3.6.3 Nodes and nodal zones
3.6.3.2 Classification of nodal zones and description of their geometries
In the definition of the geometry of nodal zones ties are assimilated to struts acting on the other side of the node in compression. At the node, the strut acting on the other side of the nodal zone corresponds to the reinforcement
At the beginning of the development of strut-and-tie models, hydrostatic nodal zones were used. The faces of the nodal zones were perpendicular and proportional to the forces acting on the node, see Figure 3.7 (a). Therefore no shear stresses were created at the node (e). However it is almost impossible to manage to have geometries assuring hydrostatic nodes in a model. For this reason, all the major codes recognize non-hydrostatic nodes nowadays (Figure 3.8). Schlaich recommended to keep stress ratios on adjacent edges of a node above 0.5, otherwise the non-uniformity of stress distribution could make the check of the node unconservative (Schlaich et al. 1987).
T C
T T C
C
CTT-node CCT-node
CCC-node C
C
C
Figure 3.7 Example of hydrostatic nodal zone (a) hydrostatic nodal zone and extended nodal zone for a CCT-node, (b) representation of the forces joining at the node, (c) equivalent representation of forces with tension considered as compression acting on the other side of the nodal zone (d) stresses acting on nodal zone, (e) Mohr’s circle for the state of stress in the nodal zone, limited to a point in this case
T
C1
Cstrut wstrut
(a) 90° θ
(b)
(e) τ
σ σ1= σ2
(d)
Cstrut
C1
T
(c)
Cstrut
C1
C2 = T σstrut
σ2
σ1
wsupport
us
Figure 3.8 Example of non-hydrostatic nodal zone (a) non-hydrostatic nodal zone and extended nodal zone for a CCT-node, (b) representation of the forces joining at the node, (c) equivalent representation of forces with tension considered as compression acting on the other side of the nodal zone (d) stresses acting on nodal zone, (e) Mohr’s circle for the state of stress in the nodal zone
Using the notations defined in Figure 3.7 and Figure 3.8, the width of the inclined strut is defined by:
θ
θ sin
cos + ⋅
⋅
= s support
strut u w
w (3.9)
The checks of hydrostatic and non-hydrostatic nodes are similar. Either the principal stresses are checked in the nodal area, or the stresses in the struts, defined by the force in the member divided by the cross sectional area, defined by Equation 3.9, are
T
C1
C
σ τ
σ2 σ1
θ
wstrut
(a) (b)
(e) (c)
σstrut
σ2
σ1 (d)
Cstrut
T
C1
C1
C2 = T
Cstrut
Nodal zone Extended nodal zone
wsupport
us
checked against strength values defined for each type of node. Some recommendation of these strength values are given in Section 3.6.3.3.
While hydrostatic nodal zones are defined by the intersection of all the joining members, non-hydrostatic nodal zones are often defined by extended nodal zones, which correspond to the intersection between the two struts in balance at the node, located inside the structure. The difference between the nodal zone and the extended nodal zone is illustrated in Figure 3.8 and nodal zone geometries for different types of nodes are detailed hereafter.
Several types of nodal zones can be found in a strut-and-tie model depending on the forces acting on them. According to the denomination defined previously, the most common cases in a two-dimensional model are: CCC, CCT and CTT, illustrated below in Figure 3.9, Figure 3.10 and Figure 3.11.
A good way to explain the iterative choice of members’ geometry and to introduce the problem of three-dimensional nodal zones treated in the next chapter (Section 4.3), is to look at how the geometry of different types of nodal zones is defined.
Figure 3.9 Example of compression-tension node with a tie in one direction (Schọfer 1999)
As it has been explained previously, to define the nodal zone, the ties are often represented as struts acting on the other side of the node. The definition of the CCT- nodal zone illustrated above in Figure 3.9 would therefore be equivalent to the one of a CCC-nodal zone with a horizontal strut acting on the left side of the node. It is quite clear in this case that the nodal zone is defined by the intersection of the struts inside the element, where the stresses from the tie deviate the stresses from the inclined strut in the nodal zone, to balance the external stresses. The width of the bearing plate and the vertical level of the node are sufficient to define the inclined strut and thus the nodal zone.
Figure 3.10 Example of compression node (Schọfer 1999)
For a CCC-node, the extended nodal zone, represented in Figure 3.10, actually includes two nodes, connected by a horizontal strut. Hence the nodal zone can be separated in two sub-areas, delimited by the vertical line passing through the intersection between the two inclined struts, and the border between the bearing areas influencing each of the struts. The horizontal strut, resulting from the action of each sub-nodal zone on the other can be considered to act at this border. Then each of the two sub-nodal zones corresponds to the elementary case of a CCC-node and the Equation 3.9 applies with us=uc in this case.
When the width of the struts has been chosen, the intersection of the struts defines the nodal zone. It is the same case for a CCT-node and a CTT-node with anchor plates, where the ties act as a strut working in compression from the other side of the node.
Figure 3.11 Example of compression-tension node with ties in more than one direction (Schọfer99)
For a CTT-node, the choice of the width of the strut, the width of the vertical tie and the bond length has to be consistent. If any one is chosen, the others have to follow, and the position of the corners of the nodal zone is determined.
In all cases the nodal zone can be defined as the intersection of all struts and ties intersecting at the node, the ties being considered as struts working in compression from the other side of the node.