Dr Martin Wieland explores the problem of tensile stresses in concrete dams
Tensile stresses in concrete dams have been a major concern for dam engineers for a number of years. It is recognised today that tensile stresses cannot be avoided during strong earthquake ground shaking and also under extreme temperature variation, especially in winter when the outside temperatures drop considerably.
In addition, there are locations with stress concentrations such as the upstream and downstream base points of concrete dams. High tensile stresses are undesirable because they cause cracks in concrete dams and because the tensile strength is only a fraction – usually less than 10% – of the uniaxial compressive strength of mass concrete. Under usual static loads such as the combination of dead load, water load of the reservoir and temperature action, the resulting tensile stresses are much smaller than the compressive stresses and the tensile strength of mass concrete is not exceeded if simplified numerical models of the dam are used. However, under seismic actions the maximum tensile and compressive stresses are almost equal due to the oscillating nature of the earthquake ground motion.
Using a linear-elastic model of a dam and its foundation, the highest static tensile stresses occur at sharp corners such as the upstream and downstream edges of the dam-foundation contact and at locations with sudden change of slope of a dam. Under earthquake action high tensile stresses also occur in the central upper portion of dams – especially in arch dams.
High dynamic tensile stresses also develop at locations where high static tensile stresses already exist. In arch dams the highest static tensile stresses occur along the upstream heal due to water load. It should be mentioned that in a linear-elastic dam model, which is welded to the elastic foundation rock, stress singularities occur at ‘negative’ (inwards) corners. Theoretically these stresses are infinite at the corner. This causes a serious problem for dam engineers as the formation of cracks has to be expected in such locations. Also, due to their simplicity, dam engineers prefer linear-elastic models for their structural analyses.
With the use of the finite element method (FEM) it has become possible to calculate stresses in dams more accurately than in the past, where, for example, for arch dams the trial load method was used to determine the stresses in arches and cantilevers, which were modelled as equivalent beams.
In gravity dams the analysis has mainly focussed on the sliding and overturning stability of concrete blocks and the location of the resultant of the applied loads (dead load and water load), which had to be located within the core of the dam base. This ensured, using a rigid body analysis, that the base section was under compression and the dam was ok.
Using FEM and a linear-elastic model of the dam-foundation system leads to stress concentrations with high (infinite) tensile stresses at corners. Moreover, high dynamic tensile stresses are calculated in the central upper portion of arch dams when a full dynamic dam analysis is carried out.
To circumvent the problem of high tensile stresses obtained from a linear-elastic analysis the following (often questionable) arguments, methods and concepts have been used by dam engineers:
• Use of FE models with coarse mesh in corner regions and calculation of stresses in stress points of finite elements rather than at the element surface. Using such models the stress concentrations become essentially ‘invisible’ and the stresses in the corners do not appear as stress singularities or zones of very high stresses.
• Calculation of stresses in stress points of finite elements rather than at the element surface. It is argued that the surface concrete layer is weakened by microcracks due to daily temperature variations and freezing and therefore stresses at the stress points are more representative than the actual surface stresses. Relatively large differences between the stresses in stress points and surface points exist if, for example only one or two FE layers are used across the thickness of an arch dam.
• Use of beam models to analyse arch dams (trial load method). Due to the assumptions of the beam theory (plane sections) no stress concentrations occur at the dam-foundation interface and at locations with sudden changes in the slope of cantilevers.
• Use of high static and dynamic tensile strength values to ensure that the calculated tensile stresses are less than the tensile strength. This has been achieved by postulating an apparent tensile strength of mass concrete, which should account for any discrepancies between the nonlinear stress-strain curve of mass concrete in tension and the linear-elastic analysis of the dam. However, the apparent tensile strength is not a physical material property of mass concrete, which can be determined experimentally, and thus shall not be used for assessing tensile stresses in dams.
• Use of mean values of tensile strength rather than 5% fractile values etc. This may be acceptable for extreme events with low probability of occurrence.
• Taking into account strength increase due to ageing of concrete, e.g. use of 10 years strength values for assessing stresses caused by an earthquake with an average return period of 100 years etc. This assumption is acceptable although it may be argued that the 100 year event might occur tomorrow.
• Reduction of the return period of design earthquakes and using optimistic (non-conservative) values of the earthquake ground motion (attenuation laws, peak ground acceleration, shape of response spectrum, duration of strong ground shaking).
• Use of high damping values for seismic analysis, e.g. for arch dams the following values have been used: 7% damping for the operating basis earthquake (OBE) and 12% or more for the safety evaluation earthquake (SEE). These values have been justified by radiation damping effects into the reservoir and the foundation. However, in situ small-amplitude vibration measurements at high arch dams where all radiation damping effects were included have shown damping ratios for the fundamental mode of vibration of 1-2% of critical damping (or even less). The damping value is the most important factor, which affects the earthquake response of a linear-elastic concrete dam.
• Use of pseudostatic seismic analysis with a seismic coefficient of 0.1 with the argument that most existing dams have been analysed with that method and no dam designed with this method has actually failed during an earthquake. The pseudostatic analysis method according to Westergaard, which disregards the dynamic response of the dam, leads to relatively small tensile stresses, therefore, the engineer does not have to cope with the tensile stress issue at all. This argument is mainly heard from engineers in regions of moderate to low seismicity, where no destructive earthquakes have taken place for centuries.
• The high tensile stresses at stress singularities (‘negative’ corners) are considered as purely theoretical in nature with no engineering relevance for dams. Stress concentrations are assumed to disappear due to the nonlinear behaviour of mass concrete and the possible opening/separation of the dam-foundation contact near the upstream heel of the dam. However, in this case a linear-elastic analysis is no longer valid for justifying this argument.
This list may be incomplete but the main issues are addressed. Using the above justifications and analysis concepts it is possible to arrive at very different tensile stresses in a concrete dam and, thus, different assessments of the safety of a dam.
Which of the above assumptions and concepts are correct or appropriate depends to a large extent on engineering judgement, as clear guidelines and concepts are still missing. However, most of the above arguments are questionable and shall not be used.
The main problem is that the application of linear-elastic static and dynamic FE analyses, which represent the state-of-the-practice in concrete dam design, leads to undesirable tensile stresses. In most cases the calculated tensile stresses are larger than the static and/or dynamic tensile strength of mass concrete. Thus the formation of cracks would be a direct consequence. As concrete dams are
constructed with numerous lifts the strength properties at the lift joints are less than in the parent mass concrete. The same applies to the grouted vertical contraction joints in arch dams, which exhibit lower strength properties than the mass concrete. Therefore, cracks in the dam body will primarily develop along lift and contraction joints.
Further weak zones are the dam-foundation contact and joints and fissures in the foundation rock, which will crack or open prior to the formation of cracks in the mass concrete and the compact rock. Once cracks have formed, all deformations will be concentrated at these cracks and joints and the other parts of the dam and the rock will be protected from increased tensile stresses if the applied loads and actions are increased. Therefore, the rational
interpretation of the results of a linear-elastic FE analysis is a major problem especially when further nonlinear analyses are not carried out.
As lift and contraction joints are expected to open in the highly stressed upper central portion of arch dams under seismic actions – even under smaller earthquakes like the OBE – the post-cracking behaviour of a dam should be assessed. In such an analysis it may have to be assumed that some concrete blocks are separated by joints and cracks.
Significance of tensile stresses in concrete dams
Mass concrete is a brittle material and cracks will first develop at locations with stress concentrations and along weak zones in
the dam body (lift joints, dam-foundation contact, contraction joints).
A distinction has to be made between cracks due to sustained loads and cracks due to short-duration seismic actions. Static cracks are expected at the upstream heel of large arch (and gravity) dams due to the effect of water load. This crack is most critical for relatively thin and very high arch dams as the hydrostatic pressure in the crack or joint will lead to further propagation or opening of any crack or joint respectively. In gravity dams, due to the large base thickness a crack at the dam-foundation interface is of much smaller concern than in an arch dam of similar height.
Under the SEE large tensile stresses are expected to occur in most large arch dams, which exceed the dynamic tensile strength of mass concrete. In this case cracks will form along a few horizontal lift joints and the vertical contraction joints. Opening of these joints can be accepted as they will close after an earthquake. Prevention of such cracks in an existing dam is almost impossible without making major (expensive) changes in the dam (adjustment of dam geometry, installation of post-tensioning anchors, etc.).
One has to accept that even in unreinforced concrete dams cracks are not necessarily signs of an impeding collapse. Cracks may be accepted if they develop along horizontal lift joints and vertical contraction joints. Because of the large size of the concrete sections, relatively large horizontal sliding movements of concrete blocks can be accepted. However, leakage along joints may develop. However, since the hydrostatic pressure is small in the upper portion of a dam damaged by an earthquake, water losses will be minor. Moreover, the watertightness of the leaking joints can be re-established after lowering the reservoir below the damaged zone.
Tensile strength of mass concrete
The uniaxial tensile strength of mass concrete is usually less than 10% of the uniaxial compressive strength. As for quality control, compression tests are mainly carried out – the tensile strength of mass concrete is estimated based on a number of empirical relations, which, however, mainly apply for normal concrete.
The tensile strength of concrete is subjected to the size effect, i.e. the larger the test sample, the smaller the tensile strength. In the case of a large concrete dam with a characteristic thickness of 20m the tensile strength drops to less than 50% of that determined in
standard tensile tests. That means the tensile strength of mass
concrete in a large concrete dam is less than 5% of the uniaxial
compressive strength. The tensile strength of lift and contraction joints is even smaller. Therefore, it can be expected that even under a moderate earthquake like the OBE cracks may develop in concrete dams.
No-tension material modelling of mass concrete
To avoid tensile stresses in mass concrete a no-tension material model has been used in static analyses (water load, temperature effects). It is assumed that at any tensile strain the resulting tensile stress is very small (i.e. almost zero). This behaviour can be modelled using a bilinear (elastic) stress-strain curve for mass concrete. However, such models do not reflect the actual behaviour of a dam as the formation of a crack cannot be modelled once the (zero) tensile strength of concrete has been exceeded.
Once a crack has formed in the concrete, it will prevent the formation of further cracks. The deformations and stresses in a cracked tensile stress zone will therefore be quite different from those of an idealised no-tension material. Therefore, this no-tension assumption is not suitable for the analysis of tensile stress zones. It is only suitable for interfaces, where tensile strains will lead to a separation of a pre-existing interface. The use of no-tension materials must therefore be confined to interfaces only and cannot be applied to the parent mass concrete.
Seismic design criteria for concrete dams
At present the following types of earthquakes are used for the seismic design of dams (icold 1989):
• Operating Basis Earthquake (OBE): average return period of ca. 145 years (50% probability of exceedance in 100 years); no structural damage is accepted; the dam shall remain operable after the OBE.
• Maximum Credible Earthquake (MCE) or Safety Evaluation Earthquake (SEE): average return period of ca. 10,000 years for a region of low to moderate seismicity like Central Europe (Note: The MCE is a deterministic event, but in practice, due to the problems involved in the estimate of the corresponding ground motion, the MCE is usually defined statistically with a typical return period of 10,000 years for countries of low to moderate seismicity.). The stability of the dam must be ensured and no uncontrolled release of water from the reservoir shall take place, however, structural damage is accepted. In the case of significant damage, the reservoir has to be lowered after the MCE.
For the design of concrete dams the OBE is very important, because it is common practice to check the ‘no structural damage’ criterion by a linear-elastic dam analysis in which it is shown that the calculated tensile stresses are less than the dynamic tensile strength of mass concrete. In this test the size effect on the tensile strength of mass concrete and the fact that the tensile strength of lift and contraction joints is even smaller than that of the parent mass concrete is ignored.
By taking into account these factors, it can be expected that under a moderate earthquake (like the OBE) cracks may develop in many concrete dams. Therefore, the ‘no structural damage’ criterion cannot be satisfied without resorting to some assumptions as discussed in the first section.
To argue that some joints may open and close after the earthquake may be correct. But such a statement does not comply with a linear-elastic analysis of a dam, which forms the basis for this test. Therefore, if the ‘no structural damage’ criterion has to be checked then contraction joint opening in arch dams has to be analysed and the possible damage has to be discussed based on the maximum joint opening displacement.
Engineers are not yet ready to jump to more sophisticated nonlinear dynamic analyses for checking the ‘no structural damage’ criterion. This may also not be necessary as this criterion may have been interpreted too rigorously in the past. One has to keep in mind that the ‘no structural damage’ criterion is not safety relevant as a dam has to be able to withstand the SEE (MCE). The ‘no structural damage’ criterion is purely an economic criterion indicating to the dam owner that up to the OBE he may not have to expect any repair works at the dam.
However, in view of all the uncertainties involved in a dynamic analysis it may not be wise to put too much emphasis on the ‘no structural damage’ criterion
The main uncertainties in the OBE analysis are as follows:
• Specification of the ground motion of the OBE.
• Evaluation of the material properties needed for a linear-elastic dynamic analysis (damping, dynamic).
• Modelling and analysis of dam-reservoir-foundation system including dynamic interaction effects (lift and contraction joint modelling, foundation modelling).
• Static and dynamic tensile strength of mass concrete and foundation rock (size effect, age of concrete, fractile values of strength etc.).
Moreover, it shall be mentioned that for dam safety agencies in some countries (e.g. Switzerland) the OBE is not a relevant safety criterion. At present ICOLD Bulletin 72 is under review and the ‘no structural damage’ criterion of the OBE, which is basically an allowable stress criterion, may have to be reviewed or modified as well.
Therefore, it is important that the SEE criterion (safety of dam) is satisfied. In this case the dynamic tensile stresses and dynamic tensile strength of mass concrete play a much smaller role than in the case of the stress criterion of the OBE.
Nevertheless, in view of the high dynamic tensile stresses in the upper central portion of arch dams it is important to select mass concrete with adequate strength properties and to make sure that the strength properties at lift joints and contraction joints with shear keys are satisfactory.
It is also problematic to select the strength properties of mass concrete in the same way as in the past, because by using the pseudostatic analysis the dynamic tensile stresses at the central upper portion of an arch dam were much smaller than in the case of a dynamic analysis. At the same time it has to be stated that the existing dams are not necessarily unsafe because lower strength concrete was used in zones of high dynamic stresses than today based on a dynamic analysis.
However, it can be expected that under the OBE cracks and joint openings are more likely than in a modern dam when subjected to the same ground motion. However, these comparisons are of a
rather speculative nature as each dam is a prototype and two earthquake ground motions at two different dam sites will never be the same.
To select the concrete strength according to the maximum (static and dynamic) tensile stresses obtained from a linear-elastic static and dynamic analysis is good advice for any dam. There is probably no need to go into too great detail of the static and dynamic strength properties of mass concrete (size effects, ageing etc.) and the strength properties of contraction and lift joints. As a reference value the static tensile strength of mass concrete shall be used, which can be obtained from a representative empirical relation. Exceedance of this reference value is accepted as it gives a relative distribution of compressive strength of mass concrete within a dam.
What can be done with tensile stresses in large concrete dams?
Under the reservoir load the high tensile stresses at the upstream heel of thick arch and gravity dams may have to be accepted. This has been the case for most existing dams. The cracks formed in this zone remain invisible as this part may be covered by sediments and any leakage caused by cracks penetrating the grout curtain may eventually be plugged by the sediments. Cracks may only be detected if they intersect with the foundation gallery.
The control of cracking at the base can be achieved by a peripheral joint at a concrete saddle, which is also referred to as Pulvino. In this case, tensile stresses due to the fluctuating reservoir level will lead to the partial opening of the peripheral joints. The advantage of this joint is that its location is well defined and it protects the other parts close to the upstream heel of the arch dam from high tensile stresses.
Instead of a full perimetral joint, a joint can be formed connecting the upstream face of the dam to a gallery in the dam near the foundation. This solution was used for relatively thin arch dams and very high arch dams. In the case of Katse dam in Lesotho a pressure control system of the artificial joint was used to cope with the effect of fluctuating reservoir level.
Again the main advantage of an artificial joint is that it can be monitored and any cracks near the upstream heel will not form in an undesirable way, which may jeopardise the safety of a dam, such as the well-known cracks in the 200m high Koelnbrein arch dam in Austria, which have caused a significant reduction in the shear resistance at the dam base.
Finally, the stress concentration at the base of the dam can be reduced and the formation of dangerous cracks controlled by changing the geometry of a dam. Usually the thickness of a dam has to be increased near the base. The critical zones are those with high hydrostatic pressures, i.e. the base of very high dams. In arch dams with a height of less than 50m, the effect of the hydrostatic pressure on crack formation is a much lesser problem than for a 250m high dam. In both dams the tensile strength of mass concrete at the base of the dam may be the same, but the effective stress, i.e. total minimum stress minus hydrostatic pressure is smaller in a 250m high dam than in a 50m high dam, as the total minimum stress near the upstream heel is roughly the same for both dams. Therefore, pure scaling laws cannot be used, when the formation of cracks at the base of dams is compared. This is a much more serious problem for very high arch dams than for smaller dams with similar geometry.
Under seismic action high tensile stresses occur in addition to those at the dam-foundation contact also in the central upper portion of arch dams. Compared to the local stress concentrations in the vicinity of the upstream and downstream base of a dam, the area of high tensile stresses in the upper central portion of a dam is much larger and covers a large area. The tensile stresses in this zone can be reduced by changing the shape of the dam. In general, a thickening of the crest region leads to a reduction in the dynamic tensile stresses.
It is often believed that the additional mass of the crest and the additional inertia forces will increase the stresses in the crest. Seismic analyses have shown that this is not the case. However, without much analysis it can be shown that by thickening the crest the mass and thus the inertia forces increase proportional to the thickness of the dam whereas the dominant flexural stresses at the dam face are reduced by the square of the dam thickness. Here it is assumed that the crest thickening does not lead to much change in the dynamic properties (eigenfrequencies) of the dam. As a consequence, the maximum seismic flexural stresses, which are the dominant ones near the crest of an arch dam, will reduce considerably if the crest thickness is increased.
Cracking near the crest can also be controlled by steel reinforcement. However, for arch dams the benefits of such skin reinforcement would have to be studied first. It is more appropriate to provide a seismic belt near the crest and/or shear keys to prevent relative movements of adjacent concrete blocks.
However, for RCC dams, it might be feasible to provide a skin reinforcement at the upstream face of the dam to control leakage of a dam after an earthquake. The same objective could also be achieved by a geomembrane.
In most cases the formation of cracks and/or joint opening in highly (seismically) stressed zones is accepted.
The experience with the dynamic behaviour of large concrete dams subjected to very strong ground shaking, similar to that expected during the SEE, is still very limited. It can be expected that after the next strong earthquake, which causes damage to a large dam, the seismic design concepts – and that includes the interpretation of dynamic tensile stresses – has to be reviewed.
Conclusions
Tensile stresses have to be accepted in concrete dams. They are a fact of life if a linear-elastic static and dynamic analysis of a concrete dam is carried out. The tensile stresses are of major concern if the current ‘no structural damage’ criterion has to be satisfied specified implicitly in the ICOLD Bulletin 72 (1989), which is currently under review.
To select the concrete strength according to the maximum (static and dynamic) tensile stresses obtained from a linear-elastic static and dynamic analysis is good advice for any dam. There is probably no need to go into too great detail of the static and dynamic strength properties of mass concrete (size effects, ageing etc.) and the strength properties of contraction and lift joints. As a reference value the static tensile strength of mass concrete shall be used, which can be obtained from a representative empirical relation. Exceedance of this reference value is accepted as it gives a relative distribution of compressive strength of mass concrete within a dam.
Despite this greatly relaxed ‘no structural damage’ criterion the dam engineer should still take care of tensile stresses and reduce them as much as possible. The no-tension material model should only be used for interfaces as it cannot account for the formation of cracks in tension zones in concrete dams.
It is hoped that this paper will stimulate some discussion among dam engineers on the significance of tensile stresses in concrete dams
.
Author Info:
The author is Dr. Martin Wieland, Chairman, ICOLD Committee on Seismic Aspects of Dam Design, c/o Electrowatt-Ekono Ltd. (Jaakko Pöyry Group), Hardturmstrasse 161, CH-8037 Zurich, Switzerland. Email: martin.wieland@ewe.ch