A dynamic response analysis method for foundation beams considering double tangential effects of foundation and beams

Abstract

The present invention discloses a foundation beam dynamic response analysis method considering the double tangential effects of foundation and beam, which relates to the field of geotechnical engineering technology, and includes the following steps: step one, establishing a double-parameter foundation control equation of the tangential effects of beam and foundation, and solving the analytical solution; step two, considering the double-parameter foundation beam degeneration solution of the tangential effects of beam and foundation; step three, verifying the analytical solution; step four, parameter analysis. The present invention adopts a foundation beam dynamic response analysis method considering the double tangential effects of foundation and beam of the above structure, based on the infinite beam vertical and horizontal coupling control equations, the vertical and horizontal coupling control equations of the beam can be converted into algebraic equations by double Fourier transform, and then the dynamic response of the infinite beam in the frequency domain is derived.

Description

Technical Field

[0001] The invention relates to the technical field of geotechnical engineering technology, and in particular to a foundation beam dynamic response analysis method considering double tangential effects of foundation and beam. Background Art

[0002] The existing mechanical model is based on the traditional viscoelastic foundation beam model, which only considers the vertical force between the beam and the foundation, and does not consider the friction between the foundation and the beam. Based on the above shortcomings, a two-parameter (Pasternak) foundation beam model considering the tangential effect of the foundation and the track beam will be introduced. It is an improvement and extension of the existing foundation model, and for the first time considers the influence of load frequency and speed on the dynamic response of the foundation beam model. On the one hand, the existence of the foundation tangential force may cause the track beam to change from a bending state to an eccentric compression state, which has a great impact on the track design; on the other hand, when the foundation is relatively hard and the contact between the foundation and the track is relatively rough, ignoring this effect will lead to a large deviation between the theoretical stress state and the measured stress state. Summary of the invention

[0003] The purpose of the present invention is to provide a foundation beam dynamic response analysis method considering the double tangential effects of the foundation and the beam, to improve the existing foundation model, and to have important engineering significance for considering the horizontal force between the foundation and the track beam.

[0004] To achieve the above object, the present invention provides a foundation beam dynamic response analysis method considering the double tangential effects of foundation and beam, comprising the following steps:

[0005] Step 1: Establish the double-parameter foundation control equation of the tangential effect of beam and foundation, and solve the analytical solution;

[0006] Step 2: Consider the tangential effect between beam and foundation and calculate the double-parameter foundation beam degeneration solution;

[0007] Step 3: Verify the analytical solution;

[0008] Step 4: Perform parameter analysis.

[0009] Preferably, the dual-parameter foundation beam control equation adopts three assumptions: (1) the horizontal spring response is assumed to be proportional to the relative horizontal displacement; (2) the Euler-Bernoulli beam is assumed to be isotropic and its deformation is a small linear elastic deformation; (3) only the deformation in the infinite beam plane is considered.

[0010] Preferably, the control equation is

[0011]

[0012] Where: k x is the stiffness of the horizontal spring; G pis the shear stiffness; C is the foundation damping; h is the height of the beam; w0(x, t) and u0(x, t) are the vertical and horizontal displacements of the neutral axis of the beam, respectively; F(x, t) is the moving simple harmonic load.

[0013] Preferably, in step 1, a response analytical solution is obtained by applying the residue theorem using double Fourier transform and its inverse transform.

[0014] Preferably, in step 3, the theoretical solution of the dynamic response of the dual-parameter foundation beam considering the tangential effect between the beam and the foundation will be compared with the numerical solution result obtained by using the quadgk function in the Matlab platform.

[0015] Preferably, in step 4, the horizontal spring stiffness k is carried out x , foundation damping C, tangent modulus G p , parameter analysis of the influence of moving harmonic load velocity v and frequency f on the dynamic response of the beam.

[0016] Therefore, the present invention adopts a foundation beam dynamic response analysis method of the above structure that considers the double tangential effects of the foundation and the beam. Based on the vertical and horizontal coupling control equations of the infinite beam, the vertical and horizontal coupling control equations of the beam can be converted into algebraic equations by double Fourier transform, and the dynamic response of the infinite beam in the frequency domain is derived; the theoretical solution of the dynamic response of the infinite double-parameter foundation beam considering the tangential effects of the foundation and the beam under the moving harmonic load is derived in this paper, and the Matlab numerical solution is in good agreement. It improves and extends the existing foundation model, and for the first time considers the influence of load frequency and speed on the dynamic response of the foundation beam model, which has important engineering significance for considering the horizontal force between the foundation and the track beam, and reduces the large deviation between the theoretical stress state and the measured stress state.

[0017] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 A force diagram of a foundation beam according to a foundation beam dynamic response analysis method considering double tangential effects of foundation and beam according to the present invention;

[0019] Figure 2 Comparison of the theoretical solution and numerical solution of the dynamic response of the beam on a dual-parameter foundation considering the tangential effect between the beam and the foundation: (a) vertical displacement; (b) rotation; (c) bending moment; (d) shear force;

[0020] Figure 3 The dynamic responses of the dual-parameter foundation beam under different horizontal spring stiffness of the present invention: (a) vertical displacement, (b) rotation angle, (c) bending moment, and (d) shear force;

[0021] Figure 4 It is the maximum vertical displacement response of the beam of the present invention with respect to the change of stiffness ratio η(lg);

[0022] Figure 5 The dynamic response of the dual-parameter foundation beam under different shear moduli of the present invention: (a) vertical displacement, (b) rotation angle, (c) bending moment, and (d) shear force;

[0023] Figure 6 Dynamic responses of the dual-parameter foundation beam under different damping: (a) vertical displacement, (b) rotation, (c) bending moment, and (d) shear force;

[0024] Figure 7 Dynamic responses of a two-parameter foundation beam with and without considering the tangential effect between the beam and foundation at different load frequencies: (a) vertical displacement, (b) rotation, (c) bending moment, and (d) shear force;

[0025] Figure 8 The maximum vertical displacement response of the beam with respect to the load frequency is shown with and without considering the tangential effect between the beam and the foundation;

[0026] Fig. 9 Dynamic responses of the dual-parameter foundation beam with and without considering the tangential effect of the foundation under different loading velocities: (a) vertical displacement, (b) rotation, (c) bending moment, and (d) shear force. DETAILED DESCRIPTION

[0027] The technical solution of the present invention is further described below through the accompanying drawings and embodiments.

[0028] 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.

[0029] Example

[0030] See also Figure 1-9 The present invention provides a foundation beam dynamic response analysis method considering the double tangential effect of foundation and beam. Step 1: Establish a double parameter foundation control equation of the tangential effect of beam and foundation and a double parameter foundation beam model. Figure 1As shown, a series of independent horizontal springs are placed at the contact surface between the foundation and the beam. This application adopts three assumptions: (1) it is assumed that the response of the horizontal spring is proportional to the relative horizontal displacement; (2) it is assumed that the Euler-Bernoulli beam is isotropic and its deformation is a small linear elastic deformation; (3) only the deformation in the plane of the infinite beam is considered.

[0031] Based on the above assumptions, the vertical and horizontal foundation reactions and soil shear force q z ,q x and q s It can be expressed as:

[0032]

[0033] In the formula, k z and k x is the stiffness of the vertical and horizontal springs, G p is the shear stiffness, C is the foundation damping; w b (x,t) and u b (x, t) are the vertical and horizontal displacements at the bottom of the beam.

[0034] like Figure 1 As shown, E, A, I, and ρ are the elastic modulus, cross-sectional area, moment of inertia about the y-axis, and beam density, respectively. The vertical and horizontal displacements of the beam can be expressed as:

[0035]

[0036] Where w0(x, t) and u0(x, t) are the vertical and horizontal displacements of the neutral axis of the beam, respectively.

[0037] The relative displacement between the beam and the foundation can be derived as:

[0038]

[0039] Where h is the height of the beam.

[0040] According to the linear elastic constitutive equation of the beam, the internal bending moment and axial force of the infinite beam can be obtained as follows:

[0041]

[0042] Under the above assumptions, the strain can be obtained through the small deformation geometry equation of the beam:

[0043]

[0044] According to Newton's second law and the moment equilibrium equation, we can get a group of equilibrium equations for the beam section:

[0045]

[0046] Where m = ρA is the mass per unit length. F(x, t) is the moving harmonic load, and its expression is as follows:

[0047] F(x,t)=F0exp(iΩt)

[0048] Among them, F0 and Ω are the amplitude and frequency of the moving simple harmonic load respectively. Combining the above formulas, the vertical and horizontal control equations of the infinite beam can be obtained:

[0049]

[0050] By taking double Fourier transform and its inverse transform to the governing equations and applying the residue theorem, analytical solutions of the vertical and horizontal displacement responses of the infinite beam are obtained.

[0051] Using the double Fourier transform and the selectivity of the Dirac-delta function, the one-dimensional infinite integral expression of the vertical and horizontal displacement response of the infinite beam can be obtained.

[0052]

[0053]

[0054] Further expanding the numerator and denominator in the integral expression yields:

[0055]

[0056]

[0057] in:

[0058] a0=(k x -Ω 2 m)(ciΩ+k z -Ω 2 m)

[0059] a1=-(4Ω 3 m 2 -(3CiΩ+2k x +2k z )mΩ+Ck x i)v

[0060]

[0061]

[0062]

[0063] a5=2EIΩmv

[0064] a6=-EI(mv 2-EA)

[0065] a7=(EA-mv 2 )

[0066] a8=2mΩv

[0067] a9=k x -mΩ 2

[0068]

[0069] When x-vt≥0, all singular points on the complex plane are first-order, and the residue of the solution can be further expressed as

[0070]

[0071] When there are not only first-order singular points but also second-order singular points on the complex plane, the contribution of the second-order singular points to the vertical displacement response must also be considered. j The residue can be calculated as follows:

[0072]

[0073] In the formula, ξ k is the second-order singular point of w0(x,t), and ξ1, ξ2, ξ3, ξ4 are the first-order singular points of w0(x,t).

[0074] When x-vt<0, the vertical displacement response solution of the infinite beam is mainly to sum the residues whose imaginary part of the first-order singular point is less than zero, as shown in the following formula:

[0075]

[0076] The solutions of all vertical displacement responses of double-parameter foundation beams under moving harmonic loads considering the tangential effects of foundation and beams have been obtained.

[0077] The integral expression of the vertical displacement response of an infinite viscoelastic beam under a moving harmonic load can be expressed as

[0078]

[0079] In the formula,

[0080] Step 2: To verify the analytical solution, the double-parameter foundation beam degeneration solution is considered for the tangential effect between the beam and the foundation;

[0081] The integral expression of the vertical displacement response of an infinite viscoelastic beam under a moving harmonic load can be expressed as

[0082]

[0083] In the formula,

[0084] Step 3: Verify the analytical solution; the theoretical solution of the dynamic response of the dual-parameter foundation beam considering the tangential effect of the beam and the foundation will be compared with the numerical solution obtained using the quadgk function in the Matlab platform. The comparison results of the theoretical solution and the numerical solution of the vertical displacement, rotation, bending moment and shear force obtained using the parameters in Table 1 are shown in Figure 2. Figure 2 As shown, Figure 2 Comparison of the theoretical solution and numerical solution of the dynamic response of the dual-parameter foundation beam considering the tangential effect between the beam and the foundation: (a) vertical displacement; (b) rotation; (c) bending moment; (d) shear force. As can be seen from the figure, the four responses are well matched with the numerical solution, and the theoretical solution of the dual-parameter foundation beam considering the tangential effect between the foundation and the beam under the action of moving harmonic load is correct.

[0085] Table 1 Calculation parameters of mobile simple harmonic force, track and foundation

[0086]

[0087] Step 4: Parameter analysis is performed based on the analytical solution. The observation point is set at the origin x = 0m, and the horizontal spring stiffness k is calculated. x , foundation damping C, tangent modulus G p , moving simple harmonic load velocity v and frequency f on the dynamic response of the beam. In addition to the vertical displacement response of the infinite beam, attention should also be paid to the results of the rotation angle, bending moment and shear force response. The rotation angle θ, bending moment M and shear force V are the different order derivatives of the vertical displacement w0(x,t) relative to the position x, namely

[0088] The horizontal spring stiffness reflects the degree of tangential effect between the beam and the foundation. Seven different horizontal spring stiffnesses, namely k x =0,3.503×10 4 ~3.503×10 9 N / m 2 , except for the change in horizontal spring stiffness, all other parameters remain unchanged. Figure 3 The dynamic response of the dual-parameter foundation beam under different horizontal spring stiffness of the present invention: (a) vertical displacement, (b) rotation angle, (c) bending moment, and (d) shear force. Figure 3 It can be observed that when k x =3.503×10 4 ~3.503×10 9 N / m 2 When , the four dynamic responses of the infinite beam (vertical displacement, rotation, bending moment and shear force) and the horizontal spring stiffness k x = 0, the response is roughly the same, and the maximum dynamic response amplitude of the beam decreases with the increase of the horizontal spring stiffness, but the decrease is very small.x >3.503×10 7 N / m 2 When the horizontal spring stiffness increases, the amplitude of the beam dynamic response decreases significantly.

[0089] In order to study the influence of the relationship between the horizontal and vertical spring stiffness on the vertical displacement response of the infinite beam, the ratio of the horizontal and vertical spring stiffness is defined as η = k x / k z . Figure 4 The relationship between the maximum vertical displacement response of the beam and the stiffness ratio η(lg) is plotted. Figure 4 It is obvious that when η(lg)<0, the vertical displacement response amplitude hardly changes, but when η(lg)>0, the vertical displacement response amplitude decays rapidly. The above phenomenon shows that k x The value results have different degrees of influence on the dynamic response of the dual-parameter foundation beam considering the horizontal tangential force. Therefore, in actual engineering, k x The value of is very important, which is significantly different from the dynamic response characteristics of the traditional two-parameter foundation beam. In addition, when the dynamic response undergoes a large mutation, the stiffness ratio η(lg)>0 is smaller than the stiffness ratio η(lg)>2 in the foundation beam model under finite boundary conditions, indicating that the dynamic response of the foundation beam under infinite boundary conditions has a large influence on the horizontal spring stiffness k. x more sensitive to changes.

[0090] When k x As k continues to increase, the influence of the foundation tangential effect on the vertical displacement response of the beam becomes more and more obvious. The relationship between the rotation angle, bending moment and shear force and stiffness ratio is similar to that of the vertical displacement. In order to save space, this chapter will not expand on the specific analysis. As the horizontal spring stiffness of the beam increases, the shear force of the beam continues to decrease. The main reason is that the model takes into account the vertical and horizontal coupled deformations of the beam unit. According to the deformation coordination condition, the increase in the horizontal spring stiffness will lead to a decrease in horizontal displacement under the same load, thereby reducing the shear force V. In summary, if k x Less than k z If the magnitude of the tangential effect between the beam and the foundation is smaller than that of the original beam (η(lg)<0), the influence of the tangential effect between the beam and the foundation can be ignored. Otherwise (η(lg)>0), the influence should be considered in the control equation. Otherwise, there will be a large deviation between the actual calculation result of the beam and the dynamic response result of the theoretical calculation.

[0091] Shear modulus (Pasternak modulus) in dual-parameter foundation G p It is usually used to indicate the degree of soil shear effect, and it also has a significant impact on the dynamic response of the infinite beam. This section considers six shear moduli, namely G p = = 0, 3.0, 3.0 × 10 8 , 3.0×10 9and 3.0×10 10 N / m 2 , in order to consider the influence of soil shear effect on the vertical displacement response of the infinite beam. Except for the change in shear modulus, the other parameters of the foundation and load are the same as those used above, and the horizontal and vertical spring stiffnesses are both 3.503×10 7 N / m 2 .

[0092] Figure 5 The dynamic response of the dual-parameter foundation beam under different shear moduli of the present invention: (a) vertical displacement, (b) rotation angle, (c) bending moment, and (d) shear force. Figure 5 As shown, when G p Less than 3.0×10 7 N / m 2 When G is , as the shear modulus increases, the vertical displacement, rotation, bending moment, and shear force response of the observation point remain basically unchanged, whether it is directly below the load or far away from the load position, and the maximum vertical displacement response is located directly below the load. p At 3.0×10 7 ~3.0×10 9 N / m 2 When G is between , the vertical displacement, rotation, moment and shear response amplitude of the infinite beam decay rapidly with the increase of shear modulus, and the vibration frequency of the vertical displacement response also decreases rapidly, making the curve smoother, and the displacement response on both sides of the load gradually becomes symmetrical about the load position. p 3.0×10 10 N / m 2 When , the vertical displacement and moment response of the infinite beam are almost in a straight line, indicating that the magnitude of the dynamic response amplitude is very small. In summary, on the dual-parameter foundation considering the tangential effect of the beam and the foundation, when the shear stiffness is less than 3.0×10 7 N / m 2 When the shear modulus is large, the influence of the foundation on the shear force of the beam can be ignored. However, when the shear modulus is large, the inhibitory effect of the shear modulus on the dynamic response of the infinite beam needs to be considered.

[0093] Damping has a certain inhibitory effect on the propagation of infinite beam vibration response, so it is also the focus of this chapter. This section considers six different foundation damping, namely C = 0, 1.7325 × 10 4 ~1.7325×10 8 N / m 2 , where the horizontal and vertical spring stiffnesses are both set to 3.503×10 7 N / m 2 , the shear modulus is set to 3.0×10 7 , all other parameters are the same as above.

[0094] Figure 6The dynamic response of the dual-parameter foundation beam under different damping: (a) vertical displacement, (b) rotation, (c) bending moment, and (d) shear force. In general, as the foundation damping increases, the maximum vertical displacement response always appears near the position directly below the moving load, and no obvious damping delay effect occurs. This phenomenon is an important difference between the dynamic response of the dual-parameter foundation beam considering the tangential effect and the traditional dual-parameter foundation beam affected by the change of foundation damping. The reason is that the existence of the tangential spring stiffness will lead to an increase in the equivalent stiffness of the foundation, weakening the delay effect of the foundation damping on the maximum displacement response. When the foundation damping C is less than 1.7325×10 5 N·s / m, the dynamic response does not change significantly, which is similar to the case when foundation damping is not considered, indicating that the influence of foundation damping can be ignored in this case. 5 N·s / m~1.7325×10 6 N·s / m, as the foundation damping increases, the amplitude at the position directly below the load drops sharply, and the four dynamic response curves begin to show an asymmetric shape. At this time, the foundation damping has a great influence on the dynamic response of the infinite beam after the moving load (x-vt<0). When the foundation damping C is 1.7325×10 6 N·s / m~1.7325×10 8 When the load is between 2000 and 2000 N·s / m, the amplitude and vibration frequency of the observation point continue to decrease directly below the load and away from the load position. After the load is moved, the four dynamic responses of the infinite beam oscillate more obviously, but they will still decay rapidly in a short time. In summary: considering the tangential effect between the beam and the foundation will lead to an increase in the equivalent stiffness of the foundation beam. When the foundation damping is small, the attenuation effect on the dynamic response and the delay effect on the vertical displacement response are not obvious.

[0095] In addition to the external load frequency of 10 Hz, three other load frequencies, namely 5, 20 and 40 Hz, are considered. In this case, except for the change in the external load frequency, the other parameters of the foundation and load remain unchanged. Among them, the horizontal and vertical spring stiffnesses are both set to 3.503×10 7 N / m 2 , the shear modulus is set to 3.0×10 7 N / m 2 , the foundation damping is set to 1.7325×10 6 N·s / m.

[0096] Figure 7 The dynamic response of the dual-parameter foundation beam with and without considering the tangential effect between the beam and the foundation under different load frequencies: (a) vertical displacement, (b) rotation, (c) bending moment, and (d) shear force. Figure 7It can be seen that as the frequency of the external load increases, the amplitude of the vertical displacement response gradually decreases, but the rotation angle, bending moment and shear force first increase and then decrease. In addition, when the frequency of the external load is less than 30Hz, there is a large difference between the dynamic responses of the two beams, but the difference between the two becomes insignificant at about 30Hz.

[0097] In order to further explore the difference in dynamic response of infinite beams with and without considering the tangential effect between beams and foundation at different load frequencies, Figure 8 The maximum vertical displacement response curve is plotted as the load frequency changes. Figure 8 It can be seen that the difference between the two foundations is very obvious when the load frequency is less than 30Hz, but when the load frequency is greater than 30Hz, the difference in the vertical displacement response between the two becomes smaller. Through the above analysis, it can be concluded that when the external load frequency is less than 30Hz, not considering the influence of the tangential effect between the beam and the foundation will cause a large difference in the dynamic response of the dual-parameter foundation beam.

[0098] In addition to the load velocity of 20 m / s (72 km / h), three other speeds are considered, namely 50, 70 and 90 m / s (= 150, 252 and 324 km / h). In this case, except for the load velocity, the other parameters of the foundation and load are the same as above, where the vertical spring stiffness is set to 3.503×10 7 N / m 2 , the horizontal spring stiffness is set to 0 and 3.503×10 7 N / m 2 , the shear modulus is 3.0×10 7 N / m 2 , the foundation damping is 1.7325×106N·s / m 2 , the load frequency is 10Hz.

[0099] Fig. 9 Figure 2 shows the dynamic response of the two-parameter foundation beam with and without considering the foundation tangential effect under different load speeds: (a) vertical displacement, (b) rotation, (c) bending moment, and (d) shear force. The dynamic response of the two-parameter foundation beam with and without considering the foundation tangential effect (i.e., k) under moving harmonic loads when the load speed v = 20, 50, 70, and 90 m / s. xWhether it is set to 0) dual-parameter foundation beam vertical displacement, rotation, bending moment and shear force diagram. In general, regardless of whether the tangential effect between the beam and the foundation is considered, the change in load velocity has limited effect on the vertical dynamic response of the infinite beam before the moving load (x-vt>0), and mainly affects the dynamic response of the infinite beam after the moving load (x-vt<0). The larger the load velocity, the larger the amplitude of the maximum vertical displacement response of the infinite beam, but the maximum response amplitudes of the rotation, bending moment and shear force gradually decrease. When the load velocity increases from 20m / s to 90m / s, the position of the maximum vertical displacement response gradually shifts to the right. This phenomenon shows that the increase in load velocity will cause the maximum vertical displacement response to appear not directly below the moving simple harmonic load, but somewhere behind it. In addition, by comparing the dynamic response of the dual-parameter foundation beam with and without considering the tangential effect with velocity, it can be seen that considering the tangential effect at the same velocity will cause the amplitude of the infinite beam dynamic response to decrease slightly, indicating that the excitation effect of the load velocity on the dynamic response of the infinite beam is weakened in this case.

[0100] Therefore, the present invention adopts a foundation beam dynamic response analysis method of the above structure that considers the double tangential effects of the foundation and the beam. Based on the vertical and horizontal coupling control equations of the infinite beam, the vertical and horizontal coupling control equations of the beam can be converted into algebraic equations by double Fourier transform, and the dynamic response of the infinite beam in the frequency domain is derived; the theoretical solution of the dynamic response of the infinite double-parameter foundation beam considering the tangential effects of the foundation and the beam under the moving harmonic load is derived in this paper, and the Matlab numerical solution is in good agreement. It improves and extends the existing foundation model, and for the first time considers the influence of load frequency and speed on the dynamic response of the foundation beam model, which has important engineering significance for considering the horizontal force between the foundation and the track beam, and reduces the large deviation between the theoretical stress state and the measured stress state.

[0101] 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 foundation beam dynamic response analysis method considering the double tangential effects of foundation and beam, characterized in that: The following steps are involved: Step 1: Establish the two-parameter foundation control equation for the tangential effect between the beam and the foundation. Place a series of independent horizontal springs at the contact surface between the foundation and the beam. Use three assumptions: (1) Assume that the horizontal spring response is proportional to the relative horizontal displacement; (2) Assume that the Euler-Bernoulli beam is isotropic and its deformation is a small linear elastic deformation; (3) Only the deformation in the plane of the infinite beam is considered: Based on the above assumptions, the vertical and horizontal foundation reactions and soil shear force q z ,q x and q x It is expressed as: In the formula, k z and k x is the stiffness of the vertical and horizontal springs, G p is the shear stiffness, C is the foundation damping, w b (x,t) and u b (x, t) are the vertical and horizontal displacements of the bottom of the beam; E, A, I, and ρ are the elastic modulus, cross-sectional area, moment of inertia about the y-axis, and beam density, respectively. The vertical and horizontal displacements of the beam are expressed as: Where w0(x, t) and u0(x, t) are the vertical and horizontal displacements of the neutral axis of the beam, respectively; The relative displacement between the beam and the foundation is derived as: Where h is the height of the beam; According to the linear elastic constitutive equation of the beam, the internal bending moment and axial force of the infinite beam are obtained as follows: Under the above assumptions, the strain is obtained through the small deformation geometry equation of the beam; According to Newton's second law and the moment equilibrium equation, we get a group of equilibrium equations for the beam section: Where m = ρA is the mass per unit length, and F(x, t) is the moving harmonic load, which is expressed as follows: F(x,t)=F0exp(iΩt) Among them, F0 and Ω are the amplitude and frequency of the moving simple harmonic load, respectively. Combining the above formula, the vertical and horizontal control equations of the infinite beam are obtained: k x is the stiffness of the horizontal spring; G p is the shear stiffness; C is the foundation damping; h is the height of the beam; w0(x, t) and u0(x, t) are the vertical and horizontal displacements of the neutral axis of the beam, respectively; F(x, t) is the moving simple harmonic load; By applying double Fourier transform and its inverse transform to the control equation and applying the residue theorem and the selectivity of Dirac-delta function, the analytical solutions of the vertical and horizontal displacement responses of the infinite beam are obtained, including: Using the double Fourier transform and the selectivity of the Dirac-delta function, the one-dimensional infinite integral expression of the vertical and horizontal displacement response of the infinite beam is obtained: Further expand the numerator and denominator in the integral expression to obtain: in: a0=(k x Oh 2 m)(ciΩ+k z -Oh 2 m) a1=-(4Ω 3 m 2 -(3CiΩ+2k x +2k z )mΩ+Ck x i)v a5=2EIΩmv a6=-EI(mv 2 -IT) a7=(EA-mv 2 ) a8=2mΩv <h2 style=";text-align:left;direction:ltr">a9=k<h2 style=";text-align:left;direction:ltr"> x <h2 style=";text-align:left;direction:ltr"> -mΩ<h2 style=";text-align:left;direction:ltr"> 2 When x-vt≥0, all singular points on the complex plane are first-order, and the residue of the solution is further expressed as: When there are not only first-order singular points but also second-order singular points on the complex plane, the contribution of the second-order singular points to the vertical displacement response must also be considered. j The residue of is calculated as follows: In the formula, ξ k is a second-order singular point of w0(x,t), ξ1,ξ2,ξ 3, ξ4 is the first-order singular point of w0(x,t); When x-vt<0, the vertical displacement response solution of the infinite beam is mainly to sum the residues whose imaginary part of the first-order singular point is less than zero as follows: The solutions of all vertical positions and responses of dual-parameter foundation beams under moving harmonic loads considering the tangential effects of foundation and beam have been obtained; The integral expression of the vertical displacement response of an infinite viscoelastic beam under a moving harmonic load is expressed as: In the formula, Step 2: Consider the tangential effect between beam and foundation and calculate the double-parameter foundation beam degeneration solution; Step 3: Verify the analytical solution; Step 4: Perform parameter analysis; In step 2, the double-parameter foundation beam degeneration solution considering the tangential effect between beam and foundation is: The integral expression of the vertical displacement response of an infinite viscoelastic beam under a moving harmonic load is expressed as: In the formula, In step 3, the theoretical solution of the dynamic response of the dual-parameter foundation beam considering the tangential effect between the beam and the foundation will be compared with the numerical solution obtained using the quadgk function in the Matlab platform.

2. The method for analyzing the dynamic response of a foundation beam considering the double tangential effects of the foundation and the beam according to claim 1, characterized in that: In step 4, the horizontal spring stiffness k is x , foundation damping c, tangent modulus G p , parameter analysis of the influence of moving harmonic load velocity v and frequency f on the dynamic response of the beam.

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