CN102507115B - Large deformation bending vibration analysis method of deep-water top-tensioned riser - Google Patents

Abstract

The invention relates to a research method of an ocean deep-water riser, in particular to a large deformation bending vibration analysis method of a deep-water top-tensioned riser. The method improves the existing large deformation bending vibration analysis technique of the deep-water top-tensioned riser, the problem of large deformation of deep-water top-tensioned riser bending vibration is considered, a large deformation bending vibration equation of the deep-water top-tensioned riser is proposed and the large deformation bending vibration analysis method of the deep-water top-tensioned riser is created based on the equation. Since the method considers the influences of large deformation to the bending vibration of the deep-water top-tensioned riser, the bending vibration analysis of the deep-water top-tensioned riser is enabled to be more accordant with the actual stress and the deformation status of the riser.

Description

A kind of analytical approach of deep water top tension type vertical pipe large deformation flexural vibrations

Technical field

The present invention relates to the research method of ocean deepwater standpipe, be specifically related to a kind of analytical approach of deep water top tension type vertical pipe large deformation flexural vibrations.

Background technology

Deep water top tension type vertical pipe is the important equipment of deep-sea oil gas exploitation, and because riser length (1000 ~ 3000m) is far longer than its sectional dimension (0.3 ~ 0.5m), therefore, its bending stiffness is less, belongs to large flexible structure.Particularly novel compliant type standpipe, its flexibility, especially far beyond the scope of traditional girder construction, belongs to super large flexible structure.At present, the Bending Vibration Analysis of deep water top tension type vertical pipe mainly adopts the complicated bend of traditional euler beam (considering the bending of axial force) theoretical, this theory supposes based on small deformation, ignores the impact on deep water top tension type vertical pipe geometric stiffness and inertial force of the gravitional force change that causes due to flexural deformation and curvature.The program sets up top tension-type vertical pipe flexural vibrations equation based on the mechanical model shown in Fig. 1.In Fig. 1, the gravity of infinitesimal section is included in tension force T.

Based on this model, prior art adopts the standpipe flexural vibrations equation of formula (1):

$m\frac{{\∂}^{2}y}{\∂t^2}+c\frac{\∂y}{\∂t}+\mathrm{EI}\frac{{\∂}^{4}y}{{\∂x}^4}-\frac{\∂}{\∂x}\left(T\frac{\∂y}{\∂x}\right)=q---\left(1\right)$

In formula: the quality of m--standpipe unit length;

EI--standpipe flexural rigidity of section;

The structure d amping coefficient of c--standpipe unit length;

Y--standpipe sag;

T--Riser tension;

The t--time;

The axial coordinate of x--standpipe;

Q--acts on the fluid load on standpipe.

The major defect of prior art is as follows:

1, do not consider that standpipe large deformation causes the vertical component of flexural vibrations acceleration

During small deformation, the acceleration vertical component of standpipe flexural vibrations is less, therefore, is left in the basket.And during large deformation, the vertical component of standpipe flexural vibrations acceleration is comparatively large, should not be left in the basket.

2, do not consider that gravity that standpipe large deformation causes is on the impact of flexural vibrations

During small deformation, owing to supposing that the distortion of standpipe occurs over just transverse direction, therefore, gravitional force does not change.But during large deformation, the cross section caused due to flexural deformation is rotated, and causes gravity to serve effect of contraction to flexural deformation.Therefore, should consider during large deformation that gravity is on diastrophic impact.

Summary of the invention

The object of the invention is to the defect for prior art, the impact of gravitional force change on flexural vibrations that the flexural vibrations acceleration of the vertical direction that the large deformation of consideration deep water top tension type vertical pipe causes and large deformation cause, sets up large flexible Deepwater Risers flexural vibrations analysis on Large Deformation method.

Technical scheme of the present invention is as follows: a kind of analytical approach of deep water top tension type vertical pipe large deformation flexural vibrations, and the deepwater jack tension riser bending vibration analytical model of foundation is as follows:

$\mathrm{EI}\frac{{\∂}^{4}v}{\∂x^4}+\mathrm{EI}{\mathrm{\κ}}^2\frac{{\∂}^{2}v}{\∂x^2}-\frac{\∂}{\∂x}\left(T\frac{\∂v}{\∂x}\right)-\mathrm{mg}\frac{\∂v}{\∂x}+m\frac{{\∂}^{2}v}{\∂t^2}+c\frac{\∂v}{\∂t}=q$

$\mathrm{EI}\frac{{\∂}^{4}u}{\∂x^4}+(\mathrm{EI}{\mathrm{\κ}}^2-T)\frac{{\∂}^{2}u}{\∂x^2}-\frac{\∂T}{\∂x}\frac{\∂u}{\∂x}-m\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}\frac{{\∂}^{2}u}{{\∂t}^2}+c\frac{\∂u}{\∂t}=\frac{\∂T}{\∂x}-\mathrm{mg}$

In formula: v--standpipe horizontal direction flexural vibrations displacement;

U--standpipe vertical direction flexural vibrations displacement;

The quality of m--standpipe unit length;

EI--standpipe flexural rigidity of section;

The structure d amping coefficient of c--standpipe unit length;

T--Riser tension;

X--standpipe axial coordinate;

The t-time;

The curvature of κ--standpipe;

G-acceleration of gravity;

Q--acts on the fluid load on standpipe;

Based on above-mentioned deepwater jack tension riser bending vibration analytical model, calculate the large deformation flexural vibrations response of deep water top tension type vertical pipe as follows:

1) set Cartesian coordinates, if the summit of deep water top tension type vertical pipe is true origin, x-axis is vertical direction coordinate axis, and y-axis is horizontal direction coordinate axis;

2) finite element equation of deep water top tension type vertical pipe large deformation flexural vibrations can will be obtained after discrete for the equation of above-mentioned deepwater jack tension riser bending vibration analytical model by Finite Element Method:

$\left[M_y\right]\left\{\stackrel{..}{v}\right\}+\left[C_y\right]\left\{\stackrel{.}{v}\right\}+\left[K_y\right]\left\{v\right\}=\left\{F_y\right\}$

$\left[M_x\right]\left\{\stackrel{..}{u}\right\}+\left[C_x\right]\left\{\stackrel{.}{u}\right\}+\left[K_x\right]\left\{u\right\}=\left\{F_x\right\}$

Wherein, [M _y]--the transverse mass matrix of standpipe;

$\left[M_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}m{\∫}_0^l{\left[N\right]}^T\left[N\right]\mathrm{dx}$

[M _x]--the vertical inertial coefficient matrix of standpipe;

$\left[M_x\right]=-\underset{e=1}{\overset{n}{\mathrm{\Σ}}}m\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}{\∫}_0^l{\left[N\right]}^T\left[N\right]\mathrm{dx}=-\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}\left[M_y\right]$

[K _y]--the lateral stiffness matrix of standpipe;

$\left[K_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}\left\{\mathrm{EI}{\∫}_0^l{\left[N^{\′\′}\right]}^T\right[N^{\′\′}]\mathrm{dx}+(\mathrm{EI}{\mathrm{\κ}}^2-T){\∫}_0^l{\left[N^{\′}\right]}^T[N^{\′}]\mathrm{dx}$

$+(\frac{\∂T}{\∂x}+\mathrm{mg}){\∫}_0^l{\left[N^{\′}\right]}^T\left[N\right]\mathrm{dx}\}$

[K _x]--the vertical stiffness matrix of standpipe;

$\left[K_x\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}\left\{\mathrm{EI}{\∫}_0^l{\left[N^{\′\′}\right]}^T\right[N^{\′\′}]\mathrm{dx}+(\mathrm{EI}{\mathrm{\κ}}^2-T){\∫}_0^l{\left[N^{\′}\right]}^T[N^{\′}]\mathrm{dx}$

$+\frac{\∂T}{\∂x}{\∫}_0^l{\left[N^{\′}\right]}^T\left[N\right]\mathrm{dx}\}$

[C _y]--the horizontal damping matrix of standpipe;

[C _y]＝α[M _y]+β[K _y]

[C _x]--standpipe vertical damping matrix;

[C _x]＝α[M _x]-β[K _x]

--standpipe transverse acceleration vector;

--standpipe transverse velocity vector;

{ v}--standpipe transversal displacement vector;

{ F _y--standpipe lateral load vector:

$\left\{F_y\right\}=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^l{\left[N\right]}_e^T\{q_y{\}}_e\mathrm{dx}$

{ ü }--the vertical vector acceleration of standpipe;

--standpipe vertical velocity vector;

{ u}--standpipe vertical displacement vector;

{ F _x--standpipe vertical load vector:

$\left\{F_x\right\}=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^l{\left[N\right]}_e^T\{q_x{\}}_e\mathrm{dx}$

[N]--standpipe unit interpolating function matrix;

[N ']--standpipe unit interpolating function matrix is to the first order derivative of x;

[N ＂]--standpipe unit interpolating function matrix is to the second derivative of x;

{ q _y--standpipe lateral load vector;

{ q _x--standpipe vertical load vector:

$\left\{q_x\right\}=\{\frac{\∂T}{\∂x}-\mathrm{mg}\}$

α, β--Rui Leizuni coefficient:

X-standpipe axial coordinate;

The curvature of κ--standpipe;

The quality of m-standpipe unit length;

G-acceleration of gravity;

L--element length;

N--element number:

E--makes the unit of accumulating operation;

3) dividing elements is carried out to standpipe;

4) parameter of each unit after dividing is substituted into step 2) in each matrix and the matrix of all unit of formulae discovery of vector and vector;

5) adopt step-by-step integral method solution procedure 2) in the finite element equation of deep water top tension type vertical pipe large deformation flexural vibrations, can obtain the large deformation flexural vibrations response of deep water top tension type vertical pipe, vibratory response comprises displacement, speed, acceleration.

Beneficial effect of the present invention is as follows: the analytical approach of the present invention to existing deep water top tension type vertical pipe large deformation flexural vibrations is improved, consider the large deformation problem of deepwater jack tension riser bending vibration, propose deep water top tension type vertical pipe large deformation flexural vibrations equation, and establish the large deformation Bending Vibration Analysis method of deep water top tension type vertical pipe based on this equation.The method considers the impact of large deformation on deepwater jack tension riser bending vibration, makes the Bending Vibration Analysis of deep water top tension type vertical pipe more meet actual loading and the deformation state of standpipe.

Accompanying drawing explanation

Fig. 1 is the small deformation mechanical model schematic diagram of prior art neutral tube infinitesimal section;

Fig. 2 is deep water top tension type vertical pipe large deformation mechanical model schematic diagram of the present invention;

Fig. 3 is the coordinate system schematic diagram of top tension-type vertical pipe;

Fig. 4 is method flow diagram of the present invention.

Embodiment

Describe the present invention below in conjunction with the drawings and specific embodiments.

Deep water top tension type vertical pipe is the right cylinder of a high-fineness ratio, and its slenderness ratio can reach 5000 ~ 6000.Therefore, its bending stiffness is less, is one and has large flexible structure.Under marine environment effect, the displacement of its flexural vibrations is comparatively large, belongs to large deformation structure.If adopt the small deformation hypothesis of prior art, then the actual loading of result of calculation and deep water top tension type vertical pipe and deformation state will have larger error.For this reason, the present invention proposes the analysis method of deepwater jack tension riser bending vibration considering large deformation, be intended to the Bending Vibration Analysis solving deep water top tension type vertical pipe large deformation.

The present invention considers the impact of gravitional force change on flexural vibrations that deep water top tension type vertical pipe large deformation causes the impact of flexural vibrations acceleration and large deformation, propose the deepwater jack tension riser bending vibration analytical model considering large deformation, and establish the analytical approach of deep water top tension type vertical pipe large deformation flexural vibrations based on this model.

Fig. 2 is the deep water top tension type vertical pipe infinitesimal section mechanical model that the present invention adopts, and the dynamic analysis based on Fig. 2 can obtain the deepwater jack tension riser bending vibration differential equation considering large deformation:

$\mathrm{EI}\frac{{\∂}^{4}v}{\∂x^4}+\mathrm{EI}{\mathrm{\κ}}^2\frac{{\∂}^{2}v}{\∂x^2}-\frac{\∂}{\∂x}\left(T\frac{\∂v}{\∂x}\right)-\mathrm{mg}\frac{\∂v}{\∂x}+m\frac{{\∂}^{2}v}{\∂t^2}+c\frac{\∂v}{\∂t}=q---\left(2\right)$

$\mathrm{EI}\frac{{\∂}^{4}u}{\∂x^4}+(\mathrm{EI}{\mathrm{\κ}}^2-T)\frac{{\∂}^{2}u}{\∂x^2}-\frac{\∂T}{\∂x}\frac{\∂u}{\∂x}-m\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}\frac{{\∂}^{2}u}{{\∂t}^2}+c\frac{\∂u}{\∂t}=\frac{\∂T}{\∂x}-\mathrm{mg}---\left(3\right)$

In formula: v--standpipe horizontal direction flexural vibrations displacement;

U--standpipe vertical direction flexural vibrations displacement;

The quality of m--standpipe unit length;

EI--standpipe flexural rigidity of section;

The structure d amping coefficient of c--standpipe unit length;

T--Riser tension;

X-standpipe axial coordinate;

The t-time;

The curvature of κ--standpipe;

G-acceleration of gravity;

Q--acts on the fluid load on standpipe;

Based on above-mentioned deepwater jack tension riser bending vibration analytical model, calculate the large deformation flexural vibrations response of deep water top tension type vertical pipe as follows, as shown in Figure 4:

1) set Cartesian coordinates, if the summit of deep water top tension type vertical pipe is true origin, x-axis is vertical direction coordinate axis, and y-axis is horizontal direction coordinate axis, as shown in Figure 3;

2) finite element equation of deep water top tension type vertical pipe large deformation flexural vibrations can will be obtained after discrete for the equation of above-mentioned deepwater jack tension riser bending vibration analytical model by Finite Element Method (known technology):

$\left[M_y\right]\left\{\stackrel{..}{v}\right\}+\left[C_y\right]\left\{\stackrel{.}{v}\right\}+\left[K_y\right]\left\{v\right\}=\left\{F_y\right\}---\left(4\right)$

$\left[M_x\right]\left\{\stackrel{..}{u}\right\}+\left[C_x\right]\left\{\stackrel{.}{u}\right\}+\left[K_x\right]\left\{u\right\}=\left\{F_x\right\}---\left(5\right)$

Wherein, [M _y]--the transverse mass matrix of standpipe;

$\left[M_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}m{\∫}_0^l{\left[N\right]}^T\left[N\right]\mathrm{dx}---\left(6\right)$

[M _x]--the vertical inertial coefficient matrix of standpipe;

$\left[M_x\right]=-\underset{e=1}{\overset{n}{\mathrm{\Σ}}}m\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}{\∫}_0^l{\left[N\right]}^T\left[N\right]\mathrm{dx}=-\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}\left[M_y\right]---\left(7\right)$

[K _y]--the lateral stiffness matrix of standpipe;

$\left[K_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}\left\{\mathrm{EI}{\∫}_0^l{\left[N^{\′\′}\right]}^T\right[N^{\′\′}]\mathrm{dx}+(\mathrm{EI}{\mathrm{\κ}}^2-T){\∫}_0^l{\left[N^{\′}\right]}^T[N^{\′}]\mathrm{dx}$ (8)

$+(\frac{\∂T}{\∂x}+\mathrm{mg}){\∫}_0^l{\left[N^{\′}\right]}^T\left[N\right]\mathrm{dx}\}$

[K _x]--the vertical stiffness matrix of standpipe;

$\left[K_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}\left\{\mathrm{EI}{\∫}_0^l{\left[N^{\′\′}\right]}^T\right[N^{\′\′}]\mathrm{dx}+(\mathrm{EI}{\mathrm{\κ}}^2-T){\∫}_0^l{\left[N^{\′}\right]}^T[N^{\′}]\mathrm{dx}$ (9)

$+\frac{\∂T}{\∂x}-{\∫}_0^l{\left[N^{\′}\right]}^T\left[N\right]\mathrm{dx}\}$

[C _y]--the horizontal damping matrix of standpipe;

[C _y]＝α[M _y]+β[K _y] (10)

[C _x]--standpipe vertical damping matrix;

[C _x]＝α[M _x]+β[K _x] (11)

--standpipe transverse acceleration vector;

--standpipe transverse velocity vector;

{ v}--standpipe transversal displacement vector;

{ F _y--standpipe lateral load vector:

$\left\{F_y\right\}=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^l{\left[N\right]}_e^T\{q_y{\}}_e\mathrm{dx}---\left(12\right)$

{ ü }--the vertical vector acceleration of standpipe;

--standpipe vertical velocity vector;

{ u}--standpipe vertical displacement vector;

{ F _x--standpipe vertical load vector:

$\left\{F_x\right\}=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^l{\left[N\right]}_e^T\{q_x{\}}_e\mathrm{dx}---\left(13\right)$

[N]--standpipe unit interpolating function matrix;

[N ']--standpipe unit interpolating function matrix is to the first order derivative of x;

[N ＂]--standpipe unit interpolating function matrix is to the second derivative of x;

{ q _y--standpipe lateral load vector;

{ q _x--standpipe vertical load vector:

$\left\{q_x\right\}=\{\frac{\∂T}{\∂x}-\mathrm{mg}\}$

α, β--Rui Leizuni coefficient;

X-standpipe axial coordinate;

The curvature of κ--standpipe;

The quality of m-standpipe unit length;

G-acceleration of gravity;

L--element length;

N--element number;

E--makes the unit of accumulating operation;

3) dividing elements (known technology) is carried out to standpipe;

Dividing elements can divide according to the method for API RP 2RD specification recommends, also can carry out relatively conservative division according to the length of the ability of computing machine and standpipe, as 1m is long.

4) parameter (comprising element length and section modulus EI, EA) of each unit after dividing is substituted into matrix and the vector of all unit of formulae discovery of each matrix in formula (6) ~ (13) and vector;

5) step-by-step integral method (known technology) is adopted, as Newmark-β method or Wilson-θ method, solution procedure 2) in equation (4) and (5), can obtain deep water top tension type vertical pipe large deformation flexural vibrations response.

The mass matrix of standpipe and stiffness matrix adopt Finite Element Method to calculate, vertical inertial force system matrix number wherein and the stiffness matrix of both direction are different from prior art, it is the content that the present invention proposes, damping matrix then adopts Rayleigh damping matrix (known technology), the lateral load vector of standpipe also calculates according to fluid dynamics and Wave Theory, and these calculating all belong to known technology for a person skilled in the art.

Obviously, those skilled in the art can carry out various change and modification to the present invention and not depart from the spirit and scope of the present invention.Like this, if belong within the scope of the claims in the present invention and equivalent technology thereof to these amendments of the present invention and modification, then the present invention is also intended to comprise these change and modification.

Claims (1)

1. an analytical approach for deep water top tension type vertical pipe large deformation flexural vibrations, is characterized in that: the deepwater jack tension riser bending vibration analytical model of foundation is as follows:

$\mathrm{EI}\frac{{\∂}^{4}v}{\∂x^4}+\mathrm{EI}{\mathrm{\κ}}^2\frac{{\∂}^{2}v}{\∂x^2}-\frac{\∂}{\∂x}\left(T\frac{\∂v}{\∂x}\right)-\mathrm{mg}\frac{\∂v}{\∂x}+m\frac{{\∂}^{2}v}{\∂t^2}+c\frac{\∂v}{\∂t}=q$

$\mathrm{EI}\frac{{\∂}^{4}u}{\∂x^4}+(\mathrm{EI}{\mathrm{\κ}}^2-T)\frac{{\∂}^{2}u}{\∂x^2}-\frac{\∂T}{\∂x}\frac{\∂u}{\∂x}-m\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}\frac{{\∂}^{2}u}{{\∂t}^2}+c\frac{\∂u}{\∂t}=\frac{\∂T}{\∂x}-\mathrm{mg}$

In formula: v--standpipe horizontal direction flexural vibrations displacement;

U--standpipe vertical direction flexural vibrations displacement;

The quality of m--standpipe unit length;

EI--standpipe flexural rigidity of section;

The structure d amping coefficient of c--standpipe unit length;

T--Riser tension;

X-standpipe axial coordinate;

The t-time;

The curvature of κ--standpipe;

G-acceleration of gravity;

Q--acts on the fluid load on standpipe;

Based on above-mentioned deepwater jack tension riser bending vibration analytical model, calculate the large deformation flexural vibrations response of deep water top tension type vertical pipe as follows:

1) set Cartesian coordinates, if the summit of deep water top tension type vertical pipe is true origin, x-axis is vertical direction coordinate axis, and y-axis is horizontal direction coordinate axis;

2) finite element equation of deep water top tension type vertical pipe large deformation flexural vibrations can will be obtained after discrete for the equation of above-mentioned deepwater jack tension riser bending vibration analytical model by Finite Element Method:

$\left[M_y\right]\left\{\stackrel{..}{v}\right\}+\left[C_y\right]\left\{\stackrel{.}{v}\right\}+\left[K_y\right]\left\{v\right\}=\left\{F_y\right\}$

$\left[M_x\right]\left\{\stackrel{..}{u}\right\}+\left[C_x\right]\left\{\stackrel{.}{u}\right\}+\left[K_x\right]\left\{u\right\}=\left\{F_x\right\}$

Wherein, [M _y]--the transverse mass matrix of standpipe;

$\left[M_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}m{\∫}_0^l{\left[N\right]}^T\left[N\right]\mathrm{dx}$

[M _x]--the vertical inertial coefficient matrix of standpipe;

$\left[M_x\right]=-\underset{e=1}{\overset{n}{\mathrm{\Σ}}}m\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}{\∫}_0^l{\left[N\right]}^T\left[N\right]\mathrm{dx}=-\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}\left[M_y\right]$

[K _y]--the lateral stiffness matrix of standpipe;

$\left[K_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}\left\{\mathrm{EI}{\∫}_0^l{\left[N^{\′\′}\right]}^T\right[N^{\′\′}]\mathrm{dx}+(\mathrm{EI}{\mathrm{\κ}}^2-T){\∫}_0^l{\left[N^{\′}\right]}^T[N^{\′}]\mathrm{dx}$

$+(\frac{\∂T}{\∂x}+\mathrm{mg}){\∫}_0^l{\left[N^{\′}\right]}^T\left[N\right]\mathrm{dx}\}$

[K _x]--the vertical stiffness matrix of standpipe;

$\left[K_x\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}\left\{\mathrm{EI}{\∫}_0^l{\left[N^{\′\′}\right]}^T\right[N^{\′\′}]\mathrm{dx}+(\mathrm{EI}{\mathrm{\κ}}^2-T){\∫}_0^l{\left[N^{\′}\right]}^T[N^{\′}]\mathrm{dx}$

$+\frac{\∂T}{\∂x}{\∫}_0^l{\left[N^{\′}\right]}^T\left[N\right]\mathrm{dx}\}$

[C _y]--the horizontal damping matrix of standpipe;

[C _y]＝α[M _y]+β[K _y]

[C _x]--standpipe vertical damping matrix;

[C _x]＝α[M _x]+β[K _x]

--standpipe transverse acceleration vector;

--standpipe transverse velocity vector;

{ v}--standpipe transversal displacement vector;

{ F _y--standpipe lateral load vector:

$\left\{F_y\right\}=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^l{\left[N\right]}_e^T\{q_y{\}}_e\mathrm{dx}$

{ ü }--the vertical vector acceleration of standpipe;

--standpipe vertical velocity vector;

{ u}--standpipe vertical displacement vector;

{ F _x--standpipe vertical load vector:

$\left\{F_x\right\}=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^l{\left[N\right]}_e^T\{q_x{\}}_e\mathrm{dx}$

[N]--standpipe unit interpolating function matrix;

[N ']--standpipe unit interpolating function matrix is to the first order derivative of x;

[N ＂]--standpipe unit interpolating function matrix is to the second derivative of x;

{ q _y--standpipe lateral load vector;

{ q _x--standpipe vertical load vector:

$\left\{q_x\right\}=\{\frac{\∂T}{\∂x}-\mathrm{mg}\}$

α, β--Rui Leizuni coefficient;

X-standpipe axial coordinate;

The curvature of κ--standpipe;

The quality of m-standpipe unit length;

G-acceleration of gravity;

L--element length;

N--element number;

E--makes the unit of accumulating operation;

3) dividing elements is carried out to standpipe;

4) parameter of each unit after dividing is substituted into step 2) in each matrix and the matrix of all unit of formulae discovery of vector and vector;

5) adopt step-by-step integral method solution procedure 2) in the finite element equation of deep water top tension type vertical pipe large deformation flexural vibrations, the large deformation flexural vibrations response of deep water top tension type vertical pipe can be obtained.