CN101886992A

CN101886992A - Determination method and application of flexural bearing capacity of non-metallic sandwich panels

Info

Publication number: CN101886992A
Application number: CN2010102154769A
Authority: CN
Inventors: 查晓雄; 张旭琛
Original assignee: Harbin Institute of Technology Shenzhen
Current assignee: Harbin Institute of Technology Shenzhen
Priority date: 2010-06-30
Filing date: 2010-06-30
Publication date: 2010-11-17
Anticipated expiration: 2030-06-30
Also published as: CN101886992B

Abstract

The invention relates to a method for determining the flexural bearing capacity of a sandwich panel with a non-metallic surface. The panel of the sandwich panel is a non-metallic material. The method for determining the flexural bearing capacity of a sandwich panel with a non-metallic surface comprises the following steps: collecting the non-metallic surface Related parameters of the sandwich panel, determine the deflection of the non-metallic sandwich panel, and determine the bending bearing capacity of the non-metallic sandwich panel. The method for determining the flexural bearing capacity of the non-metallic sandwich panel of the present invention accurately obtains the non-metallic sandwich panel under concentrated load and uniform load by considering the influence of the stiffness of the non-metallic panel on the bending deformation of the non-metallic sandwich panel. The deflection, and then determine the bending capacity of the non-metallic sandwich panel through the deflection. The invention accurately acquires the anti-bending bearing capacity of the non-metallic surface sandwich panel, thereby determining the anti-bending mechanical performance of the non-metallic surface sandwich panel, and accurately evaluating the safety performance of the non-metallic surface sandwich panel.

Description

Determination method and application of flexural bearing capacity of non-metallic sandwich panels

技术领域technical field

本发明涉及一种夹芯板抗弯承载力确定方法及应用，尤其涉及一种非金属面夹芯板抗弯承载力确定方法及应用。The invention relates to a method for determining the flexural bearing capacity of a sandwich panel and its application, in particular to a method for determining the flexural bearing capacity of a non-metallic surface sandwich panel and its application.

背景技术Background technique

随着建筑工程技术的发展，夹芯板在现代社会中越来越广泛使用。随着夹芯板的广泛使用，夹芯板技术也随之发展，从最初的金属面板的夹芯板到非金属面板的夹芯板。在现代社会中，随着非金属材料技术的提高，非金属面板的夹芯板逐渐占据了主导地位。对于非金属夹芯板的抗弯承载力确定，现有技术中还没有一个简便实用的方法，特别是缺乏考虑非金属面板刚度对非金属面夹芯板弯曲变形造成的影响进行深入地考虑，由此对于非金属面夹芯板抗弯力学性能不能精确获取，导致不能精确评估非金属面夹芯板的安全性能。With the development of construction engineering technology, sandwich panels are more and more widely used in modern society. With the widespread use of sandwich panels, the sandwich panel technology has also developed, from the original sandwich panel of metal panels to the sandwich panel of non-metal panels. In modern society, with the improvement of non-metallic material technology, the sandwich panel of non-metallic panels has gradually occupied a dominant position. For the determination of the bending capacity of non-metallic sandwich panels, there is no simple and practical method in the prior art, especially the lack of in-depth consideration of the impact of the stiffness of the non-metallic panel on the bending deformation of the non-metallic sandwich panel. As a result, the bending mechanical properties of non-metallic sandwich panels cannot be accurately obtained, resulting in the inability to accurately evaluate the safety performance of non-metallic sandwich panels.

发明内容Contents of the invention

本发明解决的技术问题是：提供一种非金属面夹芯板抗弯承载力确定方法克服现有技术中对于非金属面夹芯板抗弯力学性能不能精确获取，导致不能精确确定非金属面夹芯板的抗弯承载力的技术问题。The technical problem solved by the present invention is to provide a method for determining the flexural bearing capacity of a non-metallic sandwich panel to overcome the inability to accurately obtain the flexural mechanical properties of a non-metallic sandwich panel in the prior art, resulting in the inability to accurately determine the non-metallic panel The technical problem of the flexural bearing capacity of the sandwich panel.

本发明的技术方案是：提供一种非金属面夹芯板抗弯承载力确定方法，所述夹芯板的面板为非金属材料，非金属面夹芯板抗弯承载力确定方法包括如下步骤：The technical solution of the present invention is to provide a method for determining the flexural bearing capacity of a non-metallic sandwich panel, wherein the panel of the sandwich panel is a non-metallic material, and the method for determining the flexural bearing capacity of a non-metallic sandwich panel includes the following steps :

采集非金属面夹芯板的相关参数：采集非金属面夹芯板的跨度、非金属面板的弹性模量、非金属面夹芯板面板的宽度、非金属面夹芯板面板的厚度、非金属面夹芯板芯材厚度、非金属面夹芯板芯材的剪切模量、非金属面夹芯板芯材的有效截面面积。Collect the relevant parameters of the non-metallic sandwich panel: collect the span of the non-metallic sandwich panel, the elastic modulus of the non-metallic panel, the width of the non-metallic sandwich panel, the thickness of the non-metallic sandwich panel, the The thickness of the core material of the sandwich panel on the metal surface, the shear modulus of the core material of the sandwich panel on the non-metal surface, and the effective cross-sectional area of the core material of the sandwich panel on the non-metal surface.

确定非金属面夹芯板的挠度：非金属面夹芯板面板的刚度需要考虑，非金属面夹芯板的挠度的确定包括集中荷载下的挠度和均布荷载下的挠度，非金属面夹芯板集中荷载下的挠度采用以下公式获取：Determine the deflection of the non-metallic sandwich panel: the rigidity of the non-metallic sandwich panel needs to be considered. The determination of the deflection of the non-metallic sandwich panel includes the deflection under concentrated load and the deflection under uniform load. The deflection of the core plate under concentrated load is obtained by the following formula:

${w w}_{max max} = = - - \frac{{WL WL}^{33}}{4848 EI EI} - - \frac{0.8 0.8 WL WL}{44 A A {μ μ}^{22} G G} {((11 - - \frac{{I I}_{f f}}{I I}))}^{22}$

非金属面夹芯板均布荷载下的挠度采用以下公式获取：The deflection of the non-metallic sandwich panel under uniform load is obtained by the following formula:

${w w}_{max max} = = - - \frac{{55 qL QUR}^{33}}{384384 EI EI} - - \frac{00 . . 99 qL QUR}{88 A A {μ μ}^{22} G G} {((11 - - \frac{{I I}_{f f}}{I I}))}^{22}$

上述两公式中，各变量的表示意义如下：In the above two formulas, the meaning of each variable is as follows:

w_max表示正常使用阶段跨中位置处的挠度；w _max represents the deflection at the mid-span position during normal use;

W表示跨中集中荷载；W represents the concentrated load in the middle of the span;

q均布荷载；q Uniformly distributed load;

L表示夹芯板跨度L indicates the span of the sandwich panel

E表示非金属面夹芯板面板弹性模量E represents the elastic modulus of the non-metallic sandwich panel panel

I_f表示上下面板对其自身中和轴的惯性矩，

I _f represents the moment of inertia of the upper and lower panels about their own neutral axes,

I表示上下面板对其自身中和轴和整个夹芯板中和轴的惯性矩之和，I represents the sum of the moments of inertia of the upper and lower panels to their own neutral axis and the neutral axis of the entire sandwich panel,

$I I = = \frac{{bt bt}^{33}}{66} + + \frac{{btd btd}^{22}}{22}$

b表示非金属面夹芯板宽度b represents the width of the non-metallic sandwich panel

t表示非金属面夹芯板面板厚度t represents the thickness of the non-metallic sandwich panel panel

d表示上下面板中和轴之间的距离，d＝c+t，其中c为芯材厚度d represents the distance between the neutral axis of the upper and lower panels, d=c+t, where c is the thickness of the core material

A表示芯材的有效截面面积，

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

G表示芯材的剪切模量实测值；G represents the measured value of the shear modulus of the core material;

μ表示芯材剪切模量的换算系数，通常μ＝1.236μ represents the conversion factor of the shear modulus of the core material, usually μ = 1.236

确定非金属面夹芯板的抗弯承载力：由挠度与抗弯承载力的关系确定非金属面夹芯板的抗弯承载力。Determining the flexural capacity of the non-metallic sandwich panel: determine the flexural capacity of the non-metallic sandwich panel from the relationship between the deflection and the flexural capacity.

本发明的进一步技术方案是：在确定非金属面夹芯板的挠度步骤中，包括确定非金属面夹芯板面板刚度对夹芯板刚度的影响。A further technical solution of the present invention is: in the step of determining the deflection of the non-metallic sandwich panel, including determining the influence of the stiffness of the non-metallic sandwich panel panel on the stiffness of the sandwich panel.

本发明的进一步技术方案是：确定非金属面板刚度对夹芯板刚度的影响包括确定非金属面板刚度对夹芯板弯曲刚度的影响和确定非金属面夹芯板面板刚度对夹芯板剪应力分布的影响。The further technical scheme of the present invention is: determining the impact of the stiffness of the non-metallic panel on the stiffness of the sandwich panel includes determining the impact of the stiffness of the non-metallic panel on the bending stiffness of the sandwich panel and determining the effect of the panel stiffness of the non-metallic sandwich panel on the shear stress of the sandwich panel The influence of the distribution.

本发明的进一步技术方案是：在确定非金属面夹芯板的挠度步骤中，还包括确定非金属面夹芯板的弯曲变形和剪切变形。A further technical solution of the present invention is: in the step of determining the deflection of the non-metallic sandwich panel, it also includes determining the bending deformation and shearing deformation of the nonmetallic sandwich panel.

本发明的进一步技术方案是：确定非金属面夹芯板的弯曲变形和剪切变形包括确定非金属面板刚度对夹芯板弯曲变形的影响和确定非金属面夹芯板面板刚度对夹芯板剪切变形的影响。A further technical solution of the present invention is: determining the bending deformation and shear deformation of the non-metallic sandwich panel includes determining the influence of the stiffness of the non-metallic panel on the bending deformation of the sandwich panel and determining the effect of the stiffness of the non-metallic sandwich panel on the sandwich panel The effect of shear deformation.

本发明的技术方案是：将所述非金属面夹芯板抗弯承载力确定方法应用于非金属面夹芯板的安全评估。The technical solution of the present invention is: applying the method for determining the bending bearing capacity of the non-metallic sandwich panel to the safety assessment of the nonmetallic sandwich panel.

本发明的技术效果是：提供一种非金属面夹芯板抗弯承载力确定方法，通过考虑非金属面夹芯板的面板刚度对非金属面夹芯板弯曲变形造成的影响，精确获取非金属面夹芯板在集中荷载和均布荷载下的挠度，然后通过挠度确定非金属面夹芯板的抗弯承载力。本发明精确获取非金属面夹芯板的抗弯承载力，从而确定非金属面夹芯板的抗弯力学性能，精确评估非金属面夹芯板的安全性能。The technical effect of the present invention is: to provide a method for determining the flexural bearing capacity of a non-metallic sandwich panel, by considering the influence of the panel stiffness of the non-metallic sandwich panel on the bending deformation of the non-metallic sandwich panel, the non-metallic sandwich panel can be accurately obtained The deflection of the metal-faced sandwich panel under concentrated loads and uniformly distributed loads, and then the flexural capacity of the non-metallic-faced sandwich panel is determined by the deflection. The invention accurately acquires the anti-bending bearing capacity of the non-metallic surface sandwich panel, thereby determining the anti-bending mechanical performance of the non-metallic surface sandwich panel, and accurately evaluating the safety performance of the non-metallic surface sandwich panel.

附图说明Description of drawings

图1为本发明的流程图。Fig. 1 is a flowchart of the present invention.

图2为本发明的将非金属面夹芯板简化为夹芯梁的示意图。Fig. 2 is a schematic diagram of simplifying the non-metallic sandwich panel into a sandwich beam according to the present invention.

图3为本发明的将非金属面夹芯板简化为夹芯梁的横截面示意图。Fig. 3 is a schematic cross-sectional view of a non-metallic sandwich panel simplified into a sandwich beam according to the present invention.

图4为本发明工字梁横截面剪力分布示意图。Fig. 4 is a schematic diagram of the shear force distribution in the cross section of the I-beam of the present invention.

图5为本发明夹芯梁的剪力分布示意图。Fig. 5 is a schematic diagram of the shear force distribution of the sandwich beam of the present invention.

图6为本发明厚面板夹芯板横截面的应力分布示意图。Fig. 6 is a schematic diagram of the stress distribution in the cross section of the thick panel sandwich panel of the present invention.

图7为本发明夹芯梁集中荷载作用下变形示意图。Fig. 7 is a schematic diagram of deformation of a sandwich beam of the present invention under a concentrated load.

图8为本发明夹芯梁剪切变形示意图。Fig. 8 is a schematic diagram of the shear deformation of the sandwich beam of the present invention.

图9为本发明跨中集中荷载作用下的简支梁示意图。Fig. 9 is a schematic diagram of a simply supported beam under the mid-span concentrated load of the present invention.

图10为本发明均布荷载作用下的简支梁示意图。Fig. 10 is a schematic diagram of a simply supported beam under a uniformly distributed load in the present invention.

图11为本发明集中荷载作用下的θ与

之间的关系。Fig. 11 is θ and

The relationship between.

图12为本发明均布荷载作用下的θ与

之间的关系。Fig. 12 is θ and

The relationship between.

具体实施方式Detailed ways

下面结合具体实施例，对本发明技术方案进一步说明。The technical solutions of the present invention will be further described below in conjunction with specific embodiments.

如图1所示，本发明的具体实施方式是：提供一种非金属面夹芯板抗弯承载力确定方法，所述夹芯板的面板为非金属材料，本发明中所述夹芯板的面板以秸秆板或定向结构板(Oriented Strand board，OSB)为例进行介绍。As shown in Figure 1, the specific embodiment of the present invention is: provide a kind of method for determining the flexural bearing capacity of a non-metallic sandwich panel, the panel of the sandwich panel is a non-metallic material, the sandwich panel in the present invention The panel is introduced with straw board or Oriented Strand board (Oriented Strand board, OSB) as an example.

非金属面夹芯板抗弯承载力确定方法包括如下步骤：The method for determining the flexural bearing capacity of non-metallic sandwich panels includes the following steps:

步骤100：采集非金属面夹芯板的相关参数，即，采集非金属面夹芯板的以下参数：采集非金属面夹芯板的跨度、非金属面夹芯板面板的弹性模量、非金属面夹芯板面板的宽度、非金属面夹芯板面板的厚度、非金属面夹芯板芯材厚度、非金属面夹芯板芯材的剪切模量、非金属面夹芯板芯材的有效截面面积。这些参数中，采集非金属面夹芯板的跨度、非金属面夹芯板面板的宽度、非金属面夹芯板面板的厚度、非金属面夹芯板芯材厚度、非金属面夹芯板芯材的有效截面面积由非金属面夹芯板的形状决定，而非金属面夹芯板面板的弹性模量、非金属面夹芯板芯材的剪切模量由非金属面夹芯板的材料决定。Step 100: Collect the relevant parameters of the non-metallic sandwich panel, that is, collect the following parameters of the non-metallic sandwich panel: collect the span of the non-metallic sandwich panel, the elastic modulus of the non-metallic sandwich panel, the non-metallic The width of the metal sandwich panel panel, the thickness of the non-metal sandwich panel panel, the thickness of the non-metal sandwich panel core material, the shear modulus of the non-metal sandwich panel core material, the non-metal sandwich panel core The effective cross-sectional area of the material. Among these parameters, the span of the non-metallic sandwich panel, the width of the non-metallic sandwich panel, the thickness of the non-metallic sandwich panel, the thickness of the non-metallic sandwich core, and the The effective cross-sectional area of the core material is determined by the shape of the non-metallic sandwich panel, while the elastic modulus of the non-metallic sandwich panel panel and the shear modulus of the core material of the non-metallic sandwich panel are determined by the shape of the non-metallic sandwich panel. material decision.

步骤200：确定非金属面夹芯板的挠度，即，非金属面夹芯板挠度的确定包括集中荷载下的挠度和均布荷载下的挠度。Step 200: Determine the deflection of the non-metal sandwich panel, that is, the determination of the deflection of the non-metal sandwich panel includes the deflection under concentrated load and the deflection under uniform load.

对于非金属面夹芯板挠度的确定，首先需要考虑如下因素：非金属面夹芯板的面板刚度对非金属面夹芯板挠度的影响。For the determination of the deflection of the non-metallic sandwich panel, the following factors need to be considered first: the influence of the panel stiffness of the non-metallic sandwich panel on the deflection of the non-metallic sandwich panel.

一、非金属面夹芯板的面板刚度对非金属面夹芯板挠度的影响。1. The effect of the panel stiffness of the non-metallic sandwich panel on the deflection of the non-metallic sandwich panel.

具体而言，非金属面夹芯板的面板刚度对非金属面夹芯板挠度的影响包括非金属面夹芯板的面板刚度对夹芯板弯曲刚度的影响和非金属面夹芯板的面板刚度对夹芯板剪应力分布的影响。Specifically, the influence of the panel stiffness of the non-metallic sandwich panel on the deflection of the non-metallic sandwich panel includes the influence of the panel stiffness of the non-metallic panel on the bending stiffness of the sandwich panel and the influence of the panel stiffness of the non-metallic panel Influence of stiffness on shear stress distribution of sandwich panels.

非金属面夹芯板的面板刚度对夹芯板弯曲刚度的影响，具体如下：The effect of the panel stiffness of the non-metallic sandwich panel on the bending stiffness of the sandwich panel is as follows:

如图2、图3所示，将非金属面夹芯板简化为夹芯梁的形式，即不考虑y方向(也就是板材宽度方向)上的应力。As shown in Figure 2 and Figure 3, the non-metallic sandwich panel is simplified into the form of a sandwich beam, that is, the stress in the y direction (that is, the width direction of the plate) is not considered.

图中符号定义如下：The symbols in the figure are defined as follows:

c表示芯材厚度；c represents the thickness of the core material;

t表示面板厚度t is the panel thickness

h表示夹芯板厚度，h＝c+2th represents the thickness of the sandwich panel, h=c+2t

d表示上下面板中心线之间的距离，d＝c+td represents the distance between the centerlines of the upper and lower panels, d=c+t

b表示夹芯板宽度b represents the width of the sandwich panel

G表示芯材的剪切模量G is the shear modulus of the core material

D表示夹芯板的整体抗弯刚度D represents the overall bending stiffness of the sandwich panel

A表示夹芯板的等效横截面面积A represents the equivalent cross-sectional area of the sandwich panel

AG表示夹芯板的剪切刚度，其中A＝bd²/cAG represents the shear stiffness of the sandwich panel, where A=bd ² /c

Q表示夹芯板某一截面上的剪力Q represents the shear force on a certain section of the sandwich panel

E_f表示面板的弹性模量E _f represents the modulus of elasticity of the panel

E_c表示芯材的弹性模量E _c represents the modulus of elasticity of the core material

I表示整个截面对中性轴的惯性矩I represents the moment of inertia of the entire section about the neutral axis

I_f表示上下面板对其自身轴线的惯性矩I _f represents the moment of inertia of the upper and lower panels on their own axes

由于夹芯板由上下面板和芯材组成，如图2、图3所示夹芯梁的弯曲刚度根据材料力学刚度的计算公式得到以下公式：Since the sandwich panel is composed of upper and lower panels and core materials, the bending stiffness of the sandwich beam shown in Figure 2 and Figure 3 can be obtained from the following formula according to the calculation formula of the mechanical stiffness of the material:

$D D. = = {E E.}_{f f} \cdot &Center Dot; \frac{{bt bt}^{33}}{66} + + {E E.}_{f f} \cdot &Center Dot; \frac{bt bt {d d}^{22}}{22} + + {E E.}_{c c} \cdot &Center Dot; \frac{{bc bc}^{33}}{1212} - - - - - - ((11))$

其中第一项表示的是面板相对于其自身轴弯曲时的局部刚度；第二项代表上下两个面板相对于中轴线c-c弯曲时的产生的刚度；第三项代表芯材相对于其自身轴(同中轴线c-c)弯曲时的局部刚度。The first term represents the local stiffness of the panel when it is bent relative to its own axis; the second term represents the stiffness of the upper and lower panels when they are bent relative to the central axis c-c; the third term represents the core material relative to its own axis (Coordinate axis c-c) Local stiffness during bending.

在实际的夹芯结构中，公式(1)第二项占据了主导地位，公式(1)第一项即非金属面夹芯板面板刚度的影响不能忽略，公式(1)第三项即芯材自身刚度的影响。In the actual sandwich structure, the second term of formula (1) occupies a dominant position, the influence of the first term of formula (1), that is, the stiffness of the non-metallic sandwich panel panel, cannot be ignored, and the third term of formula (1), that is, the core Influence of material stiffness.

非金属面夹芯板的面板刚度对夹芯板剪应力分布的影响，具体如下：The effect of the panel stiffness of the non-metallic sandwich panel on the shear stress distribution of the sandwich panel is as follows:

如图4所示，由夹芯板的工作原理可以将其简化为一个工字梁的形式，得到工字梁中剪应力的分布情况。As shown in Figure 4, the working principle of the sandwich panel can be simplified into the form of an I-beam, and the distribution of shear stress in the I-beam can be obtained.

对于截面中轴线下方z处的芯材剪应力τ，根据材料力学有如下公式：For the shear stress τ of the core material at z below the central axis of the section, according to the mechanics of materials, the following formula is given:

$τ τ = = \frac{QS QS}{Ib Ib} - - - - - - ((22))$

其中：Q为所选横截面上的剪力；I为整个截面对中性轴的惯性矩；b为z₁处的宽度，S为z＞z₁部分的截面对中性轴的静距，图中z表示z处与中轴线的距离，z₁表示z₁处与中轴线的距离。Among them: Q is the shear force on the selected cross-section; I is the moment of inertia of the entire section to the neutral axis; b is the width at z ₁ , S is the static distance of the section where z>z ₁ is to the neutral axis, In the figure, z represents the distance between z and the central axis, and z ₁ represents the distance between z ₁ and the central axis.

对于夹芯结构的组合梁，考虑各部分的弹性模量，上式可以写成如下形式：For composite beams with a sandwich structure, considering the elastic modulus of each part, the above formula can be written as follows:

$τ τ = = \frac{Q Q}{Db DB} Σ Σ ((SE SE)) - - - - - - ((33))$

其中D如公式(1)所示；∑(SE)为z＞z₁的部分截面S和E的乘积之和，例如要确定芯材部分z处的剪应力，则有：where D is shown in formula (1); ∑(SE) is the sum of the products of the partial sections S and E where z>z ₁ , for example, to determine the shear stress at the core part z, then:

$Σ Σ ((SE SE)) = = {E E.}_{f f} \frac{btd btd}{22} + + \frac{{E E.}_{c c} b b}{22} ((\frac{c c}{22} - - z z)) ((\frac{c c}{22} + + z z)) - - - - - - ((44))$

因此，芯材中的剪应力：Therefore, the shear stress in the core material is:

$τ τ = = \frac{Q Q}{D D.} {{{E E.}_{f f} \cdot &Center Dot; \frac{td td}{22} + + \frac{{E E.}_{c c}}{22} ((\frac{{c c}^{22}}{44} - - {z z}^{22}))}} - - - - - - ((55))$

类似的可以得到面板中剪应力。Similarly, the shear stress in the panel can be obtained.

根据材料力学的知识，夹芯梁横截面上的剪应力的分布如图5所示：其中，(a)为夹芯梁横截面真实的剪应力分布。(b)为忽略芯材自身刚度时，夹芯梁的剪应力分布情况，

(c)为忽略芯材自身刚度及面板自身刚度时，夹芯梁的剪应力分布，

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: Among them, (a) is the real shear stress distribution of the cross-section of the sandwich beam. (b) In order to ignore the stiffness of the core material itself, the shear stress distribution of the sandwich beam,

(c) When ignoring the stiffness of the core material and the stiffness of the panel itself, the shear stress distribution of the sandwich beam,

对于低强度的泡沫芯材，可计E_c＝0，得到芯材中的剪应力常量，图5中(b)所示：For low-strength foam core materials, E _c =0 can be calculated to obtain the shear stress constant in the core material, as shown in (b) in Figure 5:

$τ τ = = \frac{Q Q}{D D.} \cdot &Center Dot; \frac{{E E.}_{f f} td td}{22} - - - - - - ((66))$

此时 $D = E_{f} \cdot \frac{{bt}^{3}}{6} + E_{f} \cdot \frac{{btd}^{2}}{2} .$ at this time $D. = {E.}_{f} &Center Dot; \frac{{bt}^{3}}{6} + {E.}_{f} &Center Dot; \frac{{btd}^{2}}{2} .$

另外，如果面板相对于其自身中轴线的抗弯刚度很小，则中的第一项也可忽略，即

则芯材中剪应力可简化为以下最简形式，如图5中(c)所示：In addition, if the bending stiffness of the panel with respect to its own central axis is small, then The first term in can also be ignored, namely

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

$τ τ = = \frac{Q Q}{bd bd} - - - - - - ((77))$

二、对于非金属面夹芯板挠度的确定，还需要考虑如下因素：非金属面夹芯板的弯曲变形和剪切变形。2. For the determination of the deflection of the non-metallic sandwich panel, the following factors also need to be considered: bending deformation and shear deformation of the non-metallic sandwich panel.

将非金属面夹芯板简化为夹芯梁的形式，材料力学中对于弯曲梁符号的规定，如公式(8)所示。Simplify the non-metallic sandwich panel into the form of a sandwich beam, and the provisions for the symbol of the curved beam in the mechanics of materials are shown in formula (8).

对于厚面板夹芯板的变形，需要明确以下几点：For the deformation of thick panel sandwich panels, the following points need to be clarified:

由于芯材为EPS-聚苯乙烯等泡沫芯材，其弹性模量很小，其自身刚度，即公式(1)中的第三项可忽略；Since the core material is a foam core material such as EPS-polystyrene, its elastic modulus is very small, and its own stiffness, that is, the third item in formula (1) can be ignored;

面板具有一定刚度，公式(1)中的第一项不能忽略；The panel has a certain stiffness, and the first term in formula (1) cannot be ignored;

由于忽略了芯材自身刚度而面板刚度不能忽略，所以芯材中的剪应力分布如图5中(b)所示，其剪应力沿芯材厚度方向为常数，大小如公式(6)所示。Since the stiffness of the core material itself is ignored and the stiffness of the panel cannot be ignored, the shear stress distribution in the core material is shown in Figure 5(b), and the shear stress is constant along the thickness direction of the core material, and its magnitude is shown in formula (6) .

由于考虑了面板的自身刚度，将会对夹芯梁的变形产生如下影响：Due to the consideration of the panel's own stiffness, it will have the following effects on the deformation of the sandwich beam:

第一种影响为使面板有两种变形方式，第一种方式为局部弯曲，相对于整个夹芯结构中轴线的弯曲变形，此时产生面板在均布应力下的拉伸和压缩，此时产生的面板中的应力为图6第一部分应力所示。第二种方式为相对于面板自身轴线而不是整个夹芯结构轴线的局部弯曲，此时产生的面板中的应力为图6第二部分应力所示。The first effect is that the panel has two deformation modes. The first mode is local bending, which is relative to the bending deformation of the central axis of the entire sandwich structure. At this time, the panel is stretched and compressed under uniform stress. At this time The resulting stress in the panel is shown in Figure 6, the first part of the stress. The second way is local bending relative to the axis of the panel itself rather than the axis of the entire sandwich structure. The stress in the panel generated at this time is shown in the second part of stress in Figure 6.

夹芯梁作为一个整体发生弯曲的变形，以承受集中荷载为例，当夹芯梁作为一个整体发生弯曲的变形如图7所示。图7中(a)为跨中受集中力的简支梁，(b)为弯曲变形，(c)为剪切变形，(d)为弯曲变形、剪切变形共同作用的结果。图7(b)中，面板同时具有以上两种变形方式。面板局部弯曲刚度对整个夹芯梁弯曲刚度的贡献值可以由公式(1)中的第一项来表示。The sandwich beam as a whole undergoes bending deformation. Taking the concentrated load as an example, the bending deformation of the sandwich beam as a whole is shown in Figure 7. In Fig. 7, (a) is a simply supported beam subjected to concentrated force in the mid-span, (b) is bending deformation, (c) is shearing deformation, and (d) is the result of joint action of bending deformation and shearing deformation. In Fig. 7(b), the panel has the above two deformation modes at the same time. The contribution of the local bending stiffness of the panel to the bending stiffness of the whole sandwich beam can be expressed by the first term in formula (1).

第二种影响为对芯材剪切变形产生的影响：The second effect is the effect on the shear deformation of the core material:

对芯材内部的剪切变形产生影响：当只考虑剪切变形，面板中轴线上的a、b、c…各点在水平方向上不产生位移(因此不会使面板上的主应力发生变化)，只沿垂直方向发生变形，如图5中(c)。其跨中位置处将会出现一个折角，此处的曲率将会无穷大，很显然这是不可能的：由材料力学的知识可知，弯矩和曲率之间有的关系，则此处弯矩无穷大。如果面板和芯材仍然要保持连接，则面板需要在跨中两侧一定距离范围内，发生局部弯曲，以使剪切变形变得平滑。此时，将会在面板中引入额外的弯矩和剪力，从而减小了剪切变形。在实际的夹芯结构中，尤其是薄面板夹芯结构，该影响很小；当面板较厚(例如石棉水泥夹芯板和非金属面夹芯板)而芯材为EPS等轻质泡沫时，该影响明显。It affects the shear deformation inside the core material: when only the shear deformation is considered, the points a, b, c... on the central axis of the panel will not be displaced in the horizontal direction (so the principal stress on the panel will not change ), deformation occurs only along the vertical direction, as shown in (c) in Figure 5. There will be a knuckle at the mid-span position, and the curvature here will be infinite, which is obviously impossible: from the knowledge of material mechanics, there is a relationship between bending moment and curvature , the bending moment here is infinite. If the panels and core are still to remain connected, the panels need to be locally bent within a certain distance on either side of the mid-span to smooth out the shear deformation. At this point, additional bending moments and shear forces are introduced in the panel, reducing shear deformation. In the actual sandwich structure, especially the thin panel sandwich structure, the effect is very small; when the panel is thick (such as asbestos cement sandwich panel and non-metallic surface sandwich panel) and the core material is lightweight foam such as EPS , the effect is obvious.

基于以上讨论，下面分两点对夹芯结构的变形进行讨论：Based on the above discussion, the following two points discuss the deformation of the sandwich structure:

(一)非金属面夹芯板的面板刚度影响下的夹芯板弯曲变形。(1) The bending deformation of the sandwich panel under the influence of the panel stiffness of the non-metallic sandwich panel.

首先考虑一个芯材剪切刚度无穷大，在均布荷载q₁作用下的夹芯梁单元。按照普通梁的弯曲理论，产生挠度w₁。该挠度与弯矩M₁及剪力Q₁有关，根据材料力学，其中剪力Q₁得出：First consider a sandwich beam element with infinite shear stiffness of the core material under the action of uniformly distributed load q ₁ . According to the bending theory of ordinary beams, a deflection w ₁ is produced. The deflection is related to the bending moment _M1 and the shear force _Q1 . According to the mechanics of materials, the shear force _Q1 is obtained as:

-Q₁＝Dw₁′″＝E_f(I-I_f)w₁′″+E_fI_fw₁′″ (9)-Q ₁ =Dw ₁ ′″=E _f (II _f )w ₁ ′″+E _f I _f w ₁ ′″ (9)

此时为忽略芯材自身刚度的影响，

因此有：At this time, in order to ignore the influence of the stiffness of the core material itself,

So there are:

$I I = = \frac{{bt bt}^{33}}{66} + + \frac{{btd btd}^{22}}{22} - - - - - - ((1010))$

${I I}_{f f} = = \frac{{bt bt}^{33}}{66} - - - - - - ((1111))$

假定面板只承受拉伸和压缩变形而不发生局部弯曲时，公式(9)右边第一项代表芯材和面板共同承担的剪力。此时暂不计面板刚度，右边第一项可按图5(c)中的应力分布进行计算：剪应力τ在芯材厚度范围内大小不变，在面板边缘处为0，在面板内部呈线性变化。因此，第一项可由-bdτ代替，其中τ为芯材中的剪应力，d为上下面板中心线之间的距离，即：Assuming that the panel only bears tensile and compressive deformation without local bending, the first item on the right side of formula (9) represents the shear force shared by the core material and the panel. At this time, the stiffness of the panel is not considered, and the first item on the right can be calculated according to the stress distribution in Figure 5(c): the shear stress τ is constant within the thickness range of the core material, 0 at the edge of the panel, and linear in the interior of the panel Variety. Therefore, the first term can be replaced by -bdτ, where τ is the shear stress in the core material and d is the distance between the centerlines of the upper and lower panels, namely:

-Q₁＝-bdτ+E_fI_fw₁′″ (12)-Q ₁ ＝-bdτ+E _f I _f w ₁ ′″ (12)

同时有：q₁＝-Q₁′，Q₁＝M₁，M₁＝-Dw₁″。At the same time: q ₁ =-Q ₁ ′, Q ₁ =M ₁ , M ₁ =-Dw ₁ ″.

(二)和确定非金属面夹芯板的面板刚度影响下的夹芯板剪切变形。(b) and determine the shear deformation of the sandwich panel under the influence of the face stiffness of the non-metallic sandwich panel.

由于上述假定芯材刚度无穷大，因此芯材中虽然存在剪应力τ，但并不会发生剪应变。因此，如果芯材剪切模量G为某一限值，则在剪应力τ的作用下，芯材产生剪应变γ＝τ/G，相当于产生一个额外的横向变形w₂。面板必须同时产生该额外变形，因此，其必须遭受一个额外的均布荷载q₂，剪力Q₂以及弯矩M₂：Due to the assumption that the stiffness of the core material is infinite, although there is a shear stress τ in the core material, no shear strain occurs. Therefore, if the shear modulus G of the core material is a certain limit value, under the action of the shear stress τ, the core material produces a shear strain γ=τ/G, which is equivalent to an additional lateral deformation w ₂ . The panel must simultaneously undergo this additional deformation and, therefore, it must be subjected to an additional uniformly distributed load q ₂ , shear force Q ₂ and bending moment M ₂ :

q₂＝-Q₂′，Q₂＝M₂，M₂＝-Dw₂″q ₂ =-Q ₂ ′, Q ₂ =M ₂ , M ₂ =-Dw ₂ ″

则总的荷载、剪力、弯矩以及变形如下所示：The total loads, shears, moments, and deformations are then as follows:

q＝q₁₊q₂ q=q ₁₊ q ₂

Q＝Q₁+Q₂ Q＝Q ₁ +Q ₂

M＝M₁+M₂ M=M ₁ +M ₂

w＝w₁+w₂ w=w ₁ +w ₂

也就是说，均布荷载q作用下的夹芯梁，将产生两组不同的变形：w₁与w₂。其中第一项代表普通弯曲变形，其与面板和芯材共同承担的剪力Q₁相关；第二项代表由于Q₁引起的芯材剪切变形：为适应芯材剪切变形需要，面板还参与了绕其自身轴线的额外弯曲变形(忽略了面板中的剪切变形，但面板仍分担剪力)；此时，需要一个额外的剪力来驱动此变形，即Q₂。Q₁与Q₂的和即为施加于梁上的总的剪力。That is to say, the sandwich beam under the uniform load q will produce two different deformations: w ₁ and w ₂ . The first term represents the ordinary bending deformation, which is related to the shear force Q ₁ shared by the panel and the core material; the second term represents the shear deformation of the core material caused by Q ₁ : in order to meet the needs of the shear deformation of the core material, the panel also participates in the additional bending deformation around its own axis (the shear deformation in the panel is ignored, but the panel still shares the shear force); at this time, an additional shear force is required to drive this deformation, namely Q ₂ . The sum of _Q1 and _Q2 is the total shear force applied to the beam.

(三)非金属面夹芯板的面板刚度影响下的夹芯板弯曲变形和剪切变形有相互关系。(3) There is a mutual relationship between the bending deformation and shear deformation of the sandwich panel under the influence of the panel stiffness of the non-metallic sandwich panel.

额外变形与芯材剪应变γ的相互关系，如图8所示。线段de的长度等于

又有线段cf的长度等于γc，由de＝cf，可得

与γ之间的关系如下：The relationship between additional deformation and core material shear strain γ is shown in Fig. 8. The length of the line segment de is equal to

And the length of the line segment cf is equal to γc, from de=cf, we can get

The relationship with γ is as follows:

$\frac{{dw dw}_{22}}{dx dx} = = γ γ \frac{c c}{d d} = = \frac{Q Q}{Gbd Gbd} \cdot &Center Dot; \frac{c c}{d d} = = \frac{Q Q}{AG AG} - - - - - - ((1313))$

其中，A＝bd²/c，AG通常指夹芯梁的剪切刚度。Wherein, A=bd ² /c, and AG usually refers to the shear stiffness of the sandwich beam.

将τ＝γG代入公式(13)，可得额外变形与剪应力之间的关系：Substituting τ=γG into formula (13), the relationship between additional deformation and shear stress can be obtained:

$τ τ = = \frac{d d}{c c} \cdot &Center Dot; G G {w w}_{22}^{' '} - - - - - - ((1414))$

将其代入公式(12)，即：Substituting it into formula (12), namely:

-Q₁＝-AGw₂′+EI_fw₁′″ (15)-Q ₁ ＝-AGw ₂ ′+EI _f w ₁ ′″ (15)

将-Q₁＝-Dw₁′″代入公式(15)，进行变化可得：Substitute -Q ₁ ＝-Dw ₁ ′″ into formula (15) and make changes to get:

${w w}_{22}^{' '} = = - - \frac{D D.}{AG AG} ((11 - - \frac{{I I}_{f f}}{I I})) {w w}_{11}^{' '' '' '} = = + + \frac{{Q Q}_{11}}{AG AG} ((11 - - \frac{{I I}_{f f}}{I I})) - - - - - - ((1616))$

由于-Q₂＝-Dw₂′″，则总的剪力：Since -Q ₂ =-Dw ₂ ′″, the total shear force:

Q＝Q₁+Q₂＝Q₁-EI_fw₂′″ (17)Q=Q ₁ +Q ₂ ＝Q ₁ -EI _f w ₂ '" (17)

将公式(16)代入到公式(17)中，可得关于Q₁的方程如下：Substituting formula (16) into formula (17), the equation about _Q1 can be obtained as follows:

Q₁″-a²Q₁＝-a²Q (18)Q ₁ ″-a ² Q ₁ ＝-a ² Q (18)

其中： $a^{2} = \frac{AG}{{EI}_{f} (1 - I_{f} / I)} - - - (19)$ in: $a^{2} = \frac{AG}{{EI}_{f} (1 - I_{f} / I)} - - - (19)$

三、非金属面夹芯板挠度的变形公式。3. The deformation formula of the deflection of the non-metallic sandwich panel.

由以上分析可知，知道夹芯梁的受力情况，即Q可以由关于x的方程给出，进而可根据公式(17)求得Q₁；再根据w₁′、w₂′与Q₁之间的关系，通过积分可最终求得w₁和w₂；最后根据w＝w₁+w₂的关系，可得简支夹芯梁在不同受力情况下，跨中位置处的最终挠度计算公式。From the above analysis, it can be seen that knowing the stress of the sandwich beam, that is, Q can be given by the equation about x, and then Q ₁ can be obtained according to formula (17); then according to the relationship between w ₁ ′, w ₂ ′ and Q ₁ The relationship between w ₁ and w ₂ can be finally obtained through integration; finally, according to the relationship of w=w ₁ +w ₂ , the calculation of the final deflection at the mid-span position of the simply supported sandwich beam under different stress conditions can be obtained formula.

(一)集中荷载情况下非金属面夹芯板挠度的变形公式。(1) Deformation formula for the deflection of non-metallic sandwich panels under concentrated loads.

如图9所示，AB段，x起始点为A，其剪力为-W/2，此时根据公式(17)的解为：As shown in Figure 9, the AB section, the starting point of x is A, and its shear force is -W/2. At this time, the solution according to formula (17) is:

$- - {Q Q}_{11} = = {C C}_{11} cosh cosh ax ax + + {C C}_{22} sinh sinh ax ax + + \frac{W W}{22} - - - - - - ((2020))$

通过积分，可得：By scoring, you can get:

${EIw wxya}_{11} = = \frac{{C C}_{11}}{{a a}^{33}} sinh sinh ax ax + + \frac{{C C}_{22}}{{a a}^{33}} cosh cosh ax ax + + \frac{{Wx wxya}^{33}}{1212} + + {C C}_{33} {x x}^{22} + + {C C}_{44} x x + + {C C}_{55} - - - - - - ((21 twenty one))$

方程(20)与方程(21)一起可得到一个关于w₂′的表达式，积分一次得到如下公式：Equation (20) and equation (21) together can obtain an expression about w ₂ ′, and integrate once to obtain the following formula:

${- - EI EI}_{f f} {w w}_{22} = = \frac{{C C}_{11}}{{a a}^{33}} sinh sinh ax ax + + \frac{{C C}_{22}}{{a a}^{33}} cosh cosh ax ax + + \frac{W W}{22 {a a}^{22}} x x + + {C C}_{66} - - - - - - ((22 twenty two))$

AB段上可找到5个边界条件，由此上述六个常数的关系如下：Five boundary conditions can be found on the AB section, so the relationship between the above six constants is as follows:

(i)x＝0，w₁＝0(任意性)(i) x=0, w ₁ =0 (arbitrary)

${C C}_{55} + + \frac{{C C}_{22}}{{a a}^{33}} = = 00$

(ii)x＝0，w₁′＝0(对称性)(ii) x=0, w ₁ '=0 (symmetry)

$\frac{{C C}_{11}}{{a a}^{22}} + + {C C}_{44} = = 00$

(iii)x＝0，w₁′″＝0(对称性)(iii) x=0, w ₁ '"=0 (symmetry)

${C C}_{11} + + \frac{W W}{22} = = 00$

$((iv iv)),, x x = = 00,, M m = = \frac{WL WL}{44}$

定义 -M＝EIw₁″+EI_fw₂″Definition -M＝EIw ₁ ″+EI _f w ₂ ″

因此， $- \frac{WL}{4} = 2 C_{3}$ therefore, $- \frac{WL}{4} = 2 C_{3}$

(v)x＝0，w₂＝0(任意性)(v) x=0, w ₂ =0 (arbitrary)

$\frac{{C C}_{22}}{{a a}^{33}} + + {C C}_{66} = = 00$

至此，各常数可按如下形式表示，其中C₂未知：So far, each constant can be expressed as follows, where C ₂ is unknown:

${C C}_{11} = = - - \frac{W W}{22};; {C C}_{33} = = - - \frac{WL WL}{88};; {C C}_{44} = = + + \frac{W W}{22 {a a}^{22}};; {C C}_{55} = = {C C}_{66} = = - - \frac{{C C}_{22}}{{a a}^{33}} - - - - - - ((23 twenty three))$

在BC段上，其中x以B点位起始点，其总剪力为0。方程(20)和(21)依然适用但其中包含W的项应消去，新的常数B₁-B₆用以取代C₁-C₆。On segment BC, where x starts from point B, the total shear force is 0. Equations (20) and (21) still apply but the terms containing W should be eliminated, and new constants B ₁ -B ₆ are used to replace C ₁ -C ₆ .

以下为四个简单的边界条件：The following are four simple boundary conditions:

(vi)x＝0，w₁＝0(任意性)(vi) x=0, w ₁ =0 (arbitrary)

${B B}_{55} + + \frac{{B B}_{22}}{{a a}^{33}} = = 00$

(vii)x＝0，w₂＝0(任意性)(vii) x=0, w ₂ =0 (arbitrary)

${B B}_{66} + + \frac{{B B}_{22}}{{a a}^{33}} = = 00$

(viii)x＝L₁，w₁″＝0(viii) x=L ₁ , w ₁ ″=0

$\frac{{B B}_{11}}{a a} sinh sinh a a {L L}_{11} + + \frac{{B B}_{22}}{a a} cosh cosh a a {L L}_{11} + + 22 {B B}_{33} = = 00$

(ix)x＝L₁，w₂″＝0(ix) x=L ₁ , w ₂ ″=0

$\frac{{B B}_{11}}{a a} sinh sinh a a {L L}_{11} + + \frac{{B B}_{22}}{a a} cosh cosh a a {L L}_{11} = = 00$

最后两个边界条件是因为自由端的弯矩M₁和M₂为0。只有当面板端部可以自由转动，并且不与刚性端部连接时，该条件才成立。以下为上述边界条件的结果：The last two boundary conditions are due to the fact that the bending moments _M1 and _M2 at the free end are zero. This condition is only true if the panel ends are free to rotate and are not connected to rigid ends. The following are the results for the above boundary conditions:

B₂＝-B₁tanhaL₁；B₃＝0；

B ₂ =-B ₁ tanhaL ₁ ; B ₃ =0;

仍然需要建立B点处的连续性。明显的，w₁′和w₂′，w₁″和w₂″应该连续；同时由公式(21)可知，w₁′″和必须连续。然而，仅用三个可提供独立方程的条件，分别为w₁′，w₂′，w₁″。Continuity at point B still needs to be established. Obviously, w ₁ ′ and w ₂ ′, w ₁ ″ and w ₂ ″ should be continuous; at the same time, it can be seen from formula (21) that w ₁ ′″ and must be consecutive. However, only three conditions, w ₁ ', w ₂ ', w ₁ ", are used to provide independent equations.

(x)w₁′在B点连续(x)w ₁ ′ is continuous at point B

$\frac{{C C}_{11}}{{a a}^{22}} cosh cosh \frac{aL aL}{22} + + \frac{{C C}_{22}}{{a a}^{22}} sinh sinh \frac{aL aL}{22} + + \frac{{WL WL}^{22}}{1616} + + {C C}_{33} L L + + {C C}_{44} = = \frac{{B B}_{22}}{{a a}^{22}} + + {B B}_{44}$

(xi)w₂′在B点连续(xi)w ₂ ′ is continuous at point B

${C C}_{11} cosh cosh \frac{aL aL}{22} + + {C C}_{22} sinh sinh \frac{aL aL}{22} + + \frac{W W}{22} = = {B B}_{11}$

(xii)w₁″在B点连续(xii) w ₁ ″ is continuous at point B

${C C}_{11} sinh sinh \frac{aL aL}{22} + + {C C}_{22} cosh cosh \frac{aL aL}{22} + + ((\frac{WL WL}{44} + + 22 {C C}_{33})) a a = = {B B}_{22} + + 22 {B B}_{33} a a$

由方程(23)和(24)可约去B₂，B₃，C₁和C₃；条件(xi)和(xii)可用于解出C₂和B₁，我们只对C₂感兴趣：B ₂ , B ₃ , C ₁ and C ₃ can be reduced by equations (23) and (24); conditions (xi) and (xii) can be used to solve for C ₂ and B ₁ , we are only interested in C ₂ :

${C C}_{22} = = {β β}_{11} \frac{W W}{22} - - - - - - ((2525))$

其中，

a由公式(19)确定。in,

a is determined by formula (19).

C₁-C₆的值全部为已知量，将其代入公式(20)与公式(21)，可解出总的变形量w，为在AB范围内由x表示的一个函数：The values of C ₁ -C ₆ are all known quantities. Substituting them into formula (20) and formula (21) can solve the total deformation w, which is a function represented by x in the range of AB:

$w w = = - - \frac{W W {x x}^{22} L L}{24 twenty four EI EI} ((33 - - \frac{22 x x}{L L})) - - \frac{WL WL}{44 AG AG} {((11 - - \frac{{I I}_{f f}}{I I}))}^{22} \times \times {{\frac{22 x x}{L L} - - \frac{22}{aL aL} [[sinh sinh ax ax + + {β β}_{11} ((11 - - cosh cosh ax ax))]]}}$

其最大值应发生在跨中位置处，即x＝L/2时：Its maximum value should occur at the mid-span position, that is, when x=L/2:

其中：

in:

通过公式(14)及对方程(21)和(22)进行两次求导，还可以得到芯材内部的剪应力及面板的法向应力。Through formula (14) and derivation of equations (21) and (22) twice, the shear stress inside the core material and the normal stress of the panel can also be obtained.

(二)均布荷载情况下非金属面夹芯板挠度的变形公式。(2) The deformation formula of the deflection of the non-metallic sandwich panel under uniform load.

如图10所示，为简支梁在均布荷载作用下的受力图。As shown in Figure 10, it is a force diagram of a simply supported beam under a uniform load.

AB部分剪力为-qx，其中x起始点为A。代入方程(18)中，结果如下：The shear force of part AB is -qx, where the starting point of x is A. Substituting into equation (18), the result is as follows:

-Q₁＝C₁coshax+C₂sinhax+qx (27)-Q ₁ ＝C ₁ coshax+C ₂ sinhax+qx (27)

通过积分，by points,

${EIw wxya}_{11} = = \frac{{C C}_{11}}{{a a}^{33}} sinh sinh ax ax + + \frac{{C C}_{22}}{{a a}^{33}} cosh cosh ax ax + + \frac{{qx qx}^{44}}{24 twenty four} + + {C C}_{33} {x x}^{22} + + {C C}_{44} x x + + {C C}_{55} - - - - - - ((2828))$

方程(27)同(28)一起，可得到关于w₂′的表达式，积分一次可得式(29)。Equation (27) together with (28) can obtain the expression about w ₂ ′, and the equation (29) can be obtained by integrating once.

$- - {E E. {I I}_{f f} w w}_{22} = = \frac{{C C}_{11}}{{a a}^{33}} sinh sinh ax ax + + \frac{{C C}_{22}}{{a a}^{33}} cosh cosh ax ax + + \frac{{qx qx}^{22}}{22 {a a}^{22}} + + {C C}_{66} - - - - - - ((2929))$

方程(28)、(29)同样适用于BC段，其中x起始点为B，并且消去包含q的项。常数C₁-C₆由B₁-B₆取代。Equations (28), (29) are also applicable to the BC segment, where the starting point of x is B, and the term containing q is eliminated. The constants C ₁ -C ₆ are replaced by B ₁ -B ₆ .

边界条件及B点的连续性要求同集中荷载作用下的厚面板夹芯梁单元，其中，跨中弯矩WL/4变为qL²/8。求解未知常数的过程同上，最终结果如下：The boundary conditions and continuity requirements of point B are the same as the thick-slab sandwich beam unit under concentrated load, in which the mid-span bending moment WL/4 becomes qL ² /8. The process of solving the unknown constant is the same as above, and the final result is as follows:

C₁＝0； $C_{3} = - \frac{{qL}^{2}}{16} + \frac{q}{2 a^{2}};$ C₄＝0； $C_{5} = C_{5} = - \frac{C_{2}}{a^{3}}$ C ₁ =0; $C_{3} = - \frac{{QUR}^{2}}{16} + \frac{q}{2 a^{2}};$ C ₄ =0; $C_{5} = C_{5} = - \frac{C_{2}}{a^{3}}$

$B_{1} = C_{2} \sinh \frac{aL}{2} + \frac{qL}{2};$ B₂＝-B₁tanhaL₁； $B_{3} = 0; B_{4} = - \frac{{qL}^{3}}{24}$ $B_{1} = C_{2} \sinh aL \frac{}{2} + \frac{QUR}{2};$ B ₂ =-B ₁ tanhaL ₁ ; $B_{3} = 0; B_{4} = - \frac{{QUR}^{3}}{twenty four}$

${C C}_{22} = = - - β β \frac{qL QUR}{22}$

其中

a由公式(19)确定。in

a is determined by formula (19).

AB段任意一点处，总的变形量由下式给出：At any point in section AB, the total deformation is given by the following formula:

$w w = = - - \frac{{qx qx}^{22} {L L}^{22}}{4848 EI EI} ((33 - - \frac{{22 x x}^{22}}{{L L}^{22}})) - - \frac{q q}{AG AG} {((11 - - \frac{{I I}_{f f}}{I I}))}^{22} \times \times {{\frac{{x x}^{22}}{22} - - \frac{{β β}_{22} {L L}^{22}}{44 θ θ} ((11 - - cosh cosh ax ax))}}$

其最大值发生在x＝L/2处：Its maximum occurs at x=L/2:

其中：

in:

通过公式(14)及对方程(28)和(29)进行两次求导，也可得到芯材内部的剪应力及面板的法向应力。The shear stress inside the core material and the normal stress of the panel can also be obtained by formula (14) and derivation of equations (28) and (29) twice.

四、非金属面夹芯板承载力的验算。4. Calculation of bearing capacity of non-metallic sandwich panels.

实际工程中夹芯板的抗弯承载力主要由正常使用极限状态时的变形控制，当受均布面荷载作用时，单跨夹芯板的抗弯承载力可按下列规定计算：In actual engineering, the flexural bearing capacity of sandwich panels is mainly controlled by the deformation at the limit state of normal use. When subjected to a uniform surface load, the flexural bearing capacity of a single-span sandwich panel can be calculated according to the following regulations:

w_max≤[f]w _max ≤ [f]

其中：[f]表示正常使用极限状态时的变形控制限限值，一般取L/200，L为夹芯板跨度。Among them: [f] indicates the deformation control limit value in the limit state of normal use, generally L/200 is taken, and L is the span of the sandwich panel.

(一)集中荷载情况下公式简化。(1) The formula is simplified in the case of concentrated load.

集中荷载作用下，非金属面夹芯板在考虑面板刚度情况下，跨中位置处在均布荷载作用下的挠度计算公式为：Under the action of concentrated load, the calculation formula for the deflection of the non-metallic sandwich panel under the uniform load at the mid-span position is as follows:

其中：

in:

θ = \frac{aL}{2},

a^{2} = \frac{AG}{{EI}_{f} (1 - I_{f} / I)}

θ = \frac{aL}{2},

a^{2} = \frac{AG}{{EI}_{f} (1 - I_{f} / I)}

由于实际工程中一般无需设置悬臂梁，此时令L₁＝0，由此上述参数可以简化成以下简单形式：Since there is generally no need to set cantilever beams in actual engineering, L ₁ =0 at this time, so the above parameters can be simplified into the following simple form:

$β_{1} = \frac{\sinh θ}{\cosh θ},$ $θ = \frac{aL}{2} = \frac{L}{2} {(\frac{AG}{{EI}_{f} (1 - I_{f} / I)})}^{1 / 2},$

β_{1} = \frac{\sinh θ}{\cosh θ},

θ = \frac{aL}{2} = \frac{L}{2} {(\frac{AG}{{EI}_{f} (1 - I_{f} / I)})}^{1 / 2},

而a²的大小实质上代表的是芯材剪切刚度与面板局部弯曲刚度的比值，由本文中所研究的几种非金属面夹芯板可知，其a²的值大约在400以上，由此计算得来的θ值一般在20左右。

与θ的关系如图11所示，其中θ为横坐标，

为纵坐标。考虑θ≥3时，

因此有：The size of a ² essentially represents the ratio of the shear stiffness of the core material to the local bending stiffness of the panel. From the several non-metallic sandwich panels studied in this paper, the value of a ² is about 400 or more. The calculated value of θ is generally around 20.

The relationship with θ is shown in Figure 11, where θ is the abscissa,

is the vertical coordinate. When considering θ≥3,

So there are:

对于实际工程中的墙面板，考虑θ值一般在20以上，此时可近似认为

即可得到墙面板在集中荷载下考虑面板刚度的最终挠度计算公式：For wall panels in actual engineering, considering that the value of θ is generally above 20, it can be approximately considered as

The calculation formula of the final deflection of the wall panel considering the panel stiffness under the concentrated load can be obtained:

${w w}_{max max} = = - - \frac{{WL WL}^{33}}{4848 EI EI} - - \frac{WL WL}{44 A A {μ μ}^{22} G G} {((11 - - \frac{{I I}_{f f}}{I I}))}^{22} - - - - - - ((3232))$

对于墙面板，估算其面板刚度与整体刚度的比值，一般在10％以上，此时θ值一般最小能达到5左右，可取

并考虑芯材剪切模量在不同试验方法下的换算关系，可得到屋面板在考虑面板刚度情况下，跨中位置处在集中荷载作用下的挠度计算公式最终化简形式为：For wall panels, it is estimated that the ratio of panel stiffness to overall stiffness is generally above 10%. At this time, the θ value can generally reach a minimum of about 5, which is desirable

And considering the conversion relationship of the shear modulus of the core material under different test methods, the final simplified form of the deflection calculation formula at the mid-span position under the concentrated load can be obtained when the roof panel stiffness is considered:

${w w}_{max max} = = - - \frac{{WL WL}^{33}}{4848 EI EI} - - \frac{88 WL WL}{44 A A {μ μ}^{22} G G} {((11 - - \frac{{I I}_{f f}}{I I}))}^{22} - - - - - - ((3333))$

其中：in:

w_max表示正常使用阶段跨中位置处的挠度。公式(32)适用于墙面板，公式(33)w _max represents the deflection at mid-span during normal use. Formula (32) applies to wall panels, formula (33)

适用于屋面板Suitable for roof panels

W表示跨中集中荷载W represents the concentrated load in the middle of the span

L表示夹芯板跨度L indicates the span of the sandwich panel

I_f表示上下面板对其自身中和轴的惯性矩，

$I I = = \frac{{bt bt}^{33}}{66} + + \frac{{btd btd}^{22}}{22}$

A表示芯材的有效截面面积，

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

G表示芯材的剪切模量实测；G represents the actual measurement of the shear modulus of the core material;

μ表示芯材剪切模量在不同实验方法之间的换算系数，μ＝1.236μ represents the conversion factor of core material shear modulus between different experimental methods, μ=1.236

(二)均布荷载情况下公式简化。(2) The formula is simplified in the case of uniform load.

均布荷载作用下，非金属面夹芯板在考虑面板刚度情况下，跨中位置处挠度计算公式为：Under the action of uniform load, the deflection calculation formula at the mid-span position of the non-metallic sandwich panel is as follows:

其中： in:

θ = \frac{aL}{2},

a^{2} = \frac{AG}{{EI}_{f} (1 - I_{f} / I)}

θ = \frac{aL}{2},

a^{2} = \frac{AG}{{EI}_{f} (1 - I_{f} / I)}

$β_{2} = \frac{1}{θ \cosh θ},$ $θ = \frac{aL}{2},$

β_{2} = \frac{1}{θ \cosh θ},

θ = \frac{aL}{2},

而a²的大小实质上代表的是芯材剪切刚度与面板局部弯曲刚度的比值，本文中所研究的几种非金属面夹芯板，其a²的值大约在400以上，由此计算得来的θ值一般在20左右。

与θ的关系如图12所示。考虑θ≥3时，因此有：The size of a ² essentially represents the ratio of the shear stiffness of the core material to the local bending stiffness of the face plate. The value of a ² of several non-metallic sandwich panels studied in this paper is about 400 or more. The resulting value of θ is generally around 20.

The relationship with θ is shown in Figure 12. When considering θ≥3, So there are:

即可得到墙面板在均布荷载下考虑面板刚度的最终挠度计算公式：For wall panels in actual engineering, considering that the value of θ is generally above 20, it can be approximately considered as

The final deflection calculation formula considering the panel stiffness under uniform load can be obtained:

${w w}_{max max} = = - - \frac{{55 qL QUR}^{33}}{384384 EI EI} - - \frac{q q {L L}^{22}}{88 A A {μ μ}^{22} G G} {((11 - - \frac{{I I}_{f f}}{I I}))}^{22} - - - - - - ((3535))$

对于墙面板，估算其面板刚度与整体刚度的比值，一般在10％以上，此时θ值一般最小能达到5左右，可取并考虑芯材剪切模量在不同试验方法下的换算关系，可得到屋面板在考虑面板刚度情况下，跨中位置处在均布荷载作用下的挠度计算公式最终化简形式为：For wall panels, it is estimated that the ratio of panel stiffness to overall stiffness is generally above 10%. At this time, the θ value can generally reach a minimum of about 5, which is desirable And considering the conversion relationship of the shear modulus of the core material under different test methods, the final simplified form of the deflection calculation formula at the mid-span position of the roof panel under the action of a uniformly distributed load can be obtained when the panel stiffness is considered:

${w w}_{max max} = = - - \frac{{55 qL QUR}^{44}}{384384 EI EI} - - \frac{0.9 0.9 {qL QUR}^{22}}{88 A A {μ μ}^{22} G G} {((11 - - \frac{{I I}_{f f}}{I I}))}^{22} - - - - - - ((3636))$

式中：In the formula:

w_max表示正常使用阶段跨中位置处的挠度，其中式(35)适用于墙面板，式(36)适用于墙面板w _max represents the deflection at the mid-span position during normal use, where formula (35) applies to wall panels, and formula (36) applies to wall panels

q表示均布荷载；q means uniform load;

L表示夹芯板跨度；L represents the span of the sandwich panel;

E表示非金属面夹芯板面板弹性模量；E represents the elastic modulus of the non-metallic sandwich panel panel;

I_f表示上下面板对其自身中和轴的惯性矩， I _f represents the moment of inertia of the upper and lower panels about their own neutral axes,

$I I = = \frac{{bt bt}^{33}}{66} + + \frac{{btd btd}^{22}}{22}$

A表示芯材的有效截面面积，

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

G表示芯材的剪切模量实测值G represents the measured value of the shear modulus of the core material

μ表示芯材剪切模量在不同实验方法之间的换算系数，通常μ＝1.236μ represents the conversion factor of core material shear modulus between different experimental methods, usually μ=1.236

步骤300：确定非金属面夹芯板的抗弯承载力：由挠度与抗弯承载力的关系确定非金属面夹芯板的抗弯承载力。Step 300: Determine the flexural capacity of the non-metallic sandwich panel: determine the flexural capacity of the non-metallic sandwich panel from the relationship between the deflection and the flexural capacity.

由于夹芯板墙面板的挠度与抗弯承载力有如下关系：Since the deflection of the sandwich panel wall panel has the following relationship with the bending capacity:

w_max≤[f]w _max ≤ [f]

[f]参见国家标准《建筑用金属面绝热夹芯板》(GB/T 23932-2009)中挠度的限值。[f] See the limit value of deflection in the national standard "Metal Surface Insulation Sandwich Panels for Buildings" (GB/T 23932-2009).

则：but:

集中荷载确定： $W \leq \frac{[f]}{\frac{L^{3}}{48 EI} + \frac{0.8 L}{4 A μ^{2} G} {(1 - \frac{I_{f}}{I})}^{2}}$ Concentrated load determination: $W \leq \frac{[f]}{\frac{L^{3}}{48 EI} + \frac{0.8 L}{4 A μ^{2} G} {(1 - \frac{I_{f}}{I})}^{2}}$

局部荷载的确定： $q \leq \frac{[f]}{\frac{{5 L}^{4}}{384 EI} + \frac{0.9 L^{2}}{8 A μ^{2} G} {(1 - \frac{I_{f}}{I})}^{2}}$ Determination of local loads: $q \leq \frac{[f]}{\frac{{5 L}^{4}}{384 EI} + \frac{0.9 L^{2}}{8 A μ^{2} G} {(1 - \frac{I_{f}}{I})}^{2}}$

从而确定非金属面夹芯板的抗弯承载力。In order to determine the flexural bearing capacity of the non-metallic sandwich panel.

本发明非金属面夹芯板抗弯承载力确定方法，通过考虑非金属面夹芯板的面板刚度对非金属面夹芯板弯曲变形造成的影响，精确获取非金属面夹芯板在集中荷载和均布荷载下的挠度，然后通过挠度确定非金属面夹芯板的抗弯承载力。本发明精确获取非金属面夹芯板的抗弯承载力，从而确定非金属面夹芯板的抗弯力学性能，精确评估非金属面夹芯板的安全性能。The method for determining the flexural bearing capacity of the non-metallic sandwich panel of the present invention accurately obtains the concentrated load of the nonmetallic sandwich panel by considering the influence of the panel stiffness of the nonmetallic sandwich panel on the bending deformation of the nonmetallic sandwich panel and the deflection under a uniform load, and then determine the flexural capacity of the non-metallic sandwich panel through the deflection. The invention accurately acquires the anti-bending bearing capacity of the non-metallic surface sandwich panel, thereby determining the anti-bending mechanical performance of the non-metallic surface sandwich panel, and accurately evaluating the safety performance of the non-metallic surface sandwich panel.

本发明的具体实施方式是：将所述非金属面夹芯板抗弯承载力确定方法应用于非金属面夹芯板的安全评估。The specific embodiment of the present invention is: applying the method for determining the bending bearing capacity of the non-metallic sandwich panel to the safety assessment of the non-metallic sandwich panel.

以上内容是结合具体的优选实施方式对本发明所作的进一步详细说明，不能认定本发明的具体实施只局限于这些说明。对于本发明所属技术领域的普通技术人员来说，在不脱离本发明构思的前提下，还可以做出若干简单推演或替换，都应当视为属于本发明的保护范围。The above content is a further detailed description of the present invention in conjunction with specific preferred embodiments, and it cannot be assumed that the specific implementation of the present invention is limited to these descriptions. For those of ordinary skill in the technical field of the present invention, without departing from the concept of the present invention, some simple deduction or replacement can be made, which should be regarded as belonging to the protection scope of the present invention.

Claims

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:

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

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{{bt}^{3}}{6} + \frac{{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.