This subsection is devoted to a discussion and comparison of the results obtained using the various design methods discussed in this section. In Table 7, the values of maximum principal stress are presented. From the table one can conclude that the results of all three methods are sufficiently close to each other in order to classify the methods as yielding equivalent results. More rigorous comparisons of the two first methods are provided in the first part of the thesis. The result using the third method carries some uncertainties related to mesh density when constructing the chart, the selection of the design chart to match the
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Table 7: Comparison of different methods for stress prediction.
Method Maximum principal stress (MPa)
FEM, solid elements 119.4
FEM, M-RESS 125.5
Design chart 124.3
actual set of parameters, parameter interpolation and reading off the chart. These effects do not seem to be significant given that the results are very close to those obtained using other methods.
7 Overview of Present Work
7.1 The Application of the M-RESS Element for Stress Evaluation of Advanced Laminated Glass Structures
In Paper 1, the M-RESS element was applied to typical examples of laminated glass struc-tures and the performance regarding accuracy and computational efficiency was evaluated.
The first example was a thin square plate subjected to biaxial bending through a uniformly distributed lateral load. The performance of the element was compared to standard three dimensional solid elements contained in the finite element program ABAQUS. Those ele-ments were a linear 8-node and a quadratic 20-node element, both quadrilateral eleele-ments with reduced integration. Convergence tests were performed in terms of the vertical dis-placement at the center point of the bottom glass surface, w, and the in-plane stress in one direction,σ. The results were normalized to the results using the 20-node element in ABAQUS and 2 millions degrees of freedom. Defining the acceptable error to be less than or equal to 5 %, both for the displacements and the stresses, Table 8 shows the num-ber of degrees of freedom required to reach acceptable levels of error for the elements investigated.
Apparently the 8-node solid element was not capable to represent the behavior of the structural problem. The computational advantage of using the M-RESS element instead of the 20-node solid element is well illustrated.
Further tests were dealing with real structures comprising commonly used types of joints in glass construction. The first example was adopted from [10], and comprised a square laminated glass plate with a bolted joint placed in the middle of the plate. A compressive force was applied to the joint. As an illustration of the structure, the geometry of the glass plate is shown in Figure 19.
The glass plate rested on a supporting steel frame and a rubber gasket protected the glass from direct contact with the steel. In [10], the fracture stress was evaluated experimentally for this structural problem. In Paper 1, finite element analyses of the structure were made using the M-RESS element for the modeling of the laminated glass part. As a comparison, a similar model was made using 20-node solid finite elements for the corresponding parts.
The model size of the model with M-RESS elements was around 10 % of that of the model with solid elements. The maximum principal stresses were located in the upper glass layer close to the bore hole as expected. In Table 9, the maximum principal stress for the different methods is shown. For the experiments, the results presented are not the maximum principal stresses occurring in the structure since it was not possible to make experimental measurements at the correct location.
Table 8: Convergence properties of different finite elements.
Finite element Min. number of DOFs for w Min. number of DOFs forσ
M-RESS 300 700
8-node solid 200000
-20-node solid 3000 3000
500
Figure 19: Geometry of glass plate.
Both of the numerical models had an error less than 15 % compared to the experimental results, and the M-RESS element predicted results that are closer to the experimental ones. With the difference in model size between the two elements, the M-RESS element is clearly a more efficient element for this kind of modeling. The 10 % error that the M-RESS element predicted is accurate enough to be used in practical design of glass structures. In Paper 1, there is a discussion relating to error sources that are related to the numerical modeling, but not to the properties of the finite elements. It is possible that the error could be further reduced without increasing the size of the model.
The second real glass structure dealt with a large glass beam that was made up by smaller glass beams through the use of adhesive joints. The structure was created and analysed experimentally and numerically in [30] and Paper 3. The aim of that work was to de-termine the shear capacity of an adhesive joint in a large dimension glass beam. The experimental test arrangement is displayed in Figure 20.
The test was a four-point bending test where the arrangement had been made so that pure shear stresses were obtained in the joints. In the original study, several adhesives were studied. For the demonstration of the applicability of the M-RESS element, an epoxy adhesive was used. As a comparison, a model was made using 20-node solid finite elements. The model size of the model using M-RESS elements was only 20 % of the model size of the model with solid elements. The variable used in the comparison was the ultimate displacement in the load direction at point 4 of Figure 20. The results are shown in Table 10.
Both the M-RESS element and the 20-node solid element yielded very accurate results for the present test case, but the model used for the M-RESS element was significantly more computationally efficient.
Table 9: Maximum principal stress close to bore hole.
Method Maximum principal stress (MPa)
Experimental (mean value) 177.1
M-RESS 159.2
20-node solid 153.4
P/2
Figure 20: Test setup of four-point bending test of glass beam with adhesive joint.
Table 10: Ultimate deformations at the mid-point of the beam.
Test Displacement (mm)
Experimental 10.00
M-RESS 10.20
20-node solid 10.24
In summary, three test cases that are relevant in the discussion regarding laminated glass structures of various support conditions have been analysed and the results consistently showed that the use of the suggested M-RESS element lead to more computationally efficient modeling at a given level of accuracy. Note that the M-RESS element is not a standard element of a commercial software package, which means that the contribution extends the abilities available in commercial finite element modeling to deal with strengh design of complex glass structure in a computationally efficient manner.