CN101886992B - Method for determining flexural capacity of sandwich plate with non-metal surfaces and application - Google Patents

Abstract

The invention relates to a method for determining the flexural capacity of a sandwich plate with non-metal surfaces. The panel of the sandwich plate is made of non-metal materials. The method for determining the flexural capacity of the sandwich plate with the non-metal surfaces comprises the following steps of: acquiring the relevant parameters of the sandwich plate with the non-metal surfaces, determining the deflection of the sandwich plate with the non-metal surfaces, and determining the flexural capacity of the sandwich plate with the non-metal surfaces. By considering the influence of the rigidity of the non-metal panel on the flexural deformation of the sandwich plate with non-metal surfaces, the method for determining the flexural capacity of the sandwich plate with the non-metal surfaces precisely obtains the deflections of the sandwich plate with the non-metal surfaces under the concentrated load and the uniformly distributed load and then determines the flexural capacity of the sandwich plate with the non-metal surfaces by the deflections. By the invention, the flexural capacity of the sandwich plate with the non-metal surfaces is precisely obtained, thereby determining the flexural mechanical properties of the sandwich plate with the non-metal surfaces and precisely estimating the safety performance of the sandwich plate with the non-metal surfaces.

Description

Flexural capacity of sandwich plate with non-metal surfaces is determined method

Technical field

The present invention relates to a kind of flexural capacity of sandwich plate and determine method, relate in particular to a kind of flexural capacity of sandwich plate with non-metal surfaces and determine method.

Background technology

Along with the development of architectural engineering technology, battenboard more and more widely uses in modern society.Along with being extensive use of of battenboard, the battenboard technology is development thereupon also, the battenboard from the battenboard of initial metal decking to nonmetal panel.In modern society, along with the raising of nonmetallic materials technology, the battenboard of nonmetal panel has occupied leading position gradually.Anti-bending bearing capacity for nonmetal battenboard is determined, go back the neither one simple and practical method in the prior art, particularly lacking the influence that flexural deformation causes to sandwich plate with non-metal surfaces of consideration non-metal surfaces panel stiffness in depth considers, can not accurately obtain for sandwich plate with non-metal surfaces anti-bending mechanics performance thus, cause can not the accurate assessment sandwich plate with non-metal surfaces security performance.

Summary of the invention

The technical matters that the present invention solves is: provide a kind of flexural capacity of sandwich plate with non-metal surfaces to determine that method overcomes in the prior art and can not accurately obtain for sandwich plate with non-metal surfaces anti-bending mechanics performance, cause accurately determining the technical matters of the anti-bending bearing capacity of sandwich plate with non-metal surfaces.

Technical scheme of the present invention is: provide a kind of flexural capacity of sandwich plate with non-metal surfaces to determine method, the panel of described battenboard is nonmetallic materials, and flexural capacity of sandwich plate with non-metal surfaces determines that method comprises the steps:

Gather the correlation parameter of sandwich plate with non-metal surfaces: the width of the span of collection sandwich plate with non-metal surfaces, the elastic modulus of nonmetal panel, sandwich plate with non-metal surfaces panel, the thickness of sandwich plate with non-metal surfaces panel, sandwich plate with non-metal surfaces core thickness, the modulus of shearing of sandwich plate with non-metal surfaces core, the effective cross-sectional area of sandwich plate with non-metal surfaces core.

Determine the amount of deflection of sandwich plate with non-metal surfaces: the rigidity of sandwich plate with non-metal surfaces panel needs to consider, the amount of deflection of sandwich plate with non-metal surfaces determine to comprise amount of deflection under the load and the amount of deflection under the evenly load,

Amount of deflection under the sandwich plate with non-metal surfaces load adopts following formula to obtain:

$w_{\max }=-\frac{{\mathrm{<figure-callout\; id="3"\; label="WL"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">WL</figure-callout>}}^3}{48\mathrm{EI}}-\frac{0.8\mathrm{WL}}{4A{\mathrm{\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2$

Amount of deflection under the sandwich plate with non-metal surfaces evenly load adopts following formula to obtain:

$w_{\max }=-\frac{{5\mathrm{<figure-callout\; id="4"\; label="qL"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">qL</figure-callout>}}^4}{384\mathrm{EI}}-\frac{0.9{\mathrm{<figure-callout\; id="2"\; label="qL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">qL</figure-callout>}}^2}{{8\mathrm{A\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2$

Determine the anti-bending bearing capacity of sandwich plate with non-metal surfaces: determine the anti-bending bearing capacity of sandwich plate with non-metal surfaces by the relation of amount of deflection and anti-bending bearing capacity, because the amount of deflection and the anti-bending bearing capacity of laminboard wall panel have following relation:

w _max≤[f]

[f] is amount of deflection,

Then:

Load is determined: $W\&le;\frac{\left[f\right]}{\frac{{\mathrm{<figure-callout\; id="3"\; label="L"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">L</figure-callout>}}^3}{48\mathrm{EI}}+\frac{0.8\mathrm{<figure-callout\; id="4"\; label="L"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">L</figure-callout>}}{{4\mathrm{A\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2}$

Determining of local load: $q\&le;\frac{\left[f\right]}{\frac{{5\mathrm{<figure-callout\; id="4"\; label="L"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">L</figure-callout>}}^4}{384\mathrm{EI}}+\frac{{0.9\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">L</figure-callout>}}^2}{{8\mathrm{A\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2}$

Thereby determine the anti-bending bearing capacity of sandwich plate with non-metal surfaces,

In the above-mentioned formula, the expression meaning of each variable is as follows:

w _MaxThe amount of deflection of representing normal operational phase span centre position;

W represents the span centre load;

Q represents evenly load;

L represents the battenboard span;

E represents sandwich plate with non-metal surfaces panel elastic modulus;

I _fThe expression upper and lower panel is to the moment of inertia of himself natural axis,

I represents the moment of inertia sum of upper and lower panel to himself natural axis and whole battenboard natural axis,

$I=\frac{{\mathrm{<figure-callout\; id="3"\; label="bt"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">bt</figure-callout>}}^3}{6}+\frac{{\mathrm{<figure-callout\; id="2"\; label="btd"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">btd</figure-callout>}}^2}{2};$

B represents the sandwich plate with non-metal surfaces width;

T represents the sandwich plate with non-metal surfaces plate thickness;

D represents the distance between the upper and lower panel natural axis, d=c+t, and wherein c is a core thickness;

A represents the effective cross-sectional area of core,

G represents the modulus of shearing measured value of core;

Description of drawings

μ represents the reduction coefficient of core modulus of shearing.

Further technical scheme of the present invention is: in the amount of deflection step of determining sandwich plate with non-metal surfaces, comprise and determine the influence of sandwich plate with non-metal surfaces panel rigidity to battenboard rigidity, promptly

Wherein, E _fThe elastic modulus of expression panel; B represents the sandwich plate with non-metal surfaces width;

T represents the sandwich plate with non-metal surfaces plate thickness;

D represents the distance between the upper and lower panel natural axis, d=c+t, and wherein c is a core thickness.

Technique effect of the present invention is: provide a kind of flexural capacity of sandwich plate with non-metal surfaces to determine method, by considering the panel rigidity influence that flexural deformation causes to sandwich plate with non-metal surfaces of sandwich plate with non-metal surfaces, accurately obtain the amount of deflection of sandwich plate with non-metal surfaces under load and evenly load, determine the anti-bending bearing capacity of sandwich plate with non-metal surfaces then by amount of deflection.The present invention accurately obtains the anti-bending bearing capacity of sandwich plate with non-metal surfaces, thereby determines the anti-bending mechanics performance of sandwich plate with non-metal surfaces, the security performance of accurate assessment sandwich plate with non-metal surfaces.

Fig. 1 is a process flow diagram of the present invention.

Fig. 2 is the synoptic diagram that sandwich plate with non-metal surfaces is reduced to the sandwich beam of the present invention.

Fig. 3 is the cross sectional representation that sandwich plate with non-metal surfaces is reduced to the sandwich beam of the present invention.

Fig. 4 is an I-beam xsect shearing distribution schematic diagram of the present invention.

Fig. 5 is the shearing distribution schematic diagram of sandwich beam of the present invention.

Fig. 6 is the stress distribution synoptic diagram of the thick panel battenboard of the present invention xsect.

Fig. 7 is out of shape synoptic diagram down for sandwich beam load of the present invention effect.

Fig. 8 is a sandwich beam detrusion synoptic diagram of the present invention.

Fig. 9 is the free beam synoptic diagram under the span centre load of the present invention effect.

Figure 10 is the free beam synoptic diagram under the evenly load effect of the present invention.

Figure 11 for load effect of the present invention down θ and

Between relation.

Figure 12 for evenly load effect of the present invention down θ and Between relation.

Embodiment

Below in conjunction with specific embodiment, technical solution of the present invention is further specified.

As shown in Figure 1, the specific embodiment of the present invention is: provide a kind of flexural capacity of sandwich plate with non-metal surfaces to determine method, the panel of described battenboard is nonmetallic materials, (Oriented Strand board OSB) is introduced for example the panel of battenboard described in the present invention with stalk plate or directional structure board.

Flexural capacity of sandwich plate with non-metal surfaces determines that method comprises the steps:

Step 100: the correlation parameter of gathering sandwich plate with non-metal surfaces, that is, gather the following parameter of sandwich plate with non-metal surfaces: the width of the span of collection sandwich plate with non-metal surfaces, the elastic modulus of sandwich plate with non-metal surfaces panel, sandwich plate with non-metal surfaces panel, the thickness of sandwich plate with non-metal surfaces panel, sandwich plate with non-metal surfaces core thickness, the modulus of shearing of sandwich plate with non-metal surfaces core, the effective cross-sectional area of sandwich plate with non-metal surfaces core.In these parameters, the effective cross-sectional area of the thickness of the span of collection sandwich plate with non-metal surfaces, the width of sandwich plate with non-metal surfaces panel, sandwich plate with non-metal surfaces panel, sandwich plate with non-metal surfaces core thickness, sandwich plate with non-metal surfaces core is by the shape decision of sandwich plate with non-metal surfaces, and the modulus of shearing of the elastic modulus of sandwich plate with non-metal surfaces panel, sandwich plate with non-metal surfaces core is by the material decision of sandwich plate with non-metal surfaces.

Step 200: determine the amount of deflection of sandwich plate with non-metal surfaces, that is, and the sandwich plate with non-metal surfaces amount of deflection determine to comprise amount of deflection under the load and the amount of deflection under the evenly load.

For determining of sandwich plate with non-metal surfaces amount of deflection, at first need to consider following factor: the panel rigidity of sandwich plate with non-metal surfaces is to the influence of sandwich plate with non-metal surfaces amount of deflection.

One, the panel rigidity of sandwich plate with non-metal surfaces is to the influence of sandwich plate with non-metal surfaces amount of deflection.

Particularly, the panel rigidity of the sandwich plate with non-metal surfaces panel rigidity that the influence of sandwich plate with non-metal surfaces amount of deflection comprised sandwich plate with non-metal surfaces is to the influence to the battenboard shearing stress distribution of the panel rigidity of the influence of battenboard bending stiffness and sandwich plate with non-metal surfaces.

The panel rigidity of sandwich plate with non-metal surfaces is to the influence of battenboard bending stiffness, and is specific as follows:

As Fig. 2, shown in Figure 3, sandwich plate with non-metal surfaces is reduced to the form of sandwich beam, promptly do not consider the stress on the y direction (sheet material Width just).

Symbol definition is as follows among the figure:

C represents core thickness;

T presentation surface plate thickness

H represents battenboard thickness, h=c+2t

D represents the distance between the upper and lower panel center line, d=c+t

B represents the battenboard width

G represents the modulus of shearing of core

D represents the whole bendind rigidity of battenboard

A represents the equivalent cross-sectional area of battenboard

AG represents the shearing rigidity of battenboard, wherein A=bd ²/ c

Q represents the shearing on a certain cross section of battenboard

E _fThe elastic modulus of expression panel

E _cThe elastic modulus of expression core

I represents the moment of inertia of whole cross section to neutral axis

I _fThe expression upper and lower panel is to the moment of inertia of himself axis

Because battenboard is made up of upper and lower panel and core, obtains following formula as Fig. 2, sandwich deflection of beam rigidity shown in Figure 3 according to mechanics of materials stiffness Calculation formula:

$D=E_f\&CenterDot;\frac{{\mathrm{<figure-callout\; id="3"\; label="bt"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">bt</figure-callout>}}^3}{6}+E_f\&CenterDot;\frac{\mathrm{<figure-callout\; id="2"\; label="bt\; d"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">bt</figure-callout>}{d}^{}{}^2}{2}+E_c\&CenterDot;\frac{{\mathrm{bc}}^3}{12}---\left(1\right)$

Wherein first expression be panel with respect to himself bending shaft the time local stiffness; The rigidity of second generation when representing the upper and lower surfaces plate crooked with respect to axis c-c; The 3rd local stiffness when representing core crooked with respect to himself axle (with axis c-c).

In the sandwich structure of reality, (1) second of formula has occupied leading position, and (1) first of formula is that the influence of sandwich plate with non-metal surfaces panel rigidity can not be ignored, and the 3rd of formula (1) is the influence of core self rigidity.

The panel rigidity of sandwich plate with non-metal surfaces is to the influence of battenboard shearing stress distribution, and is specific as follows:

As shown in Figure 4, it can be reduced to the form of an I-beam, obtain the distribution situation of shear stress in the I-beam by the principle of work of battenboard.

Core shear stress τ for z place, below, axis, cross section has following formula according to the mechanics of materials:

$\mathrm{\&tau;}=\frac{\mathrm{QS}}{\mathrm{Ib}}---\left(2\right)$

Wherein: Q is the shearing on the selected xsect; I is the moment of inertia of whole cross section to neutral axis; B is z ₁The width at place, S is z＞z ₁The cross section of part is to the quiet distance of neutral axis, and z represents the distance of z place and axis, z among the figure ₁Expression z ₁The distance of place and axis.

For the combination beam of sandwich structure, consider the elastic modulus of each several part, following formula can be write as following form:

$\mathrm{\&tau;}=\frac{Q}{\mathrm{Db}}\mathrm{\&Sigma;}\left(\mathrm{SE}\right)---\left(3\right)$

Wherein D as shown in Equation (1); ∑ (SE) is z＞z ₁Partial cross section S and the sum of products of E, for example will determine then has the shear stress at core part z place:

$\mathrm{\&Sigma;}\left(\mathrm{SE}\right)={\mathrm{<figure-callout\; id="2"\; label="E\; f\; btd"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">E</figure-callout>}}_f\frac{\mathrm{btd}}{}\frac{}{2}+\frac{E_{c}b}{2}(\frac{c}{2}-z)(\frac{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">c</figure-callout>}}{2}+z)---\left(4\right)$

Therefore, the shear stress in the core:

$\mathrm{\&tau;}=\frac{Q}{D}\{E_f\&CenterDot;\frac{\mathrm{<figure-callout\; id="2"\; label="td"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">td</figure-callout>}}{2}+\frac{E_c}{2}(\frac{{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">c</figure-callout>}}^2}{4}-z^2)\}---\left(5\right)$

Similarly can obtain shear stress in the panel.

According to the knowledge of the mechanics of materials, the distribution of the shear stress on the sandwich beam xsect as shown in Figure 5: wherein, (a) be the real shearing stress distribution of sandwich beam xsect.(b) when ignoring core self rigidity, the shearing stress distribution situation of sandwich beam,

(c) when ignoring core self rigidity and panel self rigidity, the shearing stress distribution of sandwich beam,

For low intensive foam core material, can count E _c=0, obtain the shear stress constant in the core, among Fig. 5 shown in (b):

$\mathrm{\&tau;}=\frac{Q}{D}\&CenterDot;\frac{E_f\mathrm{td}}{2}---\left(6\right)$

At this moment $D=E_f\&CenterDot;\frac{{\mathrm{<figure-callout\; id="3"\; label="bt"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">bt</figure-callout>}}^3}{6}+E_f\&CenterDot;\frac{{\mathrm{btd}}^2}{2}.$

In addition, if panel is very little with respect to the bendind rigidity of himself axis, then

In first also can ignore, promptly

Then shear stress can be reduced to following minimum form in the core, shown in (c) among Fig. 5:

$\mathrm{\&tau;}=\frac{Q}{\mathrm{bd}}---\left(7\right)$

Two, for the determining of sandwich plate with non-metal surfaces amount of deflection, also need to consider following factor: the flexural deformation of sandwich plate with non-metal surfaces and detrusion.

Sandwich plate with non-metal surfaces is reduced to the form of sandwich beam, in the mechanics of materials for the regulation of bent beam symbol, as shown in Equation (8).

For the distortion of thick panel battenboard, need clear and definite what time following:

Because core is foam core materials such as EPS-polystyrene, its elastic modulus is very little, himself rigidity, and promptly the 3rd in the formula (1) can ignore;

Panel has certain rigidity, can not ignore for first in the formula (1);

Owing to ignored core self rigidity and panel rigidity can not be ignored, thus the shearing stress distribution in the core shown in (b) among Fig. 5, its shear stress is constant along the core thickness direction, size as shown in Equation (6).

Owing to considered self rigidity of panel, will produce following influence to the distortion of sandwich beam:

First kind of influence has two kinds of modes of texturing for making panel, first kind of mode is local bending, with respect to the flexural deformation of whole sandwich structure axis, produce stretching and the compression of panel under uniform stress this moment, the stress in the panel that produce this moment is shown in Fig. 6 first stress.The second way is the local bending with respect to panel self axis rather than whole sandwich structure axis, and the stress in the panel that produce this moment is shown in Fig. 6 second portion stress.

The sandwich beam is done the as a whole distortion that bends, and is example to bear load, when the sandwich beam is done the as a whole distortion that bends as shown in Figure 7.(a) is subjected to the free beam of concentrated force for span centre among Fig. 7, (b) is flexural deformation, (c) is detrusion, (d) is flexural deformation, the coefficient result of detrusion.Among Fig. 7 (b), panel has above two kinds of modes of texturing simultaneously.Panel local bending rigidity can be represented by first in the formula (1) the contribution margin of whole sandwich beam deflection rigidity.

Second kind of influence is the influence that core detrusion is produced:

Detrusion to core inside exerts an influence: when only considering detrusion, a on the panel axis, b, c ... each point does not produce displacement (therefore the principle stress on the panel is changed) in the horizontal direction, only vertically deform, as (c) among Fig. 5.A knuckle will appear in its span centre position, and curvature herein will be infinitely great, and this is impossible obviously: by the knowledge of the mechanics of materials as can be known, have between moment of flexure and the curvature

Relation, moment of flexure infinity herein then.If panel still will keep being connected with core, then local bending need take place, so that detrusion becomes level and smooth in panel in the certain distance scope of span centre both sides.At this moment, will in panel, introduce extra moment of flexure and shearing, thereby reduce detrusion.In the sandwich structure of reality, thin panel sandwich structure especially, this influence is very little; When panel thicker (for example eternit battenboard and sandwich plate with non-metal surfaces) and core when being light foam such as EPS, this influence obviously.

Based on above discussion, divide 2 distortion to discuss below to sandwich structure:

(1) the battenboard flexural deformation under the influence of the panel rigidity of sandwich plate with non-metal surfaces.

At first consider a core shearing rigidity infinity, at evenly load q ₁Sandwich beam element under the effect.According to common bending theory of beam, produce amount of deflection w ₁This amount of deflection and moment M ₁And shearing Q ₁Relevant, according to the mechanics of materials, shearing Q wherein ₁Draw:

-Q ₁＝Dw ₁′″＝E _f(I-I _f)w ₁′″+E _fI _fw ₁′″ (9)

Be the influence of ignoring core self rigidity this moment, Therefore have:

$I=\frac{{\mathrm{<figure-callout\; id="3"\; label="bt"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">bt</figure-callout>}}^3}{6}+\frac{{\mathrm{<figure-callout\; id="2"\; label="btd"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">btd</figure-callout>}}^2}{2}---\left(10\right)$

$I_f=\frac{{\mathrm{<figure-callout\; id="3"\; label="bt"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">bt</figure-callout>}}^3}{6}---\left(11\right)$

Suppose panel only to bear stretching and compression deformation and when local bending not taking place, first shearing of representing core and panel to bear jointly in formula (9) the right.Wouldn't count panel rigidity this moment, and first on the right can be calculated by the stress distribution among Fig. 5 (c): shear stress τ size in the core thickness scope is constant, is 0 at the face plate edge place, is linear change in panel inside.Therefore, first can be replaced by-bd τ, and wherein τ is the shear stress in the core, and d is the distance between the upper and lower panel center line, that is:

-Q ₁＝-bdτ+E _fI _fw ₁′″?(12)

Have simultaneously: q ₁=-Q ₁', Q ₁=M ₁, M ₁=-Dw ₁".

(2) and the battenboard detrusion under the panel rigidity of the definite sandwich plate with non-metal surfaces influence.

Because above-mentioned supposition core rigidity infinity, though so have shear stress τ in the core, shearing strain can't take place.Therefore, if the core shear modulus G is a certain limit value, then under the effect of shear stress τ, core produces shearing strain γ=τ/G, is equivalent to produce an extra transversely deforming w ₂Panel must produce this extra distortion simultaneously, and therefore, it must suffer an extra evenly load q ₂, shearing Q ₂And moment M ₂:

q ₂＝-Q ₂′，Q ₂＝M ₂，M ₂＝-Dw ₂″

Then total load, shearing, moment of flexure and be out of shape as follows:

q＝q ₁₊q ₂

Q＝Q ₁+Q ₂

M＝M ₁+M ₂

w＝w ₁+w ₂

That is to say that the sandwich beam under the evenly load q effect will produce two groups of different distortion: w ₁With w ₂Wherein first common flexural deformation of representative, the shearing Q that itself and panel and core are born jointly ₁Relevant; Second representative is because Q ₁The core detrusion that causes: for adapting to core detrusion needs, panel has also participated in the extra flexural deformation (ignored the detrusion in the panel, but panel still being shared shearing) around himself axis; At this moment, need an extra shearing to drive this distortion, i.e. Q ₂Q ₁With Q ₂And be the total shearing that puts on the beam.

(3) battenboard flexural deformation and the detrusion correlate under the influence of the panel rigidity of sandwich plate with non-metal surfaces.

The mutual relationship of extra distortion and core shearing strain γ, as shown in Figure 8.The length of line segment de equals There is the length of line segment cf to equal γ c again,, can gets by de=cf

And the relation between the γ is as follows:

$\frac{{\mathrm{<figure-callout\; id="2"\; label="dw"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">dw</figure-callout>}}_2}{\mathrm{dx}}=\mathrm{\&gamma;}\frac{c}{d}=\frac{Q}{\mathrm{Gbd}}\&CenterDot;\frac{c}{d}=\frac{Q}{\mathrm{AG}}---\left(13\right)$

Wherein, A=bd ²/ c, AG are often referred to the shearing rigidity of sandwich beam.

With τ=γ G substitution formula (13), can get the relation between additionally distortion and the shear stress:

$\mathrm{\&tau;}=\frac{d}{c}\&CenterDot;\mathrm{<figure-callout\; id="2"\; label="G\; w"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">G</figure-callout>}{w_{}}^{}{{}_2}^{\&prime;}---\left(14\right)$

With its substitution formula (12), that is:

-Q ₁＝-AGw ₂′+EI _fw ₁′″ (15)

General-Q ₁=-Dw ₁' " substitution formula (15) changes and can get:

${{\mathrm{<figure-callout\; id="2"\; label="w"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">w</figure-callout>}}_2}^{\&prime;}=-\frac{D}{\mathrm{AG}}(1-\frac{I_f}{I}){{\mathrm{<figure-callout\; id="1"\; label="w"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">w</figure-callout>}}_1}^{\&prime;\&prime;\&prime;}=+\frac{{\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1}{\mathrm{AG}}(1-\frac{I_f}{I})---\left(16\right)$

Because-Q ₂=-Dw ₂' ", then total shearing:

Q＝Q ₁+Q ₂＝Q ₁-EI _fw ₂′″?(17)

Formula (16) is updated in the formula (17), can be about Q ₁Equation as follows:

Q ₁″-a ²Q ₁＝-a ²Q (18)

Wherein: $a^2=\frac{\mathrm{AG}}{{\mathrm{EI}}_f(1-I_f/I)}---\left(19\right)$

Three, the deformation formula of sandwich plate with non-metal surfaces amount of deflection.

By above analysis as can be known, know the stressing conditions of sandwich beam, promptly Q can be provided by the equation about x, and then can try to achieve Q according to formula (17) ₁Again according to w ₁', w ₂' with Q ₁Between relation, can finally try to achieve w by integration ₁And w ₂At last according to w=w ₁+ w ₂Relation, can get freely-supported sandwich beam under different stressing conditions, the final Calculation of Deflection formula of span centre position.

(1) deformation formula of sandwich plate with non-metal surfaces amount of deflection under the load situation.

As shown in Figure 9, AB section, x starting point are A, and its shearing is-W/2, this moment according to formula (17) separate for:

$-{\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1={\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">C</figure-callout>}}_1\cosh \mathrm{ax}+{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">C</figure-callout>}}_2\sinh \mathrm{ax}+\frac{W}{2}---\left(20\right)$

By integration, can get:

${\mathrm{<figure-callout\; id="1"\; label="EIw"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">EIw</figure-callout>}}_1=\frac{C_1}{a^3}\sinh \mathrm{ax}+\frac{C_2}{a^3}\cosh \mathrm{ax}+\frac{{\mathrm{<figure-callout\; id="3"\; label="Wx"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">Wx</figure-callout>}}^3}{12}+C_3{x}^2+C_{4}x+C_5---\left(21\right)$

Equation (20) can obtain one about w with equation (21) ₂' expression formula, integration once obtains following formula:

${-\mathrm{<figure-callout\; id="2"\; label="EI\; f\; w"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">EI</figure-callout>}}_f{w}_{}{}_2=\frac{C_1}{a^3}\sinh \mathrm{ax}+\frac{C_2}{a^3}\cosh \mathrm{ax}+\frac{W}{2a^2}x+C_6---\left(22\right)$

Can find 5 boundary conditions on the AB section, the relation of above-mentioned thus six constants is as follows:

(i) x=0, w ₁=0 (arbitrariness)

$C_5+\frac{C_2}{a^3}=0$

(ii) x=0, w ₁'=0 (symmetry)

$\frac{C_1}{a^2}+{\mathrm{<figure-callout\; id="4"\; label="C"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">C</figure-callout>}}_4=0$

(iii) x=0, w ₁' "=0 (symmetry)

${\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">W</figure-callout>}}{2}=0$

$\left(\mathrm{iv}\right),x=0,M=\frac{\mathrm{<figure-callout\; id="4"\; label="WL"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">WL</figure-callout>}}{4}$

Definition-M=EIw ₁"+EI _fw ₂"

Therefore, $-\frac{\mathrm{<figure-callout\; id="4"\; label="WL"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">WL</figure-callout>}}{4}=2{\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">C</figure-callout>}}_3$

(v) x=0, w ₂=0 (arbitrariness)

$\frac{C_2}{a^3}+C_6=0$

So far, each constant can represent by following form, wherein C ₂Unknown:

${\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">C</figure-callout>}}_1=-\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">W</figure-callout>}}{2};{{\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">C</figure-callout>}}_3=-}_{}\frac{\mathrm{<figure-callout\; id="8"\; label="WL"\; filenames="HSA00000189287800023.png"\; state="\{\{state\}\}">WL</figure-callout>}}{8};{\mathrm{<figure-callout\; id="4"\; label="C"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">C</figure-callout>}}_4=+\frac{W}{2a^2};C_5=C_6=-\frac{C_2}{a^3}---\left(23\right)$

On the BC section, wherein x is with B point position starting point, and its total shearing is 0.Equation (20) and (21) still are suitable for but the item that wherein comprises W is answered cancellation, new constant B ₁-B ₆In order to replace C ₁-C ₆

Below be four simple boundary conditions:

(vi) x=0, w ₁=0 (arbitrariness)

$B_5+\frac{B_2}{a^3}=0$

(vii) x=0, w ₂=0 (arbitrariness)

$B_6+\frac{B_2}{a^3}=0$

(viii)x＝L ₁，w ₁″＝0

$\frac{B_1}{a}\sinh a{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">L</figure-callout>}}_1+\frac{B_2}{a}\cosh a{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">L</figure-callout>}}_1+2{\mathrm{<figure-callout\; id="3"\; label="B"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">B</figure-callout>}}_3=0$

(ix)x＝L ₁，w ₂″＝0

$\frac{B_1}{a}\sinh a{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">L</figure-callout>}}_1+\frac{B_2}{a}\cosh a{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">L</figure-callout>}}_1=0$

Latter two boundary condition is because free-ended moment M ₁And M ₂Be 0.Have only when the panel end and can freely rotate, and when not being connected with rigid end, this condition is just set up.Below be the result of above-mentioned boundary condition:

B ₂＝-B ₁tanhaL ₁；B ₃＝0；?

Still need to set up the continuity at B point place.Significantly, w ₁' and w ₂', w ₁" and w ₂" should be continuous; Simultaneously as can be known, w by formula (21) ₁' " and

Must be continuously.Yet, only, be respectively w with three conditions that independent equation can be provided ₁', w ₂', w ₁".

(x) w ₁' continuous at the B point

$\frac{C_1}{a^2}\mathrm{<figure-callout\; id="2"\; label="cosh\; aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">cosh</figure-callout>}\frac{\mathrm{aL}}{}\frac{}{2}+\frac{C_2}{a^2}\mathrm{<figure-callout\; id="2"\; label="sinh\; aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">sinh</figure-callout>}\frac{\mathrm{aL}}{}\frac{}{2}+\frac{{\mathrm{<figure-callout\; id="2"\; label="WL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">WL</figure-callout>}}^2}{16}+C_{3}L+{\mathrm{<figure-callout\; id="4"\; label="C"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">C</figure-callout>}}_4=\frac{B_2}{a^2}+{\mathrm{<figure-callout\; id="4"\; label="B"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">B</figure-callout>}}_4$

(xi) w ₂' continuous at the B point

${\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">C</figure-callout>}}_1\mathrm{<figure-callout\; id="2"\; label="cosh\; aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">cosh</figure-callout>}\frac{\mathrm{aL}}{}\frac{}{2}+{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">C</figure-callout>}}_2\mathrm{<figure-callout\; id="2"\; label="sinh\; aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">sinh</figure-callout>}\frac{\mathrm{aL}}{}\frac{}{2}+\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">W</figure-callout>}}{2}={\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">B</figure-callout>}}_1$

(xii) w ₁" continuous at the B point

${\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">C</figure-callout>}}_1\mathrm{<figure-callout\; id="2"\; label="sinh\; aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">sinh</figure-callout>}\frac{\mathrm{aL}}{}\frac{}{2}+{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">C</figure-callout>}}_2\mathrm{<figure-callout\; id="2"\; label="cosh\; aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">cosh</figure-callout>}\frac{\mathrm{aL}}{}\frac{}{2}+(\frac{\mathrm{<figure-callout\; id="4"\; label="WL"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">WL</figure-callout>}}{4}+2C_3)a={\mathrm{<figure-callout\; id="2"\; label="B"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">B</figure-callout>}}_2+2B_{3}a$

By equation (23) and (24) B that can divide out ₂, B ₃, C ₁And C ₃Condition (xi) and (xii) can be used for solving C ₂And B ₁, we are only to C ₂Interested:

${\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">C</figure-callout>}}_2={\mathrm{\&beta;}}_1\frac{W}{2}---\left(25\right)$

Wherein,

A is determined by formula (19).

C ₁-C ₆Value all be known quantity, with its substitution formula (20) and formula (21), can solve total deflection w, be a function of in the AB scope, representing by x:

$w=-\frac{W{x}^{2}L}{24\mathrm{EI}}(3-\frac{2x}{L})-\frac{\mathrm{<figure-callout\; id="4"\; label="WL"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">WL</figure-callout>}}{4\mathrm{AG}}{(1-\frac{I_f}{I})}^2\&times;\{\frac{2x}{L}-\frac{2}{\mathrm{aL}}[\sinh \mathrm{ax}+{\mathrm{\&beta;}}_1(1-\cosh \mathrm{ax})\left]\right\}$

Its maximal value should occur in the span centre position, promptly during x=L/2:

Wherein:

By formula (14) and equation (21) and (22) are carried out twice differentiate, can also obtain the shear stress of core inside and the normal stress of panel.

(2) deformation formula of sandwich plate with non-metal surfaces amount of deflection under the evenly load situation.

As shown in figure 10, be the force diagram of free beam under the evenly load effect.

AB part shearing is-qx that wherein the x starting point is A.In the substitution equation (18), the result is as follows:

-Q ₁＝C ₁coshax+C ₂sinhax+qx?(27)

By integration,

${\mathrm{<figure-callout\; id="1"\; label="EIw"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">EIw</figure-callout>}}_1=\frac{C_1}{a^3}\sinh \mathrm{ax}+\frac{C_2}{a^3}\cosh \mathrm{ax}+\frac{{\mathrm{<figure-callout\; id="4"\; label="qx"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">qx</figure-callout>}}^4}{24}+C_3{x}^2+C_{4}x+C_5---\left(28\right)$

Equation (27) can obtain about w with (28) together ₂' expression formula, integration once can get formula (29).

$-{E{\mathrm{<figure-callout\; id="2"\; label="I\; f\; w"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">I</figure-callout>}}_{f}w}_2=\frac{C_1}{a^3}\sinh \mathrm{ax}+\frac{C_2}{a^3}\cosh \mathrm{ax}+\frac{{\mathrm{<figure-callout\; id="2"\; label="qx"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">qx</figure-callout>}}^2}{2a^2}+C_6---\left(29\right)$

Equation (28), (29) are equally applicable to the BC section, and wherein the x starting point is B, and cancellation comprises the item of q.Constant C ₁-C ₆By B ₁-B ₆Replace.

The continuity that boundary condition and B are ordered requires with the thick panel sandwich beam element under the load effect, and wherein, mid span moment WL/4 becomes qL ²/ 8.The process of finding the solution unknown constant is the same, and net result is as follows:

C ₁＝0； ${\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">C</figure-callout>}}_3=-\frac{{\mathrm{<figure-callout\; id="2"\; label="qL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">qL</figure-callout>}}^2}{16}+\frac{q}{2a^2};$ C ₄＝0； $C_5=C_5=-\frac{C_2}{a^3}$

${\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="HSA00000189287800042.png,HSA00000189287800043.png"\; state="\{\{state\}\}">B</figure-callout>}}_1={\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">C</figure-callout>}}_2\mathrm{<figure-callout\; id="2"\; label="sinh\; aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">sinh</figure-callout>}\frac{\mathrm{aL}}{}\frac{}{2}+\frac{\mathrm{<figure-callout\; id="2"\; label="qL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">qL</figure-callout>}}{2};$ B ₂＝-B ₁tanhaL ₁； ${\mathrm{<figure-callout\; id="3"\; label="B"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">B</figure-callout>}}_3=0;{\mathrm{<figure-callout\; id="4"\; label="B"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">B</figure-callout>}}_4=-\frac{{\mathrm{<figure-callout\; id="3"\; label="qL"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">qL</figure-callout>}}^3}{24}$

${\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">C</figure-callout>}}_2=-\mathrm{\&beta;}\frac{\mathrm{<figure-callout\; id="2"\; label="qL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">qL</figure-callout>}}{2}$

Wherein

A is determined by formula (19).

The AB section is a bit located arbitrarily, and total deflection is provided by following formula:

$w=-\frac{{\mathrm{qx}}^2{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">L</figure-callout>}}^2}{48\mathrm{EI}}(3-\frac{{2x}^2}{L^2})-\frac{q}{\mathrm{AG}}{(1-\frac{I_f}{I})}^2\&times;\{\frac{x^2}{2}-\frac{{\mathrm{\&beta;}}_2{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">L</figure-callout>}}^2}{4\mathrm{\&theta;}}(1-\cosh \mathrm{ax})\}$

Its maximal value occurs in the x=L/2 place:

Wherein:

By formula (14) and equation (28) and (29) are carried out twice differentiate, also can obtain the shear stress of core inside and the normal stress of panel.

Four, the checking computations of sandwich plate with non-metal surfaces bearing capacity.

Deformation control when the anti-bending bearing capacity of battenboard is mainly by serviceability limit state in the actual engineering, when being subjected to uniform area load to do the time spent, the anti-bending bearing capacity of single span battenboard can calculate by following regulation:

w _max≤[f]

Wherein: the Deformation control limit limit value during [f] expression serviceability limit state, generally get L/200, L is the battenboard span.

(1) simplified formula under the load situation.

Under the load effect, sandwich plate with non-metal surfaces is considering that the Calculation of Deflection formula that the span centre position is under the evenly load effect is under the panel rigidity situation:

Wherein:

$\mathrm{\&theta;}=\frac{\mathrm{<figure-callout\; id="2"\; label="aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">aL</figure-callout>}}{2},$

$a^2=\frac{\mathrm{AG}}{{\mathrm{EI}}_f(1-I_f/I)}$

Owing to generally need not to be provided with semi-girder, this seasonal L in the actual engineering ₁=0, above-mentioned parameter can be simplified to following simple form thus:

${\mathrm{\&beta;}}_1=\frac{\sinh \mathrm{\&theta;}}{\cosh \mathrm{\&theta;}},$ $\mathrm{\&theta;}=\frac{\mathrm{<figure-callout\; id="2"\; label="aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">aL</figure-callout>}}{2}=\frac{L}{2}{\left(\frac{\mathrm{AG}}{{\mathrm{EI}}_f(1-I_f/I)}\right)}^{1/2},$

And a ²Size represents in fact is the ratio of core shearing rigidity and panel local bending rigidity, by several sandwich plate with non-metal surfaces of being studied herein as can be known, its a ²Value greatly about more than 400, calculate the θ value get thus generally about 20.

With the relation of θ as shown in figure 11, wherein θ is a horizontal ordinate,

Be ordinate.Consider θ 〉=3 o'clock,

Therefore have:

For the shingle nail in the actual engineering, consider the θ value generally more than 20, can be similar to this moment thinks

Can obtain shingle nail and under load, consider the final Calculation of Deflection formula of panel rigidity:

$w_{\max }=-\frac{{\mathrm{<figure-callout\; id="3"\; label="WL"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">WL</figure-callout>}}^3}{48\mathrm{EI}}-\frac{\mathrm{WL}}{4A{\mathrm{\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2---\left(32\right)$

For shingle nail, estimate the ratio of its panel rigidity and integral rigidity, generally more than 10%, the general minimum of θ value this moment can reach about 5, and is desirable And consider the conversion relation of core modulus of shearing under the different tests method, can obtain roof boarding and consider that the final abbreviation form of Calculation of Deflection formula that the span centre position is under the load effect is under the panel rigidity situation:

$w_{\max }=-\frac{{\mathrm{<figure-callout\; id="3"\; label="WL"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">WL</figure-callout>}}^3}{48\mathrm{EI}}-\frac{8\mathrm{WL}}{4A{\mathrm{\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2---\left(33\right)$

Wherein:

w _MaxThe amount of deflection of representing normal operational phase span centre position.Formula (32) is applicable to shingle nail, formula (33)

Be applicable to roof boarding

W represents the span centre load

L represents the battenboard span

E represents sandwich plate with non-metal surfaces panel elastic modulus

I _fThe expression upper and lower panel is to the moment of inertia of himself natural axis,

I represents the moment of inertia sum of upper and lower panel to himself natural axis and whole battenboard natural axis,

$I=\frac{{\mathrm{<figure-callout\; id="3"\; label="bt"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">bt</figure-callout>}}^3}{6}+\frac{{\mathrm{<figure-callout\; id="2"\; label="btd"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">btd</figure-callout>}}^2}{2}$

B represents the sandwich plate with non-metal surfaces width

T represents the sandwich plate with non-metal surfaces plate thickness

D represents the distance between the upper and lower panel natural axis, d=c+t, and wherein c is a core thickness

A represents the effective cross-sectional area of core,

G represents the modulus of shearing actual measurement of core;

μ represents the reduction coefficient of core modulus of shearing between the different experiments method, μ=1.236

(2) simplified formula under the evenly load situation.

Under the evenly load effect, sandwich plate with non-metal surfaces is considering that span centre position Calculation of Deflection formula is under the panel rigidity situation:

Wherein:

$\mathrm{\&theta;}=\frac{\mathrm{<figure-callout\; id="2"\; label="aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">aL</figure-callout>}}{2},$

$a^2=\frac{\mathrm{AG}}{{\mathrm{EI}}_f(1-I_f/I)}$

Owing to generally need not to be provided with semi-girder, this seasonal L in the actual engineering ₁=0, above-mentioned parameter can be simplified to following simple form thus:

${\mathrm{\&beta;}}_2=\frac{1}{\mathrm{\&theta;}\cosh \mathrm{\&theta;}},$ $\mathrm{\&theta;}=\frac{\mathrm{<figure-callout\; id="2"\; label="aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">aL</figure-callout>}}{2},$

And a ²Size represents in fact is the ratio of core shearing rigidity and panel local bending rigidity, several sandwich plate with non-metal surfaces of being studied herein, its a ²Value greatly about more than 400, calculate the θ value get thus generally about 20.

With the relation of θ as shown in figure 12.Consider θ 〉=3 o'clock,

Therefore have:

For the shingle nail in the actual engineering, consider the θ value generally more than 20, can be similar to this moment thinks Can obtain shingle nail and under evenly load, consider the final Calculation of Deflection formula of panel rigidity:

$w_{\max }=-\frac{{5\mathrm{<figure-callout\; id="3"\; label="qL"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">qL</figure-callout>}}^3}{384\mathrm{EI}}-\frac{\mathrm{<figure-callout\; id="2"\; label="q\; L"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">q</figure-callout>}{L}^{}{}^2}{8A{\mathrm{\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2---\left(35\right)$

For shingle nail, estimate the ratio of its panel rigidity and integral rigidity, generally more than 10%, the general minimum of θ value this moment can reach about 5, and is desirable

And consider the conversion relation of core modulus of shearing under the different tests method, can obtain roof boarding and consider that the final abbreviation form of Calculation of Deflection formula that the span centre position is under the evenly load effect is under the panel rigidity situation:

$w_{\max }=-\frac{{5\mathrm{<figure-callout\; id="4"\; label="qL"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">qL</figure-callout>}}^4}{384\mathrm{EI}}-\frac{0.9{\mathrm{<figure-callout\; id="2"\; label="qL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">qL</figure-callout>}}^2}{8A{\mathrm{\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2---\left(36\right)$

In the formula:

w _MaxThe amount of deflection of representing normal operational phase span centre position, its Chinese style (35) is applicable to shingle nail, formula (36) is applicable to shingle nail

Q represents evenly load;

L represents the battenboard span;

E represents sandwich plate with non-metal surfaces panel elastic modulus;

I _fThe expression upper and lower panel is to the moment of inertia of himself natural axis,

I represents the moment of inertia sum of upper and lower panel to himself natural axis and whole battenboard natural axis,

$I=\frac{{\mathrm{<figure-callout\; id="3"\; label="bt"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">bt</figure-callout>}}^3}{6}+\frac{{\mathrm{<figure-callout\; id="2"\; label="btd"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">btd</figure-callout>}}^2}{2}$

B represents the sandwich plate with non-metal surfaces width

T represents the sandwich plate with non-metal surfaces plate thickness

D represents the distance between the upper and lower panel natural axis, d=c+t, and wherein c is a core thickness

A represents the effective cross-sectional area of core,

G represents the modulus of shearing measured value of core

μ represents the reduction coefficient of core modulus of shearing between the different experiments method, usually μ=1.236

Step 300: the anti-bending bearing capacity of determining sandwich plate with non-metal surfaces: the anti-bending bearing capacity of determining sandwich plate with non-metal surfaces by the relation of amount of deflection and anti-bending bearing capacity.

Because the amount of deflection and the anti-bending bearing capacity of laminboard wall panel have following relation:

w _max≤[f]

[f] is referring to the limit value of amount of deflection in the national standard " the adiabatic battenboard of metal covering for building " (GB/T 23932-2009).

Then:

Load is determined: $W\&le;\frac{\left[f\right]}{\frac{{\mathrm{<figure-callout\; id="3"\; label="L"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">L</figure-callout>}}^3}{48\mathrm{EI}}+\frac{0.8L}{4A{\mathrm{\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2}$

Determining of local load: $q\&le;\frac{\left[f\right]}{\frac{{5\mathrm{<figure-callout\; id="4"\; label="L"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">L</figure-callout>}}^4}{384\mathrm{EI}}+\frac{0.9{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">L</figure-callout>}}^2}{8A{\mathrm{\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2}$

Thereby determine the anti-bending bearing capacity of sandwich plate with non-metal surfaces.

Flexural capacity of sandwich plate with non-metal surfaces of the present invention is determined method, by considering the panel rigidity influence that flexural deformation causes to sandwich plate with non-metal surfaces of sandwich plate with non-metal surfaces, accurately obtain the amount of deflection of sandwich plate with non-metal surfaces under load and evenly load, determine the anti-bending bearing capacity of sandwich plate with non-metal surfaces then by amount of deflection.The present invention accurately obtains the anti-bending bearing capacity of sandwich plate with non-metal surfaces, thereby determines the anti-bending mechanics performance of sandwich plate with non-metal surfaces, the security performance of accurate assessment sandwich plate with non-metal surfaces.

The specific embodiment of the present invention is: described flexural capacity of sandwich plate with non-metal surfaces is determined that method is applied to the safety assessment of sandwich plate with non-metal surfaces.

Above content be in conjunction with concrete preferred implementation to further describing that the present invention did, can not assert that concrete enforcement of the present invention is confined to these explanations.For the general technical staff of the technical field of the invention, without departing from the inventive concept of the premise, can also make some simple deduction or replace, all should be considered as belonging to protection scope of the present invention.

Claims (2)

1. a flexural capacity of sandwich plate with non-metal surfaces is determined method, it is characterized in that, the panel of described battenboard is nonmetallic materials, and flexural capacity of sandwich plate with non-metal surfaces determines that method comprises the steps:

Gather the correlation parameter of sandwich plate with non-metal surfaces: the width of the span of collection sandwich plate with non-metal surfaces, the elastic modulus of nonmetal panel, sandwich plate with non-metal surfaces panel, the thickness of sandwich plate with non-metal surfaces panel, sandwich plate with non-metal surfaces core thickness, the modulus of shearing of sandwich plate with non-metal surfaces core, the effective cross-sectional area of sandwich plate with non-metal surfaces core;

Determine the amount of deflection of sandwich plate with non-metal surfaces: the rigidity of sandwich plate with non-metal surfaces panel needs to consider, the amount of deflection of sandwich plate with non-metal surfaces determine to comprise amount of deflection under the load and the amount of deflection under the evenly load,

Amount of deflection under the sandwich plate with non-metal surfaces load adopts following formula to obtain:

$w_{\max }=-\frac{{\mathrm{WL}}^3}{48\mathrm{EI}}-\frac{0.8\mathrm{WL}}{4A{\mathrm{\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2$

Amount of deflection under the sandwich plate with non-metal surfaces evenly load adopts following formula to obtain:

$w_{\max }=-\frac{{5\mathrm{qL}}^4}{384\mathrm{EI}}-\frac{0.9{\mathrm{qL}}^2}{{8\mathrm{A\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2$

Determine the anti-bending bearing capacity of sandwich plate with non-metal surfaces: determine the anti-bending bearing capacity of sandwich plate with non-metal surfaces by the relation of amount of deflection and anti-bending bearing capacity, because the amount of deflection and the anti-bending bearing capacity of laminboard wall panel have following relation:

w _max≤[f]

[f] is amount of deflection,

Then:

Load is determined: $W\&le;\frac{\left[f\right]}{\frac{L^3}{48\mathrm{EI}}+\frac{0.8L}{{4\mathrm{A\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2}$

Determining of local load: $q\&le;\frac{\left[f\right]}{\frac{{5L}^4}{384\mathrm{EI}}+\frac{{0.9L}^2}{{8\mathrm{A\&mu;}}^{2}G}{(1-\frac{I_f}{I})}^2}$

Thereby determine the anti-bending bearing capacity of sandwich plate with non-metal surfaces,

In the above-mentioned formula, the expression meaning of each variable is as follows:

w _MaxThe amount of deflection of representing normal operational phase span centre position;

W represents the span centre load;

Q represents evenly load;

L represents the battenboard span;

E represents sandwich plate with non-metal surfaces panel elastic modulus;

I _fThe expression upper and lower panel is to the moment of inertia of himself natural axis,

I represents the moment of inertia sum of upper and lower panel to himself natural axis and whole battenboard natural axis,

$I=\frac{{\mathrm{bt}}^3}{6}+\frac{{\mathrm{btd}}^2}{2};$

B represents the sandwich plate with non-metal surfaces width;

T represents the sandwich plate with non-metal surfaces plate thickness;

D represents the distance between the upper and lower panel natural axis, d=c+t, and wherein c is a core thickness;

A represents the effective cross-sectional area of core,

G represents the modulus of shearing measured value of core;

μ represents the reduction coefficient of core modulus of shearing.

2. flexural capacity of sandwich plate with non-metal surfaces according to claim 1 is determined method, it is characterized in that, in the amount of deflection step of determining sandwich plate with non-metal surfaces, comprises and determines the influence of sandwich plate with non-metal surfaces panel rigidity to battenboard rigidity, promptly

Wherein, E _fThe elastic modulus of expression panel; B represents the sandwich plate with non-metal surfaces width;

T represents the sandwich plate with non-metal surfaces plate thickness;

D represents the distance between the upper and lower panel natural axis, d=c+t, and wherein c is a core thickness.

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