Finite Element Formulation

The governing differential equation of an element provides a framework to develop a FE procedure. In the case of towed pipelines, the governing equations with lateral loads are (from Section 3.1.2)

E lj^ ^ ^ - T y " = f X x ) (4.1.a)

{m + m X ÿ + cy + EI,y^‘‘'<- T y " = f X x ,t ) (4.1.b)

In this section, an element stiffness matrix in a local coordinate system is derived using the Galerkin method. If we denote its trial solution as w , it can be written as

= (4.2)

where aj are undetermined parameters at nodes and Nj are shape functions.

Since ü (x;a) is an approximate solution, it does not exactly satisfy the governing equation and so produces a residual R(x;a) given by:

R{x;a) = - T ï ï " - (4.3)

A more accurate trial solution can be obtained as the residual approaches to zero. Many optimisation schemes were proposed for this process and the Galerkin method became the most popular because of its wider applicability compared to the variational method (Burnett, 1987). Thus the following formulation is based on the Galerkin method.

Galerkin residual equation is written as

£ R (x ;a )N i(x )d x -0 i=l, 2, . . . , n (4.4)

where f is the length of the considered element. After substituting Equation (4.3), it becomes

- T u " - f ^ ) N , { x ) d x = 0 i= I,2 ,...,n (4.5)

Integrating the first term of Equation (4.5) twice by parts and the second term once gives:

- \ j ü " N,{x)dx = £ T û 'N ',d x -{T u 'N ,)\^ (4.7)

From Equation (4.2), the derivatives of the trial solution can be written as

d u {x;a ) dNj{x)

dx ^ dx

d^u{x;a) ” d^ N j{x)

dx^ ^ ^ dx^

Also the expressions for moment M and shear force V are respectively

El^u = - M

— [ £ / ,« " ) - T u ' = - V

d x^ ^

Substituting Equations (4.6)~(4.9) into Equation (4.5) yields the following expression for the element equation.

iV,, N, dx

(4.10)

= f/y^A+iV,NX-[MoNr'

where Vo and Mq are natural boundary conditions. The above equation may be

written more compactly as

where the element stiffness matrix k is the sum of a flexural stiffiiess matrix and a geometric stiffiiess matrix k^ and the element load matrix / is the sum of an interior load matrix f j and a boundary load matrix f g, that is

k —kp + kQ and = N " d x + \ j N ' , N ' , d x f = L + f s = Î / . ^ i d x + W o N , t - { ^ o N , (4.12) (4.13)

As can be seen from Equation (4.11), the FEM can be viewed as the procedure that transforms unsolvable governing differential equation, Equation (4.1), into approximately equivalent but solvable algebraic equation which can be easily automated on computers.

Now to develop the shape function N/x). Before proceeding with the usual well- established scheme with polynomials, consider the analytical solution of the static governing Equation (4.1.a). The form of analytical solutions with and without tension T are respectively

y (x ) = Cj + c^x+ c^x^ + c^x' fo r T=0 y ( x ) - d j sinh{ccx) + cosh(ax) -\-d^x + d^ (axY (ccc)^ OOC + - — - + - — — + . . . 3! 5! 2./ + d^x -i- d^ fo r T>0

where a = f T / Ë Ï and the c’s and cTs are constants to be determined from boundary conditions. From the above analytical solutions, it can be seen that both solutions

are basically the same polynomial form with different orders and that the difference between the two solutions is negligible if ooc is less than J.O. Considering that a is less that 10~^ for conventional towed pipelines, the polynomial of order three gives sufficiently accurate results if the element length is less than 100 metres. If this is not the case, the shape functions with higher order should be used. In the present work, however, the element lengths will be kept below 100 metres to enable the influence of waves to be considered and, consequently, the polynomial with order of three is accurate enough to express the approximate solution.

The consistent (sometimes also called compatible) shape functions which satisfy both completeness and continuity requirements are used for the present problem to ensure convergence. Since the governing equation is a fourth order differential equation, the trial solution ü and its first derivative should be continuous. This requires two unknown parameters at each boundary node, hence a total of four for a one dimensional element. Four parameters require a cubic polynomial and the element trial solution takes the following form (for graphic presentation of ü , aj and N/x),

see Figure 4.4). M(x;a) = .A^,(x) (4.14) j=i where r x V r . v * .(» )■ 4 ^7) - 4 ^7) " , « 4 (7 - '

After substituting these shape functions into Equations (4.12) and (4.13), finally we obtain element property matrices. If we assume E, E, fy, and Tto be constant within an element, then these matrices are written as

* sym. 31 - 6 3£ 2 ^^ -3 £ 6 -3 £ 2 ^^ (4.16) = T 3 0 i 36 sym. 3£ -3 6 3£ -3 £ 36 -3 ê (4.17) f — f i + / g - h i 12 ' 6 _{\- y o { o y} i Mo{0) 6 _Vo{i) ' - I (4.18)

Note that the boundary load matrix is canceled whilst applying essential inter­ element and natural boundary conditions.

Following the same procedure with different governing equations, that is, different shape functions, an axial stiffness matrix and a torsional stiffness matrix k j of the beam element are respectively

X

10 AE £ 1 ~1 - 1 1 (4.20) k , = l G I , N ^ N ^ d x GI. 1 - 1 - 1 1 (4.21)

A resulting element stiffness matrix k is

k —kp + k^ + k^ kj (4.22)

where adjustment of the matrix size and superposing the components of common degrees of freedom are implicitly assumed.

4.2.2 Calculation of System Matrices

The element stiffiiess matrix derived in the previous section is based on the local coordinate system and, therefore, must be transformed to a global system to form a total stiffiiess matrix. Besides of the stiffiiess matrix, mass and damping matrices are also required for dynamic analysis. This section is concerned with the derivation of these matrices and the procedure for forming total property matrices.

Making use of the above finite element procedure, it is also possible to evaluate a mass matrix for each element of a structure. In the case of towed pipelines, an element mass matrix m consists of a structural mass matrix and an added mass matrix . Again the structural mass matrix consists of matrices in the flexural

Trisf, axial , and torsional degrees of freedom , and these can be written as follows using related consistent shape functions.

nis = ntsf + nts^ + Wg, (4.23)

where each matrix is written as follows for uniform structural mass per unit length m.

J56 221 54 -131 m i 420 sym. 156 -2 2 i (4.24) = £ m(x) JV, (%)#, (x) dx m i 420 140 70 70 140 (4.25) m s , = \ l ' ^ I ^ N , { x ) N j { x ) d x 140 70 70 140 (4.26)

The element added mass matrix, however, only consists of the matrix in the flexural degrees of freedom and can be obtained by replacing m in Equation (4.24) by the added mass per unit length, ma.

Although the above formulations are for deformation in the local ^/-direction only, the inclusion of deformation in the local z-direction is possible by following the same procedure with additional parameters (for definition, see Figure 4.5). The complete element stiffness and mass matrices for this case are shown in Tables 4.1-4.3.

In the case of damping, the FE procedure could be used again to define an element damping matrix, c .

c = \ ‘ c { x ) N , { x ) N j { x ) c k (4.27)

In practice, however, evaluation of the damping property c(x) is impracticable. For this reason, the damping is generally expressed in terms of damping ratios and the explicit damping matrix can be computed from these damping ratios as described later in this section.

The distributed load on the pipeline is assumed to be linearly varying within an element. In this case the consistent nodal load matrix f j under axial and lateral loads is shown in Table 4.4.

All the previous formulations for an element are based on the local coordinate system which is specific for each element. Therefore, all related matrices of each element must be transformed to a common global coordinate system to form a total system matrix. This process is carried out using a transformation matrix R whose component r^. is a directional cosine between a local z-axis and a globaly-axis, i.e..

P i= Z 4 4

;=i

(4.28)

where p. is a component in a local /-axis and Pj is that in a globaly-axis.

To transform the element property matrices in a local coordinate system which are 12

by 12 or 12 by 1 to those in a global system, another transformation matrix T is used.

T = R sym. 0 0 0 R 0 0 R 0 R (4.29)

The element property matrices k, c, m and / in a global coordinate system are respectively k = T k T ^ = (4.30) m = T ' m T f = T 7

The stiffness matrix K, mass matrix M, and force matrix F of the complete element assemblage are effectively obtained from the element property matrices by direct superposition.

i

M = (4.31)

i

i

where the summation / includes all elements.

There is no need to express damping by means of a damping matrix for frequency domain analysis because it is represented more conveniently in terms of the modal damping ratios In the case of time domain analysis, however, the damping cannot be expressed by the damping ratio and instead an explicit damping matrix is needed. The most effective way to determine the required damping matrix is to use Rayleigh damping - Clough and Penzien (1993) presented a detailed procedure to calculate it. Rayleigh damping is assumed to be proportional to a combination of the mass and the stiffness matrices.

Rayleigh damping leads to the following relation between damping ratio and frequency co„.