Method for calculating shear deformation component of rectangular sandwich panel with four simply supported sides under normal force

Abstract

The invention belongs to the technical field of aircraft strength and rigidity design, and particularly relates to a shear deformation component calculation method of a rectangular sandwich panel with four simply supported sides under normal force. The method includes the following steps that firstly, a coordinate system is defined based on the four-side simply-supported rectangular sandwich panel; 2, defining shearing parameters of the four-side simply-supported rectangular sandwich panel structure; 3, establishing an infinitesimal body balance equation; 4, establishing a control equation according to the infinitesimal Hooke's law; and 5, solving the shear deformation.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of aircraft strength and rigidity design, and particularly relates to a method for calculating the shear deformation component of a four-edge simply supported rectangular sandwich panel under a normal force. BACKGROUND

[0002] At present, the "structure has no harmful or permanent deformation under the limit load. The deformation under any load up to the limit load shall not affect the safe operation" in the CCAR-27 / 29-R2 Article 305 (a) of the civil helicopter airworthiness regulations is only a qualitative requirement for the deformation of the helicopter structure, and cannot directly guide the structural rigidity design. For important rigidity key structures / systems, such as cabin doors, flight control systems and support structures, transmission system support structures (such as main reduction platforms and tail inclined beams), foreign advanced helicopter companies will appropriately propose quantitative structural deformation requirements or support structure rigidity requirements, and the structural rigidity design index is relatively clear. In China, the rigidity key structure is mainly designed based on strength, the cognition of "harmful deformation" is fuzzy, and it is difficult to propose appropriate quantitative indicators for the structural deformation requirements. Over-rigidity design will cause unnecessary structural weight cost, and under-rigidity design will have flight safety risks, and the local structure must be changed or strengthened, which may damage the coordination of the overall structure, reduce the efficiency of the overall structure, and even lead to structural redesign, prolonging the development cycle. Therefore, the design of the structural rigidity is increasingly important in the current helicopter development.

[0003] In the 8.2.3 section of the book "Composite Mechanics Second Edition, Shen Guanlin, Hu Gengkai, Tsinghua University Press", the series expression of the pure bending component of the deflection surface of the four-edge simply supported composite rectangular laminated plate (special orthotropic anisotropy) under the normal pressure is derived (see the attached Figure 1 ).

[0004]

[0005] However, when the formula is used to calculate the honeycomb sandwich structure, the error is large. Because the structural characteristics of the honeycomb sandwich plate are that the upper and lower two sheets of thin and high-strength panels and the intermediate light-weight and low-rigidity honeycomb core are composed, the bending and shear rigidity of the cross section is necessarily smaller than that of the pure laminated plate with the same thickness. However, compared with the pure laminated plate with the same thickness, the loss degree of the bending rigidity of the honeycomb sandwich structure is much smaller than that of the shear rigidity. Because the materials far from the middle surface of the cross section play a major role in bending, the upper and lower panels are still subjected to a large bending action when they are supported by the honeycomb. However, in the shear, the materials close to the middle surface play a major role in shear, that is, the honeycomb plays a role in shear, and the shear capacity of the honeycomb is poor. Therefore, the out-of-plane shear deformation of the honeycomb sandwich plate is very prominent and cannot be ignored.

[0006] The calculation of shear deformation of the existing four-edge simply supported rectangular plate (such as an aviation interior plate) under normal pressure condition depends on the general mechanical software (the calculation method is a black box), and the designer cannot directly master the influence of each parameter on the final deformation. In order to better design and optimize the design, the designer needs to directly master the influence of each parameter on the final deformation, and it is very helpful for the design to obtain the analytical expression of the sandwich plate deformation correction. SUMMARY

[0007] The purpose of the application is to provide a calculation method for the shear deformation component of a four-edge simply supported rectangular sandwich plate under normal force based on basic mechanics. A pure shear series expression is added to correct the pure bending series expression. The premise of this is that the structure has small deformation, satisfies the superposition principle, and the pure shear deformation and the pure bending deformation are not coupled. The following will follow the derivation method of the pure bending series expression to derive the pure shear series expression.

[0008] TECHNICAL SCHEME The application provides a calculation method for the shear deformation component of a four-edge simply supported rectangular sandwich plate under normal force, which comprises the following steps. Step 1: Defining a coordinate system based on the four-edge simply supported rectangular sandwich plate. Step 2: Defining the shear parameters of the four-edge simply supported rectangular sandwich plate structure. Step 3: Establishing the balance equation of the micro-unit. Step 4: Establishing the control equation according to the Hook's law of the micro-unit. Step 5: Solving the shear deformation.

[0009] Further, in step 1, the plane of the rectangular sandwich plate is defined as the XY plane, the X axis is along one side of the rectangular plate, the Y axis is along the other side of the rectangular plate, the Z axis is perpendicular to the plane of the plate and downward, and the X, Y and Z axes comply with the right-hand rule.

[0010] Further, in step 2, the out-of-plane shear stiffness of the rectangular sandwich plate in the x direction is G 44 , the out-of-plane shear stiffness of the rectangular sandwich plate in the y direction is G 55 , the shear internal force field function of the rectangular sandwich plate on the cross section is , , and the normal pressure field function of the rectangular sandwich plate is q.

[0011] Further, in step 3, the micro-unit is a volume unit separated from the continuous medium, which has an infinitely small size but keeps the macroscopic material properties. The coordinate system of the micro-unit is consistent with the coordinate system of the rectangular sandwich plate.

[0012] Further, in step 3, the balance equation formula of the regular hexahedral micro-unit is as follows:

[0013] represents the difference of the shear forces along z-direction on the two planes perpendicular to x-axis on the micro-element, represents the difference of the shear forces along z-direction on the two planes perpendicular to y-axis on the micro-element, which are the internal forces of the micro-element, their sum balances with the external forces of the micro-element is the value of the normal pressure field function q in the coordinate of the micro-element.

[0014] Further, in step four, the process is as follows: Let the deflection surface function of the rectangular sandwich plate be ; Since the micro-element undergoes small deformation, the shear rotation angle is a small angle, and the radian value of the small angle is equal to the tangent value; then the shear rotation angle of the micro-element in the x-direction is , the shear rotation angle in the y-direction is ; and the following relationship can be obtained from the Hook's law of the micro-element:

[0015]

[0016] The above expressions are combined to obtain the control equation as follows:

[0018] Further, in step five, the specific process is as follows: Let the length of the rectangular sandwich plate along the x-direction be a, and the length along the y-direction be b; The normal pressure field function q of the rectangular sandwich plate is expanded by two-dimensional Fourier sine series as follows:

[0019] wherein,

[0020] The deflection surface function of the rectangular sandwich plate is expanded by two-dimensional Fourier sine series as follows:

[0021] wherein, a mn is an undetermined coefficient; The series expansion forms of the normal pressure field function q and the deflection surface function are substituted into the control equation to obtain the following formula:

[0022] The above formula is arranged as ​

[0023] Using the contrast coefficient method, we get

[0024] Substitute q mn into the above formula, the following formula is obtained:

[0025] Substitute the above formula into the two-dimensional Fourier sine series formula of the flexure surface function, the final shear deformation expression is obtained

[0027] Further, when q (x, y) is a fixed value in the whole area ;

[0028] Substitute the final shear deformation expression to obtain the following formula:

[0029] As a supplement, substitute the obtained w expression back into the micro-element Hook's law formula, and the shear stress component of any point of the flat plate can be obtained.

[0030] Based on the above, the beneficial effects of the present application are as follows: Accurate evaluation of structural stiffness and deformation behavior. The constitutive relationship of the classical theory is specially processed to better reflect the mechanical behavior of the honeycomb core. For the main source of total deformation when the sandwich plate is sheared, ignoring shear deformation will seriously underestimate the actual deflection. The stiffness design can be optimized, the degree of weakening of shear deformation on stiffness is clear, and engineers can choose more suitable core material, adjust the core material thickness or optimize the panel thickness to meet the stiffness requirements.

[0031] Predicting key failure modes and checking strength Honeycomb sandwich plate core shear failure is the main failure mode, and the derived shear stress component is the core index for evaluating this risk. A simplified engineering calculation method is established, and a practical design formula is developed. Based on the derivation, a widely used engineering practical formula is formed. The innovation of the result output provides an explicit analytical solution for the spatial distribution of shear deformation components, which can directly show the influence of various parameters on deformation. It is very beneficial for engineers to perform optimization sensitivity analysis and improve design, and is beneficial to forward design research and development and optimization design. BRIEF DESCRIPTION OF DRAWINGS

[0032] Figure 1 It is a schematic diagram of a four-edge shear support rectangular laminated plate; Figure 2 It is a schematic diagram of a honeycomb sandwich plate; Figure 3 Schematic diagram of normal load for simply supported plate Figure 4 Schematic diagram of micro-element balance DETAILED DESCRIPTION

[0033] The present application combines basic mechanics principles to derive a series expression of the deflection surface of a four-side simply supported composite honeycomb sandwich structure (see attached Figure 2 ) under normal pressure (see attached Figure 3 ). A pure shear series expression is added to the pure bending series expression for correction. The prerequisite for this is small deformation of the structure, satisfaction of the superposition principle, and no coupling between pure shear deformation and pure bending deformation. The derivation method of the pure bending series expression is followed to derive the pure shear series expression.

[0034] 1 Define coordinate system The plane of the rectangular plate is defined as the XY plane, the X axis is along one side of the rectangular plate, the Y axis is along the other side of the rectangular plate, the Z axis is perpendicular to the plane of the plate downward, the X, Y, and Z axes comply with the right-hand rule, see attached Figure 1 . 2 Define shear parameters of the honeycomb sandwich structure Let the out-of-plane shear stiffness of the sandwich plate in the x direction be G 44 , and the out-of-plane shear stiffness in the y direction be G 55 . Let the shear internal force field function on the cross section of the plate be 、 . Let the normal pressure field function of the plate be q.

[0035] 3 Establish micro-element balance equation A micro-element is a volume unit separated from a continuous medium with an infinitely small size but maintaining the macroscopic material properties. Its core function is to convert macroscopic mechanical quantities into local microscopic quantities that can be operated through calculus. The micro-element coordinate system is consistent with the rectangular plate coordinate system, see attached Figure 4 . The balance of the regular hexahedron micro-element is obtained as

[0036] In the above formula, represents the difference in z-direction shear force on the two planes perpendicular to the x axis of the micro-element, represents the difference in z-direction shear force on the two planes perpendicular to the y axis of the micro-element, which are internal forces of the micro-element, and their sum balances with the external force q of the micro-element.

[0037] 4 Establish control equation according to Hooke's law of the micro-element Let the deflection surface function of the plate be . Since the micro-element undergoes small deformation, the shear angle is a small angle, and the radian value of a small angle is equal to the tangent value. Therefore, the shear angle in the x direction of the micro-element is , The shear rotation angle in the x direction is The shear rotation angle in the x direction is

[0038]

[0039] The above expressions are combined to obtain (1) 5 Solve the shear deformation Let the rectangular plate have a length of a in the x direction and a length of b in the y direction. The pressure field function in engineering practice must satisfy the Dirichlet condition, and the normal pressure field function can be expanded using a two-dimensional Fourier sine series (2) where (3) The deflection surface function in engineering practice must satisfy the Dirichlet condition, and the deflection surface function can be expanded as a two-dimensional Fourier sine series (4) Here are undetermined coefficients. Substitute formula (4) and formula (2) into formula (1) and take the derivative of the left side to obtain the following formula

[0040] The above formula is rearranged as

[0041] Because each level of trigonometric function has the characteristic of linear independence, the coefficients of the same level of trigonometric function must be equal, so the comparison coefficient method can be used to obtain

[0042] Substitute formula (3) into the above relationship to obtain the following formula

[0043] Substitute the above formula into formula (4) to obtain the final shear series expression (5) Each term in the above formula is known, and it is the deflection surface pure shear series expression.

[0044] In the special case where is a fixed value over the entire area,

[0045] Substitute back to get (6) The above expression is the calculation method for the shear deformation component of a honeycomb sandwich structure. The above derivation process ignores the shear stiffness of the upper and lower panels because the bending stiffness of the upper and lower panels themselves is relatively small (compared to the stiffness of the entire cross section). They will cooperate with the shear deformation of the honeycomb through bending deformation. That is, in a honeycomb sandwich structure, the influence of the upper and lower panels on the pure shear deformation can be ignored.

[0046] From equation (5) or (6) of the derivation results, it can be directly seen that each parameter... , , , , The impact on deformation is very beneficial for engineering designers to conduct sensitivity analysis and improve designs, which is conducive to positive design development and optimization.

[0047] Substituting the obtained equation (5) or (6) back into the Hooke's Law equation for the infinitesimal element, the shear stress component at any point on the plate can be obtained, and the shear strength of the honeycomb can be checked.

Claims

1. A method for calculating the shear deformation components of a simply supported rectangular sandwich panel under normal force, characterized in that: Includes the following steps: Step 1: Define a coordinate system based on a simply supported rectangular sandwich panel; Step 2: Define the shearing parameters for a simply supported rectangular sandwich panel structure; Step 3: Establish the equilibrium equations for the infinitesimal element; Step 4: Establish the governing equations based on Hooke's law for infinitesimal elements; Step 5: Solve for the shear deformation.

2. The method according to claim 1, characterized in that: In step one, the plane of the rectangular sandwich panel is defined as the XY plane, with the X axis along one side of the rectangular panel, the Y axis along the other side of the rectangular panel, and the Z axis perpendicular to the flat panel and pointing downwards. The X, Y, and Z axes conform to the right-hand rule.

3. The method according to claim 2, characterized in that: In step two, let the out-of-plane shear stiffness of the rectangular sandwich panel in the x-direction be G. 44 The out-of-plane shear stiffness in the y-direction is G. 55 Let the shear force field function on the cross section of the rectangular sandwich panel be... , Let the normal pressure field function of the rectangular sandwich panel be q.

4. The method according to claim 3, characterized in that: In step three, the infinitesimal element is a volume unit that is imaginarily separated from the continuous medium, with an infinitesimal size but retaining macroscopic material properties. The coordinate system of the infinitesimal element is consistent with the coordinate system of the rectangular sandwich panel.

5. The method according to claim 4, characterized in that: In step three, the equilibrium equation for the hexahedral infinitesimal element is as follows: This represents the difference in shear force along the z-axis on two planes perpendicular to the x-axis of the infinitesimal element. This represents the difference in shear force along the z-direction on two planes perpendicular to the y-axis of the infinitesimal element. These forces are considered as internal forces within the infinitesimal element, and their sum is equal to the external forces outside the infinitesimal element. Balance, infinitesimal external forces Let q be the value of the normal pressure field function q in the coordinate system of the infinitesimal element.

6. The method according to claim 5, characterized in that: Step four involves the following process: Let the deflection surface function of the rectangular sandwich panel be... ; Since the infinitesimal element undergoes a small deformation, the shear rotation angle is a small angle, and the radian value of this small angle is equal to the tangent value; therefore, the shear rotation angle of the infinitesimal element in the x-direction is... , The shear rotation angle of the direction is From Hooke's law for infinitesimal elements, we can obtain the following relationship: The governing equations obtained by combining the above expressions are as follows: 。 7. The method according to claim 6, characterized in that: Step five involves the following steps: Let the length of the rectangular sandwich panel along the x-direction be a, and the length along the y-direction be b; The normal pressure field function q(x,y) of the rectangular sandwich panel is expanded using a two-dimensional Fourier sine series as follows: in, The deflection surface function of the rectangular sandwich panel The two-dimensional Fourier sine series expansion is as follows: Among them, a mn These are coefficients to be determined; The normal pressure field function q(x,y) and the deflection surface function Substituting the series expansion of the equation into the governing equation, we obtain the following formula: The above formula is summarized as follows: The comparison coefficient method was used to obtain... q mn Substituting into the above formula, we get the following formula: Substituting the above equation into the two-dimensional Fourier sinusoidal series formula for the deflection surface function, we obtain the final expression for the shear deformation. 。 8. The method according to claim 7, characterized in that: when A fixed value over the entire area hour; Substituting into the expression for the final shear deformation, we obtain the following formula: 。 9. The method according to claim 7, characterized in that: Substitute the obtained expression for w back into the Hooke's law formula for the infinitesimal element to find the shear stress components at any point on the plate.

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