Variability in Deflection of Reinforced Concrete Beams

With the present day use of higher strength concrete and higher grade reinforcing steel, refined /computerized method of designs of reinforced concrete beam for limit state of safety resulting sections are slender or shallower , the problem of predicting and controlling deflections of reinforced concrete flexural members under service load is becoming increasingly important. The Universally accepted verification method for structures serviceability is based on deterministic criteria and on definite initial parameters, which are variables in reality. Uncertainties exists in geometrical properties of the member like cross sectional dimensions of the member, amount of steel reinforcement, also in material properties particularly of concrete consequently structures response differs considerably in serviceability.Many researchers suggested an empirical expression for prediction of beam deflection based on large number of experimental work and further compared the calculated value of deflection from theoretical or codal expression with actual deflection of the beams. Sample results of ratio of computed /experimentally obtained deflection values are given in table 1[1]. Values in the table clearly indicate that theirexists large variation in actual deflection and computed deflections. This is mainly because of variability present in parameters contributing to deflection of reinforced concrete beams. Hence it is required to develop probabilistic approach to quantify variability in deflection of reinforced concrete beams. Literature survey on probabilistic analysis have focused on ultimate strength rather than serviceability limit states, very few probability deflection design provisions have been developed these have primarily been for structures with steel and timber[Galambos 1986, Philpot TA 1993] and on reinforced concrete [E.H.Khor 2001]. In this paper an attempt is made to suggest a distribution model for variability in deflection of reinforced concrete beams and to study the coefficient of variation. Outline of the work is as follow

• Deterministic expression for deflection

• Collection of statistical parameters involved in deflection expression

• Fool of practical range of beams

• Generate sufficient number of deflection samples using deterministic expression and variability in parameters using Monte Carlo Simulation

 Statistical analysis of generated samples for suggesting a distribution model

For present work expression for deflection given in IS 456:2000[6] is used as deterministic expression for computation of deflection. Practical range of beams with span ranging from 3m to 10m,center to center spacing of 3m,3,5m and 4m,live load of 2kn/m2 to 4kn/m2 and with /without wall load are designed as per the guidelines of IS code. In all seventy-two reinforced concrete beams are considered for the present work. Literature survey is carried out for the statistical parameters of the different variables contributing for deflection and same is presented in table 2.

Using deterministic expression for deflection and statistical parameters of variables generate large number say ten thousand samples for each beam. Specimen values of selected beam are given in figure 1(a) and (b)

Figure 1(a) deflection samples for beam 2

Sample Histograms are constructed based on the generated samples to know the range of deflection sample nature of spread etc. Specimen histograms for selected beams are shown in figure 2(a) and (b)

2468101214160 500 1000 1500 2000 2500 3000

Deflection in mm Frequency

Studies of histograms of the deflection samples give us an idea of maximum and minimum values of deflection samples, nature of frequency distribution. Histograms clearly indicate that frequency distribution is not a normal distribution as it is skewed to left. Now the PDF of deflection samples is overlapped with PDF curves of normal and lognormal distribution having same mean and standard deviation as that of deflection data. Results for selected beams are shown in figure 3(a) and (b)

246810121416180 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Deflection Density Deflection data1NormalLognormal

2345678910110 0.05 0.1 0.15 0.2 0.25 0.3

Deflection in mm Density

Deflection data 62Log_normalNormal_

Figure 3(b) Mapping normal and lognormal distribution PDF with PDF of deflection data for beam 62 In above figures it is observed that PDF deflection data matches more closely with PDF of lognormal distribution model than that of normal distribution model .Now further test is carried out by overlapping CDF of deflection data with the CDF of normal and lognormal distribution models having same mean and standard deviation as that of deflection data and results for some selected beam are presented in figures 4(a) and (b).

46810121416180 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Deflection in mm Cumulative probability

Dflection data 1LognormalNormal

2345678910110 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Deflection in mm Cumulative probability Deflection data 62Log_normalNormal_

.Figure 4(b) Mapping normal and lognormal distribution CDF with CDF of deflection data for beam 62 Above figures it is clearer that CDF of deflection data matches more closely with CDF of lognormal distribution indicated by dotted line in the figures. Further probability plot of deflection data are matched with probability p[lots of normal and lognormal probability plots .results are presented in figures 5(a) and (b) Based on study of figures 3 to figure 5 and Chi-Square goodness of fit test it is concluded that randomness in deflection follows lognormal distribution model. `

4681012141618 0.00010.00050.0010.0050.010.050.10.250.50.750.90.950.990.9950.9990.99950.9999

Deflection in mm Probability

Dflection data 1LognormalNormal

234567891011 0.00010.00050.0010.0050.010.050.10.250.50.750.90.950.990.9950.9990.9995

Deflection in mm Probability

Figure 5(b) Mapping normal and lognormal probability plots with probability plot of deflection data for beam62

After finding probability distribution model one more important parameter controlling variability of deflection of the beam is coefficient of variation for various beams are presented in Table 3. Table 3 Computation of coefficient of variation

010203040506070800 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Beam Number Coefficien of variation

From the Table 3 it is observed that the coefficient of variation for practical range of beams varies from

0.2419

From this experimentation it is observed that the deflection in the reinforced concrete beams is a random variable. The randomness in the deflection can be modeled as lognormal distribution. The coefficient of variation is consistent for practical ranges of beams with an average value of 0.2419.

6. REFERENCES

C.Q.Li and R.E.Melchers ―Reliability analysis of creep and shrinkage effects‖. E.H.Khor,D.V.Rosowsky,M.G.Stewart―Probabilistic analysis of time-dependent deflections of RC flexural members‖, Computers and Structures, 79(2001),p.p.1461-1472. Galambos TV. Ellingwood BR. ―Serviceability limit states; deflection‖, J.Struct. Engg.ASCE 1986, 107(5),p.p. 857-872. IS 456:2006 ―Indian Standard Code of Practice for Plain and Reinforced Concrete- Code of Practice(Fourth Revision) Philpot TA, Rosowsky DV, Fridley KJ ―Serviceability design in LRFD for wood‖, J.Struct.Engg.ASCE 1993, 119(12). R.Prabhakara, K. U. Muthu, and R. Meenakshi ―Allowable Span/Depth Ratio for High Strength Concrete Beams‖ The Arabian Journal for Science and Engineering, Volume 32, Number 2B R.Ranganathan; ―Structural reliability analysis and design‖ Jai Co books.

ME Student, ICOER, Wagholi