Principia presente en SMART2013

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María José Crespo presenta en París los resultados obtenidos por Principia utilizando Abaqus del benchmark SMART2013

Seismic design and best-estimate Methods Assesment for Reinforced concrete buildings subjected to Torsion and non-linear effects

Smart 2013 Benchmark

Maria J Crespo, Javier Rodriguez, Luis Lacoma, Francisco Martínez, Joaquín Martí
PRINCIPIA Ingenieros Consultores, Velázquez 94, 28006 Madrid, Spain

1. Introduction

There are considerable uncertainties involved in the prediction of the response of reinforced concrete structures to seismic events. To advance the knowledge and capabilities in that regard, CEA, EDF and IAEA organised the international benchmark named the SMART 2013 project.
The exercise involves a number of shaking table tests carried out in a mock-up, as well as predictive calculations regarding the outcome of those tests. Principia participated in the project by conducting the necessary calculations for three of the four stages and the present paper summarises the findings.

2. Description of the model (Stage #1)

Both the geometry and material properties assigned to the model were described in Stage #1. However some changes were introduced in the course of the project, as suggested by the results already obtained or by requirements of the analyses.

2.1 Materials

The mock-up involves two materials: concrete and reinforcing steel. In stage #1, the material tests were not yet available and the material properties used in the analyses were conventional design values.
There are several material models in Abaqus [SIM13] that are suitable for modelling concrete. That selected was the “smeared crack concrete model” available in the implicit code Abaqus/Standard. It provides a general capability for modelling concrete in all types of structures, including beams, trusses and solids. The model consists of an isotropically hardening yield surface that is active when the stress is dominantly compressive and an independent “crack detection surface” that determines if a point fails by cracking. It uses oriented damaged elasticity concepts (smeared cracking) to describe the reversible part of the material’s response after cracking failure.
A parabolic stress-strain law was used, with an initial modulus of 32 GPa, a peak stress of 30 MPa at 0.22% strain, and an ultimate strain of 0.35%, as presented in Figure 1. The tensile strength is 2.4 MPa with an assumed fracture energy of 180 J/m2. The resulting biaxial behaviour is shown in Figure 2.

uniaxial-behaviourDue to convergence problems experienced in some of the runs in Stage #3, a decision was made to change from implicit to explicit integration, using Abaqus/Explicit. Since this module does not support the former material model, the “concrete damaged plasticity” model was used instead. It combines non-associated multi-hardening plasticity and scalar (isotropic) damaged elasticity to describe the irreversible damage that occurs during the fracturing process. It is meant to provide an accurate approach for modelling cyclic events, allowing the user control over the stiffness recovery effects.

The reinforcing steel was modelled as an elastoplastic material with isotropic (von Mises) plasticity. Some experimental information had been provided, namely the Young’s modulus, the elastic and shear strengths, and the ultimate strain. Although the simulations of the material tests were performed with reinforcing bars embedded in the concrete elements, those of the mock-up used the “rebar layer” technique, which does not consider individual bars but the averaged stiffness effects on the concrete elements.

biaxial-behaviour-concrete2.2 Finite element mesh

The mesh used to represent the mock-up consists mainly of shell elements (8.559); a few (230) linear beam elements were also used for the pillar and the three beams on the floors.

The shaking table has 10,253 elements, as in the mesh provided by the organising committee (OC). The total number of nodes is 20,247. Figure 3 presents two views of the mesh.

abaqus_mesh

 

3. Linear Dynamic Analysis (Stage #2)

In Stage #2 linear dynamic analyses were performed. The first activity consisted in conducting modal analyses under three different hypotheses:

1.Mock-up fixed at its base with no additional masses.
2.Mock-up fixed at its base and loaded with additional masses.
3.Mock-up fixed to the shaking table and loaded with additional masses.

The frequencies corresponding to the first three natural modes calculated for each of the three above hypotheses appear in Table 1.

Table 1 Calculated and measured frequencies (Hz) under three hypotheses

Mode 1

Mode 2

Mode 3

Fixed base, no masses

19.9

35.4

65.4

Fixed base, with masses

8.40

15.4

29.7

Fixed to shaking table, with masses

5.65

8.76

19.0

Experimental

6.28

7.86

16.5

The first three modal shapes are similar under the three hypotheses. The first mode is the main one in the X-direction (see axes in Figure 3), the second one is the main one in the Y-direction, and the third one is a torsional mode around the vertical Z-axis. Figure 4 presents the three modal shapes for the third hypothesis.
The frequency differences observed between the first two hypotheses are consistent with the increase in the global mass of the model. The differences between the second and third hypothesis arise from the flexibility of the shaking table.

mock-up-modes4OC provided the participants with experimentally determined frequencies, which are also indicated in Table 1. The differences between the calculated and experimental frequencies are probably due to the material properties which, at this stage, had to be given standard values; experimentally determined properties only became available later.

This stage also included a linear time domain analysis for two time histories (identified as Runs 6 and 7), both with a PGA of 0.10g. Run 6 is a synthetic white noise, while Run 7 is a scaled design signal.
The input motion was introduced in the calculation by imposing displacements at the actuators, i.e.: creating an additional node linked with a distributing coupling to the four nodes of the shaking table surrounding the actuator. This type of connection distributes forces but does not introduce any stiffness.
Modal dynamic analyses were conducted for the different base motions and associated input motions at the actuators. The damping was assigned on a mode-by-mode basis, using the information provided in the experimental report by OC: 2.6% for the first mode, 4.2% for the second, and 5.5% for the third one. A damping ratio of 6% was assigned to all the higher modes. The results were presented in terms of absolute displacements and accelerations.
In general, for this stage the results obtained were of the same order as those provided by OC. Space limitations do not allow an exhaustive comparison, but a sample result has been selected for display, which is the horizontal X-displacement of point A in floor 3. Figure 5 provides the comparison of calculated and measured results for Run 6. The rest of the comparisons presented in the paper correspond to that same location.

4. Nonlinear Dynamic Analysis (Stage #3)

Convergence problems were experienced with Abaqus/Standard for some of the runs contemplated in this stage. The concrete cracking caused a sudden release of the internal energy inducing a reduction of the time increment to the order of 10 s. The convergence could be improved by changing the concrete fracture law or by adding numerical damping but, since such strategies would alter the structural behaviour, a decision was made to change to the explicit solver in Abaqus/Explicit [SIM13]. The time increment is now around 2 s, smaller than in Abaqus/Standard but much faster per increment since no matrix needs to be inverted or factorised. Besides, the increment is almost not reduced during the simulation.

smart2013-xdisplacements

As mentioned earlier, this required changing the material model. The main problems posed by the new formulation of the concrete behaviour are that the reversals are somewhat stiffer than expected and that, with increasing numbers of cycles, the equivalent plastic strain may ratchet up with little physical basis.
Some results are presented below for all the mandatory runs, which are no. 9, 13, 17, 19, and 23. All the results included here correspond to the same X-displacement of point A in floor 3. For each of the five runs mentioned, the comparisons appear in Figure 6 through Figure 10.
The large displacement part of the response is considered to be generally satisfactory in all cases, perhaps particularly in Runs no. 13, 17 and 19. However, there are important differences in the amplitudes of the oscillations during the second half of the history which, although their significance may not be very important, they are still not easy to explain.

SMART2013_xdisplacement-run9 SMART2013-xdisplacements-run13 SMART2013_xdisplacements-run17

SMART2013_xdisplacements-run19

SMART2013_xdisplacements-run23

 

5.Conclusions

Calculations were conducted for the first three stages of the International Benchmark SMART 2013, which involve predictions of the response of a mock-up in a series of shaking table tests. As a result of the work conducted, the following conclusions can be offered:

a) The calculated modal frequencies reproduce only moderately those determined experimentally. The differences are partly attributed to the fact that, at the time, standard rather than experimentally determined material properties had to be used.

b) The low-amplitude time history analysis did provide a reasonable match of the experimentally determined response.

c) The strong non-linearities and softening caused by the large-amplitude motions in stage #3 gave rise to convergence problems in Abaqus/Standard which could not be satisfactorily solved while maintaining the material description. This caused a migration to Abaqus/Explicit as solver, which also entailed moving from a smeared crack model to a damage plasticity model for the concrete.

d) Finally, the calculations performed in stage #3 agreed reasonably well with the experimental results at least for the large displacement part of the response, though the amplitude of the oscillations in the second half of the history is somewhat less satisfactory.

6. References

[SIM13] SIMULIA (2013) “Abaqus Analysis User’s Guide”, Version 6.13, Providence, Rhode Island.
[RIC13] Richard, B. and Chaudat, T. “Presentation of the SMART 2013 International Benchmark”, Document no. DEN/DANS/DM2S/SEMT/EMSI/ST/12-017/E, 9 September.