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

  • Abdul Kareem M. B. Al-Shammaa EMAIL logo und A. T. Alisawi
Veröffentlicht/Copyright: 23. Juni 2023
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Aus der Zeitschrift Open Engineering Band 13 Heft 1

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.

Keywords: beam design; beam stiffness; sustained deflection; immediate deflection; beam deflection; cantilever beams

Notation

b

width of beam sections (mm)

d

adequate depth of beam sections (mm)

h

total depth of beam sections (mm)

ρ

reinforcement ratio in tension equal to (p) at output file.

ρ ′

reinforcement ratio in compression.

A s

steel area in tension ( mm 2 ) .

A s ′

steel area in compression ( mm 2 ) .

l

span length of the beam and equal to (L) at output file (mm)

f c ′

ultimate compressive strength of concrete (MPa)

E s

modulus of elasticity for steel (MPa)

E c

modulus of elasticity for concrete (MPa)

n

modular ratio of elasticity.

I g

gross moment of inertia for beam sections ( mm 4 )

I e

effective moment of inertia ( mm 4 )

I cr

cracking moment of inertia ( mm 4 )

M n

nominal bending moment (kN m)

M cr

cracking moment (kN m)

M a

maximum bending moment (kN m)

IDL

initial dead loads without self-weight (kN)

W LL

unfactored live loads (kN)

W DL

unfactored dead loads (kN)

∆ DL

deflection due to dead load (mm)

( ∆ i ) LL

immediate deflection due to live load and equal to (delta_i_LL) at output file (mm)

∆ ( DL + LL )

deflection due to live plus dead loads (mm)

∆ ( cr + sh )

deflection due to creep and shrinkage of concrete (mm)

∆ T

sustained deflection and equal to (delta_T) at output file (mm)

γ

factor equal to M n b d 2 , and equal to (y) at the output file (MPa)

1 Introduction

Rectangular cross sections for concrete beams are commonly used to design reinforced concrete structures. While the design is done manually or by software, selecting the economical and strength cross sections for those beams is necessary. 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 y = 60 ksi = 414 MPa . Nelson et al. [2–7] 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 [11–23].

Figure 1 Strength of rectangular sections.
Figure 1

Strength of rectangular sections.

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

2.1 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 initial dead and live unfactored loads applied to the cantilever beams are calculated as follows:

Due to the thick slab: 0.15 m = 0.15(25) = 3.75  kN / m 2 ,

Due to the thick sand: 0.05 m = 0.05(17) = 0.85  kN / m 2 ,

Due to the thick flooring layers: 0.025 m = (20 + 24) (0.025) = 1.1 kN / m 2 .

Thus, initial dead loads (IDLs) = 5.70  kN / m 2 , while live load ( W LL ) considered equal to 3 kN / m 2 . Usually, the lengths of adjacent slabs range from 4 to 5 m. Using an average of 4.5 m yields:

IDL = 5.7 (4.5) = 25.65  kN / m , W LL = 3 (4.5) = 13.5 kN / m . By observing the constructed buildings in Iraq, these load values represent the worst case of loading excluding the concentrated loads.

  • 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) ∆ Dl = W Dl l 4 8 E c I e 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.

2.2 Calculations

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 ( M n b d 2 = γ ) will be obtained by dropping each ρ on the curve (60/4). So, the span length is expressed as follows:

(2) γ = M n b d 2 = W l 2 / 2 b d 2 , l = 2 γ b d 2 W .

Table 1

Dimensions of the proposed cross sections

γ (MPa) ρ h (mm) 250% γ (MPa) ρ h (mm) 225% γ (MPa) ρ h (mm) 200% γ (MPa) ρ h (mm) 175% γ (MPa) ρ h (mm) 150%
1.38 (200 psi) 0.0034 500 2.76 (400 psi) 0.0072 450 4.14 (600 psi) 0.0110 400 4.83 (700 psi) 0.0130 350 5.52 (800 psi) 0.0150 300 200 b (mm)
625 563 500 438 375 250
750 675 600 525 450 300
875 788 700 613 525 350
–---- –--- 800 700 600 400

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 (Table 24.2.2). So, the attempt was repeated but with ρ = 0.0034 , 0.0040 , 0.0050 , 0.0060 and γ = 1.38, 1.65, 1.96, 2.38, respectively. Appendix A presents the input file using Python 3.4 software according to the input file data shown in Table 2.

Figure 2 Flowchart to calculate beam deflections.
Figure 2

Flowchart to calculate beam deflections.

Table 2

Input file data

γ (MPa) ρ h (mm)
2.38 (345 psi) 1.96 (284 psi) 1.65 (239 psi) 1.38 (200 psi) 0.0060 0.0050 0.0040 0.0034 375 250 b (mm)
500
563
625
450 300
525
600
675
750
525 350
613
700
788
875
600 400
700
800
Table 3

Check the results

b (mm) h (mm) l (mm) Pass results (mm) ACI – provisions (mm)
( ∆ i ) LL ∆ T ( ∆ i ) LL ∆ T
250 375 1,530 2.25 4.704 4.25 6.37
438 1,828 2.63 5.920 5.07 7.61
500 2,118 2.96 7.074 5.88 8.82
563 2,410 3.28 8.212 6.69 10.0
625 2,695 3.57 9.304 7.48 11.2
300 450 2,050 3.15 7.289 5.69 8.54
525 2,429 3.59 8.913 6.74 10.1
600 2,802 4.00 10.47 7.78 11.6
675 3,171 4.38 11.99 8.80 13.2
750 3,536 4.74 13.45 9.82 14.7
350 525 2,604 4.06 10.26 7.23 10.8
613 3,070 4.58 12.24 8.52 12.8
700 3,523 5.05 14.31 9.78 14.6
788 3,973 5.49 16.23 11.0 16.5
875 4,412 5.88 18.06 12.2 18.4
400 600 2,919 4.46 10.07 8.10 12.1
700 3,422 5.04 12.13 9.50 14.2
800 3,915 5.55 14.10 10.8 16.3
Table 4

Illustrative example

b (mm) h (mm) ρ γ (MPa) l (mm) Empirical results (mm) ACI – provisions (mm)
( ∆ i ) LL ∆ T ( ∆ i ) LL ( l / 360 ) ∆ T ( l / 240 )
250 375 0.0034 1.38 1,284 1.20 1.88 O.K O.K
250 375 0.0040 1.65 1,404 1.82 3.22 O.K O.K
250 375 0.0050 1.96 1,530 2.25 4.70 O.K O.K
250 375 0.0060 2.38 1,686 2.72 6.82 O.K N.O.K
Table 5

Empirical deflections due to overload

Load increment % Empirical results (mm) ACI – provisions (mm) Remark
( ∆ i ) LL ∆ T ( ∆ i ) LL ∆ T
5.00 4.19 11.20 7.78 11.6 O.K
7.50 4.23 11.46 O.K
10.0 4.30 11.78 N.O.K
Figure 3 Optimum dimensions of R/C cantilever beams.
Figure 3

Optimum dimensions of R/C cantilever beams.

2.3 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 cantilever beams to resist loads and deflections even if the values of (( f c ′ , f y ) ) 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.

3 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 c ′ , f 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.

  1. 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.

  2. Conflict of Interest: The authors state no conflict of interest.

  3. 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.

Appendix

Appendix A. Input file

[(bh = [[(250,375),(250,438),(250.500).(250,563),(250.625

[(300,750),(300,675),(300,600),(300,525),(300,450)1

[(350,875),(350,788),(350,700),(350.613),(350,525)1

[[(400,800),(400,700),(400,600)1

[(py = [(0.0034,1.38),(0.004,1.65),(0.005. 1%),(0.006,2.38

import math

fc = 28; Es = 200,000; IDL = 25.65;WLL = I 3.5

for bhl in bh ("####################################### print ("######################################") print

for b,h in bhl

for p,y in py

lg = (b*pow(h ,3))/12

Mcr = ((0.62*math.sqrt(fc))/(h/2))*1g * 0.00000 I

(Ec = 4,700 *math.sqrt(fc n = Es/Ec

d = h-60 As = p*b*d

XI = ((-n*As)+math.sqrt((n*As)**2-4*(b/2)*(-n*As*d)))lb

X2 = ((-n*As)-math .sqrt((n*As)**2-4*(b/2)*(-n*As*d)))/b (X = max(Xl,X2

if X>O Icr = (b*pow(X.3)/3)+n*As*(d-X)**2 self_wcight = 25*b*h*O .OOOOO I WDL = self_weight+IDL W = WDLHVLL

(L = math.sqrt(2*y*b*(d**2)/W Ma = (WDL *(L**2)/2)*0.00000 I

Ie = pow((Mcr/Ma),3)*lg+( I-pow((Mcr/Ma),3))*lcr (delte_DL = WDL*pow(L,4)/(8*Ec*lc Ma2 = ((WDL+WLL)*(L**2)/2)*0.00000I

Ie2 = pow((Mcr/Ma2) ,3)*Jg+( I -pow((Mcr/Ma2),3))*1cr (delte_DL_LL = (WDL+WLL)*pow(L,4)/(8*Ec*le2 deflection#

delta i LL = delte DL LL-delte DL (pl = 0.002*b*h/(b*d --

(lcmbda = 1.4/(I+50*pl delta_cr_sh = lembda*dcltc_DL delta T = dclta i LL+dclta er sh

print(b.h ,ps,L,dclta_C.LL,dclta_T.sep = "\t")

Appendix B. Output file

b (mm) h (mm) p y L (mm) delta_i_LL (mm) delta_T (mm)
250 375 0.0034 1.38 1,284.528535 1.204772808 1.889216315
250 375 0.0040 1.65 1,404.579046 1.827000915 3.224234404
250 375 0.0050 1.96 1,530.848626 2.255624256 4.704488116
250 375 0.0060 2.38 1,686.913025 2.721597045 6.825838556
250 438 0.0034 1.38 1,534.172256 1.557049476 2.561536276
250 438 0.0040 1.65 1,677.554173 2.228283574 4.199480124
250 438 0.0050 1.96 1,828.363814 2.629206231 5.920507015
250 438 0.0060 2.38 2,014.758794 3.113463796 8.383454045
250 500 0.0034 1.38 1,777.605616 1.877419335 3.219220743
250 500 0.0040 1.65 1,943.738526 2.581116647 5.132813167
250 500 0.0050 1.96 2,118.477746 2.963310207 7.074662906
250 500 0.0060 2.38 2,334.448777 3.481715347 9.864884763
250 563 0.0034 1.38 2,022.728391 2.179406299 3.878081765
250 563 0.0040 1.65 2,211.770184 2.908877273 6.056306431
250 563 0.0050 1.96 2,410.605054 3.280388006 8.21283743
250 563 0.0060 2.38 2,656.357393 3.842227813 11.32941511
250 625 0.0034 1.38 2,261.803606 2.456848024 4.515493355
250 625 0.0040 1.65 2,473.189084 3.208436785 6.943120486
250 625 0.0050 1.96 2,695.525127 3.57580832 9.304360088
250 625 0.0060 2.38 2,970.324022 4.18477208 12.73693728
300 450 0.0034 1.38 1,720.907845 1.899491023 3.189672859
300 450 0.0040 1.65 1,881.741848 2.689818385 5.19741351
300 450 0.0050 1.96 2,050.907658 3.149179882 7.289701599
300 450 0.0060 2.38 2,259.990167 3.719227507 10.28437955
300 525 0.0034 1.38 2,038.414374 2.333452058 4.122412669
300 525 0.0040 1.65 2,228.92216 3.161096716 6.514015756
300 525 0.0050 1.96 2,429.298967 3.595194997 8.913498439
300 525 0.0060 2.38 2,676.957082 4.215627041 12.36761068
300 600 0.0034 1.38 2,351.88891 2.727181776 5.034615173
300 600 0.0040 1.65 2,571.693653 3.583057672 7.785721304
300 600 0.0050 1.96 2,802.885112 4.00502623 10.4772138
300 600 0.0060 2.38 3,088.628962 4.686845969 14.37947906
300 675 0.0034 1.38 2,661.446545 3.0876788 5.92449322
300 675 0.0040 1.65 2,910.182177 3.968212988 9.017677454
300 675 0.0050 1.96 3,171.803253 4.387258686 11.99057637
300 675 0.0060 2.38 3,495.156955 5.134825763 16.33102449
300 750 0.0034 1.38 2,967.197551 3.420295323 6.792258796
300 750 0.0040 1.65 3,244.50831 4.323955165 10.21396181
300 750 0.0050 1.96 3,536.184811 4.746234304 13.45978324
300 750 0.0060 2.38 3,896.685874 5.560589567 18.22890816
350 525 0.0034 1.38 2,185.161352 2.647921466 4.745996234
350 525 0.0040 1.65 2,389.383936 3.580501887 7.499118209
350 525 0.0050 1.96 2,604.18602 4.066801826 10.25975533
350 525 0.0060 2.38 2,869.673227 4.766943236 14.23240888
350 613 0.0034 1.38 2,576.123049 3.153073087 5.963814076
350 613 0.0040 1.65 2,816.884449 4.116546697 9.19342417
350 613 0.0050 1.96 3,070.118197 4.585419233 12.33972662
350 613 0.0060 2.38 3,383.10548 5.364770151 16.90585678
350 700 0.0034 1.38 2,956.237404 3.599176564 7.128713119
350 700 0.0040 1.65 3,232.523841 4.588461835 10.80164112
350 700 0.0050 1.96 3,523.122956 5.053681076 14.31195702
350 700 0.0060 2.38 3,882.292411 5.916382436 19.44735853
350 788 0.0034 1.38 3,334.484507 4.006319426 8.26857538
350 788 0.0040 1.65 3,646.121468 5.019921686 12.36836927
350 788 0.0050 1.96 3,973.902398 5.489907668 16.23294524
350 788 0.0060 2.38 4,379.027164 6.436849677 21.92741333
350 875 0.0034 1.38 3,702.493016 4.372149225 9.359433503
350 875 0.0040 1.65 4,048.523615 5.408836445 13.86360523
350 875 0.0050 1.96 4,412.479904 5.888618307 18.06639311
350 875 0.0060 2.38 4,862.316036 6.916459073 24.29757779
400 600 0.0034 1.38 2,670.234601 3.413739737 6.516244229
400 600 0.0040 1.65 2,919.791555 4.467862884 10.07529493
400 600 0.0050 1.96 3,182.2765 4.98213654 13.55243852
400 600 0.0060 2.38 3,506.69791 5.828305797 18.59110858
400 700 0.0034 1.38 3,130.24736 3.966187244 8.010752646
400 700 0.0040 1.65 3,422.796561 5.046467123 12.13636773
400 700 0.0050 1.96 3,730.500912 5.552493174 16.0761737
400 700 0.0060 2.38 4,110.811789 6.500408425 21.83899311
400 800 0.0034 1.38 3,580.761603 4.450992282 9.443540174
400 800 0.0040 1.65 3,915.415329 5.555043299 14.10193199
400 800 0.0050 1.96 4,267.40538 6.065308054 18.48226539
400 800 0.0060 2.38 4,702.451697 7.114153209 24.94214265

Appendix C. Example

Take any passing result from Table 3. Let it be(300 × 600 × 2802 mm as well as an algorithm parameter; f c ′ = 28 MPa , f y = 414 MPa , IDL = 25.65 kN/m, and W LL = 13.5 kN/m.

Self-weight = 25(300) (600) × 10 − 6 = 4.5 kN/m

W DL = 4.5 + 25.65 = 30.15 kN/m,

Factored ultimate loads are; W u = 1.2(30.15) + 1.6(13.5) = 57.78 kN/m

M u = W u l 2 2 = ( 57 . 78 ) ( 2802 ) 2 2 × 10 − 6 = 227 kN m

(A1) ρ = f c ′ 1 . 18 f y 1 − 1 − 2 . 36 M u 0 . 9 b d 2 f c ′ = 28 1 . 18 ( 414 ) 1 − 1 − 2 . 36 ( 227 ) * 10 6 0 . 9 ( 300 ) ( 600 − 60 ) 2 ( 28 ) .

ρ = 0.0074 > 0.005

Provided reinforcement ratio remains greater than 0.005 when any value of f y f c ′ in Figure 1 is substituted into equation (A1).

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Received: 2022-10-28
Revised: 2023-03-05
Accepted: 2023-03-11
Published Online: 2023-06-23

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/eng-2022-0428
Schlagwörter für diesen Artikel
beam design; beam stiffness; sustained deflection; immediate deflection; beam deflection; cantilever beams
Creative Commons
BY 4.0

Artikel in diesem Heft

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