WP2 / TG4
Analysis of Concrete Gravity Dam using the Finite Element Program DIANA
Dr Terry Bennett Cardiff University +44-29-2087-6322 benn nnet et t t 2@cf . ac. uk email: be
1.0 Introduction This document provides details of the analysis of a concrete gravity dam using the commercial finite element program DIANA. The model of the concrete gravity dam chosen for analysis is based upon Linsbauer and Bhattacharjee (1999), whose focus was upon the effect of uplift pressures at the dam-foundation interface on the dam safety. Here similar geometry is employed, but a perfect bond between the dam and foundation is initially assumed, followed by a frictional interface between the dam and rock foundation. 10 m
m 0 1
m 0 8
Concrete
120 m
m 0 8
60 m
120 m
Rock
Fig 1. Dam and Foundation Dimensions
2.0 Finite Element Analysis 2.1 Assumptions
The dam is 80m in height with a base width of 60m. The dam is considered long enough to be idealised as a plain strain problem. 2.2 Description of Finite Element Mesh
The mesh consists of 896 8-noded quadrilateral plain strain elements, with a total of 5702 degrees of freedom. This increases with the introduction of 10 6-noded quadratic interface elements along the dam-foundation interface to 5744 degrees of freedom (fig 2.).
Fig 2. Dam Benchmark Mesh
The model is fully fixed at the base and sides of the foundations (fig 3.).
Fig 3. Boundary Conditions 2.3 Material Properties
The continuum material properties used for the analyses are summarised below in Table 1: Material Parameters 2 Young’s Modulus (N/m ) Poisson’s Ratio 3 Mass Density (kg/m ) 2 Compressive Strength (N/m ) 2 Tensile Strength (N/m ) Mode I Fracture Energy (N/m) Tensile Softening Curve Compression Curve (Constant) Shear Retention β
Concrete 24.0e9 0.15 2400 24.0e6 1.5e6 100 Hordijk Thorenfeldt 0.01 Table 1. Material Parameters
Rock 41.0e9 0.1 2200 -
The foundation material is here treated as a simple linear elastic isotropic medium. Many choices exist within DIANA as to how materials may be modelled using the total strain based crack models. The basic parameters to be set are the isotropic, linear elastic constants, the uniaxial compressive and tensile strengths and the Mode I fracture energy Gf1. The user subsequently has
the option to alter the nature of the nonlinear behaviour through the choice of the type of crack model, tensile softening curve, compression curve and shear retention. There exist in DIANA three different implementations of the total strain based crack models, namely fixed crack, rotating crack and non-orthogonal crack models. In this investigation only the first two have been employed. The choices available for the tensile behaviour include linear elastic, ideal plastic, brittle, linear softening, exponential softening, softening after Hordijk (1991), multilinear softening and by the use of a user supplied subroutine.
The compression behaviour of a material may be modelled as linear elastic, ideal plastic, compression softening after Thorenfeldt (1987), linear hardening, multilinear hardening, saturation type hardening, parabolic softening and by the use of a user supplied subroutine.
The shear behaviour may be modelled using a constant shear retention factor, a multilinear diagram between shear stresses and shear strains, a multilinear diagram betwee n shear retention and shear strains and also by the use of a user supplied subroutine. The material properties for the soil-structure interface are given in tables 2 and 3, below: Material Parameter Coulomb Interface 2 Dummy elastic stiffness 20e12 N/m Cohesion 0.7e6 N/m2 o Friction angle 30 Dilatancy angle 10o Table 2. Frictional Interface Properties Material Parameter Discrete Crack Interface 2 Tensile Strength 0.35e6N/m Tension Softening Linear Fracture Energy 50 M/m Unloading Secant Shear Behaviour Constant Shear Modulus 8.33e9 Table 3. Discrete Crack Interface Properties
2.4 Boundary Conditions
In the interests of both computational efficiency and simplicity the foundation material is truncated horizontally 120m from each of the dam faces and vertically at 80m depth below the base of the dam. The degrees of freedom on the boundary are fully fixed.
A more realistic representation of the boundary conditions could be performed by utilising Infinite Elements (or Boundary Elements coupled to the finite element mesh) to create a linear elastic halfspace. In DIANA Infinite Elements are in a pilot state of development and are not generally available, however, consistent spring elements are available to model the stiffness of the far-field founding material. 2.5 Description of Loads
Two load cases are applied to each model, firstly a self weight analysis is performed to determine the insitu stress distribution in the dam and foundations, the displacements arising from the self weight are reset to zero prior to applying a hydrostatic load to the dam structure. A number of analyses are performed for each material model with the hydrostatic loading varied to show the effects of increasing reservoir water level. 2.6 Material Models
A number of different crack models are available in DIANA, the dam analysis is repeated for each of these models as well as with a frictional interface and a discrete crack interface between the dam and foundation material. A brief description of each of these models, from de Witte and Kikstra (2003), is given below: 2.6.1 Total Strain Based Fixed Crack Model
A constitutive model based upon total strain describes the stress as a function of the strain, i.e. the stresses are evaluated in the directions which are given by the crack directions. In the fixed stressstrain concept, the stress-strain relationships are evaluated in a fixed coordinate system which is fixed upon crack initiation. Beyond the strain at which cracking occurs, loading and unloading takes place elastically according to the secant modulus. 2.6.2 Total Strain Based Rotating Crack Model
The rotating crack model uses the coaxial stress-strain concept, whereby the stress-strain relationships are evaluated in the principal directions of the strain vector. Thus, the direction in which the stresses are evaluated is able to rotate as an analysis progresses. The elastic secant modulus is again used in loading and unloading in the softening region. 2.6.3 Frictional Interface
The soil-structure interface is modelled using interface elements (dual nodes). Relatively high dummy elastic properties are defined to minimise deformation prior to the initiation of slipping and/or gap opening behaviour. The interface between the concrete structure and the rock foundations is modelled using a Coulomb friction law. The basic input parameters for this model are the cohesion, the (tangent of the) friction angle and the (tangent of the) dilatancy angle.
In addition to the standard Coulomb model it is possible to introduce a gap criterion, whereby the normal traction is set to zero if the tensile traction normal to the interface exceeds a certain critical value. The shear behaviour of the interface after the appearance of a gap is controlled by a constant shear retention. In this investigation this feature has not been utilised. 2.6.4 Discrete Crack Interface
A discrete crack interface is introduced between the dam structure and the founding rock material. The primary input for the discrete crack model is the tensile strength of the interface and the form of the tension softening criteria (either brittle, linear, Hordijk, see Cornielissen et al (1986), or multi-linear). For the models other than brittle the fracture energy is required as input. The unloading/reloading behaviour of the interface is here modelled using the secant stiffness, however, it is also possible to use the linear elastic stiffness or include hysteresis using the function of Hordijk, see Cornielissen et al (1986). The shear behaviour after cracking maybe controlled by specifying either zero shear stiffness or a constant shear modulus. In this investigation a constant shear modulus has been employed. Crack dilatancy may also be modelled in DIANA, but is not investigated here. 2.7 Uplift Pressures
The potential for uplift pressures on the base of the dam has not been considered in these analyses. The uplift pressures could be introduced by applying pressure profiles as used by Linsbauer (1999), however, with the loading been dependant upon the extent of the opening at the dam-foundation interface, this would involve some manual incrementation.
2.8 Solution Algorithm
A regular Newton-Raphson iteration scheme is adopted as the solution algorithm. Load steps sizes are specified explicitly, generally at one tenth of the total load (decreasing near failure), however, some fluctuation away from the desired step size occurs due to application of the arc length method.
The arc length method is used to accommodate local sanp-through and snap-back behaviour. In an ordinary iteration process the predictions for the displacement increments can become very large. The Arc-length method constrains the norm of the incremental displacements to a prescribed value. This is done by simultaneously adapting the size of the increment. Note that the size is adapted within the iteration process and is not fixed at the moment the increment starts (de Witte and Kikstra 2003).
A tolerance of 1e-6 on the force norm has been used as the convergence criteria for all of the analyses performed. 2.7 Pre and Postprocessing
Pre and Postprocessing with DIANA takes place within the newly developed iDIANA. The majority of the modelling (meshing, assigning material and geometric properties, applying boundary conditions and external forces etc) and results processing is derived from the FEMGV suite of tools which has been fully integrated into a single graphical user interface along with new interactive analysis control and execution facilities.
This new fully integrated GUI is a recent development available with DIANA v8.0 (current v8.1), and represents a significant step forward in terms of usability. However, for users already familiar with the DIANA input file and command file syntax the use of DIANA from a command prompt is still, and will remain, available to users. The production of input files and command files by the GUI, even when DIANA is used in a fully interactive manner, is still very important for the purposes of quality assessment in engineering practice.
3.0 Results 3.1 Deformed Mesh
Fig 2. Fixed Crack Deformed Mesh (90m Load)
Fig 3. Rotating Crack Deformed Mesh (90m Load)
Fig 4. Frictional Soil-Structure Interface Deformed Mesh (80m Load)
Fig 5. Discrete Crack Interface Deformed Mesh (80m Load)
3.2 Crest Displacement
In order to directly compare the effect of the different material models on the dam performance, the displacement of the dam crest is chosen as a scalar indicator. The displacement of the dam crest with increasing hydrostatic loading for each model is shown in fig. 6, below:
Dam Crest Displacement for Differing Load Levels 95
90
85
) m ( l e v e L 80 r e t a W
Total Strain Crack Models Interface Models
75
70
65 7.00
9.00
11.00
13.00
15.00
17.00
19.00
Crest Displacement (mm)
3.3 Stress Distribution
The major and minor principal stress distributions are shown in figures 7 and 8 for the fixed crack model subjected to 90m hydrostatic loading.
Fig 7. Fixed Crack Major Principal Stress (90m Load)
Fig 8. Fixed Crack Minor Principal Stress (90m Load)
Figures 9 and 10 show the distribution of the major and minor principal stresses for the rotating crack model subjected to 90m hydrstatic loading.
Fig 9. Rotating Crack Major Principal Stress (90m Load)
Fig 10. Rotating Crack Minor Principal Stress (90m Load)
In this example the differences between the two total strain based crack models is not significant. Figures 11 and 12 show the major and minor principal stress distributions for the model including a frictional soil-structure interface
Fig 11. Coulomb Interface Major Principal Stress (80m Load)
Fig 12. Coulomb Interface Minor Principal Stress (80m Load)
Figures 13 and 14 show the major and minor principal stress distributions for the model including a discrete crack soil-structure interface
Fig 13. Discrete Crack Interface Major Principal Stress (80m Load)
Fig 14. Discrete Crack Interface Minor Principal Stress (80m Load)
The stresses for the interface models are shown at a 80m hydrostatic load. Due to the differing nature of the failure mode, when full bond between the structure and the founding rock material is no longer assumed, the principal stress distributions are altered. In addition the ultimate load that the structure can resist is much reduced, a 90m hydrostatic load was attempted, however, it was not possible to obtain a solution. 3.4 Crack / Interface Gap Propagation
The crack / interface strains are shown here to indicate the failure modes of the dam with the differing constitutive models employed. Each picture depicts the crack / interface strains for the maximum hydrostatic load sustained.
Fig 15. Fixed Crack – Crack Strains (90m Load)
Fig 16. Rotating Crack – Crack Strains (90m Load)
Fig 17. Frictional Interface – Interface Strains (80m Load)
Fig 18. Discrete Crack – Interface Strains (80m Load)
4.0 Conclusions This report demonstrates the capabilities of the commercial finite element code DIANA for the analysis of concrete dams. A number of constitutive models available in DIANA suitable for the analysis of concrete and soil-structure interface properties have been performed. A guide to other useful features available has been given alongside the description of the models employed.
6.0 References Linsbauer H.N. and Bhattacharjee S. Dam Safety Assessment due to Uplift Pressure Action in a th
Dam-Foundation Interface Crack , in 5 Benchmark Workshop on Numerical Analysis of
Dams, June 2-5 Denver Colorado USA, (1999).
de Witte, F.C. and Kikstra, W-P, DIANA Finite Element User’s Manual: Analysis Procedures nd
(release 8.1, 2 Ed.), TNO DIANA b.v., (2003)
Hordijk, D. A. Local Approach to Fatigue of Concrete, PhD thesis, Delft University of Technology, (1991)
Cornelissen, H. A. W., Hordijk, D. A., AND Reinhardt, H. W. Experimental determination of crack softening characteristics of normalweight and lightweight concrete. Heron 31, 2 (1986).
Thorenfeldt, E., Tomaszewicz, A., and Jensen, J. J. Mechanical properties of high-strength concrete and applications in design. In Proc. Symp. Utilization of High-Strength Concrete (Stavanger, Norway) (1987)
Filename: DIANA Analyses Directory: H:\Documents\IALAD Template: C:\Documents and Settings\Hedwig\Application Data\Microsoft\Templates\Normal.dot Title: WP2 / TG4 Subject: Author: INFOS Keywords: Comments: Creation Date: 7/25/2003 12:58 PM Change Number: 78 Last Saved On: 9/22/2003 11:29 AM Last Saved By: Hedwig Total Editing Time: 624 Minutes Last Printed On: 9/22/2003 11:29 AM As of Last Complete Printing Number of Pages: 14 Number of Words: 2,242 (approx.) Number of Characters: 12,246 (approx.)
