A Simple Method for Solving the Stress and Strain of an Infinite Plate with an Opening of Arbitrary Shape

Abstract

The present invention provides a simple method for solving the stress and strain of an infinite flat plate with an opening of any shape. By using the conformal mapping method, the unit circular opening on the mapping plane is mapped into a non-circular opening on the physical plane; a series form of the complex variable stress function is ingeniously set on the mapping plane, and directly substituting it into the free boundary conditions, and using the polynomial identity, the coefficients of the series expressions of the two complex variable stress functions can be determined by solving algebraic equations. The present invention avoids complex Cauchy integral operations, has simple operations, and clear mathematical and physical concepts.

Description

Technical Field

[0001] This application relates to the technical field of structural calculations. Specifically, it relates to a simple method for solving the stress and strain of an infinite flat plate with an opening of any shape. Background Art

[0002] In projects such as bridges, buildings, and ships, it is often necessary to set openings of various shapes on flat plates, such as inspection holes on bridges, manholes and lightening holes on ships. After opening the holes, stress concentration will inevitably be caused around the hole openings, making the stress near the hole openings rise several times that in the state without holes. People are very concerned about the stress distribution near the hole openings and need to seek effective methods for calculating the hole opening stress. Most flat plates in engineering can be approximately simplified to the plane problem of an infinite flat plate and calculated using the well-known Muskhelishvili N.I. method.

[0003] In the plane problem of a flat plate, there must be a stress function that satisfies the biharmonic condition. Solving for the stress is to determine the expression of this biharmonic function. Muskhelishvili N.I. represents the biharmonic function in complex form and transforms the problem into solving the expressions of two analytic functions. According to the Laurent expansion theorem, these two analytic functions can be represented in the form of infinite series. In this way, the problem is transformed into the problem of determining the coefficients of the infinite series.

[0004] For circular openings, the process of calculating stress using the Muskhelishvili N.I. method is relatively simple. For other shaped openings, such as elliptical openings, rounded rectangular openings, waist-shaped openings, and polygonal openings, the conformal mapping method also needs to be applied. The non-circular opening is transformed into a unit circular opening through a mapping function, and then the Muskhelishvili N.I. method is applied. The most crucial step is to list an equation with the Laurent series coefficients as unknowns by applying the boundary conditions.

[0005] After introducing the mapping function, in the process of listing equations using the Muskhelishvili N.I. method, one of the Cauchy integral calculations is relatively difficult. The denominator has high-order terms, making it more difficult to calculate the Cauchy integral using the residue theorem. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to provide a simple method for solving the stress and strain of an infinite flat plate with an opening of any shape in view of the above problems, so as to provide a pre-judgment for the safe use of structural members.

[0007] The embodiments of this application are implemented as follows:

[0008] An embodiment of the present application provides a simple method for solving the stress and strain of an infinite plate with an opening of any shape, which is characterized by including the following steps:

[0009] Step a, if the opening shape is circular, then solving the plane problem is reduced to solving two analytical complex variable stress functions and ψ1(z), and directly using the Muskhelishvili N.I. method to solve;

[0010] Step b, if the opening shape is non-circular, then find an appropriate mapping function z = ω(ξ), which can exactly map the unit circle to the opening edge, and map the infinite region outside the unit circle in the mapping plane to the infinite region outside the opening edge in the physical plane. The problem is transformed into solving two analytical complex variable stress functions and ψ2(ξ);

[0011] Step c, determine the series expression of the stress function, apply the free boundary conditions of the hole edge to list equations, and solve the series coefficients;

[0012] Step d, substitute the series coefficients into the series expression of the stress function, and calculate the stress by using the relationship expression between the stress components and the stress function;

[0013] Step e, according to the calculated stress, and then apply the constitutive equation to calculate the strain of the plate.

[0014] In some alternative embodiments, the complex variable stress functions and ψ1(z) described in step a have the following form.

[0015]

[0016]

[0017] Where z is a complex variable in the physical plane, p -n ,q -n are complex series coefficients; B, B′ and C′ are real numbers related to the stress state of the infinite plate at infinity. If the two principal stresses of the plate at infinity are σ1 and σ2 respectively, and the angle between the first principal direction and the x-axis is α, then

[0018] In some alternative embodiments, the mapping function z = ω(ξ) described in step b has the following form:

[0019]

[0020] Where ξ is a complex variable in the mapping plane, ξ = λ + iη; i is the imaginary unit, R, m1, m2, m3... are determined by the boundary shape of the holes on the physical plane.

[0021] In some alternative embodiments, the series forms of the two stress functions described in step b are as follows:

[0022]

[0023]

[0024] where a -n and b -n are series coefficients and are complex numbers; and ω′(ξ) are the conjugate complex number of the mapping function ω(ξ) and the derivative with respect to the variable ξ, respectively; is the derivative of the stress function with respect to the variable ξ.

[0025] It can also be expressed in the following form,

[0026]

[0027] where β n and p -n are both complex numbers, related not only to B, R, m1, m2, m3... but also to the series coefficients a -1 , a -2 , a -3 ...

[0028] In some alternative embodiments, the series coefficient α n of the ψ2(ξ) is equal to β n , then the second stress function can be expressed as

[0029]

[0030] In some alternative embodiments, the equations are listed by applying the free boundary conditions of the hole edge as follows:

[0031]

[0032] Taking the coefficients of each power term of ξ to be zero and listing the system of simultaneous equations, then a -n and b -n can be solved. Substituting a -n and b -n into the series expressions of the two stress functions and ψ2(ξ) described in step b, the stress function expressions can be completely determined.

[0033] In some alternative embodiments, the stress calculation process is as follows:

[0034] In the rectangular coordinate system of the physical plane, there are the following expressions for the stress component combination and the stress function:

[0035]

[0036]

[0037] According to the derivative method of composite functions and by defining new functions respectively, then

[0038]

[0039]

[0040] Substitute Equation (3) and Equation (4) into Equation (1) and Equation (2) to derive the expressions of Equation (1) and Equation (2) in the mapping plane:

[0041] σ y +σ x = 4Re[Φ(ξ)] Equation (5)

[0042]

[0043] Derive the expressions of each stress component:

[0044]

[0045]

[0046]

[0047] Re[·] represents taking the real part operation of a complex number, and Im[·] represents taking the imaginary part operation of a complex number;

[0048] In the polar coordinate system of the physical plane, the relational expressions of the stress components and the stress function are:

[0049]

[0050]

[0051] θ is the argument corresponding to the calculation point z in the physical plane, that is, θ = arg(z), z = re iθ ;

[0052] According to the properties of conformal mapping

[0053]

[0054] where ρ is the modulus of the preimage point ξ of the calculation point z in the physical plane, that is, ξ = ρe iβ ;

[0055] After substituting Equation Three, Equation Four, and Equation Twelve into Equation Ten and Equation Eleven, the expressions of Equation Ten and Equation Eleven in the mapping plane can be derived.

[0056] σ θ +σ r =4Re[Φ(ξ)] Equation Thirteen

[0057]

[0058] According to Equation Thirteen and Equation Fourteen, the expressions of the stress components in the polar coordinate system of the physical plane can be derived:

[0059]

[0060]

[0061]

[0062] In the plane problem, the calculation method of the Mises stress is as follows:

[0063]

[0064] Substituting Equation Seven, Equation Eight, and Equation Nine into Equation Eighteen can calculate the Mises stress.

[0065] In some alternative embodiments, the strain calculation process is as follows:

[0066] In the rectangular coordinate system, the strain components are expressed as

[0067]

[0068]

[0069]

[0070] Substituting Equation Seven, Equation Eight, and Equation Nine into Equation Nineteen, Equation Twenty, and Equation Twenty - One to calculate the strain component calculation formulas in the rectangular coordinate system of the physical plane:

[0071]

[0072]

[0073]

[0074] In the polar coordinate system, the strain components are expressed as:

[0075]

[0076]

[0077]

[0078] Substitute Formula XV, Formula XVI, and Formula XVII into Formula XXV, Formula XXVI, and Formula XXVII:

[0079]

[0080]

[0081]

[0082] In Formulas XIX - XXX, E and μ are the elastic modulus and Poisson's ratio of the plate respectively, and the strain under plane stress state is calculated.

[0083] If the infinitely - perforated plate is in a plane - strain state, then in Formulas XIX - XXX, use to replace E therein, and use to replace μ therein.

[0084] The beneficial effects of this application are as follows: A simple method for solving the stress and strain of an infinitely - perforated plate with an arbitrary shape provided by this application, by skillfully setting the form of the complex - variable stress function, directly substituting the free - boundary conditions, avoiding complex Cauchy integral operations, and using polynomial identity, the coefficients of the series expressions of the two complex - variable stress functions can be determined by solving algebraic equations. The operation is simple and the mathematical - physical concepts are clear. BRIEF DESCRIPTION OF THE DRAWINGS

[0085] In order to more clearly illustrate the technical solutions of the embodiments of this application, the following will briefly introduce the drawings required in the embodiments. It should be understood that the following drawings only show some embodiments of this application, and thus should not be regarded as limiting the scope. For those of ordinary skill in the art, other related drawings can also be obtained based on these drawings without creative efforts.

[0086] Figure 1 It is a schematic diagram of the transformation from the mapping plane to the physical plane in this application; DETAILED DESCRIPTION OF THE EMBODIMENTS

[0087] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the following will clearly and completely describe the technical solutions in the embodiments of this application with reference to the drawings in the embodiments of this application. Obviously, the described embodiments are some, but not all, of the embodiments of this application.

[0088] Accordingly, the following detailed description of the embodiments of the present application provided in the accompanying drawings is not intended to limit the scope of the claimed present application, but merely represents selected embodiments of the present application. All other embodiments obtained by those of ordinary skill in the art based on the embodiments in the present application without creative efforts fall within the scope of protection of the present application.

[0089] The features and performance of the present application will be further described in detail below in conjunction with embodiments.

[0090] The present invention provides a simple method for solving the stress and strain of an infinite flat plate with an opening of any shape. Specifically, after an opening of any shape is made on an infinite flat plate, the stress and strain can be solved without relying on the Cauchy integral.

[0091] In the plane problem of elasticity mechanics, there exists a stress function φ(x, y). The stress function φ(x, y) and the stresses σ x , σ y and τ xy have the following equations:

[0092]

[0093] Also, since x, y and the complex variable z and have the following expressions:

[0094] z = x + iy, then the stress function can be transformed into a function of z and , that is,

[0095]

[0096] where i is the imaginary unit,

[0097] satisfies the biharmonic equation,

[0098]

[0099] According to the composite function derivative rule, the biharmonic equation expression can be derived:

[0100]

[0101] From equations (3) and (4), it can be obtained that

[0102]

[0103] Integrating both sides of equation (5), we get

[0104]

[0105] Therefore, solving the plane problem with complex variable functions is reduced to solving two analytic functions and θ1(z). Since θ1(z) is less frequently used while its derivative θ′1(z) is more frequently used, it is renamed ψ1(z), that is,

[0106] ψ1(z) = θ′1(z) (7)

[0107] For an infinite flat plate with a circular opening under the uniform action of in-plane forces at infinity, the principal stresses are σ1 and σ2 respectively, and the angle between the first principal direction and the x-axis is α. There is no external force acting on the edge of the circular hole, and it is a free boundary. Then the two analytic functions for this type of plane problem can be set as:

[0108]

[0109]

[0110] p -n ,q -n are the coefficients of the complex series; B, B′ and C′ are real numbers related to the stress state of the infinite flat plate at infinity, and

[0111] As Figure 1 shown, in engineering, in addition to circular openings, there are also other shaped openings, such as elliptical openings, rounded rectangular openings, kidney-shaped openings and polygonal openings, etc. The complex stress functions of these opening shapes cannot directly adopt the forms of equations (8) and (9). Muskhelishvili N.I. proposed to use the method of conformal mapping: map the unit circle C in the mapping plane into the edges γ of various openings in the physical plane through the mapping function. Convert all irregular opening boundaries into the unit circular boundary in the mapping plane, and then indirectly apply the expressions of equations (8) and (9).

[0112] The mapping function can be set as

[0113]

[0114] where ξ is the complex variable in the mapping plane, ξ = λ + iη; R, m1, m2, m3... are determined by the boundary shape of the hole in the physical plane.

[0115] Substitute equation (10) into equations (8) and (9), and note that |m1| + |m2| + |m3| +... < 1, then:

[0116]

[0117]

[0118] In the mapping plane, the unit circle has free boundary conditions, i.e.:

[0119]

[0120] Muskhelishvili N.I. proposed substituting equations (10)-(12) into equation (13), and then performing Cauchy integration.

[0121]

[0122] where C is the unit circle in the mapping plane in the clockwise direction, |ξ|>1.

[0123] In fact, calculating the first and third terms of the Cauchy integral in equation (14) is relatively simple, and the integral result can be calculated using the residue theorem. During the calculation of the second term of the Cauchy integral, a high-degree polynomial appears in the denominator. The more terms are taken in equation (13), the higher the degree of the denominator, which increases the difficulty of determining the position of the isolated singularity. Finding the poles is only the first step. It is also necessary to determine whether the singularities are inside or outside the unit circle. All of these are very complex, and a simple method needs to be sought to avoid the Cauchy integral operation.

[0124] After mapping an opening of any shape into an opening of a unit circle, the stress boundary condition with no external force acting on the hole edge is transformed into equation (13). Starting from equation (13) of the boundary condition, the expression form of the third term ψ2(t) is cleverly set so that a part of it cancels out the second term in equation (13). In this way, the boundary condition is transformed into a polynomial summation operation, and its sum is zero. After combining like terms of various power functions of t, the coefficients of the polynomial must be zero, and algebraic equations can be listed to determine the coefficients of the Laurent series of the two stress functions.

[0125] For simplicity of operation, the second stress function is set as:

[0126]

[0127] where a -n and b -n are series coefficients and are complex numbers; and ω′(ξ) are the conjugate complex number of the mapping function ω(ξ) and the derivative with respect to the variable ξ, respectively; is the derivative of the stress function with respect to the variable ξ;

[0128] is used to cancel the positive power terms of ξ in because the positive power terms of ξ cannot be expressed in

[0129] can be expressed as ​

[0130]

[0131] Among them, β n and p -n are both complex numbers, which are not only related to B, R, m1, m2, m3... but also related to the series coefficients a -1 、a -2 、a -3 ...

[0132] Comparing Equation (16) with Equations (12) and (15), then α n = β n , then the second stress function can be expressed as:

[0133]

[0134] Substitute Equations (11) and (17) into the boundary condition Equation (13), and at the same time note that on the circumference C there is Then

[0135]

[0136] The coefficients of each term of the power series should be zero, and a set of simultaneous equations can be listed to solve the coefficients of each term of the series in Equations (11) and (15), and then two analytical stress functions can be determined.

[0137] The stress calculation process is as follows:

[0138] In the rectangular coordinate system of the physical plane, there are the following expressions for the stress component combination and the stress function,

[0139]

[0140]

[0141] According to the composite function derivative method and defining new functions respectively, then:

[0142]

[0143]

[0144] Substitute Formulas (21) and (22) into Formulas (19) and (20) to derive the expressions of Formulas (19) and (20) in the mapping plane:

[0145] σ y + σ x = 4Re[Φ(ξ)] (23)

[0146]

[0147] Derive the expressions for each stress component:

[0148]

[0149]

[0150]

[0151] Re[·] represents taking the real part of a complex number, and Im[·] represents taking the imaginary part of a complex number;

[0152] In the polar coordinate system of the physical plane, the relationship expressions between stress components and stress functions are:

[0153]

[0154]

[0155] θ is the argument corresponding to the calculation point z in the physical plane, i.e., θ = arg(z), z = re iθ ;

[0156] According to the properties of conformal mapping,

[0157]

[0158] where ρ is the modulus of the preimage point ξ of the calculation point z in the physical plane, i.e., ξ = ρe iβ ;

[0159] After substituting formulas (21), (22), and (30) into formulas (28) and (29), the expressions of formulas (28) and (29) in the mapping plane can be derived,

[0160] σ θ +σ r = 4Re[Φ(ξ)] (31)

[0161]

[0162] According to formulas (31) and (32), the expressions of stress components in the polar coordinate system of the physical plane can be derived:

[0163]

[0164]

[0165]

[0166] In plane problems, the calculation method of Mises stress is as follows:

[0167]

[0168] Substituting equations (25), (26), and (27) into equation (36) enables the calculation of the Mises stress.

[0169] In some alternative embodiments, the strain calculation process is as follows:

[0170] In a rectangular coordinate system, the strain components are expressed as

[0171]

[0172]

[0173]

[0174] Substitute equations (25), (26), and (27) into equations (37), (38), and (39) to calculate the strain component calculation formulas in the physical plane rectangular coordinate system:

[0175]

[0176]

[0177]

[0178] In a polar coordinate system, the strain components are expressed as:

[0179]

[0180]

[0181]

[0182] Substitute equations (33), (34), and (35) into equations (43), (44), and (45):

[0183]

[0184]

[0185]

[0186] In equations (37)-(48), E and μ are the elastic modulus and Poisson's ratio of the flat plate, respectively, and the strains are calculated under plane stress conditions.

[0187] If the infinite flat plate with an opening is in a plane strain state, then in equations (37)-(48), use to replace E therein, and use to replace μ therein.

[0188] Example 1

[0189] Taking the first three negative power terms in Equation (10) as an example, the derivation process of the simple method will be described in detail below. If the number of negative power terms taken is less than 3, only the coefficients of the higher-order negative power terms need to be set to zero; if the number of negative power terms taken is more than 3, a new derivation is required, but the method is similar, only the process is slightly more cumbersome.

[0190] 1. Laurent expansion of the function at the infinite point

[0191]

[0192]

[0193] On the unit circle C in the mapping plane, Then

[0194]

[0195] Let

[0196]

[0197] Expand u(ξ) into a Laurent series at the infinite point,

[0198]

[0199] 2. Skillfully set the expressions of two stress functions

[0200] The first stress function of the complex variable is also set in the following form:

[0201]

[0202]

[0203] The derivative function of is:

[0204]

[0205] And from Equations (51)-(53) and Equation (55), it can be derived that

[0206]

[0207] Therefore, in order for Equation (55) to have the form of Equation (12), cancel the in the positive power series of to be:

[0208]

[0209] After substituting Equation (58) into Equation (55), the second stress function of the complex variable can be obtained:

[0210]

[0211] 3. Application of the stress boundary conditions at the hole edge

[0212] Substitute Equations (54) and (59) into the free boundary condition Equation (13) at the hole edge, then Equation (13) can be expressed as:

[0213]

[0214] Take the coefficients of each power term of ξ in Equation (60) to be zero, list the simultaneous equations, and then a -n and b -n .

[0215] Solve these simultaneous equations to obtain

[0216] a -3 = -BRm3, a -2 = -BRm2, b -1 = -BR (61)

[0217]

[0218] Except for the coefficients expressed by Equations (61) and (62), the remaining series coefficients are all zero.

[0219] Substitute Equations (61) and (62) into Equations (54) and (59) to get

[0220]

[0221]

[0222] 4. Derivation of stress components

[0223] After obtaining the expressions of the two stress functions according to Equations (63) and (64), the calculation process of the stress components is directly substituted into Equations (19)-(36).

[0224] 5. Derivation of strain components

[0225] According to the stress components solved in the above step 4, substitute them into Equations (37)-(48), and the expression of the strain components can be derived.

[0226] In the above solution process, it is considered that the infinite plate with an opening is in a plane stress state. If the plate is in a plane strain state, then use to replace E in Equations (37)-(48), and use to replace μ in (70)-(72).

Claims

1. A simple method for solving the stress and strain of an infinite flat plate with an opening of any shape, characterized in that The steps include: Step a, if the opening shape is circular, then solving the plane problem is reduced to solving two analytical complex variable stress functions and , and directly applying the Muskhelishvili N.I. method to solve; the complex variable stress functions and have the following series forms: where z is a complex variable in the physical plane, z = x + iy, ; p -n , q -n are coefficients of complex series; B, and are real numbers related to the stress state of an infinite plate at infinity. If the two principal stresses of the plate at infinity are , , and the angle between the first principal direction and the x-axis is α, then , ; Step b: If the opening shape is non-circular, find an appropriate mapping function that can exactly map the unit circle to the opening edge and map the infinite region outside the unit circle in the mapping plane to the infinite region outside the opening edge in the physical plane. The problem is then transformed into solving two analytic complex variable stress functions and ; Step c, determining the series expression of the stress function, applying the free boundary conditions of the hole edge to formulate equations, and solving the series coefficients; Step d, substituting the series coefficient into the stress function series expression, and calculating the stress using the relationship expression between the stress component and the stress function; Step e: Based on the calculated stress, the constitutive equation is applied to calculate the strain of the plate.

2. A simple method for solving the stress and strain of an infinite flat plate with an opening of any shape according to claim 1, characterized in that, The mapping function described in step b has the following form: where ξ is the complex variable of the mapping plane, ; i is the imaginary unit, ; R, m1, m2, m3... are determined by the boundary shape of the hole on the physical plane.

3. A simple method for solving the stress and strain of an infinite flat plate with an opening of any shape according to claim 2, characterized in that, The series form of the two stress functions in step b is as follows: where a- n and b- n are series coefficients and are complex numbers; and are the conjugate complex number of the mapping function and the derivative with respect to the variable ξ, respectively; is the derivative of the stress function with respect to the variable ξ. It can also be expressed in the following form, where β n and p -n are both complex numbers, related not only to B, R, m1, m2, m3... but also to the series coefficients a -1 、a -2 、a -3 ...

4. A simple method for solving the stress and strain of an infinite flat plate with an opening of any shape according to claim 3, characterized in that, The series coefficient α n = β n , then the second stress function can be expressed as: 。 5. A simple method for solving the stress and strain of an infinite flat plate with an opening of any shape, according to claim 3 or 4, characterized in that The free boundary conditions at the edge of the hole can be expressed as follows: By setting the coefficients of each power term of ξ to zero and listing the simultaneous equations, a can be solved for. -n and b -n . Substituting a -n and b -n into the stress function series expression described in claim 4, the stress function expression can be completely determined.

6. A simple method for solving the stress and strain of an infinite flat plate with an opening of any shape according to claim 3, characterized in that, The stress calculation process is as follows: In the rectangular coordinate system of the physical plane, the stress component combination and stress function have the following expressions: Formula 1 Formula II According to the composite function derivation method, and define new functions respectively, then: , Formula III Formula Four Substitute Formula 3 and Formula 4 into Formula 1 and Formula 2 to derive the expressions of Formula 1 and Formula 2 on the mapping plane: Formula Five Formula VI Formula 5 and Formula 6 are combined to derive the expression of each stress component: Formula VII Formula VIII Formula Nine Re[·] means taking the real part of a complex number, and Im[·] means taking the imaginary part of a complex number; In the polar coordinate system of the physical plane, the relationship between the stress component and the stress function is expressed as: Formula Ten Formula XI θ Calculate the point for the physical plane z The corresponding argument, that is θ =arg( z ) z = re iθ ; According to the properties of conformal mapping, Formula XII Among them, ρ is the calculation point on the physical plane z of the preimage point ξ modulus, that is ξ = ρ e iβ ; After substituting Formula 3, Formula 4 and Formula 12 into Formula 10 and Formula 11, the expressions of Formula 10 and Formula 11 on the mapping plane can be derived: Formula XIII Formula Fourteen According to Formula 13 and Formula 14, the stress component expression in the physical plane polar coordinate system can be derived: Formula XV Formula XVI Formula XVII In plane problems, the Mises stress calculation method is as follows: Formula XVIII Substituting Formula 7, Formula 8 and Formula 9 into Formula 18, the Mises stress can be calculated.

7. A simple method for solving the stress and strain of an infinite flat plate with an opening of any shape according to claim 6, characterized in that, The strain calculation process is as follows: In the rectangular coordinate system, the strain components are expressed as, Formula XIX Formula XX Formula XXI Substituting Formula 7, Formula 8 and Formula 9 into Formula 19, Formula 20 and Formula 21, the strain component calculation formula in the physical plane rectangular coordinate system is derived: Formula XXII Formula XXIII Formula XXIV In polar coordinate system, the strain components are expressed as: Formula XXV Formula XXVI Formula XXVII Substitute Formula 15, Formula 16, and Formula 17 into Formula 25, Formula 26, and Formula 27: Formula XXVIII Formula 29 Formula Thirty In Formulas 19 to 30, E and μ are the elastic modulus and Poisson's ratio of the flat plate respectively, and the strain under plane stress state is calculated; If the infinitely perforated flat plate is in a plane strain state, then in Formulas 19 - 30, replace with E in them, and replace with μ .

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