In this Section; the results of the design with the four alternative strut-and-tie models are presented and commented. As it has been explained previously, in Section 6.3.2, two types of design are conducted. In the first one (Section a) below) the level of the main reinforcement as is free to be modified in order to satisfy the needs of the models. In the second type of design (Section b) below) as is fixed equal to the value used in the original design according to the code, where the main reinforcement has been placed in one layer
a) With the level of the main reinforcement as modified according to the design with the strut-and-tie model
The force distribution in two of the strut-and-tie models, at final iteration, is shown in Figure 6.11. The effect of the refinement of the nodal zones under the column on the geometry of the models is also illustrated in Figure 6.12. It should be noted, that the size of the column is fixed. In Figure 6.11 and in Figure 6.12, the dimensions of the column are the same in (a) and (b). It is the position of the nodes that is changed, the nodes being positioned closer to the edges of the column in (b), after refinement.
Figure 6.11 Force distribution in alternative strut-and-tie models: (a) Model 1, with only truss action and without refinement of nodal zones under the column, (b) Model 4, with combination of truss action and direct arch ation and with refinement of nodal zones under the column, (c) indicative colour scale of stresses
(a)
(b)
(c)
Tension
Compression 0
Figure 6.12 Top view of the strut-and-tie model at the final iteration (a) without refinement of nodal zones under the column (Model 1 and Model 3), (b) after refinement of nodal zones under the column (Model 4)
The results obtained at final iteration, for the four alternative strut-and-tie models, are given in Table 6.5 and the need for reinforcement in Table 6.6. The nomenclature of the struts and the ties used in these tables is described in Figure 6.8.
The first model is the reference one; no optimisation is done. The nodes are placed in the middle of their respective bearing areas and all the load is carried by truss action.
In the second model, the node position is refined. As can be seen in Table 6.6, the needed reinforcement is reduced along with reduction of forces transmitted into the inclined and horizontal members of the model. Model 3 reduces the needed amount of stirrups by considering that a part of the load is transmitted by direct arch action. The effect, compared to Model 1 is beneficial. The combination of the refinement of node positions and transfer of a part of the load by arch action is applied in Model 4. This model is the most advanced and efficient one. The amount of reinforcement required is reduced by 22% compared to the reference model.
(a) (b)
Table 6.5 Geometry of the alternative strut-and-tie models at final iteration and utilisation ratios of nodes and struts, with a free level as of the main reinforcement
Model 1 Model 2 Model 3 Model 4
Description of model
Truss action X X X X
Direct arch action X X
Refinement of nodal zones X X
Total load per pile (total load on pile cap: 11000kN) 1100 1100 1100 1100
Load transfered by truss action and load transfered
by arch action
Truss AD truss action 1100 1100 1081 918
arch action 0 0 18 182
Truss BD truss action 1100 1100 344 127
arch action 0 0 756 973
Truss CE truss action 1100 1100 1083 849
arch action 0 0 17 251
Level of axes of reinforcement and of
horizontal struts
as 120 100 120 125
ac 2D 70 55 70 60
ac 3D 25 35 30 50
Dimensions of the effective loading areas used under
the column
wc-x 3D 600 116 600 123
wc-y 3D 640 378 640 358
wc-x 2D 600 69 600 69
wc-y 2D 320 320 320 320
Check of nodal zones
Strut A 88% 92% 89% 91%
Strut B 72% 78% 78% 73%
Strut C 99% 98% 100% 100%
Strut D 16% 98% 18% 98%
Strut E 38% 85% 38% 81%
Strut hor-x 3D 91% 91% 92% 94%
Strut hor-y 3D 42% 91% 48% 96%
Strut hor-x 2D 97% 100% 97% 95%
Check crushing/splitting of direct struts
Arch AD 0 0 1% 15%
Arch BD 0 0 26% 57%
Arch CE 0 0 1% 22%
In the most advanced strut-and-tie model (Model 4), it can be seen that the required level of the centroid of the main reinforcement, as, reaches a value of 120 mm (Table 6.5), as as is free to be modified according to the needs of the strut-and-tie model.
However, when the same pile cap was designed using the “Concrete Handbook –
Structural Design”, as was set to 75 mm. This difference has two reasons. In the model developed, the permissible region to spread the bars above the piles is very limited. According to the definition of the parallelepiped nodal zones and in order to respect a consistent deviation of stresses, the bars should be placed within the pile width above a pile, so that the influence width of the bars is equal to the pile width.
For that reason, in some cases, the bars have to be placed in several layers, which increases the height of the nodes. The other reason, that is decisive in that case, is that the stresses in the inclined struts incoming in the nodal regions at the piles are too high. Therefore, the height of the nodes over the piles has to be increased in order to reduce the stresses.
Table 6.6 Reinforcement needed in the alternative strut-and-tie models, with a free level as of the main reinforcement
Model 1 Model 2 Model 3 Model 4 Bars:
number and length Description of
model
Truss action X X X X
Direct arch action X X
Refinement of nodal X X
Reinforcement required at each tie
T1 As 4024 3321 4905 4673 2 x 2000
Number of bars 9 ỉ25 7 ỉ25 10 ỉ25 10 ỉ25 4000
T2 As 2850 2490 2913 3035 4 x 1000
Number of bars 6 ỉ25 6 ỉ25 6 ỉ25 7 ỉ25 4000
T3 As 872 511 891 623 2 x 2000
Number of bars 2 ỉ25 2 ỉ25 2 ỉ25 2 ỉ25 4000
T4 As 2999 2545 3046 3180 2 x 750
Number of bars 7 ỉ25 6 ỉ25 7 ỉ25 7 ỉ25 1500
T5 As 872 511 1480 1008 2 x 2000
Number of bars 2 ỉ25 2 ỉ25 4 ỉ25 3 ỉ25 4000
T6 As 4026 3323 3189 2273 2 x 2500
Number of bars 9 ỉ25 7 ỉ25 7 ỉ25 5 ỉ25 5000
T7 As 872 511 863 446 2 x 1700
Number of bars 2 ỉ25 2 ỉ25 2 ỉ25 1 ỉ25 3400
T8 As 6000 4864 6000 5033 1 x 2500
Number of bars 13 ỉ25 10 ỉ25 13 ỉ25 11 ỉ25 2500
T9 As 872 511 274 62 2 x 1700
Number of bars 2 ỉ25 2 ỉ25 1 ỉ25 0 (Asmin) 3400
T10 As 2851 2491 2820 2173 4 x 500
Number of bars 6 ỉ25 6 ỉ25 6 ỉ25 5 ỉ25 2000
T11 As 3036 3036 2986 2533 4 x 1000
Number of bars 16 ỉ16 16 ỉ16 15 ỉ16 13 ỉ16 4000
T12 As 3036 3036 949 349 4 x 1000
Number of bars 16 ỉ16 16 ỉ16 5 ỉ16 2 ỉ16 4000
T13 As 3036 3036 2989 2344 2 x 1000
Number of bars 16 ỉ16 16 ỉ16 15 ỉ16 12 ỉ16 2000 Amount of
reinforcement (without considering
anchorage lengths)
Longitudinal (kg) 617 513 594 543
Transversal (kg) 114 114 132 90
Stirrups (kg) 252 252 174 133
Total (kg) 983 879 899 766
A sketch of the layout of the main reinforcement according to the design with Model 4 is given in Figure 6.13. As can be seen on this figure, some ties require two layers.
The height of the reinforcement, which is 75 mm above the piles for the first layer and 175 mm for the second layer, corresponds to the average between the two directions.
One could for instance place axis of the reinforcement along in x-direction at 62 mm and the one in y-direction at 88 mm. The reinforcement should be alternate in the two directions; it should be avoided to place two layers in one direction and two layers on top in the other direction, for the equilibrium at the node. The ties that are not over piles and do not connect singular nodes can be spread. However in this example it is only the case of Tie 8. Tie 6 could also be spread but as it has to be placed beside its exact position due to the conflict with the adjacent Tie 1, it is better to keep it concentrated, so that its axis is not too far from its position in the strut-and-tie model.
The shear reinforcement is not represented. It has to be spread to cover the entire stress field at the smeared nodes. The recommendation of Eurocode 2 can be followed, that is to spread it along the strut over the “middle three-fourth” of the distance between the pile face and the column face, and over a certain distance laterally as well.
Figure 6.13 Layout of main reinforcement for the design with the strut-and-tie Model 4 (bars ỉ25) (a) first layer 75mm above top of the piles, (b) second layer 175mm above top of the piles
b) With the level of the main reinforcement as fixed according to the design with the code
The same example has been conducted with the height of main reinforcement as fixed to 75mm, which corresponds to the average between the two directions for one layer.
The reason for that was to provide an additional comparison with the design using the codes where the reinforcement is spread in one layer. The results obtained at final iteration, for the four alternative strut-and-tie models, are given in Table 6.7 and the
(a) (b)
need for reinforcement in Table 6.8. The nomenclature of the struts and the ties used in these tables is described in Figure 6.8. In the design with as fixed, the amount of steel required is reduced, from 766 kg (Table 6.6) to 724 kg (Table 6.8), with Model 4.
It should be noted that when spreading the main reinforcement, the transverse tension induced should be checked as described in Figure 5.27. Besides, some checks of compressive stresses from the inclined struts at nodal zones over the piles are not fulfilled anymore, and would require an increase of as. However the strength value used for these nodes, corresponding to two-dimensional CCT-nodes, seems a bit conservative in the three-dimensional case and as the increase in tension is not large, the results can still be considered for a comparison.
Table 6.7 Geometry of the alternative strut-and-tie models at final iteration and utilisation ratios of nodes and struts, with a fixed level as of the main reinforcement
Model 1 Model 2 Model 3 Model 4
Description of model
Truss action X X X X
Direct arch action X X
Refinement of nodal zones X X
Total load per pile (total load on pile cap: 11000kN) 1100 1100 1100 1100
Load transferred by truss action and load transferred
by arch action
Truss AD truss action 1100 1100 1013 848
arch action 0 0 87 252
Truss BD truss action 1100 1100 310 100
arch action 0 0 790 1000
Truss CE truss action 1100 1100 1003 777
arch action 0 0 97 323
Level of axes of reinforcement and of
horizontal struts
as 75 75 75 75
ac 2D 65 55 65 55
ac 3D 25 35 30 50
Dimensions of the loading areas used at the column
wc-x 3D 600 117 600 123
wc-y 3D 640 378 640 358
wc-x 2D 600 69 600 69
wc-y 2D 320 320 320 320
Check of nodal zones
Strut A 107% 102% 112% 114%
Strut B 83% 83% 94% 87%
Strut C 115% 107% 122% 123%
Strut D 16% 97% 18% 97%
Strut E 37% 83% 37% 81%
Strut hor-x 3D 87% 88% 92% 93%
Strut hor-y 3D 40% 88% 47% 94%
Strut hor-x 2D 98% 97% 98% 97%
Check of crushing/splitting of direct struts
Arch AD - - 4% 24%
Arch BD Arch CE
- -
- -
27%
7%
59%
32%
Table 6.8 Reinforcement needed in the alternative strut-and-tie models, with a fixed level as of the main reinforcement
Model 1 Model 2 Model 3 Model 4
Bars:
number and length Description of model
Truss action X X X X
Direct arch action X X
Refinement of nodal zones X X
Reinforcement required at each tie
T1 As 3833 3233 4877 4596 2 x 2000
Number of bars 8 ỉ25 7 ỉ25 10 ỉ25 10 ỉ25 4000
T2 As 2715 2424 2946 3026 4 x 1000
Number of bars 6 ỉ25 5 ỉ25 7 ỉ25 7 ỉ25 4000
T3 As 831 498 901 621 2 x 2000
Number of bars 2 ỉ25 2 ỉ25 2 ỉ25 2 ỉ25 4000
T4 As 2834 2476 3084 3139 2 x 750
Number of bars 6 ỉ25 6 ỉ25 7 ỉ25 7 ỉ25 1500
T5 As 831 498 1435 965 2 x 2000
Number of bars 2 ỉ25 2 ỉ25 3 ỉ25 2 ỉ25 4000
T6 As 3835 3234 2831 1975 2 x 2500
Number of bars 8 ỉ25 7 ỉ25 6 ỉ25 5 ỉ25 5000
T7 As 831 498 769 390 2 x 1700
Number of bars 2 ỉ25 2 ỉ25 2 ỉ25 1 ỉ25 3400
T8 As 5670 4732 5670 4732 1 x 2500
Number of bars 12 ỉ25 10 ỉ25 12 ỉ25 10 ỉ25 2500
T9 As 831 498 235 46 2 x 1700
Number of bars 2 ỉ25 2 ỉ25 1 ỉ25 0 (Asmin) 3400
T10 As 2716 2425 2515 1900 4 x 500
Number of bars 6 ỉ25 5 ỉ25 6 ỉ25 4 ỉ25 2000
T11 As 3036 3036 2796 2342 4 x 1000
Number of bars 16 ỉ16 16 ỉ16 14 ỉ16 12 ỉ16 4000
T12 As 3036 3036 856 276 4 x 1000
Number of bars 16 ỉ16 16 ỉ16 5 ỉ16 2 ỉ16 4000
T13 As 3036 3036 2769 2143 2 x 1000
Number of bars 16 ỉ16 16 ỉ16 14 ỉ16 11 ỉ25 2000 Amount of
reinforcement (without considering
anchorage lengths)
Longitudinal (kg) 567 490 580 526
Transversal (kg) 114 114 116 75
Stirrups (kg) 252 252 164 123
Total (kg) 933 856 861 724