CN101886992A - 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

Method for determining bending resistance bearing capacity of non-metal surface sandwich board and application

Technical Field

The invention relates to a method for determining the bending resistance bearing capacity of a sandwich board and application thereof, in particular to a method for determining the bending resistance bearing capacity of a non-metal surface sandwich board and application thereof.

Background

With the development of building engineering technology, sandwich panels are increasingly widely used in modern society. With the widespread use of sandwich panels, sandwich panel technology has also evolved, from the sandwich panels of the original metal face sheets to the sandwich panels of the non-metal face sheets. In modern society, with the improvement of non-metal material technology, the sandwich board of non-metal panel gradually takes a leading position. For determining the bending resistance bearing capacity of the non-metal sandwich board, a simple, convenient and practical method does not exist in the prior art, and particularly, the influence of the rigidity of the non-metal panel on the bending deformation of the non-metal sandwich board is not considered deeply, so that the bending resistance mechanical property of the non-metal sandwich board cannot be accurately obtained, and the safety performance of the non-metal sandwich board cannot be accurately evaluated.

Disclosure of Invention

The technical problem solved by the invention is as follows: the method for determining the bending resistance bearing capacity of the non-metal surface sandwich board overcomes the technical problem that the bending resistance bearing capacity of the non-metal surface sandwich board cannot be accurately determined due to the fact that the bending resistance mechanical property of the non-metal surface sandwich board cannot be accurately obtained in the prior art.

The technical scheme of the invention is as follows: the method for determining the bending resistance bearing capacity of the sandwich plate with the non-metal surface is provided, the panel of the sandwich plate is made of a non-metal material, and the method for determining the bending resistance bearing capacity of the sandwich plate with the non-metal surface comprises the following steps:

collecting related parameters of the non-metal surface sandwich plate: collecting the span of the non-metal surface sandwich board, the elastic modulus of the non-metal surface sandwich board, the width of the non-metal surface sandwich board, the thickness of the core material of the non-metal surface sandwich board, the shear modulus of the core material of the non-metal surface sandwich board and the effective cross-sectional area of the core material of the non-metal surface sandwich board.

Determining the deflection of the non-metal surface sandwich panel: the rigidity of the non-metal surface sandwich board panel needs to be considered, the determination of the deflection of the non-metal surface sandwich board comprises the deflection under concentrated load and the deflection under uniformly distributed load, and the deflection under the concentrated load of the non-metal surface sandwich board is obtained by adopting the following formula:

<math><mrow><msub><mi>w</mi><mi>max</mi></msub><mo>=</mo><mo>-</mo><mfrac><msup><mi>WL</mi><mn>3</mn></msup><mrow><mn>48</mn><mi>EI</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>0.8</mn><mi>WL</mi></mrow><mrow><mn>4</mn><mi>A</mi><msup><mi>&mu;</mi><mn>2</mn></msup><mi>G</mi></mrow></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></math>

the deflection of the non-metal surface sandwich plate under uniform load is obtained by adopting the following formula:

<math><mrow><msub><mi>w</mi><mi>max</mi></msub><mo>=</mo><mo>-</mo><mfrac><msup><mrow><mn>5</mn><mi>qL</mi></mrow><mn>3</mn></msup><mrow><mn>384</mn><mi>EI</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>9</mn><mi>qL</mi></mrow><mrow><mn>8</mn><mi>A</mi><msup><mi>&mu;</mi><mn>2</mn></msup><mi>G</mi></mrow></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></math>

in the above two formulas, the variables have the following meanings:

w_maxrepresenting the deflection at the mid-span position during normal use;

w represents a mid-span concentrated load;

q, uniformly distributing loads;

l represents a sandwich panel span

E represents the modulus of elasticity of the face plate of the sandwich panel with the non-metal surface

I_fRepresenting the moment of inertia of the upper and lower panels about their own neutral axis,

i represents the sum of the moments of inertia of the upper and lower face plates for their own neutralization axis and the neutralization axis of the entire sandwich panel,

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

b represents the width of the sandwich panel with non-metal surface

t represents the thickness of the sandwich panel with non-metal surface

d represents the distance between the upper and lower panels and the axis, d ═ c + t, where c is the core thickness

A represents an effective cross-sectional area of the core material,

g represents an actual measured value of the shear modulus of the core material;

μ represents a conversion coefficient of the shear modulus of the core material, and is usually 1.236

Determining the bending resistance bearing capacity of the non-metal surface sandwich plate: and determining the bending resistance bearing capacity of the non-metal surface sandwich plate according to the relation between the deflection and the bending resistance bearing capacity.

The further technical scheme of the invention is as follows: and in the step of determining the deflection of the non-metal surface sandwich plate, determining the influence of the rigidity of the non-metal surface sandwich plate on the rigidity of the sandwich plate.

The further technical scheme of the invention is as follows: determining the influence of the non-metal panel stiffness on the sandwich panel stiffness comprises determining the influence of the non-metal panel stiffness on the sandwich panel bending stiffness and determining the influence of the non-metal face sandwich panel stiffness on the sandwich panel shear stress distribution.

The further technical scheme of the invention is as follows: and in the step of determining the deflection of the non-metal surface sandwich plate, determining the bending deformation and the shearing deformation of the non-metal surface sandwich plate.

The further technical scheme of the invention is as follows: determining the bending deformation and shear deformation of the non-metallic face sandwich panel includes determining the effect of the stiffness of the non-metallic face sandwich panel on the bending deformation of the sandwich panel and determining the effect of the stiffness of the non-metallic face sandwich panel on the shear deformation of the sandwich panel.

The technical scheme of the invention is as follows: the method for determining the bending resistance bearing capacity of the non-metal surface sandwich board is applied to safety evaluation of the non-metal surface sandwich board.

The invention has the technical effects that: the method comprises the steps of accurately acquiring the deflection of the non-metal surface sandwich board under concentrated load and uniform load by considering the influence of the panel rigidity of the non-metal surface sandwich board on the bending deformation of the non-metal surface sandwich board, and then determining the bending bearing capacity of the non-metal surface sandwich board through the deflection. The invention can accurately obtain the bending resistance bearing capacity of the non-metal surface sandwich board, thereby determining the bending resistance mechanical property of the non-metal surface sandwich board and accurately evaluating the safety performance of the non-metal surface sandwich board.

Drawings

FIG. 1 is a flow chart of the present invention.

FIG. 2 is a schematic view of the present invention, which simplifies the non-metal surface sandwich panel into a sandwich beam.

FIG. 3 is a schematic cross-sectional view of the present invention, which simplifies the non-metal face sandwich panel into a sandwich beam.

FIG. 4 is a schematic diagram of the cross-sectional shear distribution of an I-beam of the present invention.

FIG. 5 is a schematic diagram of the shear distribution of the sandwich beam of the present invention.

FIG. 6 is a schematic cross-sectional view of a thick panel sandwich panel showing stress distribution according to the present invention.

FIG. 7 is a schematic diagram of the deformation of the sandwich beam under the concentrated load effect.

FIG. 8 is a schematic view of the shear deformation of the sandwich beam according to the present invention.

FIG. 9 is a schematic view of a simply supported beam under the action of concentrated loads across the center of the bridge of the present invention.

FIG. 10 is a schematic view of a simply supported beam under the action of uniform load according to the present invention.

FIG. 11 shows the relationship between θ and θ under concentrated load according to the present invention

The relationship between them.

FIG. 12 shows the equation of θ and

the relationship between them.

Detailed Description

The technical solution of the present invention is further illustrated below with reference to specific examples.

As shown in fig. 1, the specific embodiment of the present invention is: the invention provides a method for determining the bending resistance bearing capacity of a sandwich plate with a non-metal surface, wherein the face plate of the sandwich plate is made of non-metal materials, and the face plate of the sandwich plate is introduced by taking a straw board or an Oriented Strand Board (OSB) as an example.

The method for determining the bending resistance bearing capacity of the sandwich plate with the non-metal surface comprises the following steps:

step 100: collecting relevant parameters of the non-metal surface sandwich plate, namely collecting the following parameters of the non-metal surface sandwich plate: collecting the span of the non-metal surface sandwich board, the elastic modulus of the non-metal surface sandwich board panel, the width of the non-metal surface sandwich board panel, the thickness of the non-metal surface sandwich board core material, the shear modulus of the non-metal surface sandwich board core material and the effective cross-sectional area of the non-metal surface sandwich board core material. Among these parameters, the span of the non-metal face sandwich panel, the width of the non-metal face sandwich panel, the thickness of the non-metal face sandwich panel core material, and the effective cross-sectional area of the non-metal face sandwich panel core material are determined by the shape of the non-metal face sandwich panel, and the elastic modulus of the non-metal face sandwich panel and the shear modulus of the non-metal face sandwich panel core material are determined by the material of the non-metal face sandwich panel.

Step 200: and determining the deflection of the non-metal surface sandwich plate, namely determining the deflection of the non-metal surface sandwich plate, wherein the deflection under concentrated load and the deflection under uniformly distributed load are included.

For determining the deflection of the sandwich plate with the non-metal surface, the following factors need to be considered firstly: the influence of the panel rigidity of the non-metal surface sandwich plate on the deflection of the non-metal surface sandwich plate.

Firstly, the influence of the panel rigidity of the non-metal surface sandwich board on the deflection of the non-metal surface sandwich board.

Specifically, the influence of the panel rigidity of the non-metal-faced sandwich panel on the deflection of the non-metal-faced sandwich panel includes the influence of the panel rigidity of the non-metal-faced sandwich panel on the bending rigidity of the sandwich panel and the influence of the panel rigidity of the non-metal-faced sandwich panel on the shear stress distribution of the sandwich panel.

The influence of the panel rigidity of the sandwich panel with the non-metal surface on the bending rigidity of the sandwich panel is as follows:

as shown in fig. 2 and 3, the non-metal surface sandwich panel is simplified into a sandwich beam form, i.e. the stress in the y direction (i.e. the width direction of the panel) is not considered.

The symbols in the figures are defined as follows:

c represents the core material thickness;

t represents the panel thickness

h represents the thickness of the sandwich panel, h ═ c +2t

d represents the distance between the center lines of the upper and lower panels, and d is c + t

b represents the width of the sandwich panel

G represents the shear modulus of the core material

D represents the overall bending stiffness of the sandwich panel

A represents the equivalent cross-sectional area of the sandwich panel

AG represents the shear stiffness of the sandwich panel, wherein A ═ bd²/c

Q represents the shear force of a certain section of the sandwich panel

E_fModulus of elasticity of the display panel

E_cExpressing the modulus of elasticity of the core Material

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

I_fRepresenting the moment of inertia of the upper and lower panels about their own axes

Because the sandwich panel consists of the upper and lower face plates and the core material, the bending rigidity of the sandwich beam shown in figures 2 and 3 obtains the following formula according to the calculation formula of the mechanical rigidity of the material:

<math><mrow><mi>D</mi><mo>=</mo><msub><mi>E</mi><mi>f</mi></msub><mo>&CenterDot;</mo><mfrac><msup><mi>bt</mi><mn>3</mn></msup><mn>6</mn></mfrac><mo>+</mo><msub><mi>E</mi><mi>f</mi></msub><mo>&CenterDot;</mo><mfrac><mrow><mi>bt</mi><msup><mi>d</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>+</mo><msub><mi>E</mi><mi>c</mi></msub><mo>&CenterDot;</mo><mfrac><msup><mi>bc</mi><mn>3</mn></msup><mn>12</mn></mfrac><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math>

wherein the first term represents the local stiffness of the panel when it is bent about its own axis; the second term represents the stiffness that results when the upper and lower panels are bent relative to the central axis c-c; the third term represents the local stiffness of the core when bent about its own axis (common axis c-c).

In an actual sandwich structure, the second term of the formula (1) occupies a dominant position, the first term of the formula (1), namely the influence of the rigidity of the non-metal surface sandwich panel cannot be ignored, and the third term of the formula (1), namely the influence of the rigidity of the core material per se.

The influence of the panel rigidity of the sandwich panel with the non-metal surface on the shear stress distribution of the sandwich panel is as follows:

as shown in fig. 4, the sandwich plate can be simplified into an i-beam form according to the working principle of the sandwich plate, so that the distribution of shear stress in the i-beam is obtained.

For the core material shear stress tau at the position z below the central axis of the section, the following formula is provided according to the mechanics of materials:

<math><mrow><mi>&tau;</mi><mo>=</mo><mfrac><mi>QS</mi><mi>Ib</mi></mfrac><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math>

wherein: q is the shear force over the selected cross section; i is the moment of inertia of the whole cross section to the neutral axis; b is z₁Width of (a), S is z > z₁The static distance of the section of the part to the neutral axis, where z denotes the distance from the neutral axis at z, z₁Denotes z₁The distance between the central axis and the outer wall of the cylinder.

For a composite beam with a sandwich structure, the above formula can be written as follows in consideration of the elastic modulus of each part:

<math><mrow><mi>&tau;</mi><mo>=</mo><mfrac><mi>Q</mi><mi>Db</mi></mfrac><mi>&Sigma;</mi><mrow><mo>(</mo><mi>SE</mi><mo>)</mo></mrow><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow></math>

wherein D is shown as formula (1); sigma (SE) is z > z₁For example, to determine the shear stress at the core portion z, the sum of the products of the partial cross sections S and E of (a) is:

<math><mrow><mi>&Sigma;</mi><mrow><mo>(</mo><mi>SE</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>E</mi><mi>f</mi></msub><mfrac><mi>btd</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><msub><mi>E</mi><mi>c</mi></msub><mi>b</mi></mrow><mn>2</mn></mfrac><mrow><mo>(</mo><mfrac><mi>c</mi><mn>2</mn></mfrac><mo>-</mo><mi>z</mi><mo>)</mo></mrow><mrow><mo>(</mo><mfrac><mi>c</mi><mn>2</mn></mfrac><mo>+</mo><mi>z</mi><mo>)</mo></mrow><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow></math>

thus, shear stress in the core material:

<math><mrow><mi>&tau;</mi><mo>=</mo><mfrac><mi>Q</mi><mi>D</mi></mfrac><mo>{</mo><msub><mi>E</mi><mi>f</mi></msub><mo>&CenterDot;</mo><mfrac><mi>td</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><msub><mi>E</mi><mi>c</mi></msub><mn>2</mn></mfrac><mrow><mo>(</mo><mfrac><msup><mi>c</mi><mn>2</mn></msup><mn>4</mn></mfrac><mo>-</mo><msup><mi>z</mi><mn>2</mn></msup><mo>)</mo></mrow><mo>}</mo><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mrow></math>

shear stresses in the panel can be similarly obtained.

According to the knowledge of material mechanics, the distribution of shear stress on the cross section of the sandwich beam is shown in figure 5: wherein, (a) is the real shear stress distribution of the cross section of the sandwich beam. (b) For sandwich beams with neglected core material stiffnessThe distribution situation of the shear stress is that,

(c) in order to neglect the self-rigidity of the core material and the self-rigidity of the panel, the shear stress distribution of the sandwich beam,

for low strength foam cores, E can be measured_cThe shear stress constant in the core material was obtained as 0, shown in fig. 5 (b):

<math><mrow><mi>&tau;</mi><mo>=</mo><mfrac><mi>Q</mi><mi>D</mi></mfrac><mo>&CenterDot;</mo><mfrac><mrow><msub><mi>E</mi><mi>f</mi></msub><mi>td</mi></mrow><mn>2</mn></mfrac><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mrow></math>

at this time <math><mrow><mi>D</mi><mo>=</mo><msub><mi>E</mi><mi>f</mi></msub><mo>&CenterDot;</mo><mfrac><msup><mi>bt</mi><mn>3</mn></msup><mn>6</mn></mfrac><mo>+</mo><msub><mi>E</mi><mi>f</mi></msub><mo>&CenterDot;</mo><mfrac><msup><mi>btd</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>.</mo></mrow></math>

In addition, if the flexural rigidity of the panel is small relative to its own central axis, thenThe first term in (1) is also negligible, i.e.

The shear stress in the core material can be simplified to the following simplest form, as shown in fig. 5 (c):

<math><mrow><mi>&tau;</mi><mo>=</mo><mfrac><mi>Q</mi><mi>bd</mi></mfrac><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mrow></math>

secondly, for the determination of the deflection of the sandwich plate with the non-metal surface, the following factors need to be considered: bending deformation and shearing deformation of the non-metal surface sandwich plate.

The non-metal surface sandwich plate is simplified into a sandwich beam form, and the specification of the bending beam symbol in material mechanics is shown in a formula (8).

For the deformation of thick panel sandwich panels, the following points need to be specified:

the core material is a foam core material such as EPS-polystyrene, the elastic modulus is very small, and the self rigidity, namely the third term in the formula (1), can be ignored;

the panel has a certain rigidity, and the first term in the formula (1) cannot be ignored;

since the stiffness of the core material itself is neglected and the stiffness of the panel cannot be neglected, the shear stress distribution in the core material is shown in (b) of fig. 5, and the shear stress is constant in the thickness direction of the core material and has a magnitude shown in formula (6).

Due to the fact that the rigidity of the panel is considered, the following influences are generated on the deformation of the sandwich beam:

the first effect is to make the panel have two deformation modes, the first mode is local bending, relative to the bending deformation of the central axis of the whole sandwich structure, at this time, the panel is stretched and compressed under uniform stress, and at this time, the stress in the panel is shown as the first part stress in fig. 6. The second way is a local bending with respect to the axis of the panel itself, instead of the whole sandwich structure axis, where the stresses in the panel are generated as shown in the second part of the stresses in figure 6.

The deformation of the sandwich beam as a whole when it is bent as shown in fig. 7, for example, when it is subjected to concentrated loads. In fig. 7, (a) shows a simply supported beam subjected to concentrated force across the span, (b) shows bending deformation, (c) shows shear deformation, and (d) shows the combined effect of bending deformation and shear deformation. In fig. 7(b), the panel has both of the above two modifications. The contribution of the panel local bending stiffness to the overall sandwich beam bending stiffness may be represented by the first term in equation (1).

The second effect is the effect on the shear deformation of the core material:

influence on shear deformation inside the core material: when considering only shear deformation, the points a, b, c … on the central axis of the panel are not displaced in the horizontal direction (and therefore do not change the principal stress on the panel), but are deformed only in the vertical direction, as shown in fig. 5 (c). A break angle will appear at the mid-span position, where the curvature will be infinite, which is obviously not possible: from knowledge of material mechanics, it can be known that there is a gap between bending moment and curvatureThe bending moment is infinite here. If the panel and core remain connected, the panel needs to be locally bent over a distance on both sides of the span to smooth out the shear deformation. At this point, additional bending moments and shear forces will be introduced into the panel, reducing shear deformation. In practical sandwich structures, especially thin panel sandwich structures, this effect is small; this effect is significant when the panels are relatively thick (e.g., asbestos cement sandwich panels and non-metal faced sandwich panels) and the core material is a lightweight foam such as EPS.

Based on the above discussion, the following discussion deals with the deformation of the sandwich structure in two points:

and (I) the sandwich plate is subjected to bending deformation under the influence of the panel rigidity of the sandwich plate with the non-metal surface.

Firstly, considering infinite shearing rigidity of a core material, and uniformly distributing load q₁And the sandwich beam unit under action. According to the bending theory of ordinary beams, the deflection w is generated₁. The deflection and bending moment M₁And shear force Q₁In terms of material mechanics, in which the shearing force Q is₁To obtain:

-Q₁＝Dw₁′″＝E_f(I-I_f)w₁′″+E_fI_fw₁′″ (9)

in order to neglect the influence of the rigidity of the core material,

therefore, there are:

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

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

assuming that the panel is only subjected to tensile and compressive deformation without local bending, the first term on the right of equation (9) represents the shear force shared by the core and the panel. At this time, the panel stiffness is temporarily not calculated, and the first term on the right can be calculated according to the stress distribution in fig. 5 (c): the shear stress tau is constant in the thickness range of the core material, is 0 at the edge of the panel and linearly changes in the panel. Thus, the first term can be replaced by-bd τ, where τ is the shear stress in the core material and d is the distance between the upper and lower panel centerlines, i.e.:

-Q₁＝-bdτ+E_fI_fw₁′″ (12)

simultaneously, the method comprises the following steps: q. q.s₁＝-Q₁′，Q₁＝M₁，M₁＝-Dw₁″。

And (II) determining the shear deformation of the sandwich plate under the influence of the panel rigidity of the sandwich plate with the non-metal surface.

Since the core stiffness is assumed to be infinite, no shear strain occurs in the core despite the shear stress τ. Therefore, if the shear modulus G of the core material is a certain limit, the core material generates a shear strain γ equal to τ/G under the action of the shear stress τ, which corresponds to an additional transverse deformation w₂. The panel must be subjected to this additional deformation at the same time, and therefore it must be subjected to an additional uniform load q₂Shear force Q₂And bending moment M₂：

q₂＝-Q₂′，Q₂＝M₂，M₂＝-Dw₂″

The total load, shear, bending moment and deformation are as follows:

q＝q₁₊q₂

Q＝Q₁+Q₂

M＝M₁+M₂

w＝w₁+w₂

that is, the sandwich beam uniformly distributed with the load q will generate two different sets of deformation: w is a₁And w₂. The first of these represents the common bending deformation, which is the shear force Q shared by the face sheet and the core₁Correlation; the second term represents the result of Q₁Induced shear deformation of the core material: to accommodate core shear deformation requirements, the panel also participates in additional bending deformation about its own axis (ignoring in the panelShear deformation, but the panels still share shear); at this point, an additional shear force is required to drive the deformation, Q₂。Q₁And Q₂Is the total shear force exerted on the beam.

And (III) the bending deformation and the shearing deformation of the sandwich plate under the influence of the panel rigidity of the sandwich plate with the non-metal surface have a mutual relation.

The additional deformation is related to the core shear strain gamma as shown in figure 8. The length of the line segment de is equal to

The length of the line segment cf is equal to gammac, and is obtained from de ═ cf

The relationship with γ is as follows:

<math><mrow><mfrac><msub><mi>dw</mi><mn>2</mn></msub><mi>dx</mi></mfrac><mo>=</mo><mi>&gamma;</mi><mfrac><mi>c</mi><mi>d</mi></mfrac><mo>=</mo><mfrac><mi>Q</mi><mi>Gbd</mi></mfrac><mo>&CenterDot;</mo><mfrac><mi>c</mi><mi>d</mi></mfrac><mo>=</mo><mfrac><mi>Q</mi><mi>AG</mi></mfrac><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>13</mn><mo>)</mo></mrow></mrow></math>

wherein A ═ bd²The/c, AG generally refers to the shear stiffness of the sandwich beam.

Substituting τ into γ G into equation (13), the relationship between the additional deformation and the shear stress can be obtained:

<math><mrow><mi>&tau;</mi><mo>=</mo><mfrac><mi>d</mi><mi>c</mi></mfrac><mo>&CenterDot;</mo><mi>G</mi><msup><msub><mi>w</mi><mn>2</mn></msub><mo>&prime;</mo></msup><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>14</mn><mo>)</mo></mrow></mrow></math>

substituting it into equation (12), i.e.:

-Q₁＝-AGw₂′+EI_fw₁′″ (15)

will be-Q₁＝-Dw₁' "into equation (15), the changes are made to obtain:

<math><mrow><msup><msub><mi>w</mi><mn>2</mn></msub><mo>&prime;</mo></msup><mo>=</mo><mo>-</mo><mfrac><mi>D</mi><mi>AG</mi></mfrac><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><msup><msub><mi>w</mi><mn>1</mn></msub><mrow><mo>&prime;</mo><mo>&prime;</mo><mo>&prime;</mo></mrow></msup><mo>=</mo><mo>+</mo><mfrac><msub><mi>Q</mi><mn>1</mn></msub><mi>AG</mi></mfrac><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>16</mn><mo>)</mo></mrow></mrow></math>

due to-Q₂＝-Dw₂' ", then the total shear force:

Q＝Q₁+Q₂＝Q₁-EI_fw₂′″ (17)

substituting equation (16) into equation (17) can yield information about Q₁The equation of (a) is as follows:

Q₁″-a²Q₁＝-a²Q (18)

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

and thirdly, a deformation formula of the deflection of the sandwich plate with the non-metal surface.

From the above analysis, it can be seen that the stress condition of the sandwich beam is known, i.e. Q can be given by the equation about x, and then Q can be obtained according to the equation (17)₁(ii) a Then according to w₁′、w₂' and Q₁The relation between w can be finally obtained by integration₁And w₂(ii) a Finally according to w ═ w₁+w₂The final deflection calculation formula of the simply supported sandwich beam at the mid-span position under different stress conditions can be obtained.

And (I) a deformation formula of the deflection of the non-metal surface sandwich plate under the condition of concentrated load.

As shown in FIG. 9, in the AB segment, where x is starting point A, the shear is-W/2, and the solution according to equation (17) is:

$-{\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1={\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">C</figure-callout>}}_1\cosh \mathrm{ax}+{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">C</figure-callout>}}_2\sinh \mathrm{ax}+\frac{W}{2}---\left(20\right)$

by integration, one can obtain:

${\mathrm{<figure-callout\; id="1"\; label="EIw"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">EIw</figure-callout>}}_1=\frac{C_1}{a^3}\sinh \mathrm{ax}+\frac{C_2}{a^3}\cosh \mathrm{ax}+\frac{{\mathrm{<figure-callout\; id="3"\; label="Wx"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">Wx</figure-callout>}}^3}{12}+C_3{x}^2+C_{4}x+C_5---\left(21\right)$

equation (20) together with equation (21) yields a correlation with w₂The expression of' is integrated once to obtain the following formula:

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

on the AB segment, 5 boundary conditions can be found, whereby the above six constants have the following relationship:

(i)x＝0，w₁either as 0 (arbitrary)

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

(ii)x＝0，w₁' As 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，w₁' 0 (symmetry)

${\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">W</figure-callout>}}{2}=0$

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

Definition of-M-EIw₁″+EI_fw₂″

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

(v)x＝0，w₂either as 0 (arbitrary)

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

To this end, the constants may be represented as follows, where C₂Unknown:

${\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">C</figure-callout>}}_1=-\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">W</figure-callout>}}{2};{{\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">C</figure-callout>}}_3=-}_{}\frac{\mathrm{<figure-callout\; id="8"\; label="WL"\; filenames="HSA00000189287800023.png"\; state="\{\{state\}\}">WL</figure-callout>}}{8};{\mathrm{<figure-callout\; id="4"\; label="C"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">C</figure-callout>}}_4=+\frac{W}{2a^2};C_5=C_6=-\frac{C_2}{a^3}---\left(23\right)$

on segment BC, where x starts at B point, its total shear is 0. Equations (20) and (21) still apply but the term containing W should be eliminated, the new constant B₁-B₆To substitute for C₁-C₆。

The following are four simple boundary conditions:

(vi)x＝0，w₁either as 0 (arbitrary)

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

(vii)x＝0，w₂Either as 0 (arbitrary)

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

(viii)x＝L₁，w₁″＝0

$\frac{B_1}{a}\sinh a{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">L</figure-callout>}}_1+\frac{B_2}{a}\cosh a{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">L</figure-callout>}}_1+2{\mathrm{<figure-callout\; id="3"\; label="B"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">B</figure-callout>}}_3=0$

(ix)x＝L₁，w₂″＝0

$\frac{B_1}{a}\sinh a{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">L</figure-callout>}}_1+\frac{B_2}{a}\cosh a{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">L</figure-callout>}}_1=0$

The last two boundary conditions are due to the bending moment M of the free end₁And M₂Is 0. This condition is only true if the panel end is free to rotate and is not connected to a rigid end. The following are the results of the above boundary conditions:

B₂＝-B₁tanhaL₁；B₃＝0；

continuity at point B still needs to be established. Obviously, w₁' and w₂′，w₁"and w₂"should be continuous; also, as shown in the formula (21), w₁' andmust be continuous. However, only three conditions are used that provide independent equations, each being w₁′，w₂′，w₁″。

(x)w₁' continuous at point B

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

(xi)w₂' continuous at point B

${\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">C</figure-callout>}}_1\mathrm{<figure-callout\; id="2"\; label="cosh\; aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">cosh</figure-callout>}\frac{\mathrm{aL}}{}\frac{}{2}+{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">C</figure-callout>}}_2\mathrm{<figure-callout\; id="2"\; label="sinh\; aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">sinh</figure-callout>}\frac{\mathrm{aL}}{}\frac{}{2}+\frac{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">W</figure-callout>}}{2}={\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">B</figure-callout>}}_1$

(xii)w₁"continuous at point B

${\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">C</figure-callout>}}_1\mathrm{<figure-callout\; id="2"\; label="sinh\; aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">sinh</figure-callout>}\frac{\mathrm{aL}}{}\frac{}{2}+{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">C</figure-callout>}}_2\mathrm{<figure-callout\; id="2"\; label="cosh\; aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">cosh</figure-callout>}\frac{\mathrm{aL}}{}\frac{}{2}+(\frac{\mathrm{<figure-callout\; id="4"\; label="WL"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">WL</figure-callout>}}{4}+2C_3)a={\mathrm{<figure-callout\; id="2"\; label="B"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">B</figure-callout>}}_2+2B_{3}a$

From equations (23) and (24), B can be reduced₂，B₃，C₁And C₃(ii) a Conditions (xi) and (xii) can be used to solve C₂And B₁We are only on C₂Of interest:

<math><mrow><msub><mi>C</mi><mn>2</mn></msub><mo>=</mo><msub><mi>&beta;</mi><mn>1</mn></msub><mfrac><mi>W</mi><mn>2</mn></mfrac><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>25</mn><mo>)</mo></mrow></mrow></math>

wherein,

a is determined by equation (19).

C₁-C₆The total deformation w can be solved by substituting the known values of (a) into equations (20) and (21), as a function of x in the AB range:

<math><mrow><mi>w</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>W</mi><msup><mi>x</mi><mn>2</mn></msup><mi>L</mi></mrow><mrow><mn>24</mn><mi>EI</mi></mrow></mfrac><mrow><mo>(</mo><mn>3</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mi>L</mi></mfrac><mo>)</mo></mrow><mo>-</mo><mfrac><mi>WL</mi><mrow><mn>4</mn><mi>AG</mi></mrow></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>&times;</mo><mo>{</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mi>L</mi></mfrac><mo>-</mo><mfrac><mn>2</mn><mi>aL</mi></mfrac><mo>[</mo><mi>sinh</mi><mi>ax</mi><mo>+</mo><msub><mi>&beta;</mi><mn>1</mn></msub><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>cosh</mi><mi>ax</mi><mo>)</mo></mrow><mo>]</mo><mo>}</mo></mrow></math>

its maximum should occur at the mid-span position, i.e. when x ═ L/2:

wherein:

by the formula (14) and the derivation of the equations (21) and (22) twice, the shear stress inside the core material and the normal stress of the panel can be obtained.

And (II) a deformation formula of the deflection of the non-metal surface sandwich plate under the condition of uniformly distributing the load.

As shown in fig. 10, the force diagram of the simply supported beam under the action of uniform load is shown.

The AB portion shear is-qx, where the x starting point is A. Substituting into equation (18), the results are as follows:

-Q₁＝C₁coshax+C₂sinhax+qx (27)

by means of the integration, the result is,

${\mathrm{<figure-callout\; id="1"\; label="EIw"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">EIw</figure-callout>}}_1=\frac{C_1}{a^3}\sinh \mathrm{ax}+\frac{C_2}{a^3}\cosh \mathrm{ax}+\frac{{\mathrm{<figure-callout\; id="4"\; label="qx"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">qx</figure-callout>}}^4}{24}+C_3{x}^2+C_{4}x+C_5---\left(28\right)$

equation (27) together with (28) mayIs obtained with respect to w₂The expression of' integrates once to obtain the expression (29).

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

Equations (28), (29) apply equally to the BC segment, where x starts at B and the terms containing q are eliminated. Constant C₁-C₆From B₁-B₆And (4) substitution.

The boundary condition and the continuity of the point B require the thick-panel sandwich beam unit under the action of the same concentrated load, wherein the mid-span bending moment WL/4 is changed into qL²/8. The process of solving the unknown constants is the same as above, and the final result is as follows:

C₁＝0； ${\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">C</figure-callout>}}_3=-\frac{{\mathrm{<figure-callout\; id="2"\; label="qL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">qL</figure-callout>}}^2}{16}+\frac{q}{2a^2};$ C₄＝0； $C_5=C_5=-\frac{C_2}{a^3}$

${\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="HSA00000189287800021.png,HSA00000189287800042.png"\; state="\{\{state\}\}">B</figure-callout>}}_1={\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">C</figure-callout>}}_2\mathrm{<figure-callout\; id="2"\; label="sinh\; aL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">sinh</figure-callout>}\frac{\mathrm{aL}}{}\frac{}{2}+\frac{\mathrm{<figure-callout\; id="2"\; label="qL"\; filenames="HSA00000189287800031.png,HSA00000189287800032.png"\; state="\{\{state\}\}">qL</figure-callout>}}{2};$ B₂＝-B₁tanhaL₁； ${\mathrm{<figure-callout\; id="3"\; label="B"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">B</figure-callout>}}_3=0;{\mathrm{<figure-callout\; id="4"\; label="B"\; filenames="HSA00000189287800043.png,HSA00000189287800051.png"\; state="\{\{state\}\}">B</figure-callout>}}_4=-\frac{{\mathrm{<figure-callout\; id="3"\; label="qL"\; filenames="HSA00000189287800043.png"\; state="\{\{state\}\}">qL</figure-callout>}}^3}{24}$

<math><mrow><msub><mi>C</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mi>&beta;</mi><mfrac><mi>qL</mi><mn>2</mn></mfrac></mrow></math>

wherein

a is determined by equation (19).

At any point in segment AB, the total deflection is given by:

<math><mrow><mi>w</mi><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mi>qx</mi><mn>2</mn></msup><msup><mi>L</mi><mn>2</mn></msup></mrow><mrow><mn>48</mn><mi>EI</mi></mrow></mfrac><mrow><mo>(</mo><mn>3</mn><mo>-</mo><mfrac><msup><mrow><mn>2</mn><mi>x</mi></mrow><mn>2</mn></msup><msup><mi>L</mi><mn>2</mn></msup></mfrac><mo>)</mo></mrow><mo>-</mo><mfrac><mi>q</mi><mi>AG</mi></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>&times;</mo><mo>{</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mfrac><mrow><msub><mi>&beta;</mi><mn>2</mn></msub><msup><mi>L</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mi>&theta;</mi></mrow></mfrac><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>cosh</mi><mi>ax</mi><mo>)</mo></mrow><mo>}</mo></mrow></math>

its maximum occurs at x ═ L/2:

wherein:

the shear stress inside the core material and the normal stress of the panel can also be obtained by the equation (14) and the derivation of the equations (28) and (29) twice.

And fourthly, checking and calculating the bearing capacity of the sandwich plate with the non-metal surface.

In practical engineering, the bending resistance bearing capacity of the sandwich plate is mainly controlled by deformation in a normal use limit state, and when the sandwich plate is subjected to uniform surface load, the bending resistance bearing capacity of the single-span sandwich plate can be calculated according to the following regulations:

w_max≤[f]

wherein: [f] the deformation control limit value in the normal use limit state is shown, and L/200 is generally taken as the sandwich plate span.

And (I) simplifying the formula under the condition of concentrated load.

Under the action of concentrated load, under the condition that the rigidity of the panel is considered, the deflection calculation formula of the sandwich panel with the non-metal surface under the action of uniformly distributed load at the midspan position is as follows:

wherein:

<math><mrow><mi>&theta;</mi><mo>=</mo><mfrac><mi>aL</mi><mn>2</mn></mfrac><mo>,</mo></mrow></math>

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

because the cantilever beam is not needed in the actual engineering, the L is made₁0, whereby the above parameters can be simplified into the following simple form:

<math><mrow><msub><mi>&beta;</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mi>sinh</mi><mi>&theta;</mi></mrow><mrow><mi>cosh</mi><mi>&theta;</mi></mrow></mfrac><mo>,</mo></mrow></math> <math><mrow><mi>&theta;</mi><mo>=</mo><mfrac><mi>aL</mi><mn>2</mn></mfrac><mo>=</mo><mfrac><mi>L</mi><mn>2</mn></mfrac><msup><mrow><mo>(</mo><mfrac><mi>AG</mi><mrow><msub><mi>EI</mi><mi>f</mi></msub><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msub><mi>I</mi><mi>f</mi></msub><mo>/</mo><mi>I</mi><mo>)</mo></mrow></mrow></mfrac><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>,</mo></mrow></math>

and a is²The magnitude of (a) is substantially representative of the ratio of the shear stiffness of the core material to the local bending stiffness of the face sheet, as can be seen in several non-metal faced sandwich panels studied herein, a²Is about 400 or more, and the value of θ calculated therefrom is generally about 20.

The relationship with theta is shown in fig. 11, where theta is the abscissa,

is the ordinate. When theta is considered to be larger than or equal to 3,

therefore, there are:

for the wall panel in the actual engineering, the theta value is considered to be generally more than 20, and at the moment, the theta value can be approximately considered to be

The final deflection of the wall panel under the concentrated load considering the rigidity of the panel can be obtainedCalculating the formula:

<math><mrow><msub><mi>w</mi><mi>max</mi></msub><mo>=</mo><mo>-</mo><mfrac><msup><mi>WL</mi><mn>3</mn></msup><mrow><mn>48</mn><mi>EI</mi></mrow></mfrac><mo>-</mo><mfrac><mi>WL</mi><mrow><mn>4</mn><mi>A</mi><msup><mi>&mu;</mi><mn>2</mn></msup><mi>G</mi></mrow></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>32</mn><mo>)</mo></mrow></mrow></math>

for the wall panel, the ratio of the panel rigidity to the integral rigidity is estimated, generally above 10%, and the theta value can reach about 5 at the minimum value, preferably

And considering the conversion relation of the shear modulus of the core material under different test methods, the final simplified form of the deflection calculation formula of the roof panel under the action of concentrated load at the midspan position under the condition of considering the rigidity of the panel can be obtained as follows:

<math><mrow><msub><mi>w</mi><mi>max</mi></msub><mo>=</mo><mo>-</mo><mfrac><msup><mi>WL</mi><mn>3</mn></msup><mrow><mn>48</mn><mi>EI</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>8</mn><mi>WL</mi></mrow><mrow><mn>4</mn><mi>A</mi><msup><mi>&mu;</mi><mn>2</mn></msup><mi>G</mi></mrow></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>33</mn><mo>)</mo></mrow></mrow></math>

wherein:

w_maxindicating the deflection at the mid-span position during normal use. Formula (II)(32) Applicable to wall faceplate, male type (33)

Adapted for roof boards

W represents the mid-span concentrated load

L represents a sandwich panel span

E represents the modulus of elasticity of the face plate of the sandwich panel with the non-metal surface

I_fRepresenting the moment of inertia of the upper and lower panels about their own neutral axis,

i represents the sum of the moments of inertia of the upper and lower face plates for their own neutralization axis and the neutralization axis of the entire sandwich panel,

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

b represents the width of the sandwich panel with non-metal surface

t represents the thickness of the sandwich panel with non-metal surface

d represents the distance between the upper and lower panels and the axis, d ═ c + t, where c is the core thickness

A represents an effective cross-sectional area of the core material,

g represents the actually measured shear modulus of the core material;

mu is a conversion coefficient of the shear modulus of the core material between different experimental methods, and mu is 1.236

And (II) simplifying the formula under the condition of uniformly distributing the load.

Under the action of uniformly distributed load, under the condition that the rigidity of the panel is considered, the deflection calculation formula at the mid-span position of the sandwich panel is as follows:

wherein:

<math><mrow><mi>&theta;</mi><mo>=</mo><mfrac><mi>aL</mi><mn>2</mn></mfrac><mo>,</mo></mrow></math> $a^2=\frac{\mathrm{AG}}{{\mathrm{EI}}_f(1-I_f/I)}$

because the cantilever beam is not needed in the actual engineering, the L is made₁0, whereby the above parameters can be simplified into the following simple form:

<math><mrow><msub><mi>&beta;</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mi>&theta;</mi><mi>cosh</mi><mi>&theta;</mi></mrow></mfrac><mo>,</mo></mrow></math> <math><mrow><mi>&theta;</mi><mo>=</mo><mfrac><mi>aL</mi><mn>2</mn></mfrac><mo>,</mo></mrow></math>

and a is²Is the ratio of the shear stiffness of the core material to the local bending stiffness of the face sheet, several non-metal faced sandwich panels studied herein, a²Is about 400 or more, and the value of θ calculated therefrom is generally about 20.

The relationship with θ is shown in fig. 12. When theta is considered to be larger than or equal to 3,therefore, there are:

for the wall panel in the actual engineering, the theta value is considered to be generally more than 20, and at the moment, the theta value can be approximately considered to be

The final deflection calculation formula of the wall panel under the condition of uniformly distributed load considering the panel rigidity can be obtained:

<math><mrow><msub><mi>w</mi><mi>max</mi></msub><mo>=</mo><mo>-</mo><mfrac><msup><mrow><mn>5</mn><mi>qL</mi></mrow><mn>3</mn></msup><mrow><mn>384</mn><mi>EI</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>q</mi><msup><mi>L</mi><mn>2</mn></msup></mrow><mrow><mn>8</mn><mi>A</mi><msup><mi>&mu;</mi><mn>2</mn></msup><mi>G</mi></mrow></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>35</mn><mo>)</mo></mrow></mrow></math>

for the wall panel, the ratio of the panel rigidity to the integral rigidity is estimated, generally above 10%, and the theta value can reach about 5 at the minimum value, preferablyAnd considering the conversion relation of the shear modulus of the core material under different test methods, the final simplified form of the deflection calculation formula of the roof panel under the action of uniformly distributed load at the midspan position under the condition of considering the panel rigidity can be obtained as follows:

<math><mrow><msub><mi>w</mi><mi>max</mi></msub><mo>=</mo><mo>-</mo><mfrac><msup><mrow><mn>5</mn><mi>qL</mi></mrow><mn>4</mn></msup><mrow><mn>384</mn><mi>EI</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>0.9</mn><msup><mi>qL</mi><mn>2</mn></msup></mrow><mrow><mn>8</mn><mi>A</mi><msup><mi>&mu;</mi><mn>2</mn></msup><mi>G</mi></mrow></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>36</mn><mo>)</mo></mrow></mrow></math>

in the formula:

w_maxindicating the deflection at the mid-span during normal use, wherein formula (35) applies to the shingle and formula (36) applies to the shingle

q represents the uniform load;

l represents a sandwich panel span;

e represents the elastic modulus of the non-metal surface sandwich panel;

I_frepresenting the moment of inertia of the upper and lower panels about their own neutral axis,

i represents the sum of the moments of inertia of the upper and lower face plates for their own neutralization axis and the neutralization axis of the entire sandwich panel,

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

b represents the width of the sandwich panel with non-metal surface

t represents the thickness of the sandwich panel with non-metal surface

d represents the distance between the upper and lower panels and the axis, d ═ c + t, where c is the core thickness

A represents an effective cross-sectional area of the core material,

g represents an actual measurement value of shear modulus of the core material

μ denotes a conversion factor of the shear modulus of the core material between different experimental methods, and is usually 1.236

Step 300: determining the bending resistance bearing capacity of the non-metal surface sandwich plate: and determining the bending resistance bearing capacity of the non-metal surface sandwich plate according to the relation between the deflection and the bending resistance bearing capacity.

The flexibility and the bending resistance bearing capacity of the sandwich board wall panel are related as follows:

w_max≤[f]

[f] see the limit value of deflection in the national standard metal surface heat insulation sandwich plate for buildings (GB/T23932-2009).

Then:

concentrated load determination: <math><mrow><mi>W</mi><mo>&le;</mo><mfrac><mrow><mo>[</mo><mi>f</mi><mo>]</mo></mrow><mrow><mfrac><msup><mi>L</mi><mn>3</mn></msup><mrow><mn>48</mn><mi>EI</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>0.8</mn><mi>L</mi></mrow><mrow><mn>4</mn><mi>A</mi><msup><mi>&mu;</mi><mn>2</mn></msup><mi>G</mi></mrow></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow></math>

determination of local load: <math><mrow><mi>q</mi><mo>&le;</mo><mfrac><mrow><mo>[</mo><mi>f</mi><mo>]</mo></mrow><mrow><mfrac><msup><mrow><mn>5</mn><mi>L</mi></mrow><mn>4</mn></msup><mrow><mn>384</mn><mi>EI</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>9</mn><msup><mi>L</mi><mn>2</mn></msup></mrow><mrow><mn>8</mn><mi>A</mi><msup><mi>&mu;</mi><mn>2</mn></msup><mi>G</mi></mrow></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow></math>

thereby determining the bending resistance bearing capacity of the sandwich plate with the non-metal surface.

According to the method for determining the bending resistance bearing capacity of the non-metal surface sandwich board, the influence of the panel rigidity of the non-metal surface sandwich board on the bending deformation of the non-metal surface sandwich board is considered, the deflection of the non-metal surface sandwich board under concentrated load and uniform load is accurately obtained, and then the bending resistance bearing capacity of the non-metal surface sandwich board is determined through the deflection. The invention can accurately obtain the bending resistance bearing capacity of the non-metal surface sandwich board, thereby determining the bending resistance mechanical property of the non-metal surface sandwich board and accurately evaluating the safety performance of the non-metal surface sandwich board.

The specific implementation mode of the invention is as follows: the method for determining the bending resistance bearing capacity of the non-metal surface sandwich board is applied to safety evaluation of the non-metal surface sandwich board.

The foregoing is a more detailed description of the invention in connection with specific preferred embodiments and it is not intended that the invention be limited to these specific details. For those skilled in the art to which the invention pertains, several simple deductions or substitutions can be made without departing from the spirit of the invention, and all shall be considered as belonging to the protection scope of the invention.

Claims (6)

1. A method for determining the bending resistance bearing capacity of a sandwich plate with a non-metal surface is characterized in that a panel of the sandwich plate is made of a non-metal material, and the method for determining the bending resistance bearing capacity of the sandwich plate with the non-metal surface comprises the following steps:

collecting related parameters of the non-metal surface sandwich plate: collecting the span of the non-metal surface sandwich board, the elastic modulus of the non-metal surface sandwich board, the width of the non-metal surface sandwich board, the thickness of the core material of the non-metal surface sandwich board, the shear modulus of the core material of the non-metal surface sandwich board and the effective cross-sectional area of the core material of the non-metal surface sandwich board;

determining the deflection of the non-metal surface sandwich panel: the rigidity of the non-metal surface sandwich panel needs to be considered, the determination of the deflection of the non-metal surface sandwich panel comprises the deflection under concentrated load and the deflection under uniformly distributed load,

the deflection of the non-metal surface sandwich plate under the concentrated load is obtained by adopting the following formula:

<math><mrow><msub><mi>w</mi><mi>max</mi></msub><mo>=</mo><mo>-</mo><mfrac><msup><mi>WL</mi><mn>3</mn></msup><mrow><mn>48</mn><mi>EI</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>0.8</mn><mi>WL</mi></mrow><mrow><mn>4</mn><mi>A</mi><msup><mi>&mu;</mi><mn>2</mn></msup><mi>G</mi></mrow></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></math>

the deflection of the non-metal surface sandwich plate under uniform load is obtained by adopting the following formula:

<math><mrow><msub><mi>w</mi><mi>max</mi></msub><mo>=</mo><mo>-</mo><mfrac><msup><mrow><mn>5</mn><mi>qL</mi></mrow><mn>3</mn></msup><mrow><mn>384</mn><mi>EI</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>9</mn><mi>qL</mi></mrow><mrow><mn>8</mn><mi>A</mi><msup><mi>&mu;</mi><mn>2</mn></msup><mi>G</mi></mrow></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><msub><mi>I</mi><mi>f</mi></msub><mi>I</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></math>

in the above two formulas, the variables have the following meanings:

w_maxrepresenting the deflection at the mid-span position during normal use;

w represents a mid-span concentrated load;

q represents the uniform load;

l represents a sandwich panel span;

e represents the elastic modulus of the non-metal surface sandwich panel;

I_frepresenting the moment of inertia of the upper and lower panels about their own neutral axis,

i represents the sum of the moments of inertia of the upper and lower face plates for their own neutralization axis and the neutralization axis of the entire sandwich panel,

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

b represents the width of the sandwich panel with non-metal surface

t represents the thickness of the sandwich panel with non-metal surface

d represents the distance between the upper and lower panels and the axis, d ═ c + t, where c is the core thickness

A represents an effective cross-sectional area of the core material,

g represents an actual measured value of the shear modulus of the core material;

μ represents a conversion coefficient of the core material shear modulus.

Determining the bending resistance bearing capacity of the non-metal surface sandwich plate: and determining the bending resistance bearing capacity of the non-metal surface sandwich plate according to the relation between the deflection and the bending resistance bearing capacity.

2. The method of claim 1, wherein the step of determining the deflection of the non-metallic face sandwich panel comprises determining the effect of the stiffness of the non-metallic face sandwich panel on the stiffness of the sandwich panel.

3. The method of claim 2, wherein determining the effect of the stiffness of the non-metallic face sandwich panel on the stiffness of the sandwich panel comprises determining the effect of the stiffness of the non-metallic face sandwich panel on the bending stiffness of the sandwich panel and determining the effect of the stiffness of the non-metallic face sandwich panel on the shear stress distribution of the sandwich panel.

4. The method for determining bending resistance and bearing capacity of a non-metal faced sandwich panel according to claim 2, wherein in the step of determining the deflection of the non-metal faced sandwich panel, further comprising determining bending deformation and shearing deformation of the non-metal faced sandwich panel.

5. The method of claim 4, wherein determining the bending and shear deformations of the non-metallic face sandwich panel comprises determining the effect of the stiffness of the non-metallic face sandwich panel on the bending deformation of the sandwich panel and determining the effect of the stiffness of the non-metallic face sandwich panel on the shear deformation of the sandwich panel.

6. A non-metallic faced sandwich panel to which the method for determining bending resistance of a non-metallic faced sandwich panel according to any one of the preceding claims is applied, wherein the method for determining bending resistance of a non-metallic faced sandwich panel is applied to safety evaluation of a non-metallic faced sandwich panel.

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