Two-dimensional finite elements
The proper mathematical method for the analysis of slabs, cited in §3.3.6 and Annex Α, despite its scientific glamour, has two intrinsic limitations.
The first limitation arises from the heterogeneity of reinforced concrete, which consists of two materials, concrete and steel. Notably, concrete cracks due to low tensile strength while steel works in plastic region.
The second limitation arises from the manual human deficiency. No matter how well trained a craftsman might be, it is very difficult for him to apply reinforcements properly to critical connections of slabs with beams and columns.
In this chapter, two basic assumptions are adopted for the modelling of slabs allowing their analysis separately from the rest of the structure. Specifically,
1) The slab supports on beams and columns provide vertical restraint.
2) The slab supports on beams and columns provide no rotational restraint.
Practical calculation of rectangular slabs is performed with the computational methods proposed by two great engineers, Marcus and Czerny.
The pi-FES module, called SLABS, for modelling and analysing isolated as well as continuous slabs using two-dimensional finite elements, yields results similar to those of the theory of elasticity as developed and tabulated by Czerny. pi-FES generates automatically both independent and adverse loadings, therefore only the results are requested from interface.
In order to understand the method, three examples are presented using the related software:
The project deals with a simple single storey building consisting of 9 columns, 12 beams and 4 slabs, as shown in the picture.
Figure 4.2.1-1 Figure 4.2.1-1
The four slabs have identical dimensions of 4.0 mx6.0 m, thickness of 150 mm, covering load g_{e}=1.0 kN/m^{2} and live load q=5.0 kN/m ^{2}. Concrete strength class: C30/37.
The usage of the software along with static analysis results are presented and commented below. It is worth noting that these results would not be achieved without the use of software. The static analyses of the following examples could also be performed by using tables, as described in subsequent paragraphs. However such calculations are limited to the extreme stress resultants, while pi-FES yields results distributed across the slab surface.
The dialogue of pi-FES appears by pressing the corresponding button in the main toolbar.
Figure 4.2.1-2: The pi-FES dialog Figure 4.2.1-2: The pi-FES dialog
Select tabs and set parameters:
"Overall size" = 0.10 m and
Figure 4.2.1-3: After the analysis the 3D model of the 4 slabs along with the actual structure are displayed Round red button "Element labels"(1) , displays the data of the structural elements.Button sequence "Quick Bar"(2) and "Mesh"(3) hides the grid. Figure 4.2.1-3: After the analysis the 3D model of the 4 slabs along with the actual structure are displayed Round red button "Element labels"(1) , displays the data of the structural elements.Button sequence "Quick Bar"(2) and "Mesh"(3) hides the grid.
Figure 4.2.1-4: Button "structure"(1) hides the structure. The slab s1 is selected by pressing button "single"(2) and left mouse click(3) on the slab. Figure 4.2.1-4: Button "structure"(1) hides the structure. The slab s1 is selected by pressing button "single"(2) and left mouse click(3) on the slab.
The model of the slabs considers hinged linear supports, represented by triangular prisms and extending along the centreline of the beams supporting the slabs.
Figure 4.2.1-5: Button “Show Selected”(1) displays the two-dimensional finite element mesh. Higher mesh density occurs nearby the supports (2) while lower in the internal of the slab . The rest of the slabs follow similar mesh generation pattern. Figure 4.2.1-5: Button “Show Selected”(1) displays the two-dimensional finite element mesh. Higher mesh density occurs nearby the supports (2) while lower in the internal of the slab . The rest of the slabs follow similar mesh generation pattern.
The discretization (meshing) of slabs of arbitrary geometry into triangular finite elements of variable size, is based on the Advancing Front method[^{1,2}]. The algorithms provide automatic fluctuation in the elements size, in order to produce adaptive mesh[^{3,4}]. The adaptation criterion is the variation in stresses due to geometry, loads and support conditions. According to this method modeling of considerable accuracy is achieved at each point of the slab. A critical advantage of the process is both ensuring high speed proper finite element creation through parallel execution of the algorithms on the available processors and high element quality (equilateral triangles) through appropriate density criteria and optimization processes.
The triangular finite element used in Advanced Solver is of shell type with 6 degrees of freedom in each node. The bending part of the finite element is based on Reissner-Mindlin theory [^{5,6}], taking into account shear deformation, while its formulation is based on hybrid model[ ^{7}]. The plane stress part of the finite element is based on assumption of stress remaining constant throughout the element. Finally, torsional part of the finite element is based on Kanok-Nukulchai[^{8}] theory. It is worth noting that the pi-FES model does not depend on the torsional stiffness of the finite elements.
Figure 4.2.1-6: Button “Show All” (1) re-displays the model. Button “Contours” shows, using color gradation,the displacements contours. Figure 4.2.1-6: Button “Show All” (1) re-displays the model. Button “Contours” shows, using color gradation,the displacements contours.
Figure 4.2.1-7 Figure 4.2.1-7
Each color (on the 3D color scale bar) corresponds to a range of displacements (mm).
For many years, the method of color gradation display of displacement contours was a 2D method of representing 3D information.
Today, provided the 3D abilities, the direct 3D or 4D display is preferable, especially for stereoscopic display.
Figure 4.2.1-8: The following button sequence displays the 3D deformation of the structure."Details"(1) in FEM results, "Diagrams at Dx=Dy=0.1m", "OK", "Selection" (2), "Displacements" "Z" & "Diagrams" (3).For better viewing "Light 2" (7) is switched on. Figure 4.2.1-8: The following button sequence displays the 3D deformation of the structure."Details"(1) in FEM results, "Diagrams at Dx=Dy=0.1m", "OK", "Selection" (2), "Displacements" "Z" & "Diagrams" (3).For better viewing "Light 2" (7) is switched on.
Figure 4.2.1-9: In the previous screen button sequence “Menu” (4), “Full Screen Mode” (5) and “4D” (6),provides stereoscopic display using “blue-red glasses” Figure 4.2.1-9: In the previous screen button sequence “Menu” (4), “Full Screen Mode” (5) and “4D” (6),provides stereoscopic display using “blue-red glasses”
Displacements which induse stresses help the engineers understand better the structural behaviour (engineering perception). When the deflected structure is concave upwards, the bending moments are positive, considering that the fibers under tension are located at the bottom face of the slabs.