Novel graph for an appropriate cross section and length for cantilever RC beams

: Whether the design is done manually or by software, the designer will have di ﬃ culty choosing the economic and strength cross section. The designer, in this case, either relies on their experience or resorts to the method of trial and error. Especially for Cantilever beams with a long span as a result of risk de ﬂ ections, it is exposed. The current theoretical study was performed on rectangular concrete cross sections of di ﬀ erent dimensions and subjected to uniformly distributed loads. Based on a previous study, the sections are reinforced with a speci ﬁ c reinforcement ratio. Through an algorithm, Python 3.4 software

Abstract: Whether the design is done manually or by software, the designer will have difficulty choosing the economic and strength cross section. The designer, in this case, either relies on their experience or resorts to the method of trial and error. Especially for Cantilever beams with a long span as a result of risk deflections, it is exposed. The current theoretical study was performed on rectangular concrete cross sections of different dimensions and subjected to uniformly distributed loads. Based on a previous study, the sections are reinforced with a specific reinforcement ratio. Through an algorithm, Python 3.4 software, and an output file, the permissible deflections for each cross section were calculated according to the ACI 318M-19. Finally, the authors could draw a graph to choose the appropriate cross section for each required beam length in less time and effort. With the large-span cantilever beams, the selection is not easy and depends on the designer's experience with trial and error. The risk associated with the cross-sectional design becomes more severe for cantilever beams usually subjected to significant deflections. In Iraq, the construction of cantilever beams with rectangular sections appears in many hotels and commercial buildings. However, the authors did not find articles that studied the problem in detail.
What is specified at the ACI 318M-19 item 24.2 with SI unit [1] was performed to calculate the deflection. Nilson et al. [2] presented a graph shown in Figure 1, which is essential to preparing the current study. From the group of curves, the common curve 60/4 was chosen, i.e., ′ = f c = 4 ksi 28 MPa and = = f 60 ksi 414 MPa y . Nelson et al. [2][3][4][5][6][7] authored textbooks that contain chapters on deflection and how to calculate it. Metwally [8] confirmed that the support location affects the immediate deflection values. Chaphalkar et al. [9] emphasized that modeling can be made to analyze the deflection of cantilever beams by finite element package. Marovic et al. [10] concluded several models for calculating the deflection of the cantilever beams with end-concentrated loads with circular and hollow cross sections. The types of deflections are most important in checking the dimensions of the selected cross sections [11][12][13][14][15][16][17][18][19][20][21][22][23].
The current study aims to create a relationship between rectangular cross sections and the span lengths of reinforced concrete cantilever beams. Finally, the authors define the form of this relationship through a graph that facilitates the selection of strength and economic cross sections.
2 Design procedure

Assumptions
• In the tension zone, all sections are reinforced with a variable reinforcement ratio, while the compression zone is reinforced with a minimum reinforcement content of ( ) ′ = ρ 0.002 . For the calculation of I cr , the effect of compression steel has been neglected due to its proximity to the neutral axis and small quantity. • The approximate load values (if any) do not significantly affect the accuracy of the deflection results. The deflection of any structure depends mainly on the span length; as a result, the length is raised to the fourth power, while the load is to the first power. For example, the deflection of the cantilever beam due to the dead load is expressed as follows: (1) • The Flange effect is not considered because it is in a tension zone. • The current study does not include deep beams with more than 900 mm depth because their reinforcement is distributed according to ACI item 9.9. • The reinforcement can be placed in one or two layers in the tension zone. The adequate depth is taken as d = h − 60. Table 1 presents typical values for cross sections bh with h taken as a percentage of b. The difference of ρ values provided to the cross sections depends on the fact that ρ is inversely proportional to the section area bh. From Figure 1, the values of ( ) = γ M bd n 2 will be obtained by dropping each ρ on the curve (60/4). So, the span length is expressed as follows:

Calculations
According to an algorithm shown in Figure 2, the authors attempted to calculate the deflection of selected cross sections. Initial calculations show that the deflection of reinforced sections with = ρ 0.015, 0.013, 0.011, 0.0072 do not match the permissible sustained deflection, as mentioned at ACI (

Analysis results
• ACI -Table 24.2.2 provides two permissible deflections that must be checked: immediate deflections equal to l ( /360) and sustained deflections equal to l ( /240). The calculated deflections listed in Appendix B have been checked with the permissible ones to know the pass lengths with their cross sections, as shown in Table 3. Finally, the cross-sectional selection against the required length was facilitated by the graphic relationship shown in Figure 3.
• Mainly, increasing the depth of the beam means increasing its rigidity and thus increasing the permissible length of the beam so that it does not exceed the specificity of deep beams. • The best span length is obtained for beams with a width of 350 mm, after which increasing the width becomes  useless. An increase in a width greater than 350 mms means an increase in the weight of the beam at the expense of its rigidity. • The sustained deflection is considered the most dangerous type, a discrepancy to what was believed in more detail in refs. [8,14,15,17,19]. All the empirical results in Appendix B agreed with the ACI conditions of immediate deflection, while sustained deflection is considered a criterion for accepting pass results. For more explanation, an example can be taken from Appendix B, as shown in Table 4.
• Referring to the output file, it is noted that the allowable deflections for all sections were obtained from a trial (ρ = 0.005) against γ = 1.96 MPa. An increase in the reinforcement ratio of more than 0.005 gives an increase in length that does not meet ACI requirements. All deflections were calculated from unfactored loads based on the ACI conditions. However, the loads must be factored in when designing, and the reinforcement ratio will increase from 0.005, as shown in Appendix C. Increasing the reinforcement ratio provided that adhering to the length adopted in the current study means forming safer   as a parameter are changed. • Concerning the loads as a parameter, it was considered the worst distributed load identified locally. Table 5 whose calculations were made on a beam model (300 × 600 × 2,802) mm.
The distributed loads should not be increased more than 7.5% in the future. This has been tested on all pass results in Table 3 and proven correct.
• All the published articles and textbooks did not conclude Figure 2 as a simplified roadmap in calculating the various deflections exactly, instead of adopting an approximate method such as finite elements as stated in ref. [9], especially for commonly used geometric sections with (ρ = 0.005) against γ = 1.96 MPa. Also, Figure 2 shows very attractive, especially for postgraduate and undergraduate students.

Conclusion
• The study focused on concluding the optimum dimensions for the reinforced concrete cantilever beams with a rectangular cross section subjected to the uniformly distributed loads commonly used in building construction, excluding the concentrated loads. Due to the significant deflections, it is not easy to select cross sections of the cantilever beams, especially with a large span.
To solve this problem, the authors plot a simplified graph to provide a cross section for the required beam length in less time and effort. The chart does not include deep beams with depths greater than 900 mm, with conditions specified in the ACI code. Also, the current study revealed an important economic aspect. The allowable span lengths are greater than the expected and locally common. Thus, a solution to an old problem has been developed that was not discussed in the published literature. • The authors strongly recommend using the results in various buildings, provided that no significant concentrated loads are applied along the beams and future uniform distributed loads do not exceed 7.5% used in the current study. • Since the increase in (ρ) increases the length of the beam, and the designer is restricted to the length adopted by this study, then any increase in (ρ) will be safer even if the values of ( ) ′ f f , c y are changed. • Sustained deflections are the most dangerous types of deflections. • The authors created a simplified algorithm for calculating deflections that are not found in any published article or textbook. • Using a beam of more than 350 mm in width is not economical.
Funding information: We declare that the manuscript was done depending on the personal effort of the author, and there is no funding effort from any side or organization, as well as no conflict of interest with anyone related to the subject of the manuscript or any competing interest.

Conflict of Interest:
The authors state no conflict of interest.
Data availability statement: Most datasets generated and analyzed in this study are in this submitted manuscript. The other datasets are available on reasonable request from the corresponding author with the attached information.