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The contraction joints of arch dams with and without shear keys are simplified to be with no-slip condition and with relative sliding condition, respectively. Based on the Lagrange multiplier method, a contact model considering the manner of independent cantilever dead load type with no-slip condition and relative sliding condition is proposed to model the nonlinearities of vertical contraction joins, which is special to the nonlinear analysis of arch dams considering the manner of dead load type. Different from the conventional Gauss iterative method, the strategy of the alternating iterative solution of normal force and tangential force is employed. The parallelization based on overlapping domain decomposition method (ODDM) and explicit message passing using distributed memory parallel computers is employed to improve the computational efficiency. An existing high arch dam with fine finite element model is analyzed to investigate the effect of shear sliding of vertical joints on seismic response of the arch dam. The result shows that the values of maximum principal tensile stress under relative sliding condition are significantly greater than those under no-slip condition.

Due to the existence of the thermal stress of mass concrete, arch dams are divided into several dam sections by contraction joints with or without shear keys. Under strong earthquake, the joint opening and sliding may happen. Since the shear keys are used to increase shear resistance, the effect of shear keys should be considered in analysis. Extensive research about the nonlinear analysis of joints has been conducted. Lau et al. [

The study on the parallel solution of dynamic finite element analysis has been widely explored. Belytschko et al. [

In this paper, the joints with and without shear keys are simplified to be with no-slip condition and with relative sliding condition, respectively. The contact model adopted is discussed with no-slip condition and with relative sliding condition for contraction joints of arch dams considering the manner of independent cantilever dead load type firstly. Secondly, the massed foundation model with viscoelastic artificial boundary and nonuniform ground motion input is explained. Thirdly, the parallelization strategy employed to increase the computational efficiency is described and the performance of parallel computation is tested. Finally, an existing high arch dam with fine finite element model is analyzed to investigate the effect of shear sliding of vertical joints on seismic response of the arch dam.

In this paper, the manner of the independent cantilever dead load type is adopted instead of staged construction and sequence of grouting during construction process of arch dams for simplification. The dead load is assumed to be applied to individual cantilevers without considering transferring force between cantilevers. After joint grouting, water load and other loads such as seasonal temperature load are applied. The equations considering the manner of the independent cantilever dead load type for arch dams including Lagrange multiplier can be written as

The detailed information of the above matrix and vector can refer to [

The contact force equation can be obtained from (

The normal and tangential contact force can be solved iteratively by (

The contraction joint has no tensile strength and can only be capable of transferring compressive force in the normal direction. For no-slip condition with shear keys, unlimited force is assumed to be transferred in the tangential direction. For relative sliding condition without shear keys, Coulomb model is suitable for tangential movement.

For no-slip condition with shear keys, the solution of each iterative step is as follows.

Firstly, the normal contact force is obtained as follows:

If

If

Secondly, the tangential contact forces are obtained as follows:

There is no limit in the tangential force, and no correction is needed. The iterative error can be got directly by

For relative sliding condition without shear keys, Coulomb friction criterion is introduced to the iteration process to model sliding of contraction joints without shear keys. The solution of each iterative step is as follows.

Firstly, the normal contact force is obtained as follows:

If

If

Secondly, the tangential contact forces are obtained as follows:

Then,

If

If

After the iteration is completed, the displacement can be obtained through substituting the contact force into the dynamic equation. Obviously, there is no need to introduce any penalty value for no-slip and relative sliding conditions during the solution process, which avoids the potential stability and uncertainty of numerical calculation caused by parameter selection. The model is special to the nonlinear analysis of arch dams considering the manner of the independent cantilever dead load type.

The viscoelastic artificial boundary is adopted to consider the effect of radiation damping of far-filed foundation [

Massed foundation model with viscoelastic boundary [

As shown in Figure

Boxed-shape model.

The basic idea of domain decomposition method (DDM) is to divide a complex computing system into several subsystems by using the strategy of partitioning. The solution of the original system is transformed into the solution of the subsystems, and data exchange is achieved among subsystems through message passing.

The DDM consists of overlapping and nonoverlapping domain decomposition method (ODDM and NODDM). The former is based on Schwarz alternation method, and the overlapping region is used to pass message among different partitions. The latter is based on substructure method, and the interface of each partition is used to pass message.

In this paper, the parallel programming based on ODDM is presented. The solution in a whole complex region can be divided into that in several overlapping simple partitions based on the idea of breaking up the whole into parts through Schwarz alternating method, which provides a mathematical basis for distributed parallel computing. The basic principle of Schwarz alternating method is illustrated with a simple static plane stress problem. The plane stress problem in an ABCD region is decomposed into that in two overlapping regions including ABGH and CDFE, and EFHG is the overlapping region shown in Figure

The diagram of overlapping domain decomposition for plane static problem. Define ABGH region:

The procedure for the alternative solution method is as follows (Figure

The basic process of Schwarz alternating method.

For the explicit wave equation in ODDM’s solution, no iteration is needed and the result of the next time step is directly solved based on the last time step owing to the decoupling of equation. The procedure is as follows:

ABGH region: based on the displacement and velocity in the ABGH region and the displacement boundary condition on the GH boundary at the last time step, the displacement and velocity in the region at the next time step are solved

CDFE region: based on the displacement and velocity in the CDFE region and the displacement boundary condition on the EF boundary at the last time step, the displacement and velocity in the region at the next time step are solved

EF boundary: the new displacement boundary condition on the EF boundary is obtained based on the new displacement in the ABGH region

GH boundary: the new displacement boundary condition on the GH boundary is obtained based on the new displacement in the CDFE region

Then go to next time step

After partitioning the finite element nodes based on the Metis program [

The process of domain decomposition based on overlapping elements.

As shown in Figure

Overlapping partition.

Node classification of subregion.

The distributed parallel computing is employed, which is suitable for large-scale cluster. It adopts master-slave programming mode and consists of one main process and several slave processes. The master process is a control program, which does not participate in calculation. It is responsible for sending data to the slave processes and receiving and sorting out the data from the slave processes. The message passing occurs among the master processes and the slave processes, and no message passing occurs among the slave processes. The slave processes are only responsible for the calculation of the corresponding subregions. The message passing is realized by blocking communication between computation and communication as shown in Figure

The diagram of parallel computing architecture.

Figure

The box model and domain decomposition into 95 partitions (different color represents different partition).

From Table

Parallel computation test value.

Number of slave processors | Running time (h) | Speedup | Parallel efficiency (%) |
---|---|---|---|

1 | 1313.1 (54d17.1 h) | 1.00 | 100 |

2 | 691.1 | 1.90 | 95 |

5 | 303.0 | 4.33 | 87 |

11 | 161.6 | 8.13 | 74 |

23 | 93.5 | 14.04 | 61 |

47 | 58.3 | 22.52 | 48 |

95 | 40.5 | 32.40 | 34 |

The dam-foundation finite element model includes 1182559 nodes, 1078026 elements, and 3547677 DOFs, respectively. The maximum dam height is 285.5 m and the crest elevation is 610 m. Figure

The dam-foundation model.

The dam model.

Vertical contraction joints.

In the present study, the vertical contraction joint is the only source of nonlinearity in the model of the system. The hydrodynamic effect is considered by the modified Westergaard added mass model [

The concrete material parameters are as follows: density

The foundation rock material parameters are as follows: density

Vertical contraction joints are as follows: for without shear keys, the frictional coefficient

The Rayleigh type damping with damping ratio of 5% is selected for analysis.

The load condition can be seen in Table

Load condition.

Upstream water level | 540 m |

Sediment level | 490 m |

Horizontal PGA | 431 gal |

Vertical PGA | 287.3 gal |

The seismic wave time histories in three directions. (a) Cross-section direction. (b) Stream direction. (c) Vertical direction.

No-slip condition of vertical contraction joints with shear keys: only opening and closing at the joints are considered

Relative sliding condition of vertical contraction joints without shear keys: opening and closing and sliding at the joints may happen

Figures

Note that the maximum principal tensile stresses around the middle and upper elevations of the upstream surface are dominated by vertical stresses under two kinds of conditions. The values of maximum principal tensile stress and vertical stress under relative sliding condition are significantly greater than those under no-slip condition. Under no-slip condition, the maximum principal tensile stress and vertical stress at these locations are 2.7 MPa and 2.3 MPa, respectively, while those under relative sliding condition are 6.5 MPa and 5.6 MPa.

Joint opening at the upstream side of the dam crest under two conditions.

Joint opening at the downstream side of the dam crest under two conditions.

Time history of joint opening at the downstream side of the dam crest for #16 joint.

Time history of joint opening at the downstream side of the dam crest for #18 joint.

Contraction joint slippage at the upstream and downstream side of the dam crest for relative sliding condition.

Figures

Envelope of maximum principal tensile stress distribution of upstream surface of the dam for no-slip condition [

Envelope of maximum static and dynamic vertical stress distribution of upstream surface of the dam for no-slip condition.

Envelope of maximum principal tensile stress distribution of upstream surface of the dam for relative sliding condition.

Envelope of maximum static and dynamic vertical stress distribution of upstream surface of the dam for relative sliding condition.

Envelope of maximum principal tensile stress distribution of downstream surface of the dam for no-slip condition [

Envelope of maximum static and dynamic vertical stress distribution of downstream surface of the dam for no-slip condition.

Envelope of maximum principal tensile stress distribution of downstream surface of the dam for relative sliding condition.

Envelope of maximum static and dynamic vertical stress distribution of downstream surface of the dam for relative sliding condition.

Note that the maximum principal tensile stress around the middle and upper elevations of the downstream surface is dominated by vertical stress under relative sliding condition, while that under no-slip condition is not dominated by vertical stress. Under relative sliding condition, the maximum principal tensile stress and vertical stress at these locations are 4.2 MPa and 3.8 MPa, respectively. From Table

Maximum principal tensile stress value and its stress component of downstream of the dam under no-slip condition (MPa).

0.35 | −0.01 | −0.34 | −1.09 | −0.02 |

Based on the Lagrange multiplier method, a contact model considering the independent cantilever dead load with no-slip condition and relative sliding condition is proposed to model the nonlinearities of vertical contraction joins, which is special to the nonlinear analysis of arch dams considering the manner of dead load type. There is no need to introduce any penalty value for no-slip and relative sliding conditions, which avoids the potential stability and uncertainty of numerical calculation caused by parameter selection. Different from the conventional Gauss iterative method, the strategy of the alternating iterative solution of normal force and tangential force is employed.

The parallelization based on ODDM and explicit message passing using distributed memory parallel computers is employed to improve the computational efficiency.

The effect of shear sliding of vertical joints on seismic response of a high arch dam in China is investigated by nonlinear comprehensive analysis model of fine finite element. The analysis results show that the shear sliding has slight effect on the joint opening characteristics and significantly changes the stress distribution.

In summary, the values of maximum principal tensile stress under relative sliding condition are significantly greater than those under no-slip condition. For the former, when the contraction joint opening occurs, the tangential movement is free, which aggravates the vibration of the corresponding dam section in the stream direction and the increase of vertical stress. For the latter, the tangential movement between dam sections is restricted, and the integrity is strengthened, which limits the vibration of the dam in the stream direction. In fact, for high arch dams in China, spherical crown-shaped shear keys are mostly adopted, whose mechanism is more complicated for seismic analysis. The two extreme conditions presented in this paper maybe provide the corresponding envelopes.

All data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This paper was supported by the National Key Research and Development Program of China (no. 2017YFC0404903), National Natural Science Foundation of China (no. 51709283), and China Three Gorges Corporation Research Project (contract no. XLD/2115).