A beam finite element for through-cracked tubular node behaviour modelling

This paper deals with the problem of a through-cracked tubular joint behaviour modelling. A new beam finite element has been created, in order to take into account the loss of stiffness of the joint due to through-thickness crack presence, and the coupling between axial force and bending momentum. This allows to study at a low computing time large damaged offshore structures, using beam elements. The mechanical model used to build the finite element is first described. An analytical identification of its internal parameters (eccentricity and stiffness) is then performed. Finally, some results are presented. Copyright © 2002 John Wiley & Sons, Ltd.


INTRODUCTION
Jacket o shore structures experience harsh loading conditions, leading to fatigue crack growth, mainly at the tubular nodes. Considering new cheap and promising underwater inspection techniques detecting only through-cracks, one may be able to predict the mechanical behaviour of such fast propagating cracks. Jacket structures involve lots of tubular members and are thus generally modelled using beam ÿnite elements to reduce computational time.
In that case, the use of classical beam ÿnite elements supposes a rigid local behaviour of the joint. Concerning the real elastic behaviour of tubular nodes, some work to model noncracked tubular joints using a global exibility matrix has already been done [1][2][3]. However, the mechanical behaviour of through-cracked nodes di er from a non-damaged one. Through-cracks are very large [4], giving an important increase in terms of exibility [5]. It also shows a new e ect due to the local geometry changes induced by the crack presence: a coupling between the bending momentum and the axial loading. In order to take into account these e ects and to study the issue of the behaviour of the complete structure, a new cracked beam ÿnite element is developed.
The ÿnite element is based on a mechanical modelling of cracked joints. The method to compute the sti ness matrix of a 2D model is presented. It is mainly based on the complementary virtual principle. Then follows the complete description of the ÿnite element building. As the model includes unknown mechanical parameters (eccentricity and sti ness), an identiÿcation scheme is proposed, using the least-squares method. Finally, results issued from the identiÿcation are presented and discussed. A T-joint with in-plane loading is chosen for illustration.

SIMPLIFIED MODEL DESCRIPTION
The local behaviour of a through-cracked node shows a bending momentum=axial load coupling, and sti ness loss at the welded connection. Considering a T-joint case with a throughcrack, only the brace tubular part is of interest: it is assumed to be clamped at the chord wall at A (Figure 1). Thus, the coupling is modelled by an eccentricity e (geometrical parameter), the loss of sti ness featured by a spiral spring with sti ness k. The behaviour of the other part (the brace beam) with length l, is the bending sti ness EI, and the axial sti ness ES. This model is an associated isostatic model of the free beam one, preventing any rigid body motion. The following developments are issued from References [6; 7]. The global sti ness matrix element K r of this isostatic model is obtained by inverting the compliance matrix S r .
Let us denote by U c the displacement vector at nodes A and C (free element displacement vector), and U r the displacement vector at node C (isostatic element displacement vector): F c is the force vector at nodes A and C (free element force vector) and F r the force vector at node C (isostatic element force vector): Figure 1. Scheme of the simpliÿed model.
The fundamental static principle gives the relationship between F c and F r : where B is a matrix. One looks now for the U c against U r relationship. When a virtual force vector F r is applied at C, we have F c = BF r : this relationship deÿnes a static admissible force ÿeld. Using this virtual force vector, the equality between the complementary virtual work for the free element T c and the associated isostatic one T r is given by and leads to The equality between the strain energy for the free beam and the isostatic beam element, using Equations (3) and (5) gives The relationship between the two sti ness matrixes is The matrix K r is related to the compliance matrix by By neglecting the shear stress-strain energy, the isostatic complementary strain energy is expressed as where M(x) is the bending momentum, N(x) the axial load, and M 1 the momentum at the clamped end A. The shape functions N i (x) of the equilibrium ÿnite element are obtained by expressing the internal forces as functions of F r : The complementary strain energy becomes and gives the compliance matrix of the isostatic element:

3D FINITE ELEMENT DESCRIPTION
The way to build the beam ÿnite element has been presented in the previous section. It was supported by a 2D example. One should now consider a 3D beam ÿnite element, with a through-crack at both nodes ( Figure 2). Cracks are oriented in the plane perpendicular to the beam element axis, leading to the introduction of two eccentricities e y i ; e z i and two sti ness springs k y i ; k z i at node i. The local element reference is (x; y; z), where x is the beam element axis. N i ; V y i ; V z i ; M y i ; M z i and M x i are, respectively, the axial force, the shear stresses along the y and z axes, the bending momenta in the y and z planes, and the torque at node i of the ÿnite element. The dual variables are the usual displacements and rotations at the nodes: U x i ; U y i ; U z i ; x i ; y i ; z i . The length of the beam is denoted by l, EI is the exural rigidity along the x-axis, ES is the axial sti ness and GJ the torsion sti ness along the x-axis. E is Young's modulus and G the shear modulus. The beams are tubes with external diameter 1 and thickness t.

Computation of the matrix B
The static equilibrium of the beam leads to:

Internal forces
The internal forces are denoted as follows: N(x) is the axial force, M y (x) and M z (x) are the bending momenta in the y and z planes, V y (x) and V z (x) are the shear stresses along the y-and z-axis, and ÿnally M x (x) is the torque along the x-axis, see Figure 2. The equilibrium of a part of the beam gives the shape force functions:

Strain energy, compliance and sti ness matrix
The total complementary strain energy is The associated isostatic compliance matrix of the ÿnite element K r is therefore The matrix K c = BK r B T is numerically computed.

Global procedure
Let us consider the case of a T-joint, clamped at both ends and loaded at the end of the cracked element (Figure 3). It is assumed that only one through-crack exists, located at the connection, near the (y; z) plane and is supposed to be symmetrical ((x; y) plane). Its length is parametered by the angle 2'. The aim is to ÿnd the mechanical parameters, here e = e y 1 and k = k z 1 , depending on '. A numerical experience for the fundamental loading cases is obtained from a local model (a three-dimensional mesh using thick quadratic shell elements). The brace tube is taken su ciently long, so that local ovalization would not a ect the displacements at the brace extremity.
Let us denote by U the displacement vector and F the force vector at node 3 of the cracked tube for the local model. Regarding the beam model, U is the displacement vector at node 3, the compliance matrix is S = S(e; k), and the relationship between displacements and external forces of the beam ÿnite element model is We look for minimizing the di erence between the two displacement vectors U and U: Q is minimal with respect to e and k when @Q @e = 0; @Q @k = 0 Remembering that the operator S and its derivatives are symmetric, the optimization problem leads to the following equations: For the beam model, one has to compute the compliance matrix S as deÿned in Equation (20). This can be achieved by static condensation of the full sti ness matrix K tot : where U i and F i are, respectively, the displacement vector and the force vector at node i. One ÿnally ÿnds that In order to have homogeneous terms in Equation (23) Their identiÿcation is done by exact analytic formulas, and at most eight parameters are needed for the complete ÿnite element. It exhibits excellent accuracy characteristics for all loading cases, particularly for very large cracks. This element is very useful for large structure modelling because of the fast computational time compared to that needed for a local shell meshing.