a=Lx '+Ly.

If with strip Directional Decomposition lotus cloth intensity than as follows:

Ly(Qy2+Qy1)/Qy1＝Lx/Lx’

(Qy2+Qy1)/Qy1＝Lx/(Lx-Ly)

LxQy2/Qy1＝Ly/(Lx-Ly)＝η

Make β=a/Lx, L=min (Lx, Ly), obtains following formula:

(1+η)/η＝1/2β

η＝αy/(1-αy)

Qy2＝ηQy1

2) Ly >=Lx, wherein, Qx, Qy all represent X and Y-direction strip lotus cloth intensity, and Qo is two-way slab evenly load, α x=Lx/Ly (i.e. α), θ=45 °, definition Lx=2b, then Ly=Ly '+2b=Lx+Ly '.

If with strip Directional Decomposition lotus cloth intensity than as follows:

(Qx2+Qx1)/Qx1＝Ly/Ly’

(Qx2+Qx1)/Qx1＝Ly/(Ly-Lx)

Qx2/Qx1＝Ly/(Lx-Ly)＝λ

Make ε=b/Ly, L=min (Lx, Ly), obtains following formula:

(1+λ)/λ＝1/2ε

λ＝αx/(1-αx)

Qx2＝λQx1

2, strip lotus cloth intensity calculates (special solution, i.e. θ=45 °)

2.1 according to theory of mechanics of materials, the elastic stiffness Bc=ED of thin plate ³/ [12* (1-μ ²)], E: the elastic modulus of material, D: thickness of slab, μ: Poisson ratio, when material be steel and reinforced concrete time, μ=0.3 and μ=1/6.In order to united analysis prerequisite, get μ=0, then the elastic stiffness Bc=ED of plate ³/ 12.The actual maximum defluxion δ of plate ₀=f*QoL ⁴/ Bc, wherein, L=min (Lx, Ly), f: (f value by identical boundary condition, can check in data in reckoner for the amount of deflection factor of plate, the maximum immunity value of two-way slab under unit evenly load), Qo is the actual evenly load of plate.For strip: Bcs=EI=EbD ³/ 12, b=1m, Bcs=ED ³/ 12.

Under 2.2 evenly load conditions, general simply supported on four sides and four limit fixed end strip load assignment (θ=45 °, strip gets unit width b=1m)

1) freely-supported condition (for Lx >=Ly)

Can obtain according to reckoner:

δymax1＝3Qy2Ly ⁴/640EI

δymax2＝5Qy1Ly ⁴/384EI

δxmax＝Qx*a2Lx2(5-2β2)/240EI，β＝a/Lx

δ ymax1, δ ymax2 are two maximum defluxions of Y-direction, and δ xmax is that X is to maximum defluxion.

According to " plan levies strip coating method " principle:

δymax＝δymax1+δymax2

δo＝fQoL ⁴/Bc＝δymax＝δxmax

Qy2＝ηQy1

δymax＝δymax1+δymax2＝(3Qy2Ly ⁴/640EI)+(5Qy1Ly ⁴/384EI)＝fQoL ⁴/Bc＝δo

(3*3Qy2L ⁴/1920EI)+(5*5Qy1L ⁴/1920EI)＝fQoL ⁴/EI→(9Qy2+25Qy1)＝1920fQo

Separately, δ o=fQoL ⁴/ Bc=δ xmax=Qx*a ²lx ²(5-2 β ²)/240EI, β=a/Lx, a=L/2, Lx=(1+1/ η) L

Above-mentioned Qx, Qy1, Qy2 through arrange after in table 1.

2) fixed end condition (for Lx >=Ly)

Can obtain according to reckoner:

δymax1＝3Qy2Ly ⁴/3840EI

Lyδymax2＝Qy1Ly ⁴/384EI

δxmax＝Qx*a ³Lx(5-4β)/480EI，β＝a/Lx

According to " plan levies strip coating method " principle:

δymax＝δymax1+δymax2

δo＝fQoL ⁴/Bc＝δymax＝δxmax

L＝min(Lx,Ly)，Qy2＝ηQy1

δymax＝δymax1+δymax2＝(3Qy2Ly ⁴/3840EI)+(Qy1Ly ⁴/384EI)＝fQoL ⁴/Bc＝δo

(3Qy2L ⁴/ 3840EI)+(10Qy1L ⁴/ 3840EI)=fQoL ⁴/ EI → (3Qy2+10Qy1)=3840fQo, Qy2=η Qy1, separately, δ o=fQoL ⁴/ Bc=δ xmax=Qx*a ³lx (5-4 β)/480EI, β=a/Lx, a=L/2, Lx=(1+1/ η) L.

Above-mentioned Qx, Qy1, Qy2 through arrange after in table 1.

Table 1

3, under evenly load condition, general simply supported on four sides and four limit fixed end strip load assignment and maximal bending moment normal form (general solution)

In fact, when 1 >=(α y=Ly/Lx) >=0.70, the desirable 45 ° of definite values of four limit fixed end two-way slab Loaded crack separated times and minor face angle theta angle; As 0.70> (α y=Ly/Lx), two-way slab Loaded crack separated time and minor face angle theta angle are departed from slightly along with the length breadth ratio of plate changes and changes and have.

More generally, a=Ly* (tg θ)/2, (Qy2+Qy1)/Qy1=Lx/ (Lx-2a), Qy2={ η tg θ/[1+ η (1-tg θ)] } Qy1, Qy2=[α ytg θ/(1-α ytg θ)] Qy1, (note: plate load angular bisector is still assumed near linear).

According to reckoner, freely-supported condition: Mx=QxLx ²/ 8-QxLx ²(3-4 β ²)/24=β ²qxLx ²/ 6, My=(Qy1+Qy2) Ly ²/ 8-Qy2Ly ²/ 12=(3Qy1+Qy2) Ly ²/ 24; Fixed end condition: in Mx=Qxa ³/ 12Lx=[η L ²/ 96 (1+ η)] in Qx, My=(Qy1+Qy2) Ly ²/ 24-Qy2Ly ²/ 32=[(4+ η) L ²/ 96] Qy1, Mx prop up=-[Qxa ²(2-β)/12]=-[(4+3 η) L ²/ 96 (1+ η)] Qx, My prop up=-[(Qy1+Qy2) Ly ²/ 12-5Qy2Ly ²/ 96]=-[(8+3 η) L ²/ 96] Qy1, through arranging as table 2 and the serial normal form of table 3, and analytic expression when providing θ=45 °.

Table 2

The situation of note: Lx<Ly and upper table are in like manner.

Specific condition is as Lx=Ly=L, α y=1, η=∞, Qx=Qy2=Qy=640f*Qo/3, Mx=My=80fQoL ²/ 9.Foregoing also can carry out analytical calculation with reference to table 5, but very close when its result of calculation and θ=45 °, for simplicity, generally can not adopt.

Table 3

The situation of note: Lx<Ly and upper table are in like manner.

Ly/Lx ratio is 0.7 ~ 1 time, and θ angle can be taken as 45 °.According to checking computations, Ly/Lx ratio and θ angle value relation are in table 4, and both can use interpolation calculation intermediate data.The length breadth ratio of plate is set in 1.0 ~ 1/3 scopes, and when Ly/Lx is 1/3, θ angle is approximately 30 °.θ angle is retracted to 30 ° from 45 °, thus it is less to make the party's support moment upwards reduce degree.Plasticity linea angulata analytic approach also demonstrates this change.Finite element analysis cloud atlas reflects four limit consolidation two-way slabs along maximal bending moment in the plate of long side direction, and along with the length breadth ratio increase of plate can become two extreme values from a middle extreme value to bilateral segmentation, and the party's support moment change is upwards limited, sees Fig. 5.

Table 4

Ly/Lx

1.00

0.95

0.90

0.85

0.80

0.75

0.70

0.65

0.60

0.55

0.50

0.45

0.40

0.35

θ angle

45°

45°

45°

45°

45°

45°

45°

43°

41°

39°

37°

35°

33°

31°

φ1

0°

0°

0°

0°

0°

0°

0°

1°

2°

3°

4°

5°

6°

7°

φ2

2°

2°

2°

2°

2°

1°

0°

0°

0°

-2°

-4°

-6°

-8°

-10°

4, the calculated value of strip coating method and comparing of exact value

" MARCUS strip coating method " when simply supported on four sides and four limit fixed ends, Qo=Qx+Qy, Qx=[Ly ⁴/ (Lx ⁴+ Ly ⁴)] Qo, Qy=[Lx ⁴/ (Lx ⁴+ Ly ⁴)] formula before Qo, the Qo lotus that can be considered as MARCUS strip coating method lotus cloth intensity divides coefficient.The result of calculation of distinct methods is listed in table 5 and table 6 and contrasts.Wherein, Precise equation numerical value comes from reckoner, and deviate is pressed: (band calculated value/Precise equation calculated value-1) % mode calculates, more than overgauge represents by force; Minus deviation represents not enough.

Table 5

Table 6

Note: in table 5 and 6, "-" indicates without numerical value, in table 5 and 6, c1-c3 represents that the lotus of two kinds of methods divides coefficient; M1-m4 represents the calculation of Bending Moment coefficient of two kinds of methods, see reckoner and " plan levies strip coating method ".

In table 5, " plan levies strip coating method " absolute value of the bias is less than 3%; In table 6, " plan levies strip coating method " absolute value of the bias is less than 5%.From the Data Comparison of table 5 and table 6, " plan levies strip coating method " has higher computational accuracy, is acceptable in the economy and security of engineering design; And the calculated value deviation of " MARCUS strip coating method " is very large, and the result of calculation of support moment has insecurity on engineering.

" plan levies strip coating method " that the present invention proposes is that a kind of grade newly of setting up within the scope of elastic properties of materials is for computing method, the method disclose there is conventional boundary condition rectangle Two-way Rc Slabs field under Uniform Load, two-way slab span centre and there is corresponding internal logical relationship between bearing maximal bending moment internal force with this plate operating characteristic, these logical relations and inner link can be expressed by simple and clear mathematical model.

1, " plan levies strip coating method " discloses lotus territory subangle is a variable with the change of two-way slab breadth length ratio, but when the breadth length ratio of plate is not less than 0.7, basic maintenance 45 degree.For the two-way slab of simply supported on four sides, subangle change in lotus territory is less on the impact of calculated value, can omit and disregard on engineering uses.

2, " plan levies strip coating method " is owing to considering the amount of deflection factor, the length breadth ratio of plate, the comprehensive relation factor such as plan shape lotus cloth intensity and lotus territory subangle etc. of two-way slab, and its computational accuracy obtained is apparently higher than " MARCUS strip coating method ".Although long side direction mid span moment form and the certain difference of physical presence, its maximal bending moment control numerical value and exact value very close.For armoured concrete slab, as long as accomplish in arrangement of reinforcement layout at the bottom of the plate of long side direction to meet bending resistance design and anchorage length, the safety operating calculating data would not be affected.

3, when Calculation of Continuous Rectangular Plates, by calculating the lotus cloth intensity of strip respectively, multispan strip system can be set up, with the analytical approach computation structure internal force of continuous plate-girder.Because precision is reliable, the utilization advantage of " plan levies strip coating method " can be given full play to.

4, strip mid span moment Mxm and Mym, calculate by Mxm=Mx+ μ * My and Mym=My+ μ * Mx formula, μ is Poisson ratio.The mid span moment obtained so is always partial to safety in engineering design.When calculating fixed end lotus cloth intensity and fixed-end moment, can consider to be multiplied by correction factor 1.05.

5, " plan levies strip coating method " is for non-rectangular quadrilaterals, if can carry out approximate envelope with rectangle quadrilateral, can use equally.

Claims (9)

1. rectangle two-way slab system lotus cloth intensity and maximal bending moment computing method, is characterized in that, the method adopts to be intended levying lotus cloth intensity and the maximal bending moment that strip coating method calculates rectangle two-way slab, specifically comprises the following steps:

The orthogonal strip that A, intercepting rectangle two-way slab Zhong Liangtiao unit are wide;

B, the breadth length ratio considering rectangle two-way slab and four limit support conditions, calculate the lotus cloth intensity of strip, described four limit support conditions comprise simply supported on four sides and four limit fixed ends;

C, calculating rectangle two-way slab maximal bending moment and relation thereof;

Described orthogonal strip meets following requirement: 1) maximum defluxion of orthogonal strip is equal with the actural deflection value of rectangle two-way slab, 2) rectangle two-way slab load carries out lotus regional partition with yield line morphogenesis characters position at the bottom of plate, 3) strip load form intends shape rectangle, triangle and trapezoidal and combination distribution substantially, 4) strip two ends boundary condition is consistent with the boundary condition of rectangle two-way slab, and meet the basis hypothesis of strip coating method, namely strip internal torque is zero.

2. rectangle two-way slab system lotus cloth intensity according to claim 1 and maximal bending moment computing method, is characterized in that, for the rectangle two-way slab of simply supported on four sides, described step B is specially:

Definition rectangle two-way slab X-direction length is Lx, and Y-direction length is Ly, carries out load subregion to rectangle two-way slab, and load blueline is lotus territory subangle line;

The formula calculating the lotus cloth intensity of strip is specially:

$\left\{\begin{array}{c}{Q}_1=\{1920/\&lsqb;(10/{\mathrm{\&alpha;}}^2-t{g}^2\mathrm{\&theta;})t{g}^2\mathrm{\&theta;}\&rsqb;\}f{Q}_o={\mathrm{\&psi;}}_1{Q}_o\\ Q_{21}=\&lsqb;1920(1-\mathrm{\&alpha;}tg\mathrm{\&theta;})/(25-16\mathrm{\&alpha;}tg\mathrm{\&theta;})\&rsqb;{\mathrm{fQ}}_o={\mathrm{\&psi;}}_{21}{Q}_o\\ Q_{22}=\&lsqb;1920\mathrm{\&alpha;}tg\mathrm{\&theta;}/(25-16\mathrm{\&alpha;}tg\mathrm{\&theta;})\&rsqb;{\mathrm{fQ}}_o={\mathrm{\&psi;}}_{22}{Q}_o\end{array}\right.$

In formula, Q ₁represent the lotus cloth intensity in direction, place, long limit in rectangle two-way slab, Q ₂₁, Q ₂₂represent two lotus cloth intensities in direction, minor face place in rectangle two-way slab, α=min (Lx, Ly)/max (Lx, Ly), θ represents lotus territory subangle, is the angle of load blueline and minor face, 30 °≤θ≤45 °, f represents the amount of deflection factor of rectangle two-way slab, Q _orepresent the evenly load value of rectangle two-way slab, ψ ₁, ψ ₂₁, ψ ₂₂for lotus divides coefficient.

3. rectangle two-way slab system lotus cloth intensity according to claim 2 and maximal bending moment computing method, it is characterized in that, in described step C, rectangle two-way slab maximal bending moment comprises the strip span centre maximal bending moment of X, Y both direction, in X, Y both direction, the strip span centre maximal bending moment M in direction, place, long limit ₁computing formula is:

M ₁＝[80/(10/α ²-tg ²θ)]ξfQ _oL ²＝m ₁Q _oL ²

The strip span centre maximal bending moment M in direction, minor face place ₂computing formula is:

M ₂＝[80(3-2αtgθ)/(25-16αtgθ)]fQ _oL ²＝m ₂Q _oL ²

In formula, L=min (Lx, Ly), ξ are correction factor, ξ=tg (θ+φ ₁), φ ₁for correction factor angle, m ₁, m ₂for bending moment coefficients.

4. rectangle two-way slab system lotus cloth intensity according to claim 3 and maximal bending moment computing method, is characterized in that, after the strip span centre maximal bending moment obtaining X or Y-direction, obtained the strip span centre maximal bending moment of other direction by following relational expression:

M ₁:M ₂＝(25-16αtgθ)ξ/[(3-2αtgθ)(10/α ²-tg ²θ)]。

5. rectangle two-way slab system lotus cloth intensity according to claim 3 and maximal bending moment computing method, is characterized in that, described lotus territory subangle θ, correction factor angle φ ₁obtain according to following table with the value of α:

α 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 θ 45° 45° 45° 45° 45° 45° 45° 43° 41° 39° 37° 35° 33° 31° φ ₁ 0° 0° 0° 0° 0° 0° 0° 1° 2° 3° 4° 5° 6° 7°

。

6. rectangle two-way slab system lotus cloth intensity according to claim 1 and maximal bending moment computing method, is characterized in that, for the rectangle two-way slab of four limit fixed ends, described step B is specially:

Definition rectangle two-way slab X is Lx to length, and Y-direction length is Ly, carries out load subregion to rectangle two-way slab, and load blueline is lotus territory subangle line;

The formula calculating the lotus cloth intensity of strip is specially:

$\left\{\begin{array}{c}{Q}_1=\{3840/\&lsqb;(5/\mathrm{\&alpha;}-2tg\mathrm{\&theta;})t{g}^3\mathrm{\&theta;}\&rsqb;\}f{Q}_o={\mathrm{\&psi;}}_1{Q}_o\\ Q_{21}=\&lsqb;3840(1-\mathrm{\&alpha;}tg\mathrm{\&theta;})/(10-7\mathrm{\&alpha;}tg\mathrm{\&theta;})\&rsqb;{\mathrm{fQ}}_o={\mathrm{\&psi;}}_{21}{Q}_o\\ Q_{22}=\&lsqb;3840\mathrm{\&alpha;}tg\mathrm{\&theta;}/(10-7\mathrm{\&alpha;}tg\mathrm{\&theta;})\&rsqb;{\mathrm{fQ}}_o={\mathrm{\&psi;}}_{22}{Q}_o\end{array}\right.$

In formula, Q ₁represent the lotus cloth intensity in direction, place, long limit in rectangle two-way slab, Q ₂₁, Q ₂₂represent two lotus cloth intensities in direction, minor face place in rectangle two-way slab, α=min (Lx, Ly)/max (Lx, Ly), θ represents lotus territory subangle, is the angle of load blueline and minor face, 30 °≤θ≤45 °, f represents the amount of deflection factor of rectangle two-way slab, Q _orepresent the evenly load value of rectangle two-way slab, ψ ₁, ψ ₂₁, ψ ₂₂for lotus divides coefficient.

7. rectangle two-way slab system lotus cloth intensity according to claim 6 and maximal bending moment computing method, it is characterized in that, for the rectangle two-way slab of four limit fixed ends, in described step C, rectangle two-way slab maximal bending moment comprises the strip span centre maximal bending moment of X, Y both direction and the strip bearing maximal bending moment of X, Y both direction, in X, Y both direction, the strip span centre maximal bending moment M in direction, place, long limit _{in 1}computing formula is:

M _{in 1}=[40 α ²/ (5-2 α tg θ)] ξ fQ _ol ²=m ₁q _ol ²

The strip span centre maximal bending moment M in direction, minor face place _{in 2}computing formula is:

M _{in 2}=40 [(4-3 α tg θ)/(10-7 α tg θ)] fQ _ol ²=m ₂q _ol ²

In X, Y both direction, the strip bearing maximal bending moment M in direction, place, long limit ₁computing formula is:

M ₁=-40{ α (4-α tg θ)/[(5-2 α tg θ) tg θ] } fQ _ol ²=m ₃q _ol ²

The strip span centre support moment M in direction, minor face place ₂computing formula is:

M ₂=-40 [(8-5 α tg θ)/(10-7 α tg θ)] fQ _ol ²=m ₄q _ol ²

In formula, L=min (Lx, Ly), ξ are correction factor, ξ=tg (θ+φ ₂), φ ₂for correction factor angle, m ₁, m ₂, m ₃, m ₄for bending moment coefficients.

8. rectangle two-way slab system lotus cloth intensity according to claim 7 and maximal bending moment computing method, it is characterized in that, after obtaining the arbitrary value in X, the strip span centre maximal bending moment of Y-direction and the strip bearing maximal bending moment of X, Y both direction, obtain other value by following relational expression:

M _{in 1}: M _{in 2}=α ²(10-7 α tg θ) ξ/[(5-2 α tg θ) (4-3 α tg θ)]

M ₁: M ₂=α (4-α tg θ) (10-7 α tg θ)/[(5-2 α tg θ) (8-5 α tg θ) tg θ]

M ₁: M _{in 1}=-(4-α tg θ)/(α ξ tg θ)

M ₂: M _{in 2}=-(8-5 α tg θ)/(4-3 α tg θ).

9. rectangle two-way slab system lotus cloth intensity according to claim 7 and maximal bending moment computing method, is characterized in that, described lotus territory subangle θ, correction factor angle φ ₂obtain according to following table with the value of α:

α 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 θ 45° 45° 45° 45° 45° 45° 45° 43° 41° 39° 37° 35° 33° 31° φ ₂ 2° 2° 2° 2° 2° 1° 0° 0° 0° -2° -4° -6° -8° -10°