4.3 Three-dimensional nodal zones
4.3.1 Geometry for consistent three-dimensional nodal zones
The three-dimensional method proposed in this work allows to define a consistent nodal geometry. The method consists in determining the shape of the undefined struts, using the known or assumed corners of the nodal zone and the direction vector parallel to the axis of the strut. Then the cross-sectional area of the strut can be calculated by the procedure described hereafter.
The parameters of the nodal zones that are supposed to be known in order to define the remaining struts are:
- the dimensions of the loading area (columns) and supporting areas (piles) - the height of the node, which is defined by the height of horizontal struts or
the height of influence areas of ties (two times the distance from the edge to the gravity centre of the bars)
Hereafter the dimensions of the supports are called a and b, respectively in the x- and y-direction. The height of the node is referred to as uc for compression nodes under the column; uc being equal to two times ac, the distance from the edge to the axis of the horizontal strut used in the strut and tie model. The height of compression-tension nodes is defined by us, which is two times the distance from the edge to the axis of the reinforcement as, see Figure 4.2 for the illustration.
Figure 4.2 2C2T-node over a pile located at the corner of a pile cap (a 2C3T-node would be detailed similarly with a vertical tie passing through the centre of the lower face of the parallelepiped nodal zone)
The method consists in identifying the corners of the nodal zone. Knowing the dimensions of the bearing plate and the height of the node, some assumptions lead to the determination of the other corners. For instance, in Figure 4.3, some typical cases are presented, that can be found in the design of pile caps. These nodal zones correspond to the ones under columns, where concentrated compression stresses are spreading in the element. This is represented by several inclined struts leaving the node towards the piles. According to the geometry of the pile cap, a number of inclined struts is chosen. In a four-pile cap, there would be four inclined struts going to each pile, as shown in Figure 4.3 (b), while for the same pile cap with an additional pile under the column, there would be also a vertical strut, as shown in Figure 4.3 (a).
Note that, in Figure 4.3 (a) and (b), four nodes would be used in the model, and therefore the general nodal zone under the column would be divided in four sub-nodal zones in Figure 4.3 (a) and (b), while eight nodes would be used in Figure 4.3 (c) corresponding to eight nodal zones. Each sub-nodal zone corresponds to the region of interaction between the external vertical stress, the stress in the inclined strut, and the stress in two perpendicular horizontal struts balancing the two first mentioned stresses. When the geometry of nodal zones is discussed hereafter it corresponds to the geometry of sub-nodal zones, also equivalent to the geometry of the 2C2T-nodal zone in Figure 4.2.
Figure 4.3 Examples of nodal zones in pile caps (a) 6C-node under the column for a 5-pile cap (three of the struts are shown), (b) 5C-node under the column with triangular horizontal struts for a 4-pile cap (four inclined struts are shown), (c) 9C-node under a column for a 8-pile cap, combination of the two previous nodes (two struts are shown at each level)
The geometry of the nodal zones in Figure 4.3 is consistent. However some problems arise to define the exact position of the nodes because of the complex geometries of the elementary nodal zones. As it has been explained previously, the position of a node should correspond to the point of intersection of the centroids of the struts balancing at the node. However, in such cases it is complicated to define the exact position of this point of intersection; moreover it is often unclear if the centroids of the four struts intersect at the same point. For this reason the use of rectangular parallelepiped nodal zones, as illustrated in Figure 4.4, seems to be an adequate solution. It will be shown hereafter that all types of nodes encountered in the design of a pile cap can be designed using this geometry, which is actually an extension of the two-dimensional geometry commonly used. The rectangular parallelepiped (or right
(a)
(b) (c)
cuboid) shape of the nodal zone will be referred to later as “parallelepiped nodal zone” or “cuboid nodal zone”.
As it has been explained in Section 3.6.3 for the two-dimensional case, in order to define the nodal zone, the ties can be considered as struts acting in compression from the opposite side of the nodal zone. Therefore in three-dimensions every nodal zone, or a partition of it, can be explained by the elementary 4C-nodal zone, which corresponds to one fourth of the parallelepiped nodal area defined in Figure 4.4.
Figure 4.4 5C-node under the column, alternative with rectangular horizontal struts (only two of the inclined struts are shown). The inclined struts have a hexagonal cross-section as for the 2C2T-node in Figure 4.2 In this manner the parallelepiped geometry can be used as well for the 2C2T-node represented in Figure 4.2, as for its extension with a vertical tie (made for instance of stirrups), that is to say a 2C3T-node. Every type of three-dimensional concentrated nodal zone can be designed using the three-dimensional elementary 4C-node (Figure 4.4), and the two-dimensional elementary 3C-node (Figure 3.8), completed by the method for combining struts defined in Section 4.3.4. In more complex cases, the method presented could be adapted.
Demonstration: forces concurrent at the elementary 4C-node
In order to fulfil moment equilibrium of the node, the forces in the struts and the ties acting on the node should be concurrent. When drawing a strut-and-tie model, this assumption is considered to be true as the struts and ties are drawn such that they intersect at the node, and the unknown forces in the struts and the ties are calculated based on this assumption. However this assumption should also be fulfilled when designing the nodal zone. Then, if one considers the stresses acting on each face of the nodal zone, showing that the forces are concurrent is equivalent to showing that the
centroidal axes of the struts acting on the nodal zone are concurrent. As the elementary nodal zone is defined by the intersection of three orthogonal struts, it is obvious that the centroid of the resultant parallelepiped belongs to the centroidal axis of each of the strut. To demonstrate that it also belongs to the centroidal axis of the inclined hexagonal strut is not as straightforward. Note that it is obvious to show it in two dimensions, when the strut has a rectangular cross section (Figure 4.5). In three- dimensions, the demonstration can be done in several ways.
Figure 4.5 Concurrency between the centroidal axis of the inclined strut and the centroid of a two-dimensional 3C-nodal zone
It should be noticed that the strut is a prismatic geometrical object formed by the translation of the parallelepiped nodal zone in the direction of the direction vector vr of the strut. Therefore any orthogonal cross-section of the strut corresponds to the projection of the parallelepiped, in the direction of vr
, in the orthogonal plane to the strut. Then the hexagonal projection can be regarded as “the view of the cube from the orthogonal plane” (Figure 4.6) (not from a point or the centroid, as this would be a perspective view). The hexagon can be divided into a parallelogram and two triangles whose centre of symmetry is the centre of the parallelogram. The two triangles compensate each other and thus the centre of the parallelogram is the centre of the hexagonal cross section, and it corresponds also to the projection of the centre of gravity of the parallelepiped, as the parallelogram results from the projection of the diagonal face of the parallelepiped. Therefore the centroidal axis of the hexagonal strut goes through the centre of the parallelepiped, hence assuring the equilibrium of moments at the node.
Another way to demonstrate it would have been to show that the centre of gravity of the parallelepiped is the centre of symmetry of the orthogonal cross section of the strut passing at this point.
w CG
Figure 4.6 Centre of gravity of the cross-section of the inclined strut