Finite element analysis calculates the response of a complex shape to any external loading by dividing the shape into lots of smaller, simpler shapes. These are the finite elements that give the method its name. The shape of each finite element is defined by the co-ordinates of its nodes. Adjoining elements with common nodes will interact.

The real-life engineering structure responds in an infinite number of ways to external forces. The way in which the finite element model will react is given by the number of degrees of freedom, which are expressed at the nodes. For example, a three-dimensional solid element has three degrees of freedom at each node representing the three global directions in which it may move. If the response of a single finite element to a particular stimulus is known, a model can be built up for the whole structure by assembling all of the simple expressions into a set of simultaneous equations with the degrees of freedom at each node as the unknowns. These are then solved using a matrix solution technique. For a structural analysis, once the displacements are known, the strains and stresses can be calculated. For a thermal analysis, the gradients and fluxes can be calculated from the temperature potentials.

Software package

A software package available from LUSAS has analysis facilities for linear, natural frequency, buckling, fatigue, nonlinear, seismic, dynamic, impact, geotechnical, thermal and blast analysis. It can be used for producing initial prototypes, designing and checking all types of civil and structural engineering structures – not just dams. Frames, space frame roofs, domes, cooling towers, multi-storey buildings, bridges, masts, tunnels, soil/structure interaction, culverts and retaining walls can all be studied.

LUSAS software has also been designed to help prove the stability of existing or new earth embankment and reinforced concrete dam designs. It plots displacement, stress vectors, stresses, strains, temperature, with associated labelled values, and calculates the extent of concrete crack propagation using an initial crack location.

The package has a 3D multi-cracking concrete model that does not need any pre-initiation to locate cracking. It calculates in-service temperature gradients within dams and allows investigation into cooling requirements during concrete placement as well as seismic stability.

The whole dam is modelled and elements can be turned on in groups to model the construction sequence. The software models rock and soil anchors. It also uses slidelines and slide surfaces to model rock joints, high-order plane strain elements to model pore water pressure for earth embankment dams, and birth and death of elements to model staged construction.

Muela dam

In 1995, LUSAS civil and structural analysis software was used by consulting engineers Mott MacDonald to prove the stability and optimum curvature and profile of the Muela dam, part of the Lesotho Highlands Water Project in Africa. The 55m high dam, with a crest length of 200m, forms the tailpond of the Muela hydro project.

The software defined a full 3D model of the dam and its rock foundation using eight-noded enhanced strain elements, with four elements across the width of the dam and at 2.5m height intervals. In all, 9528 elements were used to define the dam model including 5880 elements in the rock foundation.

The linear elastic analysis of the dam covered a range of different load cases, including reservoir full and empty, silt and temperature effects. Dynamic analysis of earthquake loadings was not considered justified because of the low seismicity of the area. However, the influence of earthquake loadings was examined by the pseudo-static method, on the basis of a ground acceleration of 0.2g. This acceleration was distributed throughout the structure on the basis of a damping ratio of three.

Monthly mean temperatures at Muela can range between 6-19°C in an average year and from as low as 3°C in a cold year. In addition, the transfer water from the Katse reservoir was expected to be at about 6°C. A thermal analysis was carried out using Schmidts method to establish the in-service temperature gradients within the dam and also the requirements for artificial cooling, on the basis of Fouriers’s law of unsteady heat flow through solids and using a step-by-step integration process. This analysis covered the initial placing temperature of the concrete, variations in air and reservoir water temperatures, heat gain in adjacent concrete pours, solar gain, lift height and intervals and heat of hydration.

The linear analyses under the various load cases indicated maximum compressive stress of 6N/mm2 under normal conditions, well within the maximum permissible stress of 8N/mm2 after allowing a safety factor of three on the characteristic concrete strength of 25N/mm2.

Tensile stresses also occurred under several load cases, primarily at the upstream heel. The extent of distribution of tensile stresses is more critical than the actual magnitude at any particular node, and this was examined further by a nonlinear ‘no tension’ analysis where a crack was allowed to propagate at areas of tensile stresses. This was carried out for the most critical load case of reservoir full and average-year winter temperatures and with uplift under the reservoir head applied at the crack. The analysis indicated that the maximum crack propagation would be solely over the upstream quarter of the dam/foundation contact, which is acceptable.

The Koyna experience

In 1967, the 103m high Koyna mass concrete dam in India was subjected to a magnitude 6.5 earthquake. Accelerometers on the site recorded the time histories of the event and a subsequent survey showed that cracking had occurred in parts of the dam. Due to an abundance of field data the dam has been the subject of a number of studies over the years. In 1995, The University of Wales in Cardiff, UK, modelled the dam using the LUSAS analysis software to investigate its response to changes in numerical concrete cracking models.

The aim of the study was to derive a concrete cracking model to allow for dynamic cyclic loadings and produce results that matched those of the experimental data. The software was used for this because it contains a material model interface which allows users to develop and research constitutive models for their own use.

During a seismic event, a variety of factors can affect the predicted response of a dam. Of particular interest is the unloading curve because, when cracks open and close, it is well established that the crack stress-strain curve does not return to zero, but to some finite tensile strain indicating that particles have wedged in the open crack.

Using the derived concrete model, three analyses were undertaken with the same finite element mesh, time step, and material data. The only difference between the analyses was the shape of the softening and unloading curves. Bilinear softening assuming no wedged particles, and exponential softening both with and without wedged particles, were investigated.

For each analysis, gravity and hydrostatic loads were applied to the structure and a dynamic analysis was run applying the combined vertical and horizontal ground motions.

The concrete model with exponential softening and wedged particles gave the lowest response overall, both in terms of response spectra and peak accelerations. For cyclic dynamic applications it appears to be important to incorporate both a realistic softening curve and crack wedging behaviour into a numerical concrete model.

Work on this study is continuing to see if other parameters also have a strong influence on dynamic structural response.