CROSS Safety Report
Understanding finite element analysis for pile caps
A reporter has encountered situations where two-dimensional finite element (2D FE) shells are used to model structural elements such as pile caps, combined bases and large ground bearing or pile foundation structures for stability cores. However, the depth or thickness of the structural element is such that the reporter questions the validity of the structural model.
Key Learning Outcomes
For civil and structural design engineers:
- The design of basic pile caps can be carried out using strut and tie methods
- Be mindful of The Concrete Centre advice that "Design using FE analysis requires engineering judgement and a feel for the behaviour of concrete"
- If a finite element method (FEM) is used, designers should know and understand the theory while being aware the selection of element type and size will affect the results
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A reporter has encountered situations where two-dimensional finite element (2D FE) shells are used to model structural elements where the depth or thickness of the structural element is such that the validity of a 2D FE surface, and in particular, the application of the underlying theory used in the formulation of the elements, is questionable. The models encountered were being used to design elements such as pile caps, combined bases, and large ground bearing or pile foundation structures for stability cores. In each case, the depth of the structural element could not be considered small.
The reporter says earlier versions of FE modelling would have been based on Kirchhoff-Love theory, often referred to as 'thin plate theory', which is the 2D extension of Euler-Bernoulli beam theory. The underlying assumption of the Kirchhoff-Love theory is that the thickness of the plate is significantly smaller than the in-plane dimensions. For this version of the theory to be relevant, the span-to-depth ratio needs to be greater than 10.
To overcome this limitation, general 2D FE structural analysis software tends to employ Mindlin-Reissner plate theory, which is the 2D equivalent of Timoshenko beam theory - often termed 'thick plate theory'. According to the reporter, various resources give slightly different limits but the span-to-depth ratio should be no lower than in the range 3-5.
modelling of structural thicknesses greater than the upper limit may give inaccurate results
In both cases, in the reporter's view, modelling of structural thicknesses greater than the upper limit may give inaccurate results. The structure may be over-constrained, and the effects of shear may be underestimated. A design based on FE elements used outside of their range of applicability may therefore give forces that are lower than they might be in the real structure.
The reporter contends that too often FE analysis is used without sufficient thought and understanding. Before using any FE analysis software, they believe the designer should know the underlying theory used for the elements being employed and understand the potential impact this may have. In general purpose structural software there is often no choice (or a very limited choice) of elements to be made, but in more specialist FE software a range of elements are used, and choosing the correct element for the problem in hand is of vital importance. In the reporter’s experience, proficiency in using FE within structural engineering is often measured as the ability to use a software package rather than the ability to understand the underlying basis of the software.
The element size should be less than span/over 10 and its width larger than the slab depth
The reporter believes the situation is not helped by the available guidance, which is either too general in nature or too specialised. In response, the reporter has resorted to researching texts and references on plate bending theory itself. The reporter found only one publication they felt dealt satisfactorily with the issue, The Concrete Centre's publication How to design reinforced flat slabs using Finite Element Analysis. This says the element size should be less than span/over 10 and its width larger than the slab depth. Therefore, the span would need to be more than 10 times the depth to comply.
The reporter goes on to say that while it is the responsibility of the user of the software to make sure they understand the analysis and limitations, the software producers could perhaps also do more. Though it is difficult to implement dimensional checks due to the relative geometric freedom that FE gives, they could perhaps give more explicit details of the FE formulation used and any limitations.
Both British Standard BS 8110 (now withdrawn) and Eurocode BS EN 1992-1-1, place limits on the ratio of beam depth to span length over which a beam is considered a deep beam. The Eurocode also provides an explicit limit on the thickness of a slab. In both codes, the standard design rules are limited to those structures not considered deep beams or slabs. In the case of the British Standard, the designer is referred to specialist literature. In the case of the Eurocode, while no direct mention is made of the design of deep beams, it does contain a reference to ‘strut and tie’ design methods. The reporter contends it appears that these code requirements, which are not specific to FE but do reflect the underlying limits of Euler-Bernoulli beam theory, are not well known by designers or perhaps reflect a mistaken belief that FE is somehow unlimited in its use.
In conclusion, the reporter says that FE analysis is perhaps used in some cases without a proper understanding of the underlying theory. They believe the focus may be on producing a photorealistic representation of the structure rather than producing a valid and appropriate model. The reporter considers that guidance is needed specific to structural engineering and aimed at the practicing engineer.
Expert Panel Comments
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It is important to emphasise that engineers should understand the nature and probable behaviour of a structure irrespective of any FE analysis. Pile caps in practice are thick stiff units, not thin slabs, and manual design with the traditional strut and tie method is generally a reliable way of proceeding.
In 2018, CROSS issued the Safety Alert Effects of scale in which the design process suggested, FE analysis being relevant to thin slabs, is inappropriate for 'thick slabs' (although what is meant is not thickness directly but short 'beams' with very high depth/ span ratio).
There is even a case on safety grounds for concern about using normal beam design here. The problem is that the high depth/span ratio will result in relatively low rebar and the capacity of the concrete section based on reasonable concrete strength may exceed the section strength based on rebar. Under overload, the section could fail as concrete cracks and the rebar, giving lower strength, does not contribute. The performance of the section has been transformed from ductile to brittle - very undesirable. However, the rules for minimum reinforcement in the Eurocode are generally formulated to give reinforcement with at least the same strength as the cracking strength of the concrete section.
what the beam analysis will not pick up is that the force is fairly constant between the two piles and will not drop off as predicted by beam theory
In most cases, the FE analysis will give a reasonable approximation of the maximum forces. For example, the maximum area of tension steel calculated for a two pile cap from an FEM may be similar to that found by using beam theory or strut and tie. However, what the beam analysis will not pick up is that the force is fairly constant between the two piles and will not drop off as predicted by beam theory.
A worrying aspect is that some software packages are starting to be used directly for detailing and, in this case, it would be very possible to get an unsafe design. To expand on this; 2D elements, whether formulated as thick or thin plate/shell, work on the assumption that the structure is essentially working in bending, but pile caps are usually of such thickness that they act as deep beams. In other words, shear behaviour is significant and they are better considered as behaving as a strut and tie.
If a pile cap is being modelled in 2D elements and the analyst is only interested in how the forces are distributed into the piles, then the mesh density makes little difference. However, if an understanding on what is happening within the pile cap is desired, then the 2D elements will give only a partial picture based on the assumption the cap is working entirely in bending and ignoring (or minimising) the effect of shear in transferring the load.
Moving on to FE modelling more generally, The Concrete Centre publication mentioned earlier lists a series of advantages and disadvantages of FE analysis on its front page. One of the disadvantages is "Design using FE analysis requires engineering judgement and a feel for the behaviour of concrete". CROSS is concerned about the number of reports being submitted about problems with the use and understanding of FEM.
There is a lack of knowledge, and the evidence suggests software is being used without a proper understanding of how it works
One Expert Panel member is also a reviewer for the Institution of Structural Engineers (IStructE) chartered membership and, when presented with multicoloured FE analysis plots in a portfolio, they typically ask candidates to explain how the structure is working and whether they have produced 'order of magnitude' checks to satisfy themselves that the FE analysis answer is reasonable (particularly when it produces reinforcement quantities).
The reviewers do not always get satisfactory answers, and these often get worse when candidates are asked about FE analysis deflection calculations and the material parameters used. There is a lack of knowledge, and the evidence suggests software is being used without a proper understanding of how it works and, significantly, without an independent check by someone more experienced in the use of these systems and their limitations.
The issue with use of FE analysis is far wider than just the appropriate selection of shell, plate or solid elements. There should be more rigorous verification and validation as well. Validation is the comparison with known results (numerical or experimental). Verification establishes that the model is not sensitive to discretisation or imperfections and the like. It is useful to reflect on the fact that the only exact formulation for a finite element is that for a beam, all other formulations are approximate.
Discussion on element types
Looking at the theory section, to quote the National Agency for Finite Element Methods and Standards (NAFEMS) publication, Finite Element Analysis for Engineers - A Primer (2013):
"In Kirchhoff theory, the out of plane normal remain straight and normal to the 2D surface. In Mindlin theory, also known as Reissner-Mindlin theory, the normal remain straight but can rotate relative to the 2D surface. Both theories allow simple bending behaviour with either the absence or presence of shear straining, respectively."
It is worth noting that a plate carries only bending, a plane stress element carries only in-plane forces, and a shell is a mathematical combination of a plate and a plane stress. This means that shells using Kirchhoff/Kirchhoff-Love formulation (known as 'thin plates' or 'thin shells') are suitable only where there is minimal shear, such in a membrane structure like a cooling tower.
shells are great for general structural modelling but they begin to lose accuracy where shear dominates
It is not thought that many structural FE analysis packages use such formulations, apart from those where they might be available as an advanced option. Shells that use Mindlin formulation ('thick shells') do include shear stiffness but they still assume that the normal remains straight, meaning that while shear deflection is included in the behaviour, deformation of the section is not.
This means shells are great for general structural modelling but they begin to lose accuracy where shear dominates. It is not a particular problem in the region around a column in a flat slab, as this is a small part of the overall structure, and the recommendation is to consider shear in this zone in a separate, more detailed model. Where shear dominates throughout, they do not capture the full behaviour of these structures.
As mentioned, this is a different consideration to the element size itself. The NAFEMS guides do not give recommendations for minimum element sizes, but the converse. For example, their Finite Element Analysis for Engineers - A Primer publication states:
"Use enough elements to provide results of sufficient accuracy, with smaller elements in areas where the physical behaviour varies most rapidly, such as near stress concentrations, and larger elements away from such areas."
Similarly, The Concrete Centre publications, How to design reinforced concrete flat slabs using Finite Element Analysis, states:
"Definitive advice cannot be given as to the ideal size mesh size, but a good starting point is for elements to be not greater than span/10 or 1000 mm, whichever is the smallest."
"…a finer mesh giving more accurate results. The engineer has to assess how fine the mesh should be; a coarse mesh may not give an accurate representation of the forces, especially in locations where the stresses change quickly in a short space e.g. at supports, near openings or under point loads. This is because there are insufficient nodes and the results are based on interpolations between the nodes."
Note that these statements are in direct opposition to those given by the reporter in their submission to CROSS. In the IStructE's Computational Engineering, there is the recommendation that:
"the element width should be at least twice its thickness"
However, this is for usefulness of result rather than accuracy. The engineering sin is not that the elements are too small, but rather are too large in areas where the stresses are changing rapidly.
The Panel agree with the reporter that all too often FE analysis is used without sufficient thought and understanding.
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