#### 010-Usage of geometry: Building shapes with curved edges or holes

The geometry of the plate is defined by the input of corner points. It can be chosen, if a straight line or a circular curved border shall be used to reach the next point.

The co-ordinates x and y of each geometry point must be given such, that a counter clock-wise orientation of the points building the plate surface is set. The build system is always updated and drawn such, that the actual last line is connected with first given point to close the system.

For example, the border 1 is a straight line. The second border is given a rotation centre at the position 500./0. so that a counter clock-wise curved border is built. (defined by +1, right hand rule)

The mesh is generated automatically.

Any holes or cut-offs can be built as well very easily. Here you have first to go into the system (border 1) defining the hole (border 2) and then to return back the same way (border 3 as 1) you have gone into the system. The last border 4 closes the system returning to the starting point. The inner circle (border 2) is built clock-wise (-1: left hand rule), the outer circle (border 4) counter clock-wise (+1: right hand rule).

#### 020-Fixings (what’s possible)

Fixings may be set at any position you want. You only have to select a point fixing of needed shape (taper shaft, disk, edge clampings or bonded disk supports without a generated hole) from the database. This selection can be located at any x-y position.

That’s all. No mesh generation by hand is needed!

You can pre-define your own often used fixings, not to key in the geometrical data every time anew.

All fixings can be supported on springs with translation and rotational rigidities. In addition, all point fixings can be fixed to a bar, which other ending is attached to the wall or loaded directly by applied forces.

Regard, that a glass pane only supported by fixings mustn’t be kinematically in any direction. Unstable behavior may arise, when contacts conditions are chosen and the pane can detach (e.g. if the pane could be pulled out of a clamp).

#### 030-Symmetry (how to use)

Symmetry conditions can be used, if the system behaves symmetrically to one or two axes. Along those symmetry lines not only the geometry must be symmetrically – as well the loading situation!

The benefit of those calculations is to save space and computation time. If the user knows, that the right of a system behaves as the left or additionally the top half as the bottom half – this knowledge can be exploited for the calculation.

As the pendulum impactor can’t be divided, pendulum impact simulations can never use this option!

A four point supported glass pane 1600x1000mm shall be calculated. This pane is horizontally arranged and only loaded by face loads (or other loads as point loads in the middle of the pane). The point fixings are all located 100x100mm from the corner.

So, instead of generating the total system, only one quarter can be consider – as we know that the other 3 quarters acts identically in the same way as the one simulated (left as right and bottom as the top half).

For this, the system has now been halved at the symmetry lines.

The border 2 now will get the „typ 2“ boundary and border 3 the „typ 3“ form the predefined settings:

This will set the conditions that for example in x-direction no deformation (u) can arise and that the rotation (φ) here equals zero.

When doing so for insulated glass units, you have to take care that those symmetry borders don’t get spacers! Those borders are located within the pane and not at the sealed outside.

For this example, instead of 5900 unknown to solve, now only 1400 d.o.f. must be regarded. But the resulting stresses and deformations for a quarter section are equal to the full solution.

#### 040-Double Glass (some explanations to climate loads)

Insulation glass can be defined in several ways: single glass, laminated glass or any combinations. Up to 3 gap’s – so 4 panes are possible. For supporting the edges, simple boundary conditions, elastically supported borders, point fittings, clamps, balustrade clips or as well free borders, where all panes are only internally fixed by spacers can be chosen.

The program is specially developed to consider any shape of DGU’s and possible load combinations together with climatic loads:

temperature changes within the sealed volume filled with gas barometric air pressure changes changes in amplitude together with any outside load like wind, line or point loads in addition

#### Temperature changes

When sun is shining the gas inside the gap is heated up and will expand. This will lead to outside bending of the panes.In winter this cooling will have the opposite effect; the glass will bend inside. These effects must be considered (in Germany) and are related to possible production situations of the DGU. The winter condition is related to a production by 27°C and a minimum temperature for the filling gas in winter of 2°C. So a difference of -25°C is regarded. The worst case for the summer condition is set to a production temperature of 19°C and a maximum heat of 38°C what results in a difference of +19°C.

#### Barometric air pressure changes

For the barometric pressure change it’s regarded in a similar way.If the reference height for the winter load case is assumed to be a production (and thus the inclusion of this condition in the intermediate space) at sea level of 990 mbar, the required pressure difference of +40 mbar results in a maximum change for summer to 1030 mbar, which will compress the panes.For the summer load case a pressure difference of -20mbar is required. Assuming a production in winter at 1030mbar, there will be a change in air pressure to 1010mbar in summer.

#### Changes in height

These are the values to consider if the production (combined with a sealed conditions inside the DGU) and installation was not on the same amplitude (height above sea level). If the previously closed DGU is installed much higher (or must be carried of mountains to reach the site), this maximum difference shall be regarded too.

For summer this is set to +600m as it reduces the outside pressure and will therefore increase the arching outside (worst case), if real values are unkown. For the winter loadcase this is set in Germany for the worst case scenario to –300m, if the real amplitude changes are not known and if possible.

Please note, that the normal outside pressure is not zero but 1013 mbar! An outside pressure set to zero will indicate, that you’re dimensioning a DGU for the space shuttle – so really in space where there is vacuum outside.

#### Calculation method

In any case MEPLA will calculate the real physical behaviour within the gap’s regarding the volume, the pressure and temperature of the filling gas for reaching the equilibrium. So this method is valid for any pane shape.

And this is done too, when no climatic load is given. All fractions of external loads are transferred via the calculated inside pressure of the filling gas from the first to the second and the next panes. If internal spacers are set, they will carry as well some fractions of this loading and bearing behaviour over the in constant distance kept borders.

This situation to solve will be much more complex, when in addition large deflections, point fittings or contact conditions are chosen, what’s all possible to use together.

#### 050-Non-Linearity (description for such effects)

Within MEPLA all calculations can be performed under the more real approach of large deformations. This geometrically non-linear behavior is able to consider the re-stiffening membrane effects arising within a plate.

A linear calculation will give only accurate results, if the plate deformations are small related to the thickness of the glass. When these deformations exceed half of the plate thickness, the membrane effect becomes more important.

For higher loads the difference in non-linear theory compared with simple linear approaches can lead to approx. 3-times smaller deflections. (see the example calculation below)

**Non-linear behaviour in combination with shear effects**

Regarding this simply supported laminated glass plate, the following results will show some of these effects:

If there is nearly no shear effect set and so each 4mm plate is acting alone, the deformations calculated in a linear way are 52mm against a very much smaller deflection of only 16.16 mm, if more realistic non-linear approaches are used for simulation.

What can be seen in the two pictures above is a totally different stress distribution. Whereas the linear calculation will certainly show the largest stresses in the middle of the plate (highest bending moment) – within the non-linear calculation the stresses are shifted to the corners. The stresses in the middle are now mainly induced by membrane forces and are only in a region of 11 N/mm² instead of 38 N/mm².

Combined shear and non-linear effects in laminated glass

The next diagram will show the influence of shear effects too. As can be seen for the same example above, a variation of the shear stiffness for the interlayer will have additional large effects onto the overall behavior.

On the x- axis the used young’s modulus of PVB is drawn in logarithmic scale. On the y-axis the maximum deformations in the middle of the plate and the maximum stresses are shown. By this, the non-linear calculated stresses must not necessarily lie in the middle of the plate.

So the effect of non-linear calculation without shear effects has for thin plates (like here 4mm) the same effect for the deformations and stresses as if a shear stiffness for the interlayer of approx. 4 N/mm² has been used in a linear calculation.

A second effect for the behavior of laminated glass (in this situation as described before) is here visible too: The highest influence can be found from 0.1 N/mm² to 50 N/mm². Below this regions a laminated glass behaves like two layers without a connection and above this range as nearly monolithic.

This will point out how important it is for designing the glass to consider already smallest gluing effects. If a young’s modulus of E = 0.6 N/mm² (G = 0.2 N/mm²) would be used, the stresses in linear or non-linear approaches are already halved.

#### 060-Signs and Units (Convention)

Sign convention

All load signs shall be given related to the global co-ordinate system. Depending upon the glass package which shall be loaded, this will define a suction or a pressure load. The z-axis is showing the positiv direction.So a positive load onto the inside (glass package 1) is a pressure load.A positive face load onto the outside (largest glass package) is a suction load.

Units

All units are in N and Millimenter. To give a face load of 1.0 kN/m² you have to write 0.001 N/mm² or using the exponential notation 1.0e-3 N/mm². This will divide your load by 1000.In this shown example al pressure load is applied onto glass package 2 of -1.0 kN/m² = -1.0e-3 N/mm² = -0.001 N/mm².