Structural glass - Structural analysis and calculation model
Now that we have summarized in our previous posts glass manufacturing process and treatments, its particularities on design and physical behaviour, as long as how it reacts to usual loads, we are now able to dive deeper into the explanation on how we model all this particularities into a mathematical model that allows us to simulate the material behaviour. We will not explain here how finite element models (FEM from now on) work, as this is beyond the scope of this post, but we will try to explain how we should adapt them to properly simulate the special behaviour of glass as a structural material.
Usual FEM models for common materials, such as steel or concrete, grow from some basic assumptions as the linearity of the structural response and the fact that, thanks to a high structure rigidity, displacements can be regarded as small.
When analysing glass elements or structures, we usually deal with thin and therefore very deformable plate elements loaded on its out-of-plane directions. This implies that displacements can no longer be regarded as small, not allowing to use basic assumptions stated before nor simplified formulation (Kirchhoff's assumptions). In addition, glass elements usually are fixed with special supports that, due to their rigidity, concentrate stresses in certain points of the elements making difficult for common FEM approaches to properly simulate this stress distributions. As a result of that, solving calculation for those glass elements will imply to solve large sets of partial differential equations (von Karman's), turning it a very complex process unsuitable for manual calculation. There is where FEM model with integrated non-linear behaviour help us to solve the calculation and design process of glass elements and structures.
As explained before and in contrast to most of other building materials, glass elements commonly experience large deformations prior to failure. This large displacements associated with out-of-plane deformations will cause a stretch on glass mid-pane and if glass is restrained, the development of the so called membrane stresses that will increase plate stiffness. Think on a long horizontal rope, tied on one side to a post and held by you on the other end. If you hang a load from the middle of the rope, it will tend to go downwards, but what you mainly feel holding the rope is a horizontal force pulling from you, that is the principle that keeps hanging bridges standing. Those (making a transition from 1D to 2D) are membrane stresses. This stiffness increase may take part even when glass ends are not restrained (e.g. when circumferential membrane stresses are set up as the plate is constrained to deform in a non-developable surface).
As a result of the presence of this membrane stresses a non-linear model has to be used, so it is able to take into account model deformation for stress redistribution allowing this membrane stresses to develop. Failure to perform a geometrically non-linear analysis for large deflection situations will result in an overestimation of the lateral deformations. This means that, for a given load, tensile stresses are lower that the ones obtained from a linear approach, that will imply using more material and therefore a non optimized design. On the contrary, for a given displacement, tensile stresses developed on the glass are higher than those indicated by a linear analysis, therefore design will be unsafe. This states the importance of considering geometrical non-linearity when assessing safety or designing glass structures.
Finite element analysis
For common geometries and uniform loading conditions, glass elements maximum stresses and deflections can easily be calculated with tables and graphs given in usual standards. For unusual glass geometries or particular support conditions or loads FEM analysis will be necessary to solve the mentioned von Karman's equations. Therefore specific software and a relatively powerful hardware will be needed. Software tools tend to user friendly interfaces more and more everyday turning modelling and calculation an easy process. With this simplification comes a high risk, as any engineer could model and calculate complex structures and obtain some results but it is a duty for every worthy engineer to be critic with those models and results, and here is where experience and knowledge become crucial skills.
Regarding the creation of FEM there are some rules or principles that must be taken into account to achieve a functional model.
When designing the mesh of elements in which our geometry will be decomposed we will have to pay special attention to adjust the density of the elements (number of FEM elements by region area) according to the expected stress distribution on the element. That way a fine mesh (high density of elements) should be created on point or regions with special stress concentrations such as places where concentrated loads act on the plate, punctual supports, geometrical discontinuities, etc. For regions with no special particularities a coarse mesh (low density of elements) will be more suitable to accelerate the calculation process. In relation to this, assumed mesh densities should be regarded correct by means of a convergence analysis, testing that a refinement (varying density) of the mesh does not affect results magnitude.
Another key principle that must be followed is to model FEM elements and supports to behave as real elements will do. Specially when modelling supports and attachments on glass structures we have to take into account that most of them are materialized through metal elements protected by liner or gaskets. Therefore, supports are not usually stuck to the glass and as a consequence they will allow glass to separate from them on the opposite direction of the one where support is acting. To simulate this behaviour usual software integrate support types known as "compression only", in which restraining acts only in one direction, allowing glass to lift from its supports if its real behaviour tends to that.
Also special care has to be taken for glass elements with in-plane loads, as buckling may occur. This phenomenon will be discussed in future posts.
Standards and normative from different countries give good approximations through different tables, graphs and simplified formulation to obtain approximate solutions for simple glass units, usually plane panes under uniform loads with standard support conditions. Common standards used in that approximations are ASTM E 1300-04, prEN 16612 or particular country standards as DTU 39 P4
To prove the difference on the results obtained through the different approaches stated, we summarize here the calculation of a theoretical example. We will calculate a square glass which dimensions are of 2x2 meters, simply supported in all its perimeter and uniformly loaded with increasing values. For the different approaches we will obtain results following prEN 16612 standard and two different FEM models will be created, one with simple lineal solving and the other with a non-linear one. To conclude, different results are compared to state its differences.
For the first approach the European standard prEN 16612 will be used as it is the valid standard for Spain. Through this method, based on the use of some parameters to adjust the model and a relatively simple formulation we will obtain the results reflected on the table (calculation process is not described, only results obtained).
A FEM model is generated on the calculation software modelling the glass geometry. Following the glass geometry a regular mesh of square elements is created. Supports are defined as simple supports locking the vertical displacement on the perimetral nodes of the model, simulating the behaviour of the glass on a theoretical steel frame. Configuration of the model can be seen on the right.
On the following images both displacements and stresses for a 5 kPa load are reflected:
Non linear model
For the non linear resolution the same model will be used, configuring software solver to use non-linear solving so membrane stresses are considered. Results for a 5 kPa uniform load can be consulted below.
Comparing both linear and non-linear results it is obvious that both stresses and displacements are lower for the non-linear solution.
On the following graphs results from the three approaches are compared:
As seen on the graphs, the phenomenon stated previously on the post are taking place. As a result of them linear solution gives higher stresses and displacements, while non-linear and standard solutions give similar results properly approximating real glass behaviour. Furthermore, for more complex geometries, load or support configurations, as there is no standards collecting methods to solve them, only FEM non-linear model will be able to achieve a proper solution for glass elements.
On the next post we will talk about fracture on glass elements, its growing processes and the main considerations that should be taking into account to properly manage this particular behaviour. We hope you enjoyed this post and see you in the next one!