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26Tamrazyan A.G., et al UDC 624.045.12
DOI: 10.54338/27382656-2021.2-3
Ashot George Tamrazyan1, Tatiana Anatoly Matseevich1
1 Moscow National Research "Moscow State University of Civil Engineering" (NRU MGSU), Moscow, RF
STIFFNESS OF POST-TENSIONED GIRDERLESS FLOORS WITH DIFFERENT COLUMN GRIDS
The paper considers models of monolithic flat floor slabs with five spans in both directions. The cell sizes are 6*6m, 6*9m, and 6*12m. The calculation method is based on the application of temperature load and rope modeling of rod elements. It is shown that post-stressing should be used for slab side lengths over 7 m, as the installation ofpre-stressed reinforcement for shorter lengths is less feasible and causes high economic costs.
Keywords: reinforced concrete slab, stiffness, deflection, column grid, post-tension. Introduction
When designing structures with a prestressed system without adhesion to concrete, it should be taken into account that prestressed reinforcement does not transfer forces to the concrete over the entire length, but only at the anchor points at the ends of the structure, as well as at the bends in the ropes. Accordingly, the prestress must be assumed in the calculation as external forces applied to the structure. The forces formed at rope bends depend primarily on the rope geometry and the forces in it.
Works [1-8] are devoted to studies of girderless structures with prestressed reinforcement. The calculation of prestressed elements in deformations (stiffness) is carried out according to the normative document1. The compression force Np is determined by taking into account all losses and ysp = 0.9. Deflections are calculated by considering the strength of concrete at different stages of loading, including the transfer of compression forces.
The deflections of reinforced concrete elements are calculated under the condition:
f < fult, (1)
where f is the deflection of the reinforced concrete element from the external load,
fult is the value of the maximum permissible deflection of a reinforced concrete element. For bendable elements of a constant cross-section, along the length of the element without cracks, the deflections are determined by the general rules of structural mechanics using the stiffness of the cross-sections determined by the formula:
D = Eb! • Ired, (2)
where Eb1 is the deformation modulus of compressed concrete, determined according to the load duration and taking into account the presence or absence of cracks,
Ired is the moment of inertia of the given cross-section in relation to its center of gravity, determined taking into account the presence or absence of cracks.
Materials and methods
The calculation method is based on the temperature load application and the modeling of ropes with rod elements [9].
1 Posobiye k SP 63.13330.2012 «Konstruktsii zhelezobetonnyye monolitnyye s napryagayemoy armaturoy bez stsepleniya s betonom. Pravila proyektirovaniya» (in Russian).
The modeling of the reinforcement of the central cell structure is carried out using a rod element in LIRA SAPR. The cross-section of the rod element is similar to the area of prestressed reinforcement in the floor slab as well as to the computer synthesis [10].
In this paper, models of flat floor slabs with five spans in both directions are investigated. The cell sizes are 6*6m, 6*9m and 6*12m, with the thickness of h = 0.3m (Fig.1). The size of the finite elements is 0.3*0.3m.
Fig. 1. Scheme of a monolithic floor with rope-mounted reinforcement of the central cell
Main characteristics of construction materials
Concrete class B30. Post-tensioned reinforcement in form of ropes (monostrends): K7O (RsP,n = 1860403 kN/m2, Esp = 1.95-105 MPa, d = 15.7mm, Asp = 1.54cm2). For calculation 5, 7 and 9 ropes are taken.
A uniformly distributed load q = 5 kN/m2 is applied to the slab.
The reinforcement of the central cell is modeled by the rod elements in the structure, the section of which is similar to the prestressed reinforcement area in the element. To simulate prestressing in the reinforcement, a temperature load is applied to it, which is calculated according to the formula:
At = s0 /a, (3)
where e0 = <x0 / E ,
Ep is the modulus of elasticity of the prestressed reinforcement, 00 is the controlled tension of the prestressed reinforcement, a is the expansion coefficient of the reinforcing steel. The deflections will be considered at characteristic points of the cell (Fig. 2).
—[
u * f
? c
Fig.2. Characteristic points of the central cell: f - deflection in the center of the cell, fa - deflection in the center of the smaller side of the cell, fb - deflection in the center of the larger side of the cell
The prestress, calculated by the formula (3), is equal:
0.7-1860 5
sn =-r = 667 -10 5
0 1.95-105
At = 667-10' 5 = 556 „c. 0.000012
The reinforcement of the structure's central cell was modeled using a rod element in LIRA SAPR 9 (Fig. 3). A 6x6m slab cell with 5 ropes along the contour is modeled using flat (floor slab) and volumetric (columns) elements. The size of the finite element is 0.3x0.3m.
The floor slab is defined as a plate with a thickness of 300mm. The section of the column is 600x600mm.
Fig. 3. Calculation scheme for a 6x6m central cell with 5 ropes
The results of the calculation are shown in Figures 4 and 5.
Fig. 4. Travel along the Z axis of the entire slab with 6x6m cells and a central cell reinforced with 5 ropes
Fig. 5. Travel along the Z axis of the central cell reinforced with 5 ropes
Results
Based on the calculations carried out for the central cells 6x6, 6x9 and 6x12m with different numbers (5, 7, 9) of ropes, the deflection values were obtained, which are summarized in Table 1.
Table. Deflections of central cells with different numbers of ropes
Cell, m Deflections (5 ropes), mm Deflections (7 ropes), mm Deflections (9 ropes), mm
a*b fi fa1 fb1 f2 fa2 fb2 f3 fa3 fb3
6x6 2.84 2.52 2.52 1.81 1.88 1.88 0.936 1.43 1.43
6x9 14.4 5.03 14.8 13 3.99 13.7 11.7 3.52 12.7
6x12 44.9 5.7 46 42.1 4.95 43.6 39.6 4.29 41.5
Table 1 shows that for a 6 x6m square cell, with an increase of prestressing ropes from 5 to 7, the deflection in the center decreases from 2.84 to 1.81mm (1.57 times or 36.3%), and for 5 to 9 ropes - from 2.84 to 0.936 (3.03 times or 67%), in the center of "a" and "b" sides with an increase of the ropes from 5 to 7, the deflection decreases from 2.52 to 1.88mm (1.34 times or 25.4%) and for 5 to 9 ropes - from 2.52 to 1.43 (1.75 times or 43%).
For a 6 x9m rectangular cell, with an increase of prestressing ropes from 5 to 7, the deflection in the center decreases from 14.4 to 13mm (1.1 times or 9.7%), and for 5 to 9 ropes - from 14.4 to 11.7 (1.23 times or 18.8%), in the center of the short side "a" with an increase of the ropes from 5 to 7, the deflection decreases from 5.03 to 3.99 mm (1.26 times or 20.7%), and for 5 to 9 ropes - from 5.03 to 3.52 (1.43 times or 30%), with an increase of the ropes from 5 to 7 in the center of long side "b", the deflection decreases from 14.8 to 13.7mm (1.08 times or 7.4%) and for 5 to 9 ropes - from 14.8 to 12.7 (1.17 times or 14.2%).
For a 6 x12m rectangular cell, with an increase of prestressing ropes from 5 to 7, the deflection in the center decreases from 44.9 to 42.1mm (1.07 times or 6.2%), and for 5 to 9 ropes - from 44.9 to 39.6 (1.13 times or 11.8%), in the center of the short side "a" with an increase of the ropes from 5 to 7, the deflection decreases from 5.7 to 4.95mm (1.15 times or 13.2%), and for 5 to 9 ropes - from 5.7 to 4.29 (1.33 times or 24.7%), in the center of long side "b" with an increase of the ropes from 5 to 7, the deflection decreases from 46 to 43.6mm (1.06 times or by 5.2%), and for 5 to 9 ropes - from 46 to 41.5 (1.11 times or by 9.8%).
The results show that with an increase of the long side of the slab cell from 5 to 7m, the effect on the deflection of the installation along the contour of a larger number of ropes is reduced significantly. For example, for the center of the cell, the effect on the deflection of installing 9 ropes compared to 5 ropes is reduced from 67% to 11.8%, for the short side "a" - from 43.3% to 24.7%, for the long side "b," - from 43.3% to 9.8%.
Conclusion
1. Prestressing is recommended to apply when the side of the slab is longer, than 7m, since with a shorter side length, the installation of prestressing reinforcement is less expedient and entails high economic costs.
2. The use of post-tensioned ropes on the shorter side of a rectangular cell is unreasonable since the deflection on the short side meets the normative value even before the prestressed reinforcement is introduced into the model.
3. It is expedient to install more ropes for longer cells. For example, according to the calculation, it is not reasonable to install less than 7 ropes for a 6*9m cell, and it is expedient to install 9 ropes or more for a 6*12m cell. However, the use of 9 or more ropes can create large flexures at the base of the column, which would require additional reinforcement and strength testing of the compressed concrete.
4. With an increase of the long side of the slab cell from 6 to 12m, the effect on the deflection of the installation along the contour of larger number of ropes decreases significantly: for the center of the cell, it reduced from 67% to 11.8%, for the short side "a" - from 43.3% to 24.7%, for the long side "b" - from 43.3% to 9.8%. This indicates that for cells with sides of 7m or more, it is more expedient to use a average number of ropes but with a higher prestressing force to reduce the financial cost of their installation.
5. The method of modeling prestressed ropes using rod reinforcement with the application of a temperature load is simple, accurate, and easy to use.
References
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Ashot George Tamrazyan, Doctor of Science (Engineering), Professor (RF, Moscow) - National Research Moscow State University of Civil Engineering (NRU MGSU), Head of the Department ofReinforced Concrete and Masonry Structures, tamrazian@mail.ru
Tatiana Anatoly Matseevich, Doctor of Science (Physics and Mathematics) (RF, Moscow) - National Research Moscow State University of Civil Engineering (NRU MGSU), Head of the Department of Higher Mathematics, matseevichta@mgsu.ru
Submitted: 13.12.2021 Revised: 16.12.2021 Accepted: 17.12.2021
