CN113722819B - Semi-analytical method for calculating bending deformation and stress of stiffening plate - Google Patents
Semi-analytical method for calculating bending deformation and stress of stiffening plate Download PDFInfo
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Abstract
The invention relates to a semi-analytical method for calculating bending deformation and stress of a stiffening plate, which comprises the steps of S1, calculating potential energy of a stiffening plate system, wherein the potential energy is the sum of strain energy of the stiffening plate and strain energy of a stiffening rib minus work done by external force; s2, a minimum potential energy principle is applied, and a bending balance equation of the stiffening plate is deduced through a variation method; s3, setting an unknown function as a double Fourier series, wherein the set Fourier series is required to meet boundary conditions, and only the constant is reserved to be determined through a bending balance equation; s4, substituting an unknown function expression containing the undetermined constant into a bending balance equation of the stiffening plate, determining the undetermined constant by adopting a Galerkin algorithm, and solving bending deformation of the stiffening plate; substituting the deformation expression into an elastic mechanical geometric equation and a physical equation, and calculating the bending stress of the reinforced plate. The method can quickly and conveniently calculate the bending deformation and the bending stress of the reinforcing plate of the ship body.
Description
Technical Field
The invention belongs to the technical field of ship structure calculation, and particularly relates to a semi-analytical method for calculating bending deformation and stress of a stiffening plate.
Background
In the field of ship structural design, whether structural design scheme is safe or not and whether the structural design scheme meets design standards or not is verified through calculation and check. The stiffener plate is the most common structural form in ship structures, and the bending of the stiffener plate is a common stress state of ship components during ship navigation and use. Although general finite element software can calculate bending deformation and bending stress of the reinforced plate, in actual design, the scheme is often changed repeatedly, and iterative optimization is gradually performed, so that the workload of finite element model modification and change is very large.
The existing analytic or semi-analytic calculation method generally equivalent the stiffening plate to an orthotropic plate or a main beam system. In order to ensure accuracy, the calculation method for equivalent reinforcing plates into orthotropic plates requires that the reinforcing ribs are small in size and densely arranged, and the load forms are uniform loads or concentrated loads acting on the reinforcing ribs, so that in practice, the conditions are difficult to meet at the same time; the method for calculating the main beam system by equivalent of the stiffening plate can only calculate the whole deformation and the local deformation can not be calculated.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a semi-analytical method for calculating the bending deformation and stress of the stiffening plate aiming at the defects in the prior art, and the method can be used for rapidly and conveniently calculating the bending deformation and bending stress of the stiffening plate of the ship body, can be used for rapidly checking the strength and the rigidity of the stiffening plate structure, and provides a theoretical basis for calculating and checking the structure of the ship body.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a semi-analytical method for calculating bending deformation and stress of a stiffening plate comprises the following steps:
s1, calculating potential energy of the stiffening plate system. Adopting a first-order shear deformation theory, listing a displacement mode of bending deformation of the stiffened plate, respectively calculating the strain and the stress of the stiffened plate by applying an elastic mechanical geometrical equation and a physical equation, and then calculating the strain energy of the stiffened plate;
the reinforcing ribs are equivalently processed into flat steel (hereinafter, no special description exists, the reinforcing ribs refer to the equivalent flat steel reinforcing ribs), the distribution of the reinforcing ribs is described by adopting a helminthic (Heaviside) function, the displacement modes of bending deformation of the reinforcing ribs are listed by adopting an iron mole Xin Ke (Timoshenko) beam theory, the strain and the stress of the reinforcing ribs are calculated respectively, and the strain energy of the reinforcing ribs is calculated;
the potential energy of the stiffening plate system is the sum of the strain energy of the stiffening plate and the strain energy of the stiffening plate minus the work done by the external force.
S2, deducing a bending balance equation of the stiffening plate. The bending balance equation of the reinforced plate is listed by a variational method by applying the principle of minimum potential energy.
And S3, providing a double Fourier series expression. The unknown function is set to be a double Fourier series, the set Fourier series needs to meet boundary conditions, only the constant is reserved to be determined through the bending balance equation, and the bending balance equation is solved by the double Fourier series method to be laid.
S4, calculating bending deformation and stress of the stiffening plate. Substituting an unknown function expression containing a double Fourier series representation of the undetermined constant into a bending balance equation of the stiffening plate, and determining the undetermined constant by adopting a Galerkin algorithm, so that the bending deformation of the stiffening plate can be solved; substituting the solved and determined deformation expression into an elastic mechanical geometric equation and a physical equation to calculate the bending stress of the reinforced plate; in order to reduce the computational effort, the process of determining the pending constants using the Galerkin algorithm is implemented using computer programming.
In step S1, a calculated coordinate system xyz of the stiffened plate is established separately p Calculated coordinate system xyz of reinforcing rib s Coordinate system xyz p The origin is positioned on the middle surface layer of the thickness of the stiffened plate, and the coordinate system xyz s The origin is positioned on the middle surface layer of the height of the reinforcing rib; wherein the x-axis is parallel to the axis of the reinforcing rib, the y-axis is perpendicular to the axis of the reinforcing rib, and the z p Axis (z) s An axis) is directed downwards perpendicular to the plane in which the stiffener plates lie (xy-plane).
The displacement of the stiffening plate in the x, y and z directions is expressed as by adopting a first-order shear deformation theory
u p (x,y,z p )＝u 0p (x,y)+z p φ xp (x,y) (1)
v p (x,y,z p )＝v 0p (x,y)+z p φ yp (x,y) (2)
w p (x,y,z p )＝w(x,y) (3)
In the formulae (1) - (3), u p (x,y,z p )，v p (x,y,z p ) And w p (x,y,z p ) Respectively represent any point (x, y, z) on the stiffened plate p ) Displacement in three directions, x, y and z; u (u) 0p (x, y) and v 0p (x, y) represents displacement of the facing layer in the stiffener panel in the x and y directions, respectively; phi (phi) xp (x, y) represents the rotation angle of the normal line of the middle surface of the stiffened plate in the zx plane, wherein the rotation angle is positive when the middle surface is turned from the x axis to the positive direction away from the z axis, and is negative otherwise; phi (phi) yp (x, y) represents the rotation angle of the normal line of the middle surface of the stiffened plate in the yz plane, the rotation angle is positive when the middle surface is turned away from the z-axis from the y-axis, otherwise, the rotation angle is negative; w (x, y) represents the displacement function of the stiffened panel in the z-direction.
The displacement of the reinforcing rib in the x, y and z directions is expressed as by adopting the iron Morgankoch beam theory
u s (x,y,z s )＝u 0s (x,y)+z s φ xs (x,y) (4)
v s (x,y,z s )＝v 0s (x,y)+z s φ ys (x,y) (5)
w s (x,y,z s )＝w(x,y) (6)
In the formulae (4) - (6), u s (x,y,z s )，v s (x,y,z s ) And w s (x,y,z s ) Respectively represent any point (x, y, z) on the reinforcing rib s ) Displacement in three directions, x, y and z; u (u) 0s (x, y) and v 0s (x, y) represents the displacement of the facing in the rib in the x and y directions, respectively; phi (phi) xs (x, y) represents the rotation angle of the normal line of the middle surface of the reinforcing rib in the zx plane, wherein the rotation angle is positive when the middle surface is turned from the x axis to the positive direction away from the z axis, and is negative otherwise; phi (phi) ys (x, y) represents the rotation angle of the normal line of the middle surface of the reinforcing rib in the yz plane, the rotation angle is positive when the middle surface is turned from the y axis to the positive direction away from the z axis, and is negative otherwise; w (x, y) represents the displacement function of the reinforcing rib in the z direction. The displacement of the reinforcing ribs in the z direction is the same as the displacement of the corresponding positions on the reinforcing plate, so that the same displacement function is adopted in the formula (3) and the formula (6).
At the interface of the plate and the rib, the displacement of the plate and the rib in the x direction and the y direction is equal, namely the displacement is coordinated
u p (x,y,t/2)＝u s (x,y,-h/2) (7)
v p (x,y,t/2)＝v s (x,y,-h/2) (8)
In the formulas (7) and (8), t represents the plate thickness of the stiffening plate; h represents the height of the reinforcing rib; u (u) p (x, y, t/2) and v p (x, y, t/2) represents the x and y displacement of the stiffened panel at any point on the panel interface, respectively; u (u) s (x, y, -h/2) and v s (x, y, -h/2) represents the x and y displacement of the stiffener at any point on the stiffener interface, respectively.
The u can be used by the formula (7) and the formula (8) 0s (x,y)，v 0s (x, y) represents u 0p (x, y) and v 0p (x, y), assumed to be phi from the flat section xp (x,y)＝φ xs (x,y)，φ yp (x,y)＝φ ys (x, y), and then substituting the formula (1) and the formula (2) to obtain
u p (x,y,z p )＝u 0s (x,y)-(h+t)φ xs (x,y)/2+z p φ xs (x,y) (9)
v p (x,y,z p )＝v 0s (x,y)-(h+t)φ ys (x,y)/2+z p φ ys (x,y) (10)
The strain of the stiffened panel is expressed as
In the formulae (11) - (15), ε xp And epsilon yp Respectively representing positive strains of the stiffening plate in the x direction and the y direction; gamma ray xyp 、γ yzp And gamma zxp The shear strain in xy, yz and zx directions of the stiffened panel are shown, respectively.
The strain of the reinforcing bars is expressed as
Epsilon in the formulas (16) and (17) xs Indicating positive x-direction strain of the reinforcing rib; gamma ray zxs The shear strain in the zx direction of the rib is shown.
The stress expression from the strain expression (11) -expression (17) can be expressed as Hooke's law, and the stress of the stiffened panel can be expressed as
In the formulae (18) - (22), σ xp Sum sigma yp Respectively representing positive x-direction stress and y-direction stress in the stiffened plate; τ xyp ，τ yzp And τ zxp Respectively representing the shearing stress of the stiffened plate in the xy direction, the yz direction and the zx direction; e, μ, κ represent the young's modulus of elasticity, poisson's ratio and shear correction coefficient, typically κ=5/6, respectively, of the stiffener plate (or stiffener) material.
The stress of the reinforcing bars is expressed as
σ xs ＝Eε xs (23)
In the formula (23) and the formula (24), sigma xs Indicating the positive stress of the reinforcing rib in the x direction; τ zxs The shear stress in the zx direction of the rib is shown.
By the above-described stress-strain expression, strain energy of the stiffened plate can be obtained. The potential energy of the stiffened plate system is the sum of the strain energy of the stiffened plate and the strain energy of the stiffener minus the work done by the external force, namely
In formula (25), pi represents the potential energy of the stiffened plate system, H () is a Helviered function, y i Is the y coordinate, t of the thickness center of the reinforcing rib s The equivalent thickness of the reinforcing ribs is represented, and N represents the number of the reinforcing ribs; v (V) 1 ，V 2 The volume domain of the space where the stiffening plate is positioned and the volume domain of the space where the stiffening rib is positioned are respectively; q (x, y) represents the external load function.
In a stable system, the potential energy pi of the stiffener should be kept to a minimum. In the formula (25), in order to enable the potential energy pi of the stiffening plate to reach the minimum value, a principle of variation is adopted. In step S2, the variation expression of the principle of the minimum potential energy is,
δΠ＝0 (26)
delta in the formula (26) represents a variation;
the Euler-Poisson equation, namely the bending equilibrium differential equation (27) -equation (31) of the reinforced plate, is solved by using a variational method.
The external load q (x, y) normally carried by the stiffener plates can be equivalently a concentrated load or an accumulation (integral) of the concentrated load.
For k concentrated loads, the load is expressed as
Wherein p is i To act on point (x i ,y i ) Concentrated load of locations, δ () is a Dirac (Dirac) pulse function.
For uniformly distributed loads, the load is expressed as
Wherein p is uniformly distributed load intensity, and D is the area acted by the load.
For a load that varies linearly in the x-direction, the load is expressed as
Wherein p is 1 Is x 1 Load intensity at position, p 2 Is x 2 Load intensity at the location.
The boundary condition of the stiffening plate is generally a four-side simple branch or a four-side solid branch. If the stiffening plate is in the four-side simply supported boundary condition, the boundary condition is expressed as
At x=0 or a
At y=0 or b
If the reinforcing plate is in the four-side clamped boundary condition, the boundary condition is expressed as
At x=0 or a
At y=0 or b
The solution problem consisting of equation (27) -equation (31) and boundary conditions equation (32) and equation (33) [ or boundary conditions equation (34) and equation (35) ] can be solved by a double fourier series method. In step S3, for the simple branch boundary conditions, the double Fourier series solution is assumed to be
In step S3, for the clamped boundary condition, the double Fourier series solution is assumed to be
In the formulae (36) - (45), a represents that the stiffening plate is long in the x directionThe degree, b, represents the length of the stiffening plate along the y direction; m and n represent the number of terms of the Fourier series, respectively; u (u) mn ，v mn ，φ xmn ，φ ymn And w mn For the constant to be determined, it is impossible to calculate an infinite number of terms in actual calculation, and only the number of convergence terms is calculated.
In practice, the two fourier series expressions (36) -40) of the unknown function in the above method satisfy the boundary condition expressions (32) and (33), and the expressions (41) -45 satisfy the boundary condition expressions (34) and (35). In step S4, formula (36) -formula (40), or formula (41) -formula (45) is substituted into formula (27) -formula (31), and the above-mentioned undetermined constant is determined by using a galkin algorithm; after the determination of the undetermined constant, the deformation of the reinforcing plate is determined, and the bending stress of the reinforcing plate can be calculated by substituting the deformation into the formula (18) -formula (24).
The invention has the beneficial effects that:
1. the calculation method provided by the invention can be used for quickly and conveniently calculating the bending deformation and bending stress of the reinforcing plate of the ship body. The method is very suitable for rapid calculation of repeated change of the structural scheme, and once the scheme is changed, the bending deformation and the bending stress of the new scheme can be obtained by only modifying corresponding parameters in the scheme and recalculating the parameters once. And the common finite element software is adopted to calculate the stiffening plate of the multiple-change structural scheme, so that the model needs to be repeatedly modified, parameter setting is modified, and the modification workload is large. Compared with the finite element method, the calculation method has small calculation error and high efficiency for calculating and checking the scheme for multiple changes, and the modification workload is small.
2. Compared with the existing calculation method for equivalent of the stiffening plate into the orthotropic plate, the method has no limitation on the size, arrangement and load form of the stiffening plate, the precision is still higher, and the stress distribution of the stiffening plate can be calculated.
3. Compared with the existing calculation method for equivalent of the stiffening plate into the main beam system, the method can calculate the superposition result of the whole deformation and the local deformation.
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
FIG. 1 is a coordinate system of the calculation method of the present invention. In the figure, 1 is a stiffening plate, and 2 is a stiffening rib; solid line coordinate line xyz p The coordinate system of the stiffened plate is that the origin of coordinates is positioned on the thickness middle plane of the stiffened plate, and the dotted line coordinate line xyz s The origin of coordinates is positioned on the middle height surface of the reinforcing rib for the coordinate system of the reinforcing rib; the length of the reinforcing plate along the x direction is a, the length of the reinforcing plate along the y direction is b, the thickness of the reinforcing plate is t, the height of the reinforcing ribs is h, and the spacing is l s Equivalent thickness t s 。
FIG. 2 shows the results of bending deformation calculated by two methods (the method of the present invention and the finite element method) in the examples of the present invention. Wherein:
FIG. 2 (a) is a schematic four-sided bracing plate with a=2 m and b=1.5 m, the reinforcing bars are flat steel 10×100, the spacing is 0.5m, and the central position acts 2×10 4 Deformation distribution at x=a/4 position when concentrating force of N;
FIG. 2 (b) is a=2 m, b=1.5 m four-sided simply supported reinforcing bars, the reinforcing bars are flat steel 10×100, the spacing is 0.5m, and the central position acts 2×10 4 Deformation profile at y=b/4 when force is concentrated in N.
FIG. 3 shows the results of bending stresses calculated by two methods (the method of the present invention and the finite element method) in the examples of the present invention. Wherein:
FIG. 3 (a) is a schematic four-sided bracing plate with a=2 m and b=1.5 m, the reinforcing bars are flat steel 10×100, the spacing is 0.5m, and the central position acts 2×10 4 The stress distribution of the lower surface y direction of the stiffening plate at the position x=a/4 when the force is concentrated;
FIG. 3 (b) is a=2 m, b=1.5 m four-sided simply supported reinforcing bars, the reinforcing bars are flat steel 10×100, the spacing is 0.5m, and the central position acts 2×10 4 And the stress distribution is in the x direction on the lower surface of the stiffening plate at the position y=b/4 when the force is concentrated.
Detailed Description
For a clearer understanding of technical features, objects and effects of the present invention, a detailed description of embodiments of the present invention will be made with reference to the accompanying drawings.
And establishing a coordinate system. Calculating sitting position by respectively establishing stiffening plates as shown in figure 1Standard xyz p And a stiffener calculation coordinate system xyz s The x-axis is parallel to the axis of the reinforcing rib, the y-axis is perpendicular to the axis of the reinforcing rib, and z p Axis (z) s The shaft) downward. Under the coordinate system, the y coordinates of the central position of the thickness of the web plate of the reinforcing rib are read and respectively marked as y 1 ，y 2 ，y 3 ……y n 。
The calculated known parameters are obtained. Acquiring the following known condition parameters, wherein the length a of the stiffening plate in the x direction and the length b of the stiffening plate in the y direction; the plate thickness of the stiffening plate is t. The spacing between the reinforcing ribs is l s The height of the reinforcing rib is h.
The reinforcing ribs in different forms are equivalent to flat steel. The reinforcing ribs in different forms are equivalently processed into flat steel with equivalent thickness t s Calculated by the method of moment of inertia equivalence, i.e. by the equation t s ＝12I s /h 3 Find, wherein I s The self neutral axis moment of inertia of the reinforcing rib.
And calculating potential energy of the stiffening plate system. In xyz, the first-order shear deformation theory is adopted p The displacement mode of bending deformation of the stiffened plate is listed in a coordinate system, an elastic mechanical geometrical equation and a physical equation are applied, strain and stress of the stiffened plate are calculated respectively, and strain energy of the stiffened plate is calculated;
in xyz s Describing the distribution of the reinforcing ribs by adopting a Helvezid function in a coordinate system, listing the displacement modes of bending deformation of the reinforcing ribs by adopting an iron Morganic corbel theory, respectively calculating the strain and the stress of the reinforcing ribs, and then calculating the strain energy of the reinforcing ribs;
and adding the strain energy of the reinforcing plate and the strain energy of the reinforcing rib to obtain the strain energy of the reinforcing plate. The potential energy of the stiffening plate system is the work of subtracting the external force from the strain energy of the stiffening plate.
Deriving a stiffened plate bending balance equation. And the potential energy of the stiffened plate system reaches the minimum value by applying the minimum potential energy principle and by a variation method, and a bending balance equation of the stiffened plate is listed.
The expression form of the external load is determined. The usual load form on the stiffened panel can always be equivalent to a concentrated load or an accumulation (integration) of concentrated loads.
And determining boundary conditions of the stiffening plates. The common boundary conditions in the stiffening plate of the ship body are four-side simple supports and four-side solid supports.
A dual fourier series expression is presented. The unknown function is set to be a double Fourier series, the set Fourier series needs to meet boundary conditions, only the constant is reserved to be determined through the bending balance equation, and the bending balance equation is solved by the double Fourier series method to be laid.
And calculating bending deformation and stress of the stiffening plate. Substituting an unknown function expression containing a double Fourier series representation of the constant to be determined into a bending balance equation of the reinforced plate, and determining the constant to be determined by adopting a Galerkin algorithm to solve the bending deformation of the reinforced plate; substituting the solved and determined deformation expression into an elastic mechanical geometric equation and a physical equation to calculate the bending stress of the reinforced plate. In order to reduce the calculation workload, the process of determining the undetermined constant by using the Galerkin algorithm can be realized by adopting computer programming, and when the structural scheme is changed, only the corresponding parameters are needed to be modified, and the program is restarted, thereby being convenient and quick.
The embodiments of the above-described calculation method may be implemented by using computer programming, such as Matlab programming, matheca programming, or map programming, where the above-described galkin algorithm program is a subroutine. Each time the structural design scheme is changed, only the initialization parameters such as a, b, t and l are required to be modified s And t s And parameters such as deformation and stress of a new scheme can be calculated by re-running the program, and the method is convenient and quick. The number of terms (i.e., the values of m and n) of the dual fourier series is set in the program, which indicates that 5mn fourier coefficients need to be calculated. The Galerkin algorithm is realized by adopting 4-time circulation, namely, a bending differential equation expressed by Fourier series is multiplied by a corresponding trigonometric function (weight function) item by item and then integrated, so that the differential equation is converted into an algebraic equation, and the equation can be solved by exactly 5mn non-odd algebraic equations. In order to save program run time, the orthogonality of trigonometric functions should be noted in programming.
To verify the effectiveness of the method of the invention, the bending deformation of the stiffened plate example model is calculated by a finite element methodAnd bending stress, wherein the model parameters of the stiffening plate example are a=2 m, b=1.5 m, the boundary condition is four-side simple support, the stiffening ribs are flat steel 10×100, the spacing is 0.5m, and the central position is acted by 2×10 4 And N is concentrated by force P. Comparison of the calculation results of fig. 2 and 3 shows that the calculation error of the method of the present invention is small compared with the finite element method, and the effectiveness of the method is verified.
The embodiments of the present invention have been described above with reference to the accompanying drawings, but the present invention is not limited to the above-described embodiments, which are merely illustrative and not restrictive, and many forms may be made by those having ordinary skill in the art without departing from the spirit of the present invention and the scope of the claims, which are to be protected by the present invention.
Claims (8)
1. The semi-analytical method for calculating bending deformation and stress of the stiffening plate is characterized by comprising the following steps of:
s1, calculating potential energy of a stiffening plate system: adopting a first-order shear deformation theory, listing a displacement mode of bending deformation of the stiffened plate, respectively calculating the strain and the stress of the stiffened plate by applying an elastic mechanical geometrical equation and a physical equation, and then calculating the strain energy of the stiffened plate; the reinforcing ribs are equivalently processed into flat steel, the distribution of the reinforcing ribs is described by adopting a helminthic (Heaviside) function, the displacement mode of bending deformation of the reinforcing ribs is listed by adopting an iron friction Xin Ke (Timoshenko) beam theory, the strain and the stress of the reinforcing ribs are respectively calculated, and the strain energy of the reinforcing ribs is calculated; the potential energy of the reinforcing plate system is the sum of the strain energy of the reinforcing plate and the strain energy of the reinforcing rib minus the work done by the external force;
in step S1, a calculated coordinate system xyz of the stiffened plate is respectively established p Calculated coordinate system xyz of reinforcing rib s Coordinate system xyz p The origin is positioned on the middle surface layer of the thickness of the stiffened plate, and the coordinate system xyz s The origin is positioned on the middle surface layer of the height of the reinforcing rib; wherein the x-axis is parallel to the axis of the reinforcing rib, the y-axis is perpendicular to the axis of the reinforcing rib, and the z p Axis (z) s Axis) perpendicular to the plane of the stiffener plateThe face (xy plane) points to the lower side;
the displacement of the stiffening plate in the x, y and z directions is expressed as by adopting a first-order shear deformation theory
u p (x,y,z p )＝u 0p (x,y)+z p φ xp (x,y) (1)
v p (x,y,z p )＝v 0p (x,y)+z p φ yp (x,y) (2)
w p (x,y,z p )＝w(x,y) (3)
In the formulae (1) - (3), u p (x,y,z p )，v p (x,y,z p ) And w p (x,y,z p ) Respectively represent any point (x, y, z) on the stiffened plate p ) Displacement in three directions, x, y and z; u (u) 0p (x, y) and v 0p (x, y) represents displacement of the facing layer in the stiffener panel in the x and y directions, respectively; phi (phi) xp (x, y) represents the rotation angle of the normal line of the middle surface of the stiffened plate in the zx plane, wherein the rotation angle is positive when the middle surface is turned from the x axis to the positive direction away from the z axis, and is negative otherwise; phi (phi) yp (x, y) represents the rotation angle of the normal line of the middle surface of the stiffened plate in the yz plane, the rotation angle is positive when the middle surface is turned away from the z-axis from the y-axis, otherwise, the rotation angle is negative; w (x, y) represents a displacement function of the stiffened plate in the z direction;
the displacement of the reinforcing rib in the x, y and z directions is expressed as by adopting the iron Morgankoch beam theory
u s (x,y,z s )＝u 0s (x,y)+z s φ xs (x,y) (4)
v s (x,y,z s )＝v 0s (x,y)+z s φ ys (x,y) (5)
w s (x,y,z s )＝w(x,y) (6)
In the formulae (4) - (6), u s (x,y,z s )，v s (x,y,z s ) And w s (x,y,z s ) Respectively represent any point (x, y, z) on the reinforcing rib s ) Displacement in three directions, x, y and z; u (u) 0s (x, y) and v 0s (x, y) represents the displacement of the facing in the rib in the x and y directions, respectively; phi (phi) xs (x, y) represents the angle of rotation of the normal of the middle surface of the reinforcing rib in the zx plane, and the middle surface is far turned from the x axisThe rotation angle is positive when the rotation angle is away from the positive direction of the z axis, otherwise, the rotation angle is negative; phi (phi) ys (x, y) represents the rotation angle of the normal line of the middle surface of the reinforcing rib in the yz plane, the rotation angle is positive when the middle surface is turned from the y axis to the positive direction away from the z axis, and is negative otherwise; w (x, y) represents a displacement function of the reinforcing rib in the z direction; the displacement of the reinforcing rib in the z direction is the same as the displacement of the corresponding position on the reinforcing plate, so that the same displacement function is adopted in the formula (3) and the formula (6);
at the interface of the plate and the rib, the displacement of the plate and the rib in the x direction and the y direction is equal, namely the displacement coordination condition
u p (x,y,t/2)＝u s (x,y,-h/2) (7)
v p (x,y,t/2)＝v s (x,y,-h/2) (8)
In the formulas (7) and (8), t represents the plate thickness of the stiffening plate; h represents the height of the reinforcing rib; u (u) p (x, y, t/2) and v p (x, y, t/2) represents the x and y displacement of the stiffened panel at any point on the panel interface, respectively; u (u) s (x, y, -h/2) and v s (x, y, -h/2) represents the x and y displacement of the stiffener at any point on the stiffener interface, respectively;
the u can be used by the formula (7) and the formula (8) 0s (x,y)，v 0s (x, y) represents u 0p (x, y) and v 0p (x, y), assumed to be phi from the flat section xp (x,y)＝φ xs (x,y)，φ yp (x,y)＝φ ys (x, y), and then substituting the formula (1) and the formula (2) to obtain
u p (x,y,z p )＝u 0s (x,y)-(h+t)φ xs (x,y)/2+z p φ xs (x,y) (9)
v p (x,y,z p )＝v 0s (x,y)-(h+t)φ ys (x,y)/2+z p φ ys (x,y) (10)
The strain of the stiffened panel is expressed as
In the formulae (11) - (15), ε xp And epsilon yp Respectively representing positive strains of the stiffening plate in the x direction and the y direction; gamma ray xyp 、γ yzp And gamma zxp Respectively representing the shear strain of the stiffened plate in the xy direction, the yz direction and the zx direction;
the strain of the reinforcing bars is expressed as
Epsilon in the formulas (16) and (17) xs Indicating positive x-direction strain of the reinforcing rib; gamma ray zxs Represents the shear strain in the zx direction of the reinforcing rib;
s2, deducing a bending balance equation of the stiffening plate: the minimum potential energy principle is applied, and a bending balance equation of the reinforced plate is listed through a variation method;
s3, a double Fourier series expression is provided: setting an unknown function as a double Fourier series, setting the Fourier series to meet boundary conditions, and only reserving a constant number to be determined through a bending balance equation to prepare a bedding for solving the bending balance equation by the double Fourier series method;
s4, calculating bending deformation and stress of the stiffening plate: substituting an unknown function expression containing a double Fourier series representation of the undetermined constant into a bending balance equation of the reinforced plate, and determining the undetermined constant by adopting a Galerkin (Galerkin) algorithm to solve the bending deformation of the reinforced plate; substituting the solved and determined deformation expression into an elastic mechanical geometric equation and a physical equation to calculate the bending stress of the reinforced plate.
2. The semi-analytical method for calculating bending deformation and stress of a stiffened plate according to claim 1, wherein the stress expression from the strain expression (11) -expression (17) can be expressed as Hooke's law, and the stress of the stiffened plate is expressed as
In the formulae (18) - (22), σ xp Sum sigma yp Respectively representing positive x-direction stress and y-direction stress in the stiffened plate; τ xyp ，τ yzp And τ zxp Respectively representing the shearing stress of the stiffened plate in the xy direction, the yz direction and the zx direction; e, mu, kappa represent respectively the Young's bullets of the stiffening plates or the stiffening rib materialsModulus of nature, poisson's ratio and shear correction coefficient, typically κ=5/6;
the stress of the reinforcing bars is expressed as
σ xs ＝Eε xs (23)
In the formula (23) and the formula (24), sigma xs Indicating the positive stress of the reinforcing rib in the x direction; τ zxs The shear stress in the zx direction of the rib is shown.
3. The semi-analytical method for calculating bending deformation and stress of a stiffened plate according to claim 2, wherein the strain energy of the stiffened plate is calculated according to the formulas (11) - (24), and the potential energy of the stiffened plate system is the sum of the strain energy of the stiffened plate and the strain energy of the stiffened plate minus the work done by the external force, namely
In formula (25), pi represents the potential energy of the stiffened plate system, H () is a Helviered function, y i Is the y coordinate, t of the thickness center of the reinforcing rib s The equivalent thickness of the reinforcing ribs is represented, and N represents the number of the reinforcing ribs; v (V) 1 ，V 2 The volume domain of the space where the stiffening plate is positioned and the volume domain of the space where the stiffening rib is positioned are respectively; q (x, y) represents the external load function.
4. A semi-analytical method for calculating bending deformation and stress of a stiffened plate according to claim 3, wherein in a stable system, the potential energy n of the stiffened plate should be kept to a minimum, and in formula (25) in order to minimize the potential energy n of the stiffened plate, the principle of variation is adopted, that is, in step S2, the expression of variation of the principle of minimum potential energy is
δΠ＝0 (26)
Delta in the formula (26) represents a variation;
the Euler-Poisson equation, namely the bending equilibrium differential equation (27) -formula (31) of the reinforcing plate is solved by using a variational method,
5. the method for calculating bending deformation and stress half-resolution of a stiffened plate according to claim 4, wherein the external load q (x, y) borne by the stiffened plate in the formula (31) can be equivalently represented as a concentrated load or an accumulation of concentrated loads, a plurality of concentrated loads, uniform loads and linear loads are respectively represented as follows,
for k concentrated loads, the load is expressed as
Wherein p is i To act on point (x i ,y i ) Concentrated load of location, delta () is a Dirac (Dirac) pulse function;
for uniformly distributed loads, the load is expressed as
Wherein p is uniformly distributed load intensity, and D is the area acted by the load;
for a linear load that varies linearly along the x-direction, the load is expressed as
Wherein p is 1 Is x 1 Load intensity at position, p 2 Is x 2 Load intensity at the location.
6. The method for calculating bending deformation and stress half-resolution of a stiffened plate according to claim 4, wherein the boundary condition of solving equation (27) -formula (31) is a stiffened plate four-side simply supported or four-side firmly supported, and if the stiffened plate is in the four-side simply supported boundary condition, the boundary condition is expressed as
At x=0 or a
At y=0 or b
If the reinforcing plate is in the four-side clamped boundary condition, the boundary condition is expressed as
At x=0 or a
At y=0 or b
7. The method according to claim 6, wherein when solving the solution problem consisting of equations (27) - (31) and boundary condition (32) and (33) or boundary condition (34) and (35) by using the double fourier series method, in step S3, the double fourier series solution is assumed to be
In step S3, for the clamped boundary condition, the double Fourier series solution is assumed to be
In the formulas (36) - (45), a represents the length of the stiffening plate along the x direction, and b represents the length of the stiffening plate along the y direction; m and n represent the number of terms of the Fourier series, respectively; u (u) mn ，v mn ，φ xmn ，φ ymn And w mn For the constant to be determined, it is impossible to calculate an infinite number of terms in actual calculation, and only the number of convergence terms is calculated.
8. The half-resolution method for calculating bending deformation and stress of a stiffened plate according to claim 7, wherein the double fourier series expression (36) -expression (40) of the unknown function satisfies the boundary condition expression (32) and expression (33), and the expression (41) -expression (45) satisfies the boundary condition expression (34) and expression (35); in step S4, formula (36) -formula (40), or formula (41) -formula (45) is substituted into formula (27) -formula (31), and the above-mentioned undetermined constant is determined by using a galkin algorithm; after the determination of the undetermined constant, the deformation of the reinforcing plate is determined, and the bending stress of the reinforcing plate can be calculated by substituting the deformation into the formula (18) -formula (24).
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