CN102289536A - Method for analyzing transverse force bending with parameters of vertical pipe of deep water drilling well - Google Patents

Abstract

The invention relates to a research method of an ocean deepwater riser, in particular to a parameter transverse force bending analysis method of a deepwater drilling riser. This method takes into account the influence of the vertical movement of the deepwater drilling riser on the bending deformation, which is more in line with the actual stress and deformation state of the deepwater drilling riser than the existing method, overcomes the unsafe shortcomings of the existing technology, and enables deepwater drilling The riser parametric transverse force bending analysis is more realistic.

Description

A parametric lateral force bending analysis method for deepwater drilling riser

technical field

The invention relates to a research method of an ocean deepwater riser, in particular to a parameter transverse force bending analysis method of a deepwater drilling riser.

Background technique

Deepwater drilling riser is a main type of deepwater marine riser. The lower end of the riser is connected to the wellhead on the seabed. The tension of the pipe wall directly acts on the wellhead, and the fluid in the pipe continuously flows from the seabed through the riser to the floating platform. . The two ends of the deepwater drilling riser are flexibly connected to the subsea wellhead and the drilling platform through flexible joints. The deepwater drilling platform has two structures: semi-submersible and boat-shaped. the vertical displacement. Due to the load of the marine environment, the drilling riser will generate a transverse force (the acting direction of the wave current is perpendicular to the axis of the riser) and bend. Therefore, part of the vertical displacement of the top of the riser will cause the axial deformation of the riser, and the other will cause Riser bending deformation (similar to the pressure bar stability problem).

The existing deepwater drilling riser bending analysis method is based on the theory of complex bending of beams (transverse force bending considering axial force), that is, the bending deformation of the riser is only caused by the environmental load (wave force) perpendicular to the riser. Bending deformation caused by transverse force, namely:

$\frac{{<span\; class="notranslate">\; \&PartialD;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; EI</span>}\frac{{<span\; class="notranslate">\; \&PartialD;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; the\; y</span>}}{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; T</span>}\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; q</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

In the formula: y is the lateral bending displacement of the riser;

x is the axial coordinate of the riser;

EI is the bending rigidity of the riser cross section;

T is the riser tension;

q(x) is the fluid load acting on the riser.

Equation (1) is the bending differential equation of the deepwater drilling riser, which does not consider the bending deformation caused by the vertical displacement of the top of the riser. Such considerations are unreasonable for deepwater drilling risers with large slenderness ratios and thus greater flexibility.

As we all know, when one end of the rod produces axial displacement, the rod will bend under two conditions. The displacement is all converted into the strain energy of the rod; the second is that the bending stiffness of the rod is small or has initial bending deformation.

The main defects of the existing deepwater drilling riser parameter transverse force bending analysis method are as follows:

和弯曲刚度

均较小。而梁的复杂弯曲理论研究的杆件长细比(10～50)远远小于深水钻井立管，且传统的梁弯曲理论忽略轴向变形，因为，传统梁结构所受的轴向作用(力或位移)较小。但深水钻井立管的受力特点和边界条件与传统的梁结构和压杆稳定问题不同，深水钻井立管受轴向张力和横向荷载的作用，如果根据受力条件分析，除弯曲引起的两端相向位移外，不会产生其它因素引起的两端相向位移。但深水钻井立管的顶端却由于浮式平台的垂荡运动而产生较大的竖向位移，使立管两端产生相向位移(类似于受压杆)，且竖向位移的幅度已经超出了小变形范围(可达几米)。因此，现有技术不考虑立管顶端的竖向位移引起的弯曲，而采用梁的复杂弯曲理论分析深水钻井立管的弯曲问题，其分析结果是偏于不安全的。The slenderness ratio of the deepwater drilling riser is large (1000-5000), therefore, its axial stiffness

and bending stiffness

Both are small. However, the slenderness ratio (10-50) of the rods studied by the complex bending theory of beams is much smaller than that of deepwater drilling risers, and the traditional beam bending theory ignores the axial deformation, because the axial action (force) on the traditional beam structure or displacement) is small. However, the stress characteristics and boundary conditions of the deepwater drilling riser are different from the traditional beam structure and pressure rod stability. The deepwater drilling riser is affected by axial tension and lateral load. Except for the relative displacement of the two ends, there will be no relative displacement of the two ends caused by other factors. However, the top of the deepwater drilling riser has a relatively large vertical displacement due to the heave motion of the floating platform, causing the two ends of the riser to have a relative displacement (similar to a compression bar), and the magnitude of the vertical displacement has exceeded Small deformation range (up to several meters). Therefore, the existing technology does not consider the bending caused by the vertical displacement of the top of the riser, but uses the complex bending theory of the beam to analyze the bending problem of the deepwater drilling riser, and the analysis result is unsafe.

Contents of the invention

The object of the present invention is to aim at the defects of the prior art, and provide a method for analyzing the transverse force bending of the deepwater drilling riser, so as to make the bending analysis of the deepwater drilling riser more realistic.

The technical scheme of the present invention is as follows: a deepwater drilling riser parameter transverse force bending analysis method, which considers the environmental load and the vertical displacement of the top of the riser at the same time, and provides the bending differential equation of the deepwater drilling riser as follows:

$\frac{{<span\; class="notranslate">\; \&PartialD;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; EI</span>}\frac{{<span\; class="notranslate">\; \&PartialD;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; the\; y</span>}}{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; T</span>}\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; q</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; GA</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; u</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; x</span>}}$

In the formula: y is the lateral bending displacement of the riser;

x is the axial coordinate of the riser;

EI is the bending rigidity of the riser cross section;

T is the riser tension;

q(x) is the fluid load acting on the riser;

GA is the shear stiffness of the riser section;

εx为轴向应变；u is the vertical displacement of the riser, where,

εx is the axial strain;

Using the above equation, calculate the bending stress of the riser.

Further, the above-mentioned deepwater drilling riser parameter transverse force bending analysis method, the specific process of calculating the riser bending stress in this method is as follows:

(1) Divide the riser into several units;

(2) Express the displacement function of the unit as an interpolation function:

u=[S]{η},

y=[N]{a},

In the formula, [S] is the axial displacement interpolation function,

{η} is the column vector of node axial displacement,

[N] is the lateral displacement interpolation function,

{a} is the column vector of node lateral displacement;

(3) Using the Galerkin method, the deepwater drilling riser bending differential equation is transformed into a matrix equation as follows:

[K]{a}={F}

In the formula:

stiffness matrix

load vector

{a} is the column vector of node lateral displacement;

l is the unit length;

n is the number of units;

(4) Substituting the tension T of standpipe into the expression of stiffness matrix [K] in step (3), calculate the stiffness matrix of standpipe;

(5) Calculate the vertical displacement vector {η} at the top of the riser based on the heave motion of the floating platform;

(6) the vertical displacement vector {η} at the top of the riser calculated in step (5) is substituted into the expression calculation load vector of the load vector {F};

(7) Substituting the stiffness matrix [K] and the load vector {F} of the riser into the matrix equation in step (3), calculate the node lateral displacement column vector {a} of the riser;

(8) Substituting the node lateral displacement column vector {a} of the riser and the unit interpolation function [N] into the following formula, the lateral bending deflection line of the riser can be calculated:

y=[N]{a};

(9) Substituting the lateral bending deflection line of the riser obtained in step (8) into the following formula to calculate the bending stress:

$<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; \&sigma;</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; EI</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; N</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; \}</span>$

D为立管外径。In the above formula: W is the bending modulus of the riser section,

D is the outer diameter of the riser.

The beneficial effects of the present invention are as follows: the present invention considers the influence of the vertical movement of the deep-water drilling riser on the bending deformation in the analysis of the horizontal force bending of the deep-water drilling riser, and is more in line with the actual force of the deep-water drilling riser than the existing method and deformation state, which overcomes the unsafe shortcoming of the prior art, and makes the parameter transverse force bending analysis of the deepwater drilling riser more realistic.

Description of drawings

Fig. 1 is a flow chart of the analysis method of the deepwater drilling riser parameter transverse force bending according to the present invention.

Detailed ways

The present invention will be described in detail below in conjunction with the accompanying drawings and specific embodiments.

The two ends of the deepwater drilling riser are connected to the subsea wellhead and the drilling platform through flexible joints. The deepwater drilling platform has two structures: semi-submersible and boat-shaped. Vertical displacement. Due to the load of the marine environment, the drilling riser will generate a transverse force (the acting direction of the wave current is perpendicular to the axis of the riser) and bend. Therefore, part of the vertical displacement of the top of the riser will cause the axial deformation of the riser, and the other will cause Riser bending deformation (similar to the pressure bar stability problem). The existing technology does not consider the bending deformation caused by the vertical displacement of the top of the riser. Therefore, the current bending analysis of the drilling riser is completely based on the traditional transverse force bending theory. Due to the large flexibility of the deepwater drilling riser and the large heave motion of the drilling platform, it is unreasonable and unsafe to conduct the transverse force bending analysis of the drilling riser only, because it underestimates the bending deformation and thus the bending stress. The invention considers the influence of the large displacement of the top of the deepwater drilling riser on the bending of the transverse force, and proposes a parameter transverse force coupling bending analysis method of the riser of the deepwater drilling.

The present invention adopts the bending differential equation of the deepwater drilling riser that considers both the environmental load and the vertical displacement of the riser top, specifically as follows:

$\frac{{<span\; class="notranslate">\; \&PartialD;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; EI</span>}\frac{{<span\; class="notranslate">\; \&PartialD;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; the\; y</span>}}{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; T</span>}\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; q</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; GA</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; u</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

In the formula: y is the lateral bending displacement of the riser;

x is the axial coordinate of the riser;

EI is the bending rigidity of the riser cross section;

T is the riser tension;

q(x) is the fluid load acting on the riser;

GA is the shear stiffness of the riser section;

ε_x为轴向应变；u is the vertical displacement of the riser, u=(x), here,

ε _x is the axial strain;

为立管顶端竖向位移引起的横向力。In formula (2),

is the lateral force caused by the vertical displacement of the top of the riser.

The specific steps of the riser parametric transverse force bending analysis are as follows:

(S1) dividing the riser into several units;

(S2) Express the displacement function of the unit as an interpolation function:

u＝[S]{η} (3)

y=[N]{a} (4)

In the formula, [S] is the axial displacement interpolation function,

{η} is the column vector of node axial displacement,

[N] is the lateral displacement interpolation function,

{a} is the column vector of node lateral displacement;

(S3) using the Galerkin method (this is common knowledge in the art),

(该公式已修改)

(the formula has been modified)

In the above formula, L is the total length of the standpipe, δy is the variation of the bending displacement, and the function in the formula (5) is a weight function, which is common knowledge;

The deepwater drilling riser bending differential equation (formula (2)) is transformed into a matrix equation as follows:

[K]{a}＝{F}             (6)

In the formula:

stiffness matrix

load vector

{a} is the column vector of node lateral displacement;

l is the unit length;

n is the number of units;

(S4) Substituting the tension T of the riser (obtained according to the design top tension coefficient, which is a known technology) into the expression (formula (7)) of the stiffness matrix [K] in the step (S3), calculate the stiffness matrix of the riser ;

(S5) Calculate the vertical displacement vector {η} at the top of the riser based on the heaving motion of the floating platform (this is a known technology);

(S6) the expression (formula (8)) of the standpipe top vertical displacement vector {η} that is calculated in the step (S5) is substituted into the load vector {F} to calculate the load vector;

(S7) Substituting the stiffness matrix [K] and the load vector {F} of the riser into the matrix equation (formula (6)) in the step (S3), calculate the node lateral displacement column vector {a} of the riser;

(S8) Substituting the node lateral displacement column vector {a} of the riser and the unit interpolation function [N] into the formula (4), the lateral bending deflection curve of the riser can be calculated:

y=[N]{a};

(S9) Substituting the riser lateral bending deflection line obtained in the step (S8) into the following formula (9) to calculate the bending stress:

$<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; \&sigma;</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; EI</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; N</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

D为立管外径。In the above formula: W is the bending modulus of the riser section,

D is the outer diameter of the riser.

The invention considers the influence of the vertical movement of the deepwater drilling riser on the bending deformation, is more in line with the actual force and deformation state of the deepwater drilling riser than the existing method, and overcomes the unsafe defect of the prior art.

Obviously, those skilled in the art can make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and equivalent technologies, the present invention also intends to include these modifications and variations.

Claims (2)

1. A deepwater drilling riser parameter transverse force bending analysis method is characterized in that: the method considers environmental load and riser top vertical displacement simultaneously, and the deepwater drilling riser bending differential equation provided is as follows: $\frac{{<span\; class="notranslate">\; \&PartialD;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; EI</span>}\frac{{<span\; class="notranslate">\; \&PartialD;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; the\; y</span>}}{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; T</span>}\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; q</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; GA</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; u</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; x</span>}}$ In the formula: y is the lateral bending displacement of the riser; x is the axial coordinate of the riser; EI is the bending rigidity of the riser cross section; T is the riser tension; q(x) is the fluid load acting on the riser; GA is the shear stiffness of the riser section; u is the vertical displacement of the riser, where, ε _x is the axial strain; Using the above equation, calculate the bending stress of the riser. 2. deep-water drilling riser parameter transverse force bending analysis method as claimed in claim 1, is characterized in that: the method calculates the specific process of standpipe bending stress as follows: (1) Divide the riser into several units; (2) Express the displacement function of the unit as an interpolation function: u=[S]{n}, y=[N]{a}, In the formula, [S] is the axial displacement interpolation function, {η} is the column vector of node axial displacement, [N] is the lateral displacement interpolation function, {a} is the column vector of node lateral displacement; (3) Using the Galerkin method, the deepwater drilling riser bending differential equation is transformed into a matrix equation as follows: [K]{a}={F} In the formula: stiffness matrix

load vector

{a} is the column vector of node lateral displacement; l is the unit length; n is the number of units; (4) Substituting the tension T of standpipe into the expression of stiffness matrix [K] in step (3), calculate the stiffness matrix of standpipe; (5) Calculate the vertical displacement vector {η} at the top of the riser based on the heave motion of the floating platform; (6) the vertical displacement vector {η} at the top of the riser calculated in step (5) is substituted into the expression calculation load vector of the load vector {F}; (7) Substituting the stiffness matrix [K] and the load vector {F} of the riser into the matrix equation in step (3), calculate the node lateral displacement column vector {a} of the riser; (8) Substituting the node lateral displacement column vector {a} of the riser and the unit interpolation function [N] into the following formula, the lateral bending deflection line of the riser can be calculated: y=[N]{a}; (9) Substituting the lateral bending deflection line of the riser obtained in step (8) into the following formula to calculate the bending stress: $<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; \&sigma;</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; EI</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; N</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; \}</span>$ In the above formula: W is the bending modulus of the riser section,

D is the outer diameter of the riser.

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