SOAP — a computer program package for the static analysis of offshore structures

S O A P - - a computer program package for the static analysis of offshore structures C. SUNDARARAJAN, T. MURAKI and A. K. VAISH Keith. Feih,sch Associates. Engineers. San Francisco. Cal!/brnia. USA

A new computer program package, SOA P (Structural Offshore Analysis Package), developed by Keith, Feibusch Associates, Engineers, for the static analysis and code compliance evaluation of offshore jacket structures subjected to wave and current forces is described. The program calculates the hydrodynamic forces due to wave and current using an appropriate linear or non-linear wave theory and the Morrison's equation, and performs the structural analysis by the finite element method, it has the capability to analyse both fixed base structures and pile-supported structures. While the superstructure is assumed to behave linearly, the non-linear flexural behaviour of the piles, as well as the non-linear response of the soil may be included in the analysis. In addition to calculating the joint displacements and member stresses, the program also compares the calculated stresses against AP! Code allowables, and identifies those members that do not satisfy the code criteria.

INTRODUCTION SOAP (Structural Offshore Analysis Packagel is a computer program package developed by Keith, Feibusch Associates, Engineers for the static analysis and code compliance evaluation of offshore, steel jacketed structures subjected to wave and current forces. The structure may be fixed at the base or supported on piles. While the superstructure is assumed to behave linearly, the program can handle the non-linear flexural behaviour of the piles, as well as the non-linear response of the soil. The program has a library of linear and non-linear wave theories, and any one of the theories may be used to calculate the water particle velocities and accelerations. The drag and inertial forces on the structure due to the water particle velocities and accelerations, respectively, are calculated using the Morrison's equation;. The displacements and stresses of the structure and piles due to the above hydrodynamic forces, self-weight and other user-input loads are calculated by the finite element method. The maximum stresses in the tubt, lar structural members are compared against the API Code-' allowables, and members that fail to satisfy the code requirements are identified. This paper presents a detailed overview of the SOAP program package, including the method of analysis, program capabilities and organization. 0141 11958[),010003 0552.00 © 1980 CML Publications

M E T H O D O F ANALYSIS Analysis of offshore structures subjected to hydrodynamic forces consists of two distinct phases.

Hydrodynamic force calculations Submerged members of offshore structures are subjected to hydrodynamic forces due to waves and currents. In steel jacketed structures, the cross-sectional dimensions of the members are small compared to the wave length; and the hydrodynamic forces on such members may be calculated by the semi-empirical Morrison's equation ~. The force intensity at any point on a circular member of a fixed structure is given by:

F= Co p5_ Avlr1-4-CMpVa

(1)

where F = hydrodynamic force per unit length, C o = d r a g coefficient, C.~l=inertia coefficient, p = m a s s density of fluid, r = w a t e r particle velocity normal to the member axis, a = water particle acceleration normal to the member axis. A =projected area of the member per unit length (=DI,), V=volume of the member per unit length (=rcD~/4). Dh=hydraulic diameter of the member cross-section. In the projected area and volume calculations, the hydraulic diameter of the member which is equal to the outside diameter plus twice the marine growth thickness should be used. Since the Morrison's equation is applicable to circular members only, an equivalent hydraulic diameter should be specified for non-circular members. The horizontal and vertical components of the waveinduced water particle velocities and accelerations may be calculated by one of the following wave theories: (i) Airy's linear theory3; (ii) Stoke's third order theory'*; Off) Stoke's fifth order theoryS; (iv) Cnoidal theory6; or (v) stream function theory 7. Each wave theory has its own limitations and range of applicability. Though there is no universally accepted method for choosing the appropriate wave theory, the guidelines developed by Dean s for the choice of the wave theory as a function of the wave height, period and the mean water depth has been used by many analysts. The current velocities, if any, should be added vectorially to the wave-induced velocities. The velocities and accelerations thus obtained are resolved normal and parallel to the member axis, and the normal components are used in the Morrison's equation. The parallel components do not contribute any significant hydrodynamic forces, and hence are discarded.

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STRUCTURAL0DISPLACEMENTS , STRESSES,CODECHECK Fig,re 1. lntelfacing of mod,les Jbr the analysis of structtu'es supported on piles Struct,ral analysis The structure may be either fixed at the base or supported on piles. In the former case, a linear structural analysis is performed by the finite element method. The SOAP program package has the latest multi-level substructure analysis capability. Hence even very large structures with many thousands of degrees-of-freedom may be divided into substructures, and analysed 'in-core' efficiently. In the analysis of pile-supported structures, the nonlinear flexural behaviour of the piles, as well as the nonlinear response of the soil, represented by axial, transverse and torsional springs may be considered. But the structural behaviour is assumed to be linearly elastic. Performing a non-linear analysis of the structure-pilesoil system using a conventional finite element, iterative procedure would be very costly, because of the large number ofdegrees-of-freedom ofthe system. However, the cost may be considerably reduced, if the non-linear, iterative analysis is performed only for the non-linear pile-soil system with relatively few degrees-of-freedom, and a linear analysis is performed for the much larger (super) structure. The SOAP program uses such a substructuring approach which involves the following six stepsg: Step 1: Condense the linear (super)structure and the loads acting on it down to the pile-structure interface. Since it is a linear analysis, this step is outside the nonlinear iterative analysis loop. Step 2: Condense each pile and the attached soil springs up to the interface, and add the resulting stiffness matrices to the condensed structural stiffness matrix. (Since the flexural behaviour of the pile, as well as the response of the soil springs are non-linear, a reasonable

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value for the pile flexural stiffness and the soil spring stiffnesses should be assumed in the first iteration.) Step 3: Solve the interface equilibrium equation:

= {p}

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where [KI is the combined, condensed stiffness matrix of the structure and the pile-soil system, {P} is the condensed load vector, and {x} is the displacement vector of the interface nodes. The above matrix and vectors are of the order 6N, where N is the number of interface nodes. Step 4: Back-substitute the above interface displacements to the pile-soil system and calculate the pile and soil displacements. Step 5: Calculate the pile and soil spring stiffnesses corresponding to the displacements calculated in step 4. If the calculated stiffnesses, and the stiffnesses used in step 2 are within the allowable tolerance, the necessary convergence has been achieved; so proceed to step 6. If the convergence is not achieved, R E P E A T STEPS 2 to 5, using the newly-calculated stiffnesses. Step 6: Back-substitute the converged interface displacements calculated in step 3 to the structure, and calculate the joint displacements and member forces.

PROGRAM ORGANIZATION The program package consists of three independent, but fully compatible modules. The first module, WAVE, generates the hydrodynamic loads on the structure due to waves and currents. The second module, SUPER, performs the (super)structural analysis and the API code compliance evaluation. If the structure is fixed at the base, the structural displacements, forces and stresses are calculated, and the stresses compared with the code allowables. If the structure is supported on piles, the structural stiffness and loads are condensed to the pile interface. The third module, PILE, performs a threedimensional, non-linear pile analysis, in which the condensed structural stiffness and loads are included. The forces acting on the pile and soil, as well as the structurepile interface displacements are calculated. The interface displacements are fed back to the second moduls, SUPER, which back-substitutes these displacements and calculates the structural displacements, forces and stresses. The stresses are compared with the code allowables. The interfacing of the modules are structured with user convenience in mind. A complete analysis, from the hydrodynamic load evaluation to the code compliance check may I~e performed in a single run, or each module may be run separately. Transfer of data from one module to another is through tapes, and it makes the interfacing very easy on the part of the user. A schematic representation of the interaction between the modules is given in Figs. 1 and 2. The capabilities of these modules are described below.

WAVE module The WAVE module is used to generate the hydrodynamic loads on the structure due to waves and currents. The 'design wave' is specified by the wave height, period and direction of propagation. User has the option to use one of the following wave theories to calculate the wave-

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points of the member or, if either value is zero, at the point(s) of immersion are calculated, and a linear distribution is assumed in between. These distributed loads are calculated in the local member coordinate system and are given by A i, A2, F1, F2, and GI, G2 (see Fig. 3). Loads on non-structural members are also calculated in the same way. But, since these members are not included in the structural analysis, the loads acting on them are transferred to the structural joints to which they are attached, as joint loads and moments. The hydrodynamic force calculations are carried out for discrete positions of the wave with respect to the structure, as the wave is 'stepped' through the structure (Fig. 4). The structural data and the generated hydrodynamic forces for each wave step are stored on tapes (TAPE D, TAPE 3, TAPE 4 and T A P E 7) for later use in the structural analysis module, SUPER. At each wave step, the base shears and moments in the two horizontal directions, as well as the resultant base shear and moment are also calculated and printed out. This information may be useful to the designer in identifying the wave steps which produce the more critical loading conditions. The hydrodynamic forces, the wave profile, and the water particle velocities and accelerations for each wave step may also be printed out at user option.

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induced fluid particle velocities and accelerations: (a) Airy's linear wave theory, (b) Stoke's third order wave theory, (c) Stoke's fifth order wave theory, (d) Cnoidal wave theory, or (e) variable order stream function wave theory (order = 1 to 23). Influence of the water currents may be included in the force calculations by specifying the direction of the current, and the velocities at the mud level and the mean water level. Velocities at intermediate levels are calculated by linear interpolation, and the velocities thus obtained are added vectorially to the wave-induced fluid particle velocities. The hydrodynamic forces are calculated using the Morrison's equation (1). User may specify different drag and inertia coefficients, C o and CM, respectively, for different members or member groups. Non-structural members which may not contribute to the structural stiffness (significantly), but may transfer hydrodynamic forces because of their exposure to wave and current may also be included in the force calculations. Boat landings and railings are typically treated as non-structural members. Only the portion of the member that is below the actual water level (not the mean water level) is subjected to hydrodynamic forces. The load densities at the two end

The SUPER module performs the linear, (super)structural analysis and the API Code 2 compliance evaluation. This module may be used in three modes: (a) mode 1 - - analysis of fixed base structures, and code compliance evaluation, (b) mode 2 - - c o n d e n s a t i o n of structural stiffness and loads to the structure-pile interface, (c) mode 3 - - back-substitution of the structurepile interface displacements (calculated by the third module PILE) to evaluate the structural displacements, forces and stresses, and the code compliance evaluation. The structural members should be either threedimensional pipes or general three-dimensional beams. The former are specified by the outside diameter and thickness, and the latter by the cross-sectional areas and

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Figure 4. Illustration o f wave stepping. (a) W a v e steps in time domain, T = wave period, N = number o f wave steps; (b) wave steps in space domain, L = wave length, N = number o f wave steps; (c) wave position relative to structure, - - - - , wave step 1, , wave step 2

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inertias. Influence of the shear deformations may be included in the structural analysis. The piles often extend above the mudlevel through the jacket legs. These piles may be either grouted or ungrouted. In case of grouted piles, the composite member consisting of the jacket leg, pile and concrete may be treated as a three-dimensional beam member. In ease of ungrouted piles, the pile inside the jacket leg is restrained by centralizers. While it can move independent of the jacket leg in the direction of its longitudinal axis, it has to move with the leg in the directions perpendicular to the above axis. The SUPER module has the capability to model this behaviour properly. User may specify the 'wave step' for which the structural analysis is to be performed, or specify that the wave step corresponding to the maximum base shear or moment be analysed; in the latter case, the program automatically chooses the appropriate wave step. The hydrodynamic loads from the particular wave step are read from tapes generated by the WAVE module. Selfweight of the structural members are automatically calculated from user-specified weight density per unit length. The buoyancy, flooded water weight and/or marine growth weight may be included in the weight density specification. Additional loads due to nonstructural merfibers, deck equipments, wind, etc. may be input as joint loads and moments. If the structure is fixed at the base, the joint displacements and member stresses are calculated by a linear, finite element approach (mode 1 analysis). SUPER module has the latest multi-level substructure analysis capability, and hence the structure may be input as a single system or as a number of subsystems (substructures), depending on the number of degrees-of-freedom of the structure. If the structure is supported on piles, the structural stiffness and loads are condensed to the structure-pile interface (mode 2 analysis)*. The condensed stiffness matrix and load vector are written on tape (TAPE 11), to be used in the PILE module. In mode 3 analysis, the structure-pile interface displacements are read from the tape (TAPE 12) created by PILE, and back-substituted to the structure, to calculate the joint displacements and member stresses. In most structural analysis programs, the forces and moments are calculated only at the member ends (joints). But the end moments need not necessarily be the maximum, and there are many members in offshore jacket structures (particularly near the water level) for which the maximum moments, and hence the maximum stresses, occur within the span. This is properly considered by the SUPER module. The moments are calculated at tile ends and nine intermediate points, duly including the influence of the distributed dead and hydrodynamic loads; and from these the maximum resultant moment value is identified. The bending moments about the two local member axes, and the axial force are included in the stress calculations. The axial stress is uniform over the crosssection. The magnitude of the bending stress is a maximum at 90 ° from the direction of the resultant bending moment in case ofdircular members, and at a corner of the * The structure itselfmay be subdivided into a number of substructures, and each substructure successively condensed down to the structure-pile interface.

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cross-section in case of box or girder members. An API Code 2 compliance evaluation is carried out for all pipe members in the (super)structure. The allowable direct and shear stresses and the allowable slenderness ratio for each member is automatically calculated by the SUPER module, using the user-input yield stress and member classification. These allowables are compared with the calculated maximum stresses and the slenderness ratio, and the ratio of the allowables to the calculated values are printed out. If any one of the ratios is greater than 1.0, the member is flagged as a 'FAILED' member.

PILE module The PILE module perform a three-dimensional, nonlinear analysis of the pile-soil system. The condensed structural matrix and load vector, generated by the SUPER module, are included in the above analysis. The piles are tubular members and are attached to tile (super)structure at the structure-pile interfaces. The pile orientation is specified by the structural joint to which it is attached, its vertical length and the horizontal offsets. Each pile is divided into a number of segments of specified length and constant cross-section, and each segment is divided into a number of equal elements. The axial and torsional resistances of the pile are assumed to be linear with deformations; but the flexural resistance may be non-linear. Tile non-linear (multilinear) moment-curvature relationship is input as a function of tile ratio (P/Py), where P is the current axial force and P~.~.is the axial force at yield. The soil is modelled by axial, translational and/or torsional springs lumped at the nodes between elements, and their force-displacement relationships may be nonlinear (multi-linear). Different soil springs may be specified at the top and bottom of segments, and the springs at the intermediate element nodes are obtained by linear interpolation)The springs at tile bottom of a segment may be different from the spring at the top of the next segment, thus.permitting abrupt changes in the soil properties, For example, to model stratified soil media. Linear tip springs are used to model the resistance of rocks or stiff soil on which the pile rests. These springs represent the axial, transverse, flexural and torsional resistances provided by the rock or soil. The non-linear pile-soil analysis is performed by the secant stiffness method. Assuming reasonable values for the pile and soil spring stiffnesses, the stiffness matrix of each pile and the attached soil springs is condensed to the structure-pile interface, and combined with the condensed structural stiffness matrix generated by the SUPER modul~.. Using this combined interface stiffness matrix and the condensed load vector generated by the SUPER module, the interface displacements are evaluated, and are back-substituted to the pile-soil system to calculate the pile-soil displacements and forces. The pile and soil spring stiffnesses corresponding to these displacements are then evaluated, and the iteration is continued with these new stiffnesses used to recalculate the interface stiffness matrix and displacements. The iterations are repeated until the pile and soil spring stiffnesses converge to a user-specified tolerance, or the maximum number of iterations is exceeded. At the end of the final iteration, the forces and displacements of the piles and soil springs are printed out. In addition, tile interface displacements are stored on tape

CONCLUSIONS

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Figure 5. Internal data and tape transfers. (TAPE 12) for use in the mode 3 analysis by the SUPER module.

Interfacing of the modtdes The complete solution, from the hydrodynamic load generation to the stress analysis and code compliance evaluation may be performed in a single run ('single-step analysis'), or each phase of the analysis, namely: (a) hydrodynamic load generation, (b) condensation of the structural stiffness and load vector to the pile interface, (c) pile analysis, and (d) the back-substitution of pi!e interface displacements to calculate the structural displacements, forces and stresses, and the code compliance evaluation may be performed in stages ('multiple-step analysis'). In a single-step analysis, the complete SOAP program package, consisting of the WAVE, SUPER and PILE modules, is used as if it were a single program. All the data cards (DATA I for the WAVE module, DATA II for the SUPER module (mode 2), DATA 3 for the PILE module, and DATA IV for the SUPER module (mode 3)) are input together, and all the outputs from the three modules are obtained together at the end of the execution. All tape transfers and interfaces are internal and automatic. The advantage of the single-step procedure is that the complete analysis can be performed in a single run, and hence it may be used to perform the complete analysis over a weekend, at low-cost. A line diagram indicating the internal data and tape transfers in the SOAP program package is given in Fig. 5. Some users may prefer to perform the analysis in stages (multiple-step analysis). In such a case, each module is run separately; and in each run, in addition to the appropriate data cards (DATA I, DATA II, DATA III or DATA IV), the necessary tapes generated by the previous runs are also input. The advantage of this approach is that the results from each phase of analysis may be checked before proceeding with the next phase of analysis.

The SOAP program package provides the state-of-the-art technology for the static analysis of offshore jacket structures subjected to wave and current forces, Features of this program package include: (1) choice of a variety of wave theories (Airy's linear theory, Stoke's third order theory, Stoke's fifth order theory, Cnoidal theory and variable order stream function theory), (2) arbitrary directions of wave and current travel, (3) realistic force distribution on members near the water surface, (4) inclusion of non-structUral members such as boat landings in the hydrodynamic force calculations, (5) latest substructure analysis capabilities, which allow for the fast 'in-core' solution of even very large structures, (6) realistic representation of grouted and ungrouted piles in the jacket legs, (7) three-dimensional, non-linear pile-soil analysis by an efficient substructuring approach, (8) calculation of the maximum stresses in structural members, duly accounting for thedistributed nature of the hydrodynamic forces, and (9) automatic API code compliance evaluation. The program package is operational in CDC CYBER 175 computer. It may be easily converted to run in other machines also. ACKNOWLEDGEMENTS The SUPER module and the PILE module in the SOAP program package are based on a linear substructure analysis program and a non-linear pile analysis program developed by Professor E. L. Wilson and Mr. J. Dickens, and Professor G. H. Powell and Mr. R. Rigs (all of University of California, Berkeley), respectively, for Keith, Feibusch Associates, Engineers. Their cooperation and assistance in the development of SOAP are greatly appreciated. The authors are also grateful to Mr. S, R. Caldwell (Keith, Feibusch Associates, Engineers, Dallas, Texas) for the many useful discussions during the course of the program development.

REFERENCES 1 Morrison, J. R., O'Brien, M. P., Johnson, J. W. and Schaaf, S. A. The force exerted by surface waves on piles, Petrol. Trans. 1950, 189, 149 2 API Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms, API RP 2A, American Petroleum Institute, Washington, DC, November 1977 3 Wiegel, R. L. Oceanographical Engineering, Prentice-Hall, Englewood Cliffs, N J, 1964, pp. 13-21 4 Skjelbreia, L. Gravity Waves: Stoke's Third Order Approximation, California Research Corporation, 1959 5 Skjelbreia, L. and Hendrickson, J. Fifth order gravity wave theory, Proc. Seventh Col~ Coastal Eng. 1961, I, 184 6 Wiegel, R. L. Oceanographical Engineering, Prentice-Hall, Englewood Cliffs, NJ, 1954, p. 40. 7 Dean, R. G. Stream function representation of nonlinear ocean waves, J. Geophys. Res. 1965, 70, 4561 8 Dean, R. G. Relative validities of water wave theories, Proc. A s c E Cm~ Civil Eng. Oceans, San Francisco, 1967, p. l 9 Bryant, L. M. and Matlock, H. Three-dimensional analysis offramed structures with nonlinear pile foundations, Proc. Ninth A. Offshore~ Technol. C m f 1977, 3, 599

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