42nd Brazilian Concrete Congress
The optimization for the lowest cost, of concrete beams with rectangular section with simple or double reinforcement, is studied.
Through a program, the most economical total height and width are determined considering the constructive provisions: minimum section dimensions, longitudinal and transverse reinforcement rates, minimum horizontal and vertical spacing between bars, skin reinforcement and cover. To determine the useful height, the normalized nominal diameters of the bars and their distribution in multiple layers for tensioned and compressed reinforcement are considered. Ultimate limit states and serviceability limit states are also considered. The optimization results are compared using the restrictions in a combined way, analyzing the influence of each one. It is concluded that the non-consideration of the constructive provisions in the optimization leads to non-optimal results when executing the project.
The design of reinforced concrete beams is an iterative process, where for each section studied there are several conditions to be met, in addition to the variation of the final calculation loads, considering the self-weight. It is then considered that the problem is solved when a section is found that satisfies the static conditions and constructive dispositions, not taking into account the cost of the adopted solution.
Even with a commercial structural calculation program, the process of determining optimal sections becomes very slow, as it is not done automatically, and the engineer must carry out the entire survey of the costs of the calculated beam for each desired section.
There is therefore a need to create tools so that structural projects are developed optimally from the beginning.
The use of beam section optimization is fully applicable in works even where there are restrictions on their dimensions, such as impositions of the architectural project. Even if these restrictions do not allow choosing the optimal section, through the cost curves as a function of the dimensions of the section, it is possible to choose the lowest cost section that meets all the restrictions.
As for the diversity of cross-sections in use on the same floor, execution control can be improved by preparing a Forms Project, detailing the manufacture and assembly of each panel that will make up each beam.
One can group beams with similar loading and then determine the optimal section that satisfies the most stressed beam in the group and use an optimized frame for those with lower stress. In certain cases, it is even possible to use an optimal section for each beam in the project, such as in foundation beams or in those where there is no limitation in terms of height or width.
When preparing an optimized project, it is necessary that the costs of the materials used represent the reality of who will execute it. In addition, it is important to identify strategic differentials that the builder may have, such as having its own equipment for formwork, adoption or not of a Formwork Project or direct suppliers of certain materials.
The builder’s level of control will determine the degree of complexity that can be achieved in the optimization. Pre-manufactured industries, construction companies that have a formwork and frame center, purchase of ready-made frames, for example, allow optimizing the frame and formwork as much as possible without compromising execution time and quality.
2. Considerations on constructive provisions
All construction provisions comply with NBR 6118 of 1980. The cover adopted is 2cm for all reinforcements and the maximum dimension of the aggregate is 19mm (gravel 1).
When there is no longitudinal compression reinforcement, assembly reinforcements are inserted. Minimum steel areas are stipulated according to the Standard for longitudinal tensile and transverse shear reinforcement, as well as for skin reinforcement in cross sections with a height greater than 60cm. The shear reinforcement has constant steel area and spacing along the entire span of the beam, with the required number of branches always even.
The cost of concrete, steel and formwork was considered as the total cost of the beam, already including labor, without social charges and BDI. The costs of these materials are an average of the costs appropriated by some local construction companies, in works of vertical buildings, for residential or commercial use.
3. Dimensioning and detailing
The calculation of the longitudinal tension and compression reinforcement is done through an iterative process, arranging the bars in layers, when necessary, depending on the maximum spacing between the bars and the useful space to arrange them in the section. The center of gravity of the reinforcements is calculated, having then the exact values of d (useful height) and d’ for this situation, making the calculation of the steel areas again with these values. The process ends when the steel areas of successive iterations are equal.
The program uses the simplification allowed by NBR 6118 of considering the efforts concentrated in the center of gravity of the longitudinal reinforcements, provided that the distance from this center to the point of the reinforcement section furthest from the neutral line, normally measured to it, is less than 5% of the total height of the section. If this condition is not met, this cross section is discarded for optimization purposes. The program calculates the minimum spacing between horizontal and vertical bars considering the maximum dimension of the aggregate, the diameter of the bar or spacing not smaller than 2cm. There is also the option of leaving spaces for the passage of the vibrator, both in the tension and compression armature.
If there is no need for compression reinforcement, mounting reinforcement is added for the stirrups with the same diameter as these. When calculating the shear reinforcement, the tension in the compressed connecting rod is verified.
The deflections are calculated considering the effects of cracking and creep in mid-span, as well as the crack width.
For the generation of Cost x Total Height curves, for several section widths, the program allows choosing which criteria will be considered in the optimization, as well as the initial and final height and width.
4. Analysis of results
For this work, a simply supported beam with a span of 6.0 m, permanent load of 10.8 kN/m excluding self-weight and accidental load of 12 kN/m was considered. The beam’s own weight is automatically calculated by the program for each studied section.
The costs of materials, including labor, are as follows: concrete fck 20 Mpa, 164.85 R$/m3; CA-50 steel, 1.58 R$/kg; shapes, 18.69 R$/m2.
Initially, the Cost x Total Height curves were generated (Figure 1), for several section widths considered feasible, taking into account only the ultimate limit states, not considering any constructive arrangement.
Next, the Cost x Total Height curves (Figure 2) were generated considering the ultimate limit states and the construction arrangements. Sections considered unenforceable or that could not meet a constructive provision were discarded.
Then, in addition to sections that were unfeasible or that could not meet any constructive disposition, those that did not meet the Limit State of Excessive Deformation were discarded, that is, they had arrows greater than the admissible arrows (Figure 3).
The next step was to consider the previous restrictions, but increasing the tensile reinforcement of the beam so that the acting stresses were reduced until reaching the admissible deflections (Figure 4).
By adopting such an artifice, there was a very sharp increase in the costs of sections that previously did not meet the arrow criterion, notably in sections of smaller width. In addition, the minimum costs were not affected since the optimal total heights previously found already satisfy the deformation criterion.
Trying to satisfy the criterion of admissible arrows, few sections managed to be assembled without disobeying the constructive dispositions, all of them with a high cost.
The influence of the gauge used in the longitudinal reinforcement on the minimum cost of the sections was analyzed (Figure 5). For the example studied, it was found that increasing the gauge significantly reduces the minimum cost of the beam.
The exception for the beam with a width of 10 cm was already expected due to the high number of layers needed to accommodate the bars with the largest diameter, even so the ideal gauge was not the one with the smallest diameter.
Table 1 briefly shows the minimum costs for each width studied and the corresponding height, for the various considerations made in the Work. The first restriction (ELU) only takes into account the resistance to bending moment and shear force. A big difference is then noticed when constructive dispositions are introduced as a restriction. The difference in optimal height ranges from 16 to 48%.
Adding the other restrictions, deformations and cracking, it is noticed that there was practically no change in the optimal height. This is due to the fact that the optimal sections given the constructive provisions, meet the other criteria.
As for the Influence of double reinforcement, analyzing the results it was found that the optimal total height for each width studied was only equal to the minimum height for single reinforcement in one case, although it was always greater than this. In other cases, the optimal height is up to 6% greater than the minimum height for simple reinforcement. It is necessary to analyze whether there are values of bending moments where the double reinforcement becomes more economical than the single reinforcement.
It was found that when the reinforcement does not meet the requirement that the distance from the center of gravity of the same to the point of the section of the reinforcement farthest from the neutral line must be less than 5% of the total height of the section, it also does not meet the criterion of maximum arrow.
The main conclusion of this work was that the optimal values determined without taking into account the constructive dispositions differ significantly from those that consider them.
Therefore, for a practical application in a project, if the first results are used, the beams will not be optimized since their construction takes into account the constructive dispositions. It can also be said that the choice of the section through curves without considering the constructive dispositions will lead to a less economical sizing despite presenting lower costs.
In practical application in design, it was found to be simpler to use cost curves than to generate 1 or 2 optimal values for each section. The curves allow searching for the lowest cost section that meets the external constraints (architecture, installations, etc.) of the project, since these curves already consider the internal constraints (stability, deformation, and others).
6. Future work
This work shows the need to create models that take into account the restrictions to which the beams are subject in the design and execution and that the optimization studies developed from now on take these restrictions into account, leading to more realistic results.
It is also necessary to extend this technique to a larger number of applications and situations, determining the influence of the parameters involved in the determination of optimal sections.
7. Future considerations and improvements
As for the program, the calculation algorithms must be optimized for a faster generation of cost curves. Choice of gauges that lead to the most favorable arrangement of the layers possible, with the option of using one or more different gauges, depending on the manufacturing control of the frame. Consider the information of the slab that supports the beam, in order to determine precisely its real shape area. In the case of this work, no discount was considered, which is the case with beams for direct support of prefabricated slabs.
As for the dimensioning and detailing method, the reduction of the longitudinal reinforcement ratio in the sections with reduced bending moments, as well as the anchorage lengths in the supports, must be considered. Calculation of deformations by a more exact method, considering cracking along the entire span. Adoption of less conservative expressions for the evaluation of the cracking limit state.
As for the parameters involved, the cost of the form adopted is an average value of the entire form of the work, that is, pillar, beam and slab. The ideal would be to appropriate the cost of the formwork intended only for making the beams, which has certain particularities in relation to the rest of the formwork.
8. Bibliographical References
1. ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS, Projeto e execução de obras de concreto armado, NBR 6118. Rio de Janeiro, 1980.
2. ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS, Barras e fios de aço destinados a armadura para concreto armado. NBR 7480. Rio de Janeiro, 1996.
3. COELLO, Carlos Coello, Farrera, F. A. Use of genetic algorithms for the optimal design of reinforced concrete beams. New Orleans, Tulane University, [199-]
4. COELLO, Carlos Coello. Uso de algoritimos genéticos para para el deseño óptimo de armaduras. New Orleans, Tulane University, 1994.
5. FUSCO, P. B. Técnicas de armar as estruturas de concreto. São Paulo: Pini, 1995
6. SHEHATA, Ibrahim A. E. M., Grossi, Breno F. Otimização de viga de concreto armado. In: CONGRESSO BRASILEIRO DO CONCRETO, 40, 1998, Rio de Janeiro, 1998.
7. SUSSEKIND, J. C. Curso de Concreto, v. 1, Editora Globo, Porto Alegre, 1981.