Transactions of the 17th International Conference on
Structural Mechanics in Reactor Technology (SMiRT 17)
Prague, Czech Republic, August 17 –22, 2003
Paper # B01-5
Super-Elements…a Remedy for Non-Linear Analyses of Large-Sized Models
S.M.Palekar1), K.V.Subramanian1), M.S.Bavare1)
1) TCE Consulting Engineers Limited, Mumbai, India
ABSTRACT
FE analysts encounter large-sized mathematical models posing challenges in terms of inadequate disk-space, memory and exceedingly high solution-time on computers. Some-times compromises on modelling like splitting the model or reducing the model-size by coarser mesh or opting for lower-order of elements are necessary to fit the analytical problem within the capabilities of computer. Such compromises may entail significant loss of accuracy in the analysis solution results. The situations worsen if non-linearities are modelled. Enhancing hardware and software capacities, providing higher space and faster clock-speeds may be necessary. However, this mostly may not be economically viable. Technological advances have made it feasible for the analysts to take recourse to powerful analysis techniques which could fit the problem size in relatively less resources, yet solving the analysis problem without appreciable loss in accuracy. Sub-structuring technique using super-elements is a method whereby such problems can be tackled. The current paper deals with analyses involved in large-sized modelling tasks, complexities of analysis procedures and application of sub-structuring technique to alleviate the analysis hurdles.
KEY WORDS: Sub-structuring, Super-elements (SE), Soil-Structure-Interaction (SSI), RC structures, Nuclear power plant (NPP), Non-linear static analysis (NLSA), Raft.
INTRODUCTION
The analysis and design of Nuclear Power Plant (NPP) structures involves handling of large size complex FE models. This is more so when, particularly to meet the functional requirements and optimisation of building-layout, the various buildings are to be amalgamated in the plant complex rather than isolating them by expansion joints. For the 500 MWe Prototype Fast Breeder Reactor (PFBR) presently under detailed engineering, a combined structural complex of Nine buildings viz. Reactor Containment Building (RCB), Steam-Generator Buildings (2 nos), Fuel Building, Control Building, Service Building, Radio-active Waste Building, Electrical Buildings (2 nos.), results into a large building admeasuring about 102.7 m in North-South (NS) direction and 95.2 m in East-West (EW) direction in plan. All the nine component buildings are monolithic with a common 3 m thick concrete raft. The raft admeasures about 115 m by 105.5 m in plan.
When the structure consists of RC walls emerging from raft, the stiffness of the foundation system is influenced. Such raft foundation of buildings below ground level tends to exhibit partial lift-off during certain load-combinations. This calls for analysis in Non-linear domain. Sub-structuring technique can be used for solution of such problems where super-elements represent a condensed group of large number of finite-elements. This paper presents an application of sub-structuring technique in the analysis and design of foundation raft including SSI and effect of super-structure.
MODELING AND ANALYSIS COMPLEXITIES
Structural analyses are carried out on global Finite element model. The FE model involves 3-D super-structure (SS) comprising of RC floors, roofs, columns, beams and walls; steel trusses supporting floors, RC raft-foundation resting on rock as founding strata. The mathematical model consists of 3-D beam elements for RC columns and beams, isoparametric thick shell elements for RC walls, floors in RCB and raft foundation, thin shell elements for floor slabs in other buildings and 3-D spar elements for trusses. To account for SSI modelling of Soil-springs is necessary in the analysis domain. The foundation medium is modelled by a system of Soil-springs attached to finite nodes of raft foundation. Calculation of Soil-stiffness accounting for semi-infinite extent of stratified foundation-medium is not a closed-form solution; hence, parallel calculations are involved using Boussinesq’s theory with end-result of set of appropriate values of Soil-spring constants for sub-grade modelling.
Besides, during high Ground Water Table conditions, the structure as a whole is acted upon by buoyant forces. Such behaviour calls for modelling the Soil-springs capable of bearing only compression and ineffective in tension.
The Global model involves 1,62,645 nodes, about 9,75,870 nodal Degrees Of Freedom, about 9,999 beam-elements, 53,516 thick-shell beam-elements, 2,778 thin-shell beam-elements, 658 truss-elements and about 6,342 Soil-spring (linear or nonlinear stiffness) elements.
Due to a large sized mathematical model, structural analysis particularly in the domain of Non-Linear Static Analysis (NLSA) can not be handled with the current computational resources or may take long solution and post-processing times besides high disc-space and subsequent handling issues of post-post-processing data. The overall process would take further long when number of load cases involved is high. Therefore, within limited resources of space and time, the only rational approach is to opt for sub-structuring wherein groups of finite-elements were condensed into super-elements. These super-elements are connected to each other and rest of the non-SE regions by Master Degrees of Freedom (MDOF) at nodes of common interfaces between them. Thus, generalised stiffness-matrix elements are formed for each of the super-elements, which are functions of MDOF at interface nodes. Loading on various elements of the Super-elements is also transformed into load-vectors in terms of MDOF. Such a model with super-elements and non-super-elements is used for the FE solution. The results are obtained at MDOF at nodes of the non-SE regions. Finally, the SE are expanded, results of all the constituent finite elements are calculated for subsequent post-processing. The elements to be condensed to form Super-elements are to be selected judiciously. For a building structure model, having raft foundation supported on soil springs, only the stiffness at the nodes, common to the raft and soil springs undergo change from one iteration to another to account for loss of contact with soil. The stiffness at all those nodes which are not common to soil springs, do not undergo any change. The number of such nodes belonging to the super-structure above the raft is very large. After condensation of these nodes, the problem size reduces drastically in proportion to the number of raft nodes in the global model. This reduced problem is solved iteratively for non-linear behaviour due to loss of contact with soil.
STRATEGY FOR STRUCTURAL ANALYSIS
A NLSA for portion of the model is planned, once by adopting the option of structuring (i.e. Sub-structured model) and later without resorting to Sub-structuring (i.e. original model) and the nodal-displacements, elemental stress-resultants particularly for the raft foundation of the structure are compared. The raft is selected since it is the first one to be greatly influenced by the non-linearity of the Soil-springs. The application presented in this paper deals with only single super-element (i.e. super-structure above raft foundation) linked with non-SE portion (i.e. raft foundation).
Mathematical Model
For the purpose of this paper, this portion of Global-model is called as original model. In the overall global-model of Nine Buildings, a component building called as Electrical-Building (admeasuring about 26 m by 32.5 m in plan) is targeted for this study. This component building has all the element types of the global model like 3-D beam-elements for beams and columns, quadratic thick shell beam-elements for walls and raft foundation, linear thin shell beam-elements for floor-slabs and Compression-only elements for Soil-springs. The mathematical model has Global X, Y and Z-axes along NS, WE and vertically upward directions respectively. All the forces and displacements are positive along the positive senses of global-axes. All the component materials of the model are assigned respective weight-density like that of concrete, steel, soil-stiffness.
Loading for Analysis
The structure is analysed for following loading. 1. Gravity weight of structure acting downwards.
2. NS-horizontal forces (along Global X-axis) on the structure pro rate with 0.25g times weight-distribution across the structure.
3. WE-horizontal forces (along Global Y-axis) on the structure pro rate with 0.25g times weight-distribution across the structure.
Sub-structuring Process
During the planning stage, the regions of overall global model are divided into three parts, viz. The super-structure (SS) constituted by the structural system above raft foundation which exhibits linear behaviour; the RC raft whose behaviour is linear; and entirely non-linear region constituted by Soil-springs. It is planned that, the SS would be condensed into super-element (SE) having linkages with raft through its interface nodes as MDOFs of SE. Despite being linear, the raft is not condensed into super-element since the attached Soil-springs are non-linear and their interaction with raft model is the focus of the analysis. The sequence involved in the structural analysis is presented below.
(a) Mathematical model is developed for the Super-Structure (SS) portion. (b) Loading is applied on SS model.
(c) Master degrees of freedom are assigned at the common interface between SS and raft.
2. Mathematical model is developed for raft and Soil-springs keeping the compatibility of nodal-configuration with SE developed in Step-1.
3. SE developed in Step-1 is linked with the raft-model developed in Step-2 with common nodes in the interface plane serving as linkage between superelement and non-superelement regions.
4. Loading is applied on raft portion modelled in Step-2 corresponding to respective static-load cases or combinations in Step-1(b).
5. NLSA as planned is carried out for overall model wherein raft, Soil-springs are linked with the Super-element (which represents the major linear structure in a condensed form of superelement) providing analysis results directly for raft. These results are supplied to design module of RC design of Raft foundation, where the reinforcement areas for RC raft are computed. Nodal displacements are used for lift-off area determination. Nodal spring-forces are input to bearing pressure calculations at various nodes of raft. The bearing pressures below raft are compared with allowable pressures under normal / abnormal / severe / extreme environmental conditions. If acceptable, the next step no.6 is taken. If the design conditions in this para are not met with, then modifications to the raft structure are carried out such as increase in plan-area of raft, increase in depth of raft, using higher grade of concrete for raft, increase in additional dead weight to counter tendency of lift-off etc. 6. Since SS is substructured into SE, its results are available only at MDOF. The Super-element is expanded into
Finite elements, wherein the displacements at its nodes are computed from the solution results at MDOF. Elemental stress resultants of the elements of the super-structure region are extracted in the expansion phase.
Structural Analysis
Analysis of the model was performed for simulation of non-linear static (NLSA) behaviour. NLSA involves all the linear element types like beams, shell elements and nonlinear element type for Soil-springs below foundation which can bear only compression and zero-stiffness in tension to simulate lift-off condition below raft-foundation during equilibrium iterations.
The combined loading as discussed before is generated on the structure but to account for non-linear behaviour like partial lift off below raft foundation the calculations for axial deformation of Soil-spring elements are scanned and those springs exhibiting tension are detached. The remaining springs those are in compression at the end of the iteration present initial Soil-spring configuration for the following equilibrium iteration. Newton-Raphson’s method of solution convergence is used. The analyses proceed through equilibrium iterations until the status of Rock Springs (like number of springs, configuration of Soil-springs, magnitudes of forces and nodal displacements after equilibrium iteration) cease to alter beyond tolerance limits with reference to last iteration. Then it is construed that lift-off status has converged and equilibrium with external loading is established. Current analysis needed three equilibrium iterations for convergence.
Results and Observations
On completion of both the types of analyses on original and sub-structured models, solution results comprising of nodal-displacements (m), elemental stress-resultants for beam and shell elements (in kN, m units) are obtained. It was aimed at checking the accuracy of results with respect to those in Original model and sub-structured model. The tabulations are presented below for only 10 to 14 raft-nodes having maximum downward (negative) and upward (positive) displacements. Peak displacements are underlined in the Table:-1 and Table:-2.
Table:-1
Peak Downward Displacement (m)
Node number Original Model Uz Substructured Model Uz Ratio of Uz displacements
179441 -4.9210E-03 -4.9208E-03 1.0000410
179444 -4.9139E-03 -4.9138E-03 1.0000200
179445 -4.9044E-03 -4.9042E-03 1.0000410
179443 -4.8933E-03 -4.8932E-03 1.0000210
179446 -4.8637E-03 -4.8635E-03 1.0000410
179442 -4.8518E-03 -4.8517E-03 1.0000210
179439 -4.7970E-03 -4.7969E-03 1.0000210
Node number Original Model Uz Substructured Model Uz Ratio of Uz displacements
191041 -4.7762E-03 -4.7760E-03 1.0000420
179440 -4.7751E-03 -4.7749E-03 1.0000420
Table:-1 above presents Uz displacements (negative since vertically downwards) with highest 10 values. It appears that both the analyses show close agreement as the ratio is close to 1.0.
Table:-2 below presents Uz displacements (vertical) being upwards with highest 14 values. It appears that both the analyses show close agreement as the ratio is close to 1.0. These nodes are lifted off from Soil.
Table:-2
Peak Upward Displacement (met)
Node number Original Model Uz Substructured Model Uz Ratio of Uz displacements
191435 9.2404E-04 9.2325E-04 1.0008560
191478 9.3194E-04 9.3153E-04 1.0004400
191426 9.3376E-04 9.3320E-04 1.0006000
191434 9.3578E-04 9.3511E-04 1.0007170
191019 9.4685E-04 9.4569E-04 1.0012270
191479 9.6554E-04 9.6508E-04 1.0004770
191017 9.8773E-04 9.8665E-04 1.0010950
191480 9.9745E-04 9.9695E-04 1.0005020
191433 1.0144E-03 1.0134E-03 1.0009870
191432 1.0257E-03 1.0247E-03 1.0009760
191481 1.0296E-03 1.0290E-03 1.0005830
191431 1.0362E-03 1.0353E-03 1.0008690
191430 1.0440E-03 1.0432E-03 1.0007670
191425 1.0475E-03 1.0469E-03 1.0005730
191429 1.0506E-03 1.0498E-03 1.0007620
191428 1.0546E-03 1.0539E-03 1.0006640
191427 1.0583E-03 1.0576E-03 1.0006620
Elemental stress resultants for the raft elements are tabulated for few maximum values of Moments (Mx, My in kNm/m elemental width), Out of plane shear forces (Nx, Ny in kN/m elemental width). It appears from the results of both the models that, elemental forces are in agreement with each other. This indicates that the sub-structured model closely simulates the non-linear behaviour exhibited by original model.
Table-3
Stress-Resultants in Raft (Hogging-Moments and Shear-forces)
Element Mx-Orig Mx-Subs Element My-Orig My-Subs Element Nx-Orig Nx-Subs Element Ny-Orig Ny-Subs
52658 4727.300 4727.200 52665 5282.500 5282.400 52661 3401.500 3401.400 52661 3629.300 3629.300
52656 4483.800 4483.800 52576 4569.600 4569.500 52662 2284.400 2284.400 52217 2720.400 2720.500
Element Mx-Orig Mx-Subs Element My-Orig My-Subs Element Nx-Orig Nx-Subs Element Ny-Orig Ny-Subs
52659 4043.900 4043.800 52657 4412.600 4412.500 52574 1448.300 1448.300 52662 1874.800 1874.700
52663 4014.100 4014.000 52578 4314.300 4314.200 52576 1345.600 1345.600 52667 1844.800 1844.700
52657 3872.800 3872.800 52662 4300.000 4299.900 52566 987.100 987.110 52660 1802.900 1802.800
52578 3761.000 3760.900 52579 4067.200 4067.000 52296 889.420 889.380 52659 1792.100 1792.100
52664 3578.400 3578.300 52656 3826.100 3826.100 52499 867.060 867.140 52666 1576.000 1575.900
52579 3482.500 3482.400 52588 3818.800 3818.600 52364 837.840 837.920 52673 1543.800 1543.700
52575 3404.600 3404.500 52663 3700.200 3700.100 52665 803.700 803.680 52218 1237.800 1237.800
52580 3319.800 3319.700 52664 3458.700 3458.600 52559 693.600 693.620 52223 992.590 992.600
52660 3299.200 3299.200 52587 3447.300 3447.200 52550 691.900 691.960 52426 965.120 965.080
52581 3137.700 3137.700 52581 3441.200 3441.100 52500 675.080 675.150 52414 915.040 915.010
52588 3041.700 3041.600 52585 3400.100 3399.900 52363 673.200 673.270 52594 900.510 900.420
Maxim 4727.300 4727.200 Maxim 5282.500 5282.400 Maxim 3401.500 3401.400 Maxim 3629.300 3629.300
Table-4
Stress-Resultants in Raft (Sagging-Moments and Shear-forces)
Element Mx-Orig Mx-Subs Element My-Orig My-Subs Element Nx-Orig Nx-Subs Element Ny-Orig Ny-Subs
52550 -1402.800 -1402.600 52317 -3159.400 -3159.100 52515 -789.730 -789.810 52479 -1063.000 -1062.900
52519 -1407.400 -1407.000 52462 -3165.700 -3165.500 52195 -800.090 -799.570 52568 -1104.900 -1104.900
52649 -1409.100 -1408.900 52438 -3178.000 -3177.800 52194 -833.290 -832.810 52569 -1144.000 -1144.000
52639 -1441.000 -1440.800 52427 -3229.100 -3229.000 52669 -888.830 -888.790 52584 -1283.500 -1283.500
52640 -1457.300 -1457.100 52461 -3231.400 -3231.100 52591 -929.140 -929.150 52478 -1321.000 -1320.900
52638 -1463.500 -1463.300 52430 -3261.800 -3261.600 52587 -937.750 -937.720 52589 -1330.700 -1330.600
52521 -1475.800 -1475.500 52316 -3392.500 -3392.200 52635 -947.550 -947.440 52580 -1334.200 -1334.100
52637 -1521.100 -1520.900 52298 -3448.200 -3447.900 52666 -949.270 -949.250 52583 -1360.600 -1360.600
52530 -1576.500 -1576.300 52459 -3474.600 -3474.300 52475 -962.060 -961.970 52574 -1429.000 -1429.000
52636 -1606.000 -1605.700 52440 -3530.200 -3530.000 52680 -969.230 -969.170 52573 -1570.200 -1570.200
52439 -1656.700 -1656.500 52315 -3585.100 -3584.800 52639 -1054.500 -1054.300 52572 -1576.900 -1576.900
52299 -1674.000 -1673.900 52299 -3654.200 -3653.900 52607 -1185.500 -1185.400 52582 -1600.900 -1600.900
52319 -1696.200 -1695.800 52467 -3854.600 -3854.300 52667 -1243.800 -1243.800 52477 -1725.900 -1725.800
52469 -1748.800 -1748.500 52439 -3960.500 -3960.300 52590 -1601.100 -1601.100 52577 -1849.200 -1849.200
Minim -1748.800 -1748.500 Minim -3960.500 -3960.300 Minim -1601.100 -1601.100 Minim -1849.200 -1849.200
resorting to substructuring that was about 415 seconds long process for obtaining raft-element results. Whereas once substucturing was resorted to, the generation pass of super-element required about 1408 seconds, followed by use pass and solution pass taking 139 seconds together; on the whole, solution of raft model was obtained in 1547 seconds. Here the super-element (i.e. superstructure) was not expanded since the priority was for analysis and design of raft and not superstructure. So for single trial-set (raft’s thickness etc) substucturing has taken longer, but for subsequent multiple trials or several runs of geometrical non-linear analyses for the same load-vector it would be beneficial in terms of saving in overall analysis time. Efficacy of substructuring lies in such repetitious analysis runs.
CONCLUSION
On NLSA, the solution results of sub-structure model are reasonably close to original model. But overall solution time for the sub-structured model would be appreciably less than original model particularly when number of steps are higher than one. This has got major practical significance because, condensed stiffness matrix and load-vector of the super-element which is function of few MDOF at the interface nodes is utilised by the analysis process in all the iterations of the NLSA. Thus size of global stiffness matrix and nodal displacement vector is reduced during the solution phase of mathematical process, besides time required for solution of equilibrium equations is reduced since the unknown displacement vector is limited to nodes of raft-elements, Soil-springs and MDOF of the Super-elements.
Thus this paper has demonstrated efficacy of Sub-structuring technique that it can be used conveniently in cases where particular portion of interest (raft foundation in current case) in the overall structure is retained as non-super-element and rest of the linear portion (Superstructure) is sub-structured into non-super-element. During intermediate analysis runs of sub-structured model non-SE portion could be redesigned for structural design, sizing etc. And only in final analysis run, the super-element region is expanded back to its original form and element-stress resultants and nodal results could be extracted.
For the analyst of large-sized mathematical models, it is prudent to explore whether substructuring would provide a cost-effective solution when compared with overall time required for conventional analysis. Once it is observed that on the whole substructuring would prove more economical then, option of multiple superelements could also be explored. Such studies for simplified loading could prove beneficial before large sized loading input.
Thus, the sub-structuring technique as discussed above, presents a feasible and cost-effective computational recourse enabling the incorporation of effect of super-structure in the analysis and design of raft-foundation of such large size structural systems with simulation of SSI.
REFERENCES
