A method for determining the effective length of a beam with two parameters and infinite foundation under moving loads

Abstract

The present invention discloses a method for determining the effective length of a dual-parameter infinite foundation beam under a moving load, which relates to the field of geotechnical engineering technology, and includes the following steps: step one, solving the displacement response of an infinite foundation beam model using a hypothetical exponential solution; step two, deriving a theoretical formula for predicting the effective length using the mathematical characteristics of an exponential function; step three, verifying the feasibility of the infinite beam displacement response theory; and step four, comparing with the existing calculation formula for calculating the effective length of an infinite beam, and analyzing the influence of different moving speeds on the effective length of the infinite beam. The present invention adopts a method for determining the effective length of a dual-parameter infinite foundation beam under a moving load using the above structure. The displacement response of the infinite foundation beam calculated by the hypothetical exponential solution is basically consistent with the result obtained by the Fourier transform and residue theorem method. The displacement response of the infinite foundation beam obtained by the hypothetical exponential solution eliminates the influence of imaginary numbers, making it superior to the residue theorem method in deriving the effective length.

Description

Technical Field

[0001] The invention relates to the technical field of geotechnical engineering, and in particular to a method for determining the effective length of a dual-parameter infinite foundation beam under a moving load. Background Art

[0002] The effective length is the range of action of the train wheel load when it moves on the infinite track surface. It aims to simulate the dynamic response of the infinite beam by using a finite beam with an effective length. On the one hand, the length of railway tracks is usually hundreds or thousands of kilometers, and it is not an effective method to use an infinite beam model to simulate the track response; on the other hand, the moving load only affects a very limited area of ​​the infinite track. Therefore, it is time-consuming and laborious to use an infinite beam to simulate the response of a part of the area considering the entire railway line. In the analysis based on the finite element method (FEM), when the change of foundation properties needs to be considered, engineers use "finite" elements more efficiently and quickly than "infinite" elements. This method has the advantages of saving time, high efficiency, and reducing computational costs. However, existing studies have shown that this method is only applicable to elastic foundation beam models, and cannot be accurately used for complex foundation models such as viscoelastic foundation beams and dual-parameter foundation beams. In view of the above shortcomings, a method for calculating the effective length of infinite beams based on axially loaded dual-parameter foundation beam models is proposed. Summary of the invention

[0003] The purpose of the present invention is to provide a method for determining the effective length of a dual-parameter infinite foundation beam under a moving load. The displacement response of the infinite foundation beam calculated by the assumed exponential solution is basically consistent with the result obtained by the Fourier transform and residue theorem method, which shows the correctness of the method and can be accurately used for complex foundation models such as viscoelastic foundation beams and dual-parameter foundation beams.

[0004] To achieve the above object, the present invention provides a method for determining the effective length of a dual-parameter infinite foundation beam under a moving load, comprising the following steps:

[0005] Step 1: Use the assumed exponential solution method to solve the displacement response of the infinite foundation beam model;

[0006] Step 2: Using the mathematical characteristics of the exponential function to derive the theoretical formula for predicting the effective length;

[0007] Step 3: Verify the feasibility of infinite beam displacement response theory;

[0008] Step 4: Compare with the existing calculation formula for the effective length of the infinite beam and analyze the influence of different moving speeds on the effective length of the infinite beam.

[0009] Preferably, step one includes the following steps:

[0010] S1 establishes the control equation of the beam as:

[0011]

[0012] Where: E is Young's modulus, I is the area moment of inertia, m is the mass per unit length, w1 = w1(x, t) is the lateral displacement of the beam observed in the stationary coordinate system, N is the axial load borne by the beam, c is the foundation damping coefficient, k is the foundation stiffness, G p is the Pasternak modulus, the external load P moves at a constant velocity v, and δ(·) is the Dirac function.

[0013] S2 solves the closed-form solution of the displacement response of the beam with underdamped and overdamped foundation;

[0014] S3 will derive the effective length of the infinite beam based on the dynamic response of the foundation under underdamping and overdamping.

[0015] Preferably, under underdamping, the effective length of the infinite beam on both sides of the load is calculated as follows:

[0016]

[0017] Where: L + and L - They represent the effective lengths of the front wave (beam response before moving load) and the rear wave (beam response after moving load) of the infinite beam displacement response curve, respectively, and γ is a positive real root;

[0018] The effective length expression under overdamping is:

[0019]

[0020] Where: L - represents the effective length of the infinite beam back wave, meaning.

[0021] Preferably, Fourier transform and residue theorem are used to obtain the beam displacement response for verification.

[0022] Preferably, in step three, the ratio of the deflection at the effective length of the viscoelastic foundation beam to the maximum deflection is verified by moving the load, beam and foundation parameters.

[0023] Preferably, in step four, the elastic foundation and the elasto-viscous foundation are compared.

[0024] Therefore, the present invention adopts a method for determining the effective length of a dual-parameter infinite foundation beam under a moving load using the above-mentioned structure, and provides a method for calculating the effective length of an infinite beam using a dual-parameter foundation beam model. The displacement response of the infinite foundation beam calculated by the assumed exponential solution is basically consistent with the result obtained by the Fourier transform and residue theorem method, verifying its accuracy; the displacement response of the infinite foundation beam obtained by the assumed exponential solution eliminates the influence of imaginary numbers, making it better than the residue theorem method when deriving the effective length; under different foundation stiffness, the change trend of the effective length is almost unaffected by the foundation damping, while increasing the load movement speed and enhancing the foundation stiffness will significantly affect the effective length.

[0025] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 The axial force diagram of the foundation beam of an embodiment of a method for determining the effective length of a dual-parameter infinite foundation beam under a moving load of the present invention;

[0027] Figure 2 is the deflection curve of the viscoelastic foundation beam under underdamping: (a) α = 0.44, β = 0.05 (β cr =1.64); (b)α=1.2,β=0.05(β cr =1.05);

[0028] Figure 3 is the deflection curve of the viscoelastic foundation beam under overdamping: (a) α = 0.44, β = 1.7, (β cr =1.64); (b)α=1.2,β=1.7(β cr =1.05);

[0029] Figure 4 The deflection curve of the dual-parameter foundation beam under axial load under underdamping: (a) α = 0.44, β = 0.05, η P =0.5,η N =0.04(β cr =1.07); (b) α = 1.2, β = 0.05, η P =0.5,η N =0.04(β cr =0.90);

[0030] Figure 5 The deflection curve of the double-parameter foundation beam under axial load under overdamping: (a) α = 0.44, β = 1.1, η P =0.5,η N =0.04(βcr=1.07); (b)α=1.2,β=1.1,η P =0.5,ηN =0.04(β cr =0.90);

[0031] Figure 6 Comparison of existing effective length calculation formulas for different types of foundations: (a) elastic foundation (α = 0.44); (b) viscoelastic foundation (α = 1.2, β = 0.05); (c) axially loaded dual-parameter foundation beam (α = 1.2, β = 0.05, η P =0.5,η N =0.04);

[0032] Figure 7 The deflection curves of the dual-parameter foundation beam under axial load at different speed ratios;

[0033] Figure 8 is the effective length of the rear wave, front wave and total wave at different speed ratios. DETAILED DESCRIPTION

[0034] The technical solution of the present invention is further described below through the accompanying drawings and embodiments.

[0035] Unless otherwise defined, the technical terms or scientific terms used in the present invention should be understood by people with ordinary skills in the field to which the present invention belongs. The words "first", "second" and similar words used in the present invention do not indicate any order, quantity or importance, but are only used to distinguish different components. "Include" or "comprise" and similar words mean that the elements or objects appearing before the word include the elements or objects listed after the word and their equivalents, without excluding other elements or objects. "Connect" or "connected" and similar words are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. "Up", "down", "left", "right" and the like are only used to indicate relative positional relationships. When the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0036] Example

[0037] See also Figure 1-3 The present invention provides a method for determining the effective length of a dual-parameter infinite foundation beam under a moving load, comprising the following steps:

[0038] Step 1: Use the assumed exponential solution method to solve the displacement response of the infinite foundation beam model;

[0039] S1 Figure 1 As shown in the figure, the concentrated load moves along the axially loaded double-parameter foundation Euler-Bernoulli beam, and the control equation of the beam is established as:

[0040]

[0041] Where: E is Young's modulus, I is the area moment of inertia, m is the mass per unit length, w1 = w1(x, t) is the lateral displacement of the beam observed in the stationary coordinate system, N is the axial load borne by the beam, c is the foundation damping coefficient, k is the foundation stiffness, G p is the Pasternak modulus, the external load P moves at a constant velocity v, and δ(·) is the Dirac function.

[0042] In order to solve the above problem, the moving coordinates s = vt, t = t are used. Therefore, the chain rule of displacement with respect to time derivative can be expressed as:

[0043]

[0044] By substituting equation (2) into equation (1), we can obtain the fourth-order constant coefficient differential equation in the moving coordinate system:

[0045]

[0046] For steady-state response, all partial derivatives involving time t in equation (4.3) are not considered. In addition, in order to solve the non-homogeneous differential equation (4.3), its homogeneous solution should be found first. Therefore, the homogeneous equation of equation (4.3) can be expressed as:

[0047]

[0048] Since the time-varying terms vanish, the steady-state response physically means that the beam deformation is stationary relative to the moving coordinate system. In order to make equation (4) dimensionless, the following dimensionless parameters are introduced:

[0049]

[0050] Where λ represents the static displacement of the elastic foundation beam caused by the dead load P; η N Indicates the axial load ratio, η P is the dimensionless shear modulus of the foundation; α is the velocity ratio, vcr is the critical velocity of the moving load on the Winkler foundation beam (G p =0, N = 0); β is the foundation damping ratio, is the critical damping of a single degree of freedom spring-mass system.

[0051] Based on the above dimensionless parameters, the homogeneous equation (4) can be updated as:

[0052]

[0053] The general solution of equation (6) can be expressed as:

[0054] w(s)=e ξs (7)

[0055] Substituting formula (7) into formula (6) yields the following characteristic equation:

[0056] ξ 4 +4λ 2 [α 2 -η p +η N ]ξ 2 -8αβλ 3 ξ+4λ 4 =0 (8)

[0057] S2 solves the closed-form solution of the displacement response of the beam with underdamped and overdamped foundation;

[0058] Underdamped

[0059] Assume that when (β<β crp ), the foundation is underdamped, and the four complex roots of equation (8) can be assumed to be in the following form:

[0060] ξ1=p+iq1, ξ2=p-iq1, ξ3=-p+iq2, ξ4=-p-iq1 (9)

[0061] Where: p, q1 and q2 are positive real numbers.

[0062] The roots of equation (8) must satisfy the relationship (ξ-ξ1)(ξ-ξ2)(ξ-ξ3)(ξ-ξ4)=0. By substituting this relationship into equation (9), the following fourth-order polynomial equation can be derived:

[0063]

[0064] By combining equations (8) and (10), we can obtain the following set of equations:

[0065]

[0066] Combining equations (11a) and (11b), we can solve for the positive real numbers q1 and q2:

[0067]

[0068] Substituting formula (12) into formula (11c) yields:

[0069] p 6 +2λ 2 (α 2 -η p +η N ) 4 +λ 4 [(α 2 -η p +η N) 2 -1]p 2 -α 2 β 2 λ 6 =0 (13)

[0070] Then, let p = γλ, equation (13) can be simplified as:

[0071] γ 6 +2(α 2 -η p +η N )γ 4 +[(α 2 -η p +η N ) 2 -1]γ 2 -α 2 β 2 =0 (14)

[0072] Where γ is the positive real root of equation (14). Equation (14) is a sixth-order polynomial equation. In order to better find the roots of the solution equation, it can be degenerated into a cubic equation.

[0073] To ensure that the slope and deflection of the beam are zero at infinity, the solution to equation (8) can be expressed as:

[0074] w(s)=w + (s) = e -ps (C1sinq2s+C2cosq2s)s≥0 (15)

[0075] w(s)=w - (s) = e ps (C3sinq1s+C4cosq1s)s≤0 (16)

[0076] Among them C i (i=1,2,3,4) are four position parameters determined by the infinite beam boundary conditions.

[0077] According to the deflection, slope, moment continuity and shear force discontinuity of the moving concentrated load at the starting point (s=0), the following four boundary conditions can be obtained:

[0078]

[0079] The superscript (`) represents the differentiation with respect to s.

[0080] Based on the four boundary conditions provided by equation (17), four unknown constants C can be obtained: i The relationship between (i=1,2,3,4):

[0081]

[0082] Substituting the solutions of q1 and q2 in equation (12) into equation (4.18), and setting p = λγ, the displacement response of the infinite beam under underdamping can be rewritten as:

[0083]

[0084]

[0085] Parameter C i It can be further rewritten as:

[0086]

[0087] in

[0088]

[0089] and

[0090]

[0091] where γ represents the positive real root of equation (14); W st is the coefficient related to static deflection, A D is a coefficient introduced to further simplify the displacement response expression of the dual-parameter foundation beam. and These are the two frequencies of a dual-parameter foundation beam under underdamping.

[0092] Overdamping

[0093] It can be seen from formula (22d) that as the foundation damping ratio β gradually increases, the coefficient A 1- becomes infinite or even an imaginary number, and the displacement response of the beam after the moving load (s<0) expression (20) is no longer applicable, and the foundation enters an over-damped state. However, in this case, the displacement response of the beam before the moving load ((s>0)) is always a real number as the damping ratio increases. Therefore, the displacement response of the beam after the moving load (s<0) needs to be further derived, while the displacement response of the beam before the moving load (s>0) can still be calculated using equation (19). In order to obtain the critical damping ratio, we first consider equation (22d) The denominator is 0:

[0094]

[0095] By combining equations (14) and (24), we can solve the square of the coefficient γ:

[0096]

[0097] The coefficient γ in formula (25) 2Substituting into equation (24), the critical damping coefficient of the axially loaded dual-parameter foundation beam can be obtained:

[0098]

[0099] From the relationship between damping coefficient and damping ratio, that is, The critical damping coefficient of the foundation beam model in this chapter can be obtained:

[0100]

[0101] As the damping ratio continues to increase, when it exceeds the critical damping ratio, that is, β>β cr,p , the coefficient in formula (23) and will become an imaginary number. Using the hyperbolic function relationship isinis = sinhs and cos = coshs, the displacement response of the infinite beam after moving load under overdamping can be rewritten as:

[0102]

[0103] in:

[0104]

[0105] All other parameters are the same as in equation (23).

[0106] S3 will derive the effective length of the infinite beam based on the dynamic response of the foundation under underdamping and overdamping.

[0107] Under critical damping, the dynamic response of the beam will propagate to infinity, indicating that the infinite beam has no effective length in this case. The effective length of the infinite beam will be derived from the dynamic response of the foundation under underdamping and overdamping.

[0108] Underdamped

[0109] Through trigonometric transformation, the infinite beam displacement response theoretical solutions (19) and (20) can be further rewritten as:

[0110]

[0111]

[0112] The coefficient and corners It can be expressed as:

[0113]

[0114]

[0115] It is obvious that there is the following relationship between equations (30) and (31):

[0116]

[0117]

[0118] It can be seen from equations (34) and (35) that the displacement response on both sides of the load gradually decreases with the increase of the distance from the load, and the maximum value is less than and The effective length of the infinite beam is based on the fact that the displacement response is quite small. Therefore, if the distance s from the load is long enough, the displacement response at that location can be ignored. For the displacement response of the beam before the moving load, assuming that the distance between a certain location and the load is s = A / λγ, the corresponding exponential term is e -A When , the displacement response at this position can be ignored. In addition, the displacement response of the beam after the moving load also has a similar trend. In summary, under underdamping, the calculation formula for the effective length of the infinite beam before and after the moving load is as follows:

[0119]

[0120] Where L + and L - They represent the effective lengths of the front wave (beam response before moving load) and the rear wave (beam response after moving load) of the infinite beam displacement response curve, respectively, and γ represents the positive real root of equation (14).

[0121] Overdamping

[0122] Under the condition of overdamping of foundation, the closed-form solution of displacement response of the front wave can still be calculated by equation (19), so the effective length of the front wave can still be calculated by equation (36). However, due to the change of displacement response expression, the effective length of the rear wave needs further discussion. Based on hyperbolic function transformation: coshx = (e x +e -x ) / 2 and sinhx=(e x -e -x ) / 2, the following exponential function can be obtained by transforming formula (28):

[0123]

[0124] From equation (37), it can be concluded that the beam displacement response is mainly composed of the exponential term and The displacement response will gradually decay as the distance s increases. Due to the inequality Always holds true, and the same is true when deriving formula (36). Assume that when the exponential term for e -A , the position response becomes quite small and can be almost ignored.

[0125] Therefore, in this case the effective length can be exponentially To deduce, the effective length expression is:

[0126]

[0127] Where L- represents the effective length of the infinite beam back wave, is the vibration frequency under overdamping, and the other parameters are the same as in formula (28).

[0128] In general, the effective length is calculated by equations (36) and (38) with a certain accuracy and is used to estimate the range of action of the moving load on the infinite beam. The prediction accuracy of the effective length can be adjusted by changing the numerator A. The numerator in equations (4.36) and (4.38) is 4, that is, A = 4. In addition, the total wave effective length is the sum of the effective lengths of the front wave and the rear wave.

[0129] For an undamped foundation, it has been shown that when an external load moves on an infinite beam at a critical velocity, the beam displacement response will propagate to infinity. This phenomenon indicates that the displacement wave will propagate to infinity and the effective length cannot be calculated. Therefore, the effective length of an infinite beam without damping is not considered in this chapter.

[0130] Step 2: Use the mathematical properties of the exponential function to derive the theoretical formula for predicting the effective length.

[0131] The beam displacement response obtained by Fourier transform and residue theorem will be used to verify the above theoretical derivation. The displacement response of the infinite foundation beam in the complex domain is derived by Fourier transform and residue theorem (residue theorem method), which shows that the closed solution contains not only the real part but also the imaginary part, but the real solution is difficult to separate from the complex solution. In contrast, the displacement response obtained by the solution method assuming exponential form adopted by the present invention not only does not contain the imaginary part, but also has a more obvious rule, and the effective length of the infinite beam can be obtained by mathematical methods.

[0132] Step 3: Verify the feasibility of the infinite beam displacement response theory by moving loads, beams and foundation parameters.

[0133] The moving load, beam and foundation parameters used in the numerical verification are listed in Table 1. The foundation stiffness is considered as a constant and its nonlinear characteristics are not considered.

[0134] Table 1 Load, beam and foundation parameters

[0135]

[0136] Select two speed ratios, namely 0.44 and 1.2, corresponding to the speed v = α vcr =90 and 247.5m / s. In addition, two damping ratios are selected, namely 0.05 and 1.1, corresponding to the damping coefficients and 13207.3N·s / m 2 , representing the underdamped and overdamped dual-parameter foundation beam.

[0137] Regarding the moving load problem, Sun used Fourier transform and residue theorem to study the displacement response of beams on viscoelastic foundations. When the foundation is underdamped and overdamped, the results obtained by calculating the degenerate solution of the beam displacement response are compared with the results obtained by Sun. Figure 2 Moderate underdamping and Figure 3 The moderate and over damping cases are compared. Figure 2 is the deflection curve of the viscoelastic foundation beam under underdamping: (a) α = 0.44, β = 0.05 (β cr =1.64); (b)α=1.2,β=0.05(β cr =1.05), Figure 3 is the deflection curve of the viscoelastic foundation beam under overdamping: (a) α = 0.44, β = 1.7, (β cr =1.64); (b)α=1.2,β=1.7(β cr =1.05), the effective lengths of the front and rear waves are theoretically calculated by equations (36) and (38) and are also plotted on Figure 2 and 3 (marked with red dots in the figure). Figure 2 and 3 It can be seen that in each case, the two solutions agree well with each other, verifying the validity of the degenerate solution of the infinite beam displacement response. In addition, the effective length can be used to estimate the approximate range of the moving load on the infinite viscoelastic foundation beam in both underdamped and overdamped conditions. Figure 2 It can be clearly observed in both (b) and 3(b) that the calculated effective lengths on both sides of the moving load under overdamping are different. Figure 2 (a) and 3(a) produce the same effective length on the left and right sides of the moving load under underdamped foundation. The ratio of the deflection at the effective length of the viscoelastic foundation beam to the maximum deflection is shown in Table 2. It can be seen from Table 2 that the deflection at the effective length position of the front wave and the rear wave is less than 5% compared with the maximum deflection, which is almost negligible.

[0138] Table 2 Ratio of deflection at effective length to maximum deflection of viscoelastic foundation beam under underdamping and overdamping

[0139]

[0140] Verify the accuracy of the theoretical solution of the axially loaded dual-parameter foundation beam. The numerical verification will use two speed ratios, namely 0.44 and 1.2, corresponding to the speed v = α vcr =90 and 247.5m / s, representing subcritical and supercritical velocities. In addition, two foundation damping ratios, 0.05 and 1.1, are used, corresponding to the foundation damping coefficients: and 8544.8N·s / m 2 , representing the underdamped and overdamped dual-parameter foundation beam.

[0141] The solutions of (19) and (20) for underdamped foundation and (19) and (28) for overdamped foundation are compared with the displacement response equations derived by Fourier transform and residue theorem. Figure 4 The deflection curve of the dual-parameter foundation beam under axial load under underdamping: (a) α = 0.44, β = 0.05, η P =0.5,η N =0.04(β cr =1.07); (b) α = 1.2, β = 0.05, η P =0.5,η N =0.04(β cr =0.90), Figure 5 The deflection curve of the double-parameter foundation beam under axial load under overdamping: (a) α = 0.44, β = 1.1, η P =0.5,η N =0.04(β cr =1.07); (b) α = 1.2, β = 1.1, η P =0.5,η N =0.04(β cr =0.90), the deflection response and effective length theoretical solution of the axially loaded dual-parameter foundation beam calculated by two different theories ( Figure 4 and 5 The infinite beam theoretical solutions calculated by the two methods are in good agreement, which verifies the reliability of the theoretical solution of the deflection response of the axially loaded dual-parameter foundation beam.

[0142] Under different foundation damping, the ratio of the deflection at the effective length of the axially loaded dual-parameter foundation beam to the maximum deflection is shown in Table 3. The effective length theoretical solution performs well in estimating the load influence range. The accuracy of the deflection response and effective length of the axially loaded dual-parameter foundation beam is verified.

[0143] Table 3 Ratio of deflection to maximum deflection at effective length of dual-parameter foundation under axial load under underdamping and overdamping

[0144]

[0145]

[0146] Step 4: Compare with the existing calculation formula for the effective length of the infinite beam and analyze the influence of different moving speeds on the effective length of the infinite beam.

[0147] By calculating the effective length formula (36) and (38) and L wang+ =Lwang- =4 / λ,L Ioa+ =L Ioa- =3π / 2λ for comparison.

[0148] Figure 6 Comparison of existing effective length calculation formulas for different types of foundations: (a) elastic foundation (α = 0.44); (b) viscoelastic foundation (α = 1.2, β = 0.05); (c) axially loaded dual-parameter foundation beam (α = 1.2, β = 0.05, η P =0.5,η N =0.04). Figure 6 It can be seen that under elastic foundation, although the effective lengths obtained by the three methods are not exactly the same, the displacement response of the infinite beam at the three effective length positions is very small and can be almost ignored. Table 4 shows the ratio of deflection to maximum deflection at the effective length of different types of foundations under underdamping and overdamping. As shown in Table 4, regardless of the type of foundation, equations (36) and (38) can predict the effective length of the infinite beam, and the ratio of deflection to maximum deflection at the effective length is less than 5%. In summary, the effective length calculation formula can predict the load range of beams with more complex foundations.

[0149] Table 4.4 Ratio of deflection at effective length to maximum deflection for different types of foundations under underdamping and overdamping

[0150]

[0151]

[0152] The effective length of the infinite beam is inevitably affected by the moving load velocity. In this section, the moving load velocity effect is evaluated, considering that the velocity ratio range is large enough, that is, from 0.4 to 2.0, with an interval of 0.4, of which 0.4 to 1.2 (81.8 m / s to 245.5 m / s) is the subcritical velocity, and 1.2 to 2.0 (245.5 m / s to 409 m / s) is the supercritical velocity.

[0153] Figure 7 The deflection curves of the axially loaded dual-parameter foundation beam under different speed ratios were plotted. Figure 8 The effective lengths of the rear wave, front wave and total wave for different velocity ratios are plotted in . Except for the load moving velocity, all other parameters of the moving load, beam and foundation are the same as those adopted in the above. Figure 7 It can be seen that as the load velocity ratio increases from 0.4 to 1.2, the maximum value of the beam deflection response increases, but when it increases from 1.2 to 2.0, the phenomenon is the opposite. In addition, the amplitude of the deflection response increases sharply with the increase of the velocity ratio, which indicates that the deflection response can propagate over a longer distance.

[0154] Figure 8It shows that when the velocity ratio exceeds the critical velocity ratio α = 1.20 (or v = 245.5m / s), the effective lengths of the rear wave, front wave and total wave increase sharply, which indicates that the displacement wave propagation capacity is significantly enhanced when the load moves beyond the critical velocity. This critical velocity is very consistent with the formula for predicting the critical velocity of the load derived by Dimitrová based on the axially loaded dual-parameter foundation beam model, that is,

[0155] Combination Figure 7 and 8 It can be seen that when the load velocity is greater than the critical velocity (supercritical velocity), the back and front wave deflection responses are obviously asymmetric with respect to the moving load, but it is interesting that the effective lengths of the back and front waves have the same trend as the velocity ratio increases. In addition, higher load velocities significantly enhance the deflection response transmission distance, resulting in a sharp increase in the effective lengths of the front and back waves. However, when the load movement velocity is lower than the critical velocity (subcritical velocity), higher velocities slightly increase the effective length of the infinite beam.

[0156] Therefore, the present invention adopts a method for determining the effective length of a dual-parameter infinite foundation beam under a moving load using the above-mentioned structure, and provides a method for calculating the effective length of an infinite beam using a dual-parameter foundation beam model. The displacement response of the infinite foundation beam calculated by the assumed exponential solution is basically consistent with the result obtained by the Fourier transform and residue theorem method, verifying its accuracy; the displacement response of the infinite foundation beam obtained by the assumed exponential solution eliminates the influence of imaginary numbers, making it better than the residue theorem method when deriving the effective length; under different foundation stiffness, the change trend of the effective length is almost unaffected by the foundation damping, while increasing the load movement speed and enhancing the foundation stiffness will significantly affect the effective length.

[0157] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A method for determining the effective length of a dual-parameter infinite foundation beam under moving load, characterized in that: The following steps are involved: Step 1: Use the assumed exponential solution method to solve the displacement response of the infinite foundation beam model; The concentrated load S1 moves along the axially loaded dual-parameter foundation Euler-Bernoulli beam, and the governing equation of the beam is established as: Where: E is Young's modulus, I is the area moment of inertia, m is the mass per unit length, w1 = w1(x, t) is the lateral displacement of the beam observed in the stationary coordinate system, N is the axial load borne by the beam, c is the foundation damping coefficient, k is the foundation stiffness, G p is the Pasternak modulus, the external load P moves at a constant speed v, δ(·) is the Dirac function; Using the moving coordinates s = x-vt, t = t, the chain rule of displacement with respect to the time derivative can be expressed as: By substituting equation (2) into equation (1), we can obtain the fourth-order constant coefficient differential equation in the moving coordinate system: For steady-state response, all partial derivatives containing time t in equation (3) are not considered. In order to solve the non-homogeneous differential equation (3), its homogeneous solution should be found first. The homogeneous equation of equation (3) can be expressed as: In order to make equation (4) dimensionless, the following dimensionless parameters are introduced: Where λ represents the static displacement of the elastic foundation beam caused by the dead load P; η N Indicates the axial load ratio, η P is the dimensionless shear modulus of the foundation; α is the velocity ratio, v cr is the critical speed of the moving load on the Winkler foundation beam; β is the foundation damping ratio, is the critical damping of a single degree of freedom spring-mass system; Based on the above dimensionless parameters, the homogeneous equation (4) can be updated as: The general solution of equation (6) can be expressed as: w(s)=e ξs (7) Substituting formula (7) into formula (6) yields the following characteristic equation: x 4 +4min 2 [a 2 -or p -or N ]ξ 2 -8a 2 see 3 x+4l 4 =0(8) S2 solves the closed-form solution of the displacement response of the beam with underdamped and overdamped foundation; Underdamped Assume that when β<β crp When , the foundation is underdamped, and the four complex roots of equation (8) can be assumed to be in the following form: Where: p, q1 and q2 are positive real numbers; The roots of equation (8) must satisfy the relationship (ξ-ξ1)(ξ-ξ2)(ξ-ξ3)(ξ-ξ4)=0. By substituting this relationship into equation (9), the following fourth-order polynomial equation can be derived: By combining equations (8) and (10), we can obtain the following set of equations: Combining equations (11a) and (11b), we can solve for the positive real numbers q1 and q2: Substituting formula (12) into formula (11c) yields: Then, let p = γλ, equation (13) can be simplified as: Where γ is the positive real root of equation (14), which is a sixth-order polynomial equation, degenerated into a cubic equation; To ensure that the slope and deflection of the beam are zero at infinity, the solution to equation (8) can be expressed as: w(s)=w + (s)=e -ps (C1 sinq2s+C2cosq2s)s≥0 (15) w(s)=w - (s)=e ps (C3sinq1s+C4cosq1s)s≤0 (16) Among them C i (i=1,2,3,4) are four position parameters determined by the infinite beam boundary conditions; According to the deflection, slope, moment continuity and shear force discontinuity of the moving concentrated load at s=0 at the starting point, the following four boundary conditions can be obtained: The superscript (′) represents the differential with respect to s; Based on the four boundary conditions provided by equation (17), four unknown constants C can be obtained: i The relationship between (i=1,2,3,4): Substituting the solutions of q1 and q2 in equation (12) into equation (18), and setting p = λγ, the displacement response of the infinite beam under underdamping can be rewritten as: Parameter C i It can be further rewritten as: in and where γ represents the positive real root of equation (14); W st is the coefficient related to static deflection, A D is a coefficient introduced to further simplify the displacement response expression of the dual-parameter foundation beam. and are the two frequencies of the dual-parameter foundation beam under underdamping; Overdamping It can be seen from formula (22d) that as the foundation damping ratio β gradually increases, the coefficient becomes infinite or even an imaginary number. The displacement response of the beam after the moving load s<0 expression (20) is no longer applicable. At this time, the foundation enters an over-damped state. In this case, the displacement response of the beam before the moving load s>0 is always real as the damping ratio increases. The displacement response of the beam after the moving load s<0 needs to be further derived, while the displacement response of the beam before the moving load s>0 can still be used in equation (19). In order to obtain the critical damping ratio, first consider equation (22d) The denominator is 0: By combining equations (14) and (24), we can solve the square of the coefficient γ: The coefficient γ in formula (25) 2 Substituting into equation (24), the critical damping coefficient of the axially loaded dual-parameter foundation beam can be obtained: From the relationship between damping coefficient and damping ratio, that is, The critical damping coefficient of the foundation beam model can be obtained: As the damping ratio continues to increase, when it exceeds the critical damping ratio, that is, β>β cr,p , the coefficients in formula (22) and formula (23) and will become an imaginary number; using the hyperbolic function relationship isinis = sinhs and cos = coshs, the displacement response of the infinite beam after moving load under overdamping can be rewritten as: in: All other parameters are the same as in formula (23); S3 will derive the effective length of the infinite beam based on the dynamic response of the foundation under underdamping and overdamping; Under critical damping, the dynamic response of the beam will propagate to infinity, indicating that the infinite beam has no effective length in this case. The effective length of the infinite beam will be derived from the dynamic response of the foundation under underdamping and overdamping. Underdamped Through trigonometric transformation, the infinite beam displacement response theoretical solutions (19) and (20) can be further rewritten as: The coefficient and corners It can be expressed as: It is obvious that there is the following relationship between equations (30) and (31): It can be seen from equations (34) and (35) that the displacement response on both sides of the load gradually decreases with the increase of the distance from the load, and the maximum value is less than and The effective length of the infinite beam is based on the fact that the displacement response is quite small; for the displacement response of the beam before the moving load, assuming that the distance between a certain position and the load is s = A / λγ, the corresponding exponential term is e -A When , the displacement response at this position can be ignored; the displacement response of the beam after moving load also has a similar trend; Under underdamping, the effective length of the infinite beam before and after the moving load is calculated as follows: Where L + and L - They represent the effective lengths of the infinite beam displacement response curve before the front wave moving load and after the back wave moving load, respectively. γ represents the positive real root of equation (14); Overdamping Under the condition of overdamping of foundation, the closed-form solution of displacement response of the front wave can still be calculated by equation (19), and the effective length of the front wave can still be calculated by equation (36). Due to the change of displacement response expression, the effective length of the rear wave needs further discussion. Based on the hyperbolic function transformation: coshx = (e x +e -x ) / 2 and sinhx=(e x -e -x ) / 2, the following exponential function can be obtained by transforming formula (28): From equation (37), it can be concluded that the beam displacement response is mainly composed of the exponential term and Determined by, the displacement response will gradually decay as the distance s increases; due to the inequality Always holds true, and the same is true when deriving formula (36). Assume that when the exponential term for e -A When , the position response becomes quite small and can be almost ignored; In this case the effective length can be indexed To derive, the effective length expression is: Where L- represents the effective length of the infinite beam back wave, is the vibration frequency under overdamping, and the other parameters are the same as those in formula (28); The effective length is calculated by equations (36) and (38) with a certain accuracy and is used to estimate the range of action of the moving load on the infinite beam. The prediction accuracy of the effective length can be adjusted by changing the numerator A. The numerator in equations (36) and (38) is 4, that is, A = 4. The total wave effective length is the sum of the effective lengths of the front wave and the rear wave. Step 2: Using the mathematical characteristics of the exponential function to derive the theoretical formula for predicting the effective length; Step 3: Verify the feasibility of infinite beam displacement response theory; Step 4: Compare with the existing calculation formula for the effective length of the infinite beam and analyze the influence of different moving speeds on the effective length of the infinite beam.

2. A method for determining the effective length of a dual-parameter infinite foundation beam under a moving load according to claim 1, characterized in that: The beam displacement response is obtained using Fourier transform and residue theorem for verification.

3. A method for determining the effective length of a dual-parameter infinite foundation beam under a moving load according to claim 1, characterized in that: In step 3, the ratio of the deflection at the effective length of the viscoelastic foundation beam to the maximum deflection is verified by moving the load, beam and foundation parameters.

4. A method for determining the effective length of a dual-parameter infinite foundation beam under a moving load according to claim 1, characterized in that: In step 4, the elastic foundation and the elasto-viscous foundation are compared.

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