allowable spandepth ratio for high strength

R. Prabhakara, K. U. Muthu, and R. Meenakshi

October 2007 The Arabian Journal for Science and Engineering, Volume 32, Number 2B 349

ALLOWABLE SPAN/DEPTH RATIO FOR HIGH STRENGTH CONCRETE BEAMS

R. Prabhakara, K. U. Muthu*, and R. Meenakshi

Department of Civil Engineering, M S Ramaiah Institute of Technology, Bangalore, India

1. INTRODUCTION

Deflection is an important serviceability limit state to be satisfied in the design of concrete structures. In recent times, high strength concrete and steel have been used widely in construction. This has resulted in the design of beams of smaller depths, which may undergo greater deflections. This has been investigated in references [112].

Basically, there are two approaches used in the design offices: viz. (1) control of deflection approach; and (2) computation of deflections approach. In the first approach, span to effective depth ratios are specified in the codes of practices to control the deflections. Control is obtained by providing a suitable effective depth of the beam which is obtained based on support conditions, span, area and type of tension reinforcement, area of compression reinforcement, and flanges for flanged beams. An important parameter, i.e. ratio of sustained load to total load (ws/w) on the beam which influences the creep deflections does not find a place in several cases. A few researchers have incorporated this effect [13, 14]. The control approach is preferred by many due to its simplicity. However, several investigators [10, 13, 1518] pointed out the discrepancies and disparities. Recently Scanlon and Lee [19] proposed a unified span to depth ratio equation for control of deflections in one and two way concrete construction. The above method requires further examination with respect to the high strength concrete beams and slabs.

An attempt has been made in the present study to obtain the spanto effectivedepth ratio for singlyreinforced and doublyreinforced highstrength concrete beams. The method is based on ACI 318-05 and the effect of the ratio of the sustained load to total load and the breadth of the beam has been included in the analysis. A total of 747 singlyreinforced beams and 263 doublyreinforced beams were used in developing the proposed equation. Design charts are presented for ready use.

* Address for correspondence Professor and Head, Dept. of Civil Engg, MSRIT, Bangalore 560054, India Tel: +91-080-23600822 Fax: +91-080-23603124 Email: [email protected]

Key words: beams, cracked inertia, deflection, high strength concrete, serviceability, ultimate strength

Paper Received 12 August 2005; Revised 3 October 2006; Accepted 18 March 2007


The Arabian Journal for Science and Engineering, Volume 32, Number 2B October 2007 350

1. 1. Proposed Method for Singly Reinforced Beams

Figure 1 (a) shows the cross section of a singly reinforced beam. The short-term deflection is computed as

4

sc e

w lE I = , (1)

where s = short term deflection = elastic deflection coefficient ( = 5/384) for a simply supported beams subjected to a uniformly distributed load,

l = span of the beam

Ec = modulus of elasticity of concrete (= 4730 fc) Ie = effective moment of inertia as per ACI 318-05 [6].

Dd

a) Singly Reinforced Beam

b

D

a) Doubly Reinforced Beam

b

Ast

d

Ast

d'

Asc

Figure 1. Sectional details of rectangular beams

1. 2. Effective Moment of Inertia: Ie

The effective moment of inertia is specified as

( )( )3e c r g c r c r aI I I I M M= + (2) where Ig and Icr are the gross and the cracked moment of inertia of the section, Mcr and Ma are the cracking moment and actual moment respectively.

3

1 2gb DI = (3)

The cracked moment of inertia is expressed as 3

1crI k b d= (4) where

32

1 (1 )3Xk m X= + (4 a)

(a) (b)

b b



The value of the modulus of elasticity of steel is taken as 200 GPa and the total depth is taken as 1.1 times the effective depth d :

31.331

12grbdI = (5)

1. 3. Cracking Moment : Mcr The cracking moment is calculated using the flexural formula as

gcr r

t

IM f

y= (6)

Where fr is the modulus of rupture taken as;

'0.62r cf f= (7) And yt is the distance from the centroidal axis and Ig is the gross moment of inertia of the section. Taking yt as half of the total depth and

2 '0 .1 2 5c r cM b d f= (8) 1. 4. Service Load Moment : Ms

The ultimate moment of resistance of the section given as

'0.9 (1 0.59 )st y

u st yc

A fM A d f

b d f= (9)

Assuming the partial safety factor as 1.5 , the moment at the service loads is taken as

1.5u

sMM = (10)

Substituting the expression for Icr from Equation (4) and Ig from Equation (5) in Equation (2). The effective moment of inertia function is expressed as:

32eI k bd= (11)

where

32 1 1[ (0.111 )( ) ]crs

Mk k kM

= + . (12) The values of k2 depends on k1, Mcr , and Ms. These quantities in turn depends on the mechanical properties of the

materials viz. modulus of elasticity of concrete and steel respectively, sectional properties viz. breadth , effective depth of the beam, steel ratio and strength properties of concrete, and yield stress of steel. Hence, the computation of k2 is quite involved and is evaluated in the subsequent sections. 1.5. Shrinkage and Creep Deflections The additional long-term deflection (l) due to shrinkage and creep is given by

4w l

E Isl

c e = (13)

where = long time multiplier for sustained loads; ws = sustained load

1 50 ' = + (14)

where = time dependent factor taken as 2.0 and ' = compression steel ratio. '

Ascbd

= (15) where Asc is the area of compression steel



1.6. Total Defection: t The short-term deflections were computed for eighty high-strength concrete beams reported in the literature. This

include nine different investigations. Table 1 gives the ratio of calculated to experimental deflections at service loads

Table 1. Ratio of Calculated to Experimental Deflection at Service Load for High Strength Concrete Beams

(cal)/(exp) fc' IS 456 ACI 318 CEB-FIP Nayak et.al. Sl.No Author

No. of beams

MPa Mean CV Mean CV Mean CV Mean CV

1 Lakshmikantha [20] 20 40-100 1.38 23.3 1.35 22.6 1.63 30.2 1.4 25.4

2 Anima [21] 4 45-60 1.22 8.9 1.16 8.7 0.76 8.6 1.29 9.3

3 Ko & Kim [22] 8 65-80 1.14 9.7 1.02 21.6 1.36 13.1 1.15 15.5

4 Keith E Leslie [ 23] 3 70-80 1.32 11.4 1.31 9.0 1.41 13.5 1.31 15.5

5 Wafa & Ashour [23] 3 75-90 1.36 18.5 1.28 13.5 1.76 25.1 1.13 21.3

6 Ghosh [25 ] 4 >100 1.36 20.7 1.29 23.5 1.77 12.2 1.19 17.5

7 Sarkar [26] 12 75-110 1.12 16.6 1.13 15.8 1.3 19.7 1.11 17.9

8 Ahmad & Barkar [27] 5 60-80 1.09 25.6 1.09 26.7 1.59 34.2 1.1 24.8

9 Bernardo & Lopes [28] 18 65-85 1.19 24.4 1.21 22.4 1.22 23.1 1.26 22.3

10 Ahmad & Batts [29] 3 65-90 1.41 27.6 1.00 14.4 1.56 10.6 0.99 27.5

Total No. of beams = 80 1.24 22.0 1.2 21.9 1.42 29.1 1.24 22.9

It is noted that the best average ratio of calculated to experimental deflections is 1.2 with the least coefficient of variation of 21.9. Hence, a multiplier constant was introduced to ACI 318-02 and, assuming the same trend exists for additional long term deflections also, the total deflection is estimated as.

1.2 ( )t s l = + . (16)

1.7. Limiting Span to Effective Depth Ratio: (l/d)

The total deflection is limited to a target value of (l/250) and is equated as

44

3 32 2

1.2250

s

c c

w ll w lE k bd E k bd

= + (17)

substituting the values of and for long term deflections (vide equations (1) and (19)) 0.33 0.33

23.906(1 2c s

l w kd bE w w

= + (18)

1.8. Evaluation of k2

The constant k2 is dependent on , fy and fc. The values of k2 was calculated for the following variations. (1) The cylinder compressive strength was varied from 40 to 100 Mpa in increments of 10 Mpa.

(2) The steel ratio is varied from min to max and the values are taken as



min'f c

fy = (19)

and 0.75max b = (20)

where ( )'0.85 600

600c y

by

f f

f

= + (21)

and = 0.85 for 'cf 27.58 Mpa (21a) = 0.85 0.0073 ( 'cf 27.6) 0.65

for 'cf > 27.58 Mpa (21b) where min, b, and max are the minimum, balanced and maximum steel ratios of tension reinforcement. The

computed values of k2 are shown in Figure 2.

Figure 2. Variation of k2 with for singly reinforced beams

The statistical best fit line was drawn and the regression line is

2 1.96 0.02k = + (22) and the correlation coefficient was 0.94. The above equation holds good for 0.006 0.04.

1.9. Computation of (l/d) Ratios

The limiting span to effective depth ratio for singly reinforced rectangular beam can be calculated as follows. The value of k2 is calculated using Equation (22) for the designed steel ratio. Substituting the values of modulus of elasticity of concrete, breadth of beam, and the ratio sustained load to service load in Equation (18) and the (l/d) ratio can be obtained for a calculated service load w. Alternatively, a graphical representation is made from Equation (18) and is shown in Figure 3. While using the design chart (vide Figure 3 ) , the units for l and b are in mm, w in kN/m2 and fc in MPa.



Design chart for singly reinforced beam

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ws/w

L/d*

(w/b

Ec)

**0.

33p=0.0025 p=0.005

p=0.0075 p=0.01

p=0.0125 p=0.015

p=0.0175 p=0.02

p=0.0225 p=0.025

p=0.0275 p=0.03

Figure 3. Design chart for singly reinforced beams

1.10. Doubly Reinforced Beams

The proposed method for singly reinforced beams is extended to doubly reinforced beams ( beams with compression reinforcement). Figure 1(b) shows the cross section of a doubly reinforced beam. The neutral axis factor X(=x/d) for doubly reinforced beam can be written as,

( ) { } ( ) ( )2( 1) ' ( 1) ' 2 ( 1) ' 'xX m m m m m m d dd = = + + + + + . (23) The cracked moment of inertia about the neutral axis is taken as,

( ) ( )2'23 3 / 3 1 1 '

.cr

dI bd X m X m xd

= + +

(24)

Equation (24) reduces to Equation (4) for singly reinforced beam when = 0. The value of the ultimate moment Mu for doubly reinforced beam was obtained using the sectional analysis. As both tensile steel and compression steel ratios contribute to the sectional properties of the cross section, the total steel ratio t is denoted as,

't = + . (25) A total of 263 doubly reinforced beam sections were obtained by various combinations of compression steel ratios

and tensile steel ratios. The total combination of steel ratios vary from 0.005 to 0.04. The concrete grades of 50 to 100 Mpa were used. The two grades of steel 415 and 500 Mpa were considered in addition to the above variables.



Figure 4. Variation of k2 with for doubly reinforced beams The effective moment of inertia is thus expressed in a simplified form as

2 2.1 0.009tk = + (26) The above equation holds for 0.005 t 0.04 was used to determine the effective moment of inertia of doubly

reinforced beams ( vide Equation (11) and (18)).

Design Chart for doubly reinforced beams

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ws/w

l/d(w

/bE

c)**

0.33

p=0.0075

p=0.0135

p=0.021

p=0.036

p=0.0285

Figure 5. Design chart for doubly reinforced beams



1.11. Comparison of Effective Depth Values The effective depths were determined using the proposed method for a typical singly reinforced beam of 6 metres

and subjected to a service load of 10 kN/m. the breadth of the beam was taken as 230mm. M50 grade concrete and Fe415 steel was considered. The effective depths obtained are compared with the effective depths computed from the rupture limit state and deflection limit state as per ACI 318-05. Table 2 gives the comparison of the same.

Table 2. Comparison of Effective Depths (mm) for a Singly Reinforced Beam ACI Proposed method for ws/w of

t Strength Deflection limit limit 0.2 0.4 0.6 0.8 1.0 mm mm mm mm mm mm mm

0.25 460.6 375 417.74 456.73 489.35 515.11 539.45 0.5 327.7 375 391.48 428.18 456.73 482.46 503.75

0.75 269.3 375 370.32 403.00 430.88 456.73 479.09 1 234.7 375 354.97 380.61 412.71 436.37 456.73

1.25 211.3 375 342.55 370.32 396.01 417.74 439.16 1.5 194.1 375 326.24 360.58 382.74 403.00 422.90

1.75 180.9 375 311.41 346.01 370.32 391.48 410.24 2 170.4 375 301.80 342.55 358.69 378.51 398.31

Data: b = 230mm, l=6000mm, w =10KN/m, M50 and Fe415 It is inferred that for low percentage of steel and for low ratio of sustained to service loads, the effective depths

obtained from rupture limit governs the design. On the other hand, for HSC beams with the higher percentage of steel and higher ratio of sustained load to service load, the effective depths obtained from the code being less, would not be sufficient to limit the deflection to permissible values. Hence for such cases of singly reinforced HSC beams, the effective depths would have to be increased to the values calculated by the proposed method. A similar computation was done for a typical doubly reinforced beam of breadth 230mm , span being 8m and a service load of 30 kN/m. Concrete grade M50 and steel grade Fe415 considered. The results in Table 3 show that the existing codal provisions of ACI 318-05 gives lesser effective depth , while the proposed method requires more effective depth to satisfy the deflection limit.

Table 3. Comparison of Effective Depths (mm) for a Doubly Reinforced Beam t c =t+c ACI Proposed method for ws/w of Strength Deflection 0.2 0.4 0.6 0.8 1.0 limit limit mm mm mm mm mm mm mm

0.8 0.549 1.35 470.7 500.00 681.9 740.8 791.5 836.4 876.8 1.176 0.92 2.1 385.1 500.00 606.6 659.1 704.2 744.1 780.1 1.55 1.29 2.85 337.5 500.00 556.5 604.6 646.0 682.6 715.7 1.92 1.67 3.6 306.9 500.00 519.8 564.7 603.4 637.6 668.4

Data: b = 230mm, l=8000mm, w =30KN/m, M50 and Fe415. 2. CONCLUSIONS

1. A method is proposed to obtain the effective depth of singly reinforced and doubly reinforced high strength concrete beams using the modified ACI318-05 procedure; by incorporating the effect of ratio of sustained load to the service load.

2. The design charts are presented for ready use and a comparison has been made between the effective depths obtained by the proposed method and those from the strength limit and deflection limit of ACI 318-05.

3. The present study shows that the effective depths obtained by ACI 318-05 are found to be inadequate to satisfy a limiting deflection of span/250 viz. singly reinforced beams with higher percentage of steel subjected to higher ratio of sustained to service load ( vide Table 1) and in case of doubly reinforced beams the proposed method shows that effective depths larger than those obtained from strength and deflection limit as per ACI318-05 provisions are required for higher percentage of steel and various proportions of sustained to service loads.



NOTATION

Asc , Ast Area of compression and tension steel

As Area of steel

b Breadth of the beam

d Effective depth to the center of tensile reinforcement

D Total depth of beam

d Effective cover to the compression steel Ec Modulus of elasticity of concrete (4730 fc) Es Modulus of elasticity of steel

fc Cylinder compressive strength of concrete fr Modulus of rupture of concrete

fy Yield strength of steel

Icr Cracked moment of inertia of the section

Ie Effective moment of inertia of the section

Ig Gross moment of inertia of the section

k1, k2 Constants in Equation (10) and (16) respectively

l Span length

m Modular ratio

Ma Actual moment

Mcr Cracking moment

Ms Service load moment

Mu Ultimate moment of resistance

w Intensity of load

ws Intensity of sustained load

X Neutral axis factor

yt Distance from centroidal axis furthest fibre

Elastic deflection coefficient Constant Central deflection r Short term deflection l Additional long term deflection Tensile steel ratio (=Ast/bd) b Balanced steel ratio t Total tensile steel ratio (=+) min Minimum steel ratio max Maximum steel ratio Compression steel ratio (=Asc /bd) Long time multiplier for sustained loads Time dependent factor

REFERENCES

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[9] CEB FIP, CEB FIP Model Code (MC-90), Comit Euro International du Beton (CEB). London: Thomas Telford, 1993.



[10] R. I. Gilbert, Deflection Calculation for RC StructuresWhy We Sometimes Get It Wrong, ACI Structural Journal, 96 (1999), pp. 10271032.

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[18] K. B. Bondy, ACI Code Deflection Requirements Time for a Change?, Serviceability of Concrete , SP225 eds., F. Barth, R. Frosch, H. Nassif, and A. Scanlon. Farmington Hills, Michigan: American Concrete Institute, pp.133145.

[19] S. Andrew and H. L. Young, Unified Span-to-Depth Ratio Equation for Nonprestressed Concrete Beams and Slabs, ACI Structural Journal, 103 (2006), pp. 142148.

[20] B. A. Lakshmikantha, Experimental Investigations on High Performance Concrete Beams, M. Tech. Thesis, M.S. Ramaiah Institute of Technology, 2004.

[21] Anima, Flexural Behaviour of High Strength Concrete Beams, M. Tech. Thesis, M.S. Ramaiah Institute of Technology, 2002.

[22] M. Y. Ko, S. W. Kim, and J. K. Kim, Experimental Study on the Plastic Rotation Capacity of Reinforced HSC Beams, Magazine of Concrete Research, 2001.

[23] K. E. Leslie, K. S. Rajagopalan, and N. J. Everard, Flexural Behaviour of HSC Beams, ACI Journal , 1976, pp. 517521.

[24] F. F. Wafa and S. A. Ashour, Minimum Flexural Reinforcements of HSC Beams, SP 172-30, 1996.

[25] S. W. Shin, S. Gosh, and J. Moreno, Flexural Ductility of Ultra High Strength Concrete Members, ACI Structural Journal, 1989, pp. 394400.

[26] S. S. Adwan and J. G. L. Munday, High Strength Concrete: An Investigation of the Flexural Behaviour of HSC Beams, Structural Engineering, 75 (1997), pp. 155121.

[27] S. H. Ahmad and R. Barker, Flexural Behavior of Reinforced High-Strength Lightweight Concrete Beams, ACI Structural Journal, 88 (1991), pp. 6977.

[28] L. F. A. Bernardo and S. M. R. Lopes, Flexural Ductility of High Strength Concrete Beams, Structural Concrete, 4(2003), pp. 135152.

[29] S. H. Ahmad and B. Jamie, Flexural Behavior of Doubly Reinforced High Strength Lightweight Concrete Beams with Web Reinforcement, ACI Structural Journal, 88(3) (1991), pp. 351358.

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