160

CHAPTER 7

FORMULATION OF DESIGN EQUATIONS USING REGRESSION

ANALYSIS

7.1 GENERAL

In this chapter regression analysis of the experimental data is

discussed in brief. Regression analysis has been carried out for channel and

trapezoidal sections separately so that the existing equations of code of

practice for wall and compression members may also be applied for singly

symmetric open section compression members with web stiffener. The main

objective of the regression analysis is to find suitable buckling factors for

channel and trapezoidal sections through optimisation of the experimental

data.

7.2 MATHEMATICAL EXPRESSION

Mathematical expressions are formulated for the proposed

modifications in the short column design formula of IS 456 – 2000 and design

expression of ACI 318 – 2008 for concrete wall. It is proposed to introduce

minimal changes without altering the basic form of these popular equations so

that they may also be used as design equations for thin-walled open section

compression members.

7.2.1 IS 456 Short Column Expression

As the columns tested are neither too long nor too short, a

mathematical expression similar to Rankine Gordan formula, applicable for

161

intermediate columns, is used to predict the buckling load of channel and

trapezoidal sections. Moreover the failure mode of the columns is global

buckling either with nominal torsional deformation or large torsional

deformation and seldom fails by local buckling failure. The basic form of

Rankine‟s equation used to predict the buckling failure load Puc is

(1/Puc) = (1/Pc) + (1/ Pb) (7.1)

where Pc = Crushing Load and

Pb = Crippling load or buckling load

The crushing load corresponds to the material strength of the

member and can be taken from IS 456 – 2000, clause 39.3, design expression

of reinforced concrete compression members with λ less than 12 subjected to axial load with minimum eccentricity. Hence the crushing failure load Pc is

given by

Pc = [0.40 fck Ac + 0.67 fy Asc] (7.2)

where fck = Characteristic ompressive strength of concrete cube in MPa

Ac = Cross sectional area of concrete in mm2

fy = Yield strength of reinforcement steel in MPa

Asc = Cross sectional area of reinforcement steel in mm2

The crippling load corresponds to the load supported by the

member found through Euler‟s expression. But the crippling load in case of

thin-walled open sections corresponds to the reduction in load capacity. The

above loss in load bearing capacity is due to slenderness and singly symmetric

open section geometry of the member and minimum eccentricity of the load.

Hence the effect of buckling on load capacity is accounted by slenderness

factor „a‟ and buckling factor „b‟, which are similar to Rankine‟s coefficient.

Hence Equation 7.1 is modified as

162

Puc = Pc [b /{1 + a (λ2 )}] (7.3)

The slenderness and buckling factors, a and b are evaluated by

optimisation programme written for channel and trapezoidal sections in

MATLAB using Particle Swarm Optimisation (PSO).

7.2.2 ACI 318 Concrete Wall Expression

The ACI committee 318 – 2008 in Section 14.5.2, gives an

empirical equation for the design axial load strength of reinforced concrete

wall as

Puc = 0.55 Φ f‟c Ag [1 – (kH/32t)2] (7.4)

where Ф = Capacity reduction factor and shall be taken as 0.70

f‟c = Crushing strength of concrete cylinder taken as 0.446fcu

Ag = Gross area of concrete

k = factor for support condition taken as 0.80 for restrained

against rotation and 1 for fully unrestrained.

H and t = Height and thickness of wall

It is proposed to introduce two factors to take care of buckling,

slenderness and other incompatibility that are associated with using the above

expression for thin-walled open section compression members. The modified

ACI 318 wall expression is given below

Puc = 0.55 Φb f‟c Ag [1 – (H/χ32t)2] (7.5)

where Φb = Buckling Factor

χ = Slenderness Factor

The buckling and slenderness factors, Φb and χ are evaluated by

optimisation programme written for channel and trapezoidal sections in

MATLAB using Particle Swarm Optimisation (PSO).

163

7.3 OPTIMISATION TECHNIQUES

Since 1970 structural optimization has been the subject of intensive

research and different approaches for optimal design of structures have been

advocated. Mathematical programming methods make use of the derivatives

of original function with respect to the design variables. On the other hand the

application of combinatorial optimization methods based on probabilistic

searching do not need gradient information and therefore avoid to perform the

computationally expensive sensitivity analysis step. Gradient based methods

present a satisfactory local rate of convergence, but they cannot assure that

the global optimum can be found. Stochastic process techniques can be used

to analyze problems described by a set of random variables having known

probability distributions.

Large scale problems are often computationally demanding,

requiring significant resources in time and hardware to solve. Engineering

optimization problems are often plagued by multiple local optima and

numerical noise, requiring the use of global search methods such as

population based algorithms to deliver reliable results.

During the last three decades, there has been a growing interest in

problem solving systems based on algorithms that rely on analogies to natural

processes called Evolutionary Algorithms. The best known algorithms in this

class include Genetic algorithms and Evolution Strategies. The genetic

algorithms are search technique based on the mechanics of natural selection

and natural genetics. Neural network methods are based on solving the

problem using the efficient computing power of the network of interconnected

neuron processors.

7.3.1 Particle Swarm Optimisation (PSO)

The PSO is a recent addition to the list of global search methods. It

has been successfully applied to large scale problems in several engineering

164

disciplines and readily parallelisable. It also has fewer algorithm parameters

than genetic algorithm. Several modifications have been made to the original

swarm algorithm to improve and adapt it to specific type of problems.

7.3.2 PSO Algorithm

Particle swarm optimization algorithm, which is tailored for

optimizing difficult numerical functions and based on metaphor of human

social interaction, is capable of mimicking the ability of human societies to

process knowledge (Kennedy et al 2001). It has roots in two main component

methodologies: artificial life (such as bird flocking, fish schooling and

swarming); and, evolutionary computation. Its key concept is that potential

solutions are flown through hyperspace and are accelerated towards better or

more optimum solutions. Its paradigm can be implemented in simple form of

computer codes and is computationally inexpensive in terms of both memory

requirements and speed. It lies somewhere in between evolutionary

programming and the genetic algorithms. As in evolutionary computation

paradigms, the concept of fitness is employed and candidate solutions to the

problem are termed particles or sometimes individuals, each of which adjusts

its flying based on the flying experiences of both itself and its companion. It

keeps track of its coordinates in hyperspace which are associated with its

previous best fitness solution, and also of its counterpart corresponding to the

overall best value acquired thus far by any other particle in the population.

Vectors are taken as presentation of particles since most

optimization problems are convenient for such variable presentations. In fact,

the fundamental principles of swarm intelligence are adaptability, diverse

response, proximity, quality, and stability. It is adaptive corresponding to the

change of the best group value. The allocation of responses between the

individual and group values ensures a diversity of response. The higher

dimensional space calculations of the PSO concept are performed over a

165

series of time steps. The population is responding to the quality factors of the

previous best individual values and the previous best group values. The

principle of stability is adhered to since the population changes its state if and

only if the best group value changes (Clerc and Kennedy 2002, Zahiria and

Seyedin 2007). As it is reported in (Yu et al 2004), this optimization

technique can be used to solve many of the same kinds of problems as GA,

and does not suffer from some of GAs difficulties. It has also been found to

be robust in solving problem featuring nonlinearity, non-differentiability and

high-dimensionality. PSO is the search method to improve the speed of

convergence and find the global optimum value of fitness function.

7.3.3 Global Best and Local Best

PSO starts with a population of random solutions „„particles‟‟ in a

D-dimension space. The ith particle is represented by Xi = (xi1,xi2, . . . ,xiD).

Each particle keeps track of its coordinates in hyperspace, which are

associated with the fittest solution it has achieved so far. The value of the

fitness for particle i (pbest) is also stored as Pi = (pi1, pi2, . . . ,piD). The

global version of the PSO keeps track of the overall best value (gbest), and its

location, obtained thus far by any particle in the population. PSO consists of,

at each step, changing the velocity of each particle toward its pbest and gbest.

The velocity of particle i is represented as Vi= (vi1, vi2. . . viD). Acceleration

is weighted by a random term, with separate random numbers being generated

for acceleration toward pbest and gbest. The position of the ith particle is then

updated according to Equation 7.7 (Kennedy et al 2001).

Vid = w x vid + c1 x rand ( ) x (Pid – xid) + c2 x rand ( ) x (Pgd – xid) (7.6)

xid = xid + c vid (7.7)

where, Pid and Pgd are pbest and gbest. Several modifications have been

166

proposed in the above literature to improve the PSO algorithm speed and

convergence toward the global minimum. One modification is to introduce a

local-oriented paradigm (lbest) with different neighborhoods. It is concluded

that gbest version performs best in terms of median number of iterations to

converge. However, Pbest version with neighborhoods of two is most

resistant to local minima. PSO algorithm is further improved via using a time

decreasing inertia weight, which leads to a reduction in the number of

iterations. Figure 7.1 shows the flowchart of the PSO algorithm adopted in the

present work.

7.3.4 Advantages of PSO

This new approach features many advantages; it is simple, fast and

easy to be coded. Also, its memory storage requirement is minimal.

Moreover, this approach is advantageous over evolutionary and genetic

algorithms in many ways. First, PSO has memory. That is, every particle

remembers its best solution (local best) as well as the group best solution

(global best). Another advantage of PSO is that the initial population of the

PSO is maintained, and so there is no need for applying operators to the

population, a process that is time and memory-storage-consuming. In

addition, PSO is based on „„constructive cooperation‟‟ between particles, in

contrast with the genetic algorithms, which are based on „„the survival of the

fittest‟‟.

7.4 MATHEMATICAL FORMULATION

The objective function of the present problem, the ultimate load

capacity of thin-walled open section RC compression members is formulated

as follows.

Objective function = z = min ∑ ( Puei – Puci)2 (7.8)

167

where Puei = Experimental buckling load of ith

column

Puci = predicted buckling load of ith

column using Equation 7.3

The above mathematical formulation leads to the evaluation of

slenderness and buckling factors „a‟ and „b‟ in such a way that the sum of

squared error is minimum if not zero.

Start

Select parameters of PSO: 1/a and b

Generate the random positions

and velocities of particles

Initialize, pbest with a copy of the position for particle, determine gbest

Update velocities and positions

according to Eqs. (7.1 and 7.2)

Evaluate the fitness of each particle

Update pbest and gbest

Optimal value of the parameters

End

YES

NO Satisfying

Stopping

Criterion

Figure 7.1 Flow Chart of PSO Programme

168

7.5 EVALUATION OF BUCKLING FACTORS

The slenderness and buckling factors „a‟ and „b‟ proposed for IS 456 - 2000 short column design expression and the buckling and slenderness

factors Фb and χ proposed for ACI 318 – 2008 concrete wall design

expression are evaluated by PSO technique described in section 7.3.

Programmes were written separately for IS and ACI factors in MATLAB and

executed to find the most optimum values for the above factors. The above

two programmes used are given in the Appendix. The evaluated values of

factors are given in Table 7.1. The slenderness factor „a‟ of IS 456 is very small and hence its value is given in terms of (1/a). The ultimate load

predicted by the above two equations using the proposed factors are

computed. The Puc thus calculated is compared with the experimental ultimate

load Pue to check the compatibility of these buckling factors for channel and

trapezoidal sections. The comparison made for channel specimens using IS

456 – 2000 slenderness and buckling factors are given in Table 7.3 and Table

7.4 gives the same for trapezoidal specimens. The above optimised factors

found from regression analysis are not simple to remember and use. Hence

after checking the compatibility of the design equations with the values

available in Table 7.1, the slenderness and buckling factors are rounded off in

Table 7.2 to a suitable whole number, so that they can be remembered easily

and employed in design equations, which will predict ultimate load in the

range of 90 to 95% of Pue.

Table 7.1 Buckling and Slenderness Factors of C and T Series from PSO

Analysis

Sl.

No

Design

Equation

Specimen

IS 456 – 2000 Factors ACI 318 – 1989

Factors

Slenderness

1/a

Buckling

b

Buckling

Фb

Slenderness

χ 1. Channel 3343 0.8066 0.9292 1.6795

2. Trapezoidal 8897 0.6762 0.8066 1.9295

169

Table 7.2 Buckling and Slenderness Factors of C and T Series for 95%

Design Curve

Sl.

No

Design

Equation

Specimen

Design Curve of

Modified IS 456 – 2000 Equation

Design Curve of ACI

318 – 1989

Equation

Slenderness

1/a

Buckling

b

Buckling

Фb

Slenderness

χ

1. Channel 3300 0.75 0.90 1.60

2. Trapezoidal 8900 0.65 0.75 1.80

7.5.1 Compatibility of Modified IS Equation

The degree of compatibility achieved by the modified short column

formula of IS 456 – 2000 with the experimental data of C series of channel

sections is given in Table 7.3. The slenderness and buckling factors employed

in this table to check the compatibility are from Table 7.1. The mean and

standard deviations achieved are quite satisfactory. The combined mean and

standard deviation for the entire C series found to be 1.003 and 0.089

respectively. The minimum and maximum values of (Puc/Pue) ratio for C30

specimens are 0.851 and 1.050. The same for C25 series is 0.919 and 1.268

respectively. The above values indicate the consistent estimation of Puc

through PSO analysis. The good agreement between the Puc predicted by the

PSO, Puc estimated by design curve and the experimental Pue is clearly

illustrated in Figure 7.2. The mean, standard deviation and coefficient of

correlation of the modified IS 456 – 2000 design curve are 0.935, 0.090 and

0.978 respectively.

The degree of compatibility achieved by the short column formula

of IS 456 – 2000 after modification for T series specimens is given in Table

7.4. The mean and standard deviations achieved are satisfactory. The

170

combined mean and standard deviation for the entire T series found to be

0.944 and 0.088 respectively. The minimum and maximum values of (Puc/Pue)

ratio for T30 specimens are 0.855, 1.169. The same for T25 series is 0.828 and

1.000 respectively. The above values indicate the consistent estimation of Puc

by PSO technique. The good agreement between the Puc predicted by the

PSO, Puc estimated by design curve and the experimental Pue can be seen in

Figure 7.3. The mean, standard deviation and coefficient of correlation of the

modified IS 456 – 2000 design curve for T Series specimens are 0.935, 0.090

and 0.978 respectively. The buckling coefficient in Tables 7.3 to 7.6 is

explained in Section 7.5.3.

Table 7.3 Puc Predicted by Modified IS Short Column Formula - Channel

Sl.

No

Sp

ecim

en

Pu

c i

n k

N

(Pu

c/P

ue)

Ra

tio

Bu

ckli

ng

Co

effi

cien

t

Sp

ecim

en

Pu

c i

n k

N

(Pu

c/P

ue)

Ra

tio

Bu

ckli

ng

Co

effi

cien

t

1. C30 - 1.1 281 0.947 0.700 C25 - 1.1 189 1.268 0.638

2. C30 - 1.2 298 0.957 0.726 C25 - 1.2 207 1.182 0.675

3. C30 - 1.3 304 0.941 0.749 C25 - 1.3 221 1.095 0.711

4. C30 - 1.4 307 0.941 0.768 C25 - 1.4 232 1.077 0.742

5. C30 - 2.1 221 1.035 0.633 C25 - 2.1 121 1.134 0.472

6. C30 - 2.2 231 0.965 0.672 C25 - 2.2 134 1.017 0.531

7. C30 - 2.3 247 0.976 0.708 C25 - 2.3 141 0.989 0.593

8. C30 - 2.4 259 0.907 0.740 C25 - 2.4 164 1.017 0.655

9. C30 - 3.1 128 0.995 0.467 C25 - 3.1 109 1.117 0.479

10. C30 - 3.2 143 1.050 0.525 C25 - 3.2 118 1.089 0.537

11. C30 - 3.3 156 1.000 0.588 C25 - 3.3 131 0.976 0.598

12. C30 - 3.4 175 0.951 0.651 C25 - 3.4 145 1.045 0.659

13. C30 - 4.1 87 0.851 0.349 C25 - 4.1 69 0.927 0.358

14. C30 - 4.2 106 0.934 0.411 C25 - 4.2 79 0.919 0.420

15. C30 - 4.3 124 1.001 0.483 C25 - 4.3 89 0.928 0.492

16. C30 - 4.4 144 0.905 0.565 C25 - 4.4 105 0.952 0.572

Mean 0.960 Mean 1.046

Standard Deviation 0.0503 Standard Deviation 0.0997

171

Table 7.4 Puc Predicted by Modified IS Short Column Formula –

Trapezoidal Specimens S

l.N

o

Sp

ecim

en

Pu

c i

n k

N

(Pu

c/P

ue)

Ra

tio

Bu

ckli

ng

Co

effi

cien

t

Sp

ecim

en

Pu

c i

n k

N

(Pu

c/P

ue)

Ra

tio

Bu

ckli

ng

Co

effi

cien

t

1. T30 - 1.1 261 1.074 0.624 T25 - 1.1 151 1.000 0.588

2. T30 - 1.2 269 1.000 0.636 T25 - 1.2 155 0.901 0.610

3. T30 - 1.3 272 0.941 0.648 T25 - 1.3 168 0.954 0.628

4. T30 - 1.4 289 0.855 0.658 T25 - 1.4 169 0.828 0.646

5. T30 - 2.1 222 1.099 0.594 T25 - 2.1 123 0.946 0.537

6. T30 - 2.2 225 1.000 0.615 T25 - 2.2 127 0.901 0.568

7. T30 - 2.3 236 0.937 0.632 T25 - 2.3 132 0.852 0.597

8. T30 - 2.4 239 0.895 0.648 T25 - 2.4 143 0.883 0.623

9. T30 - 3.1 166 1.169 0.535 T25 - 3.1 85 0.885 0.444

10. T30 - 3.2 177 1.054 0.565 T25 - 3.2 94 0.847 0.490

11. T30 - 3.3 183 0.984 0.597 T25 - 3.3 102 0.864 0.533

12. T30 - 3.4 192 0.910 0.623 T25 - 3.4 109 0.886 0.576

Mean 0.993 Mean 0.896

Standard Deviation 0.0925 Standard Deviation 0.0497

Figure 7.2 Modified IS Equation Design Curve for C Series Specimens

172

Figure 7.3 Modified IS Equation Design Curve for T Series Specimens

7.5.2 Compatibility of Modified ACI Equation

The degree of compatibility achieved by the modified concrete wall

formula of ACI 318 - 2008 with C30 and C25 series of channel sections is

given in Table 7.5. The mean and standard deviations achieved are

satisfactory. The combined mean and standard deviation of the entire C series

found to be 1.016 and 0.0998 respectively. The minimum and maximum

values of (Puc/Pue) ratio for C30 specimens are 0.919 and 1.276. The same for

C25 series is 0.775 and 1.109 respectively. The above values indicate the

consistent estimation of Puc by the PSO analysis employed for ACI concrete

wall formula. The good correlation between the Puc predicted by the PSO, Puc

estimated by design curve and the experimental Pue can be seen in Figure 7.4.

The mean, standard deviation and coefficient of correlation of the modified

ACI 318 design curve for C Series specimens are 0.940, 0.112 and 0.980

respectively.

The degree of compatibility achieved by the modified concrete wall

formula of ACI 318 - 2008 with the T30 and T25 series of trapezoidal sections

173

is given in Table 7.6. The mean and standard deviations achieved are

satisfactory. The combined mean and standard deviation of the entire T series

found to be 0.973 and 0.1085 respectively. The minimum and maximum

values of (Puc/Pue) ratio for T30 specimens are 0.934 and 1.218. The same for

T25 series is 0.810 and 0.962 respectively. The above values indicate

consistent estimation of Puc by the modified concrete wall formula of ACI 318

through the PSO technique. The degree of agreement between the Puc

predicted by the PSO, Puc estimated by design curve and the experimental Pue

is clearly illustrated in Figure 7.5. The mean, standard deviation and

coefficient of correlation of the modified ACI 318 design curve for T Series

specimens are 0.876, 0.106 and 0.971 respectively.

Table 7.5 Puc Predicted by Modified ACI Wall Formula - Channel

Sl.

No

Sp

ecim

en

Pu

c i

n k

N

(Pu

c/P

ue)

Ra

tio

Bu

ckli

ng

Co

effi

cien

t

Sp

ecim

en

Pu

c i

n k

N

(Pu

c/P

ue)

Ra

tio

Bu

ckli

ng

Co

effi

cien

t

1. C30 - 1.1 273 0.919 0.612 C25 - 1.1 149 0.997 0.381

2. C30 - 1.2 297 0.956 0.656 C25 - 1.2 180 1.030 0.518

3. C30 - 1.3 308 0.954 0.689 C25 - 1.3 206 1.019 0.607

4. C30 - 1.4 313 0.961 0.714 C25 - 1.4 224 1.043 0.667

5. C30 - 2.1 216 1.013 0.535 C25 - 2.1 83 0.775 -0.198

6. C30 - 2.2 232 0.972 0.607 C25 - 2.2 115 0.873 0.260

7. C30 - 2.3 254 1.005 0.659 C25 - 2.3 134 0.936 0.479

8. C30 - 2.4 270 0.943 0.696 C25 - 2.4 165 1.024 0.602

9. C30 - 3.1 133 1.034 0.307 C25 - 3.1 102 1.046 0.236

10. C30 - 3.2 157 1.156 0.479 C25 - 3.2 119 1.105 0.444

11. C30 - 3.3 174 1.114 0.586 C25 - 3.3 138 1.027 0.567

12. C30 - 3.4 194 1.053 0.656 C25 - 3.4 154 1.109 0.646

13. C30 - 4.1 96 0.943 0.031 C25 - 4.1 61 0.830 -0.198

14. C30 - 4.2 132 1.164 0.350 C25 - 4.2 85 0.989 0.260

15. C30 - 4.3 158 1.276 0.520 C25 - 4.3 101 1.054 0.479

16. C30 - 4.4 178 1.121 0.623 C25 - 4.4 119 1.085 0.602

Mean 1.036 Mean 0.996

Standard Deviation 0.1023 Standard Deviation 0.0963

174

Table 7.6 Puc Predicted by Modified ACI Wall Formula - Trapezoidal S

l.N

o

Sp

ecim

en

Pu

c i

n k

N

(Pu

c/P

ue)

Ra

tio

Bu

ckli

ng

Co

effi

cien

t

Sp

ecim

en

Pu

c i

n k

N

(Pu

c/P

ue)

Ra

tio

Bu

ckli

ng

Co

effi

cien

t

1. T30 - 1.1 269 1.106 0.675 T25 - 1.1 140 0.926 0.495

2. T30 - 1.2 284 1.056 0.732 T25 - 1.2 149 0.868 0.615

3. T30 - 1.3 291 1.008 0.776 T25 - 1.3 169 0.962 0.703

4. T30 - 1.4 316 0.934 0.811 T25 - 1.4 173 0.848 0.767

5. T30 - 2.1 229 1.134 0.596 T25 - 2.1 113 0.871 0.316

6. T30 - 2.2 237 1.053 0.680 T25 - 2.2 123 0.875 0.506

7. T30 - 2.3 255 1.010 0.743 T25 - 2.3 133 0.857 0.638

8. T30 - 2.4 260 0.973 0.791 T25 - 2.4 149 0.917 0.730

9. T30 - 3.1 173 1.218 0.451 T25 - 3.1 78 0.810 -0.069

10. T30 - 3.2 191 1.137 0.588 T25 - 3.2 94 0.850 0.297

11. T30 - 3.3 200 1.073 0.686 T25 - 3.3 108 0.911 0.522

12. T30 - 3.4 212 1.005 0.757 T25 - 3.4 117 0.952 0.667

Mean 1.059 Mean 0.887

Standard Deviation 0.080 Standard Deviation 0.046

Figure 7.4 Modified ACI Equation Design Curve for C Series Specimens

175

Figure 7.5 Modified ACI Equation Design Curve for T Series Specimens

7.5.3 Buckling Coefficients

The buckling coefficients are computed through the PSO

programme and given in Tables 7.3 to 7.6. The buckling coefficients can also

be calculated as ratio between the ultimate load (Puc) predicted by un-

modified equation and the ultimate load estimated by the modified equation.

These buckling coefficients correlate well with those found through the PSO

programme.

IS 456 – 2000 buckling coefficient

Cb (IS) = Puc / {(0.40 fck Ac + 0.67 fy Asc) [b /{1 + a (λ2 )}]} (7.9)

ACI 318 – 2008 Buckling Coefficient

Cb (ACI) = Puc / {0.55 Φb f‟c Ag [1 – (H/χ32t)2]} (7.10)

176

The buckling coefficients can be used for evaluation of ultimate

load of open channel or trapezoidal section thin-walled members, once the

dimensions, percentage reinforcement and material strength are known,

thereby detailed calculation can be avoided. The variation of IS 456 buckling

coefficients Cb(IS) with slenderness ratio for channel section is given in Figure

7.6. The same for ACI 318 buckling coefficient Cb(ACI) is given in Figure 7.7.

The above plots shall be used to find the value of buckling coefficient of

channel sections for given slenderness ratio. The governing expression given

in the above plot for the variation of Cb (IS) or Cb(ACI) can also be used for

finding the value of buckling coefficient. The product of the above buckling

coefficient and the original unmodified IS 456 or ACI 318 design equation

gives the ultimate load capacity of the thin-walled open section compression

member. The above method of estimating the ultimate load using buckling

coefficients shall be employed in the absence of data on slenderness and

buckling factors for given type of compression member. The variation of the

buckling coefficients Cb (IS) and Cb(ACI) with slenderness ratio for trapezoidal

section is given in Figure 7.8 and Figure 7.9 respectively.

Figure 7.6 Buckling Coefficients for IS Equation – C Series Specimens

177

Figure 7.7 Buckling Coefficients for ACI Equation – C Series Specimens

Figure 7.8 Buckling Coefficients for IS Equation – T Series Specimens

178

Figure 7.9 Buckling Coefficient for ACI Equation – T Series Specimens

7.6 COMPARISON OF PREDICTED LOADS

The ratio of theoretical ultimate load to experimental ultimate load

calculated based on the proposed empirical formula and the modified IS 456

and ACI 318 formulae using design curve slenderness and buckling factors

for channel specimens are compared in Table 7.7. The comparison of

experimental ultimate load and the ultimate load predicted by the above three

design equations is also made in Figure 7.10 for the C series specimens. The

above two comparisons are made for T30 and T25 trapezoidal specimens in

Table 7.8 and in Figure 7.11 respectively.

In case of C series specimens all equations predict the ultimate load

in excellent correlation with the experimental load as indicated by the values

of mean and standard deviation. Among the three, the proposed empirical

equation is little conservative in predicting the ultimate load for the entire

range of slenderness ratio. All the three equations predict the ultimate load in

179

an equally good and consistent manner for flexural buckling failure

specimens. The best mean value is possessed by the IS 456 modified short

column equation and the best standard deviation is possessed by the proposed

empirical equation. The best coefficient of correlation is possessed by the

modified ACI – 318 concrete wall equation.

Table 7.7 (Puc/Pue) Ratio Predicted by Proposed Equations - Channel

Sl.

No

Sp

ecim

en (Puc/Pue) Ratio Predicted

by Proposed equations

Sp

ecim

en (Puc/Pue) Ratio Predicted

by Proposed equations

Empirical IS

2000

ACI

1989 Empirical

IS

2000

ACI

1989

1. C30 - 1.1 0.751 0.946 0.919 C25 - 1.1 1.011 1.268 1.000

2. C30 - 1.2 0.823 0.958 0.955 C25 - 1.2 1.024 1.183 1.029

3. C30 - 1.3 0.861 0.941 0.954 C25 - 1.3 1.015 1.094 1.020

4. C30 - 1.4 0.908 0.942 0.960 C25 - 1.4 1.055 1.079 1.042

5. C30 - 2.1 0.864 1.038 1.014 C25 - 2.1 1.023 1.131 0.776

6. C30 - 2.2 0.849 0.967 0.971 C25 - 2.2 0.961 1.015 0.871

7. C30 - 2.3 0.905 0.976 1.004 C25 - 2.3 0.960 0.986 0.937

8. C30 - 2.4 0.881 0.906 0.944 C25 - 2.4 1.020 1.019 1.025

9. C30 - 3.1 0.876 0.992 1.031 C25 - 3.1 0.953 1.112 1.041

10. C30 - 3.2 0.963 1.051 1.154 C25 - 3.2 0.987 1.093 1.102

11. C30 - 3.3 0.942 1.000 1.115 C25 - 3.3 0.927 0.978 1.030

12. C30 - 3.4 0.924 0.951 1.054 C25 - 3.4 1.030 1.043 1.108

13. C30 - 4.1 0.696 0.853 0.941 C25 - 4.1 0.791 0.932 0.824

14. C30 - 4.2 0.858 0.938 1.168 C25 - 4.2 0.858 0.919 0.988

15. C30 - 4.3 0.968 1.000 1.274 C25 - 4.3 0.907 0.927 1.052

16. C30 - 4.4 0.899 0.906 1.119 C25 - 4.4 0.957 0.955 1.082

Mean 0.920 1.003 1.016

Standard Deviation 0.085 0.089 0.100

Coefficient of Correlation 0.974 0.971 0.983

In case of T series specimens all equations predict the ultimate load

in reasonable agreement with the experimental load as indicated by the values

of mean and standard deviation. Among the three, the proposed empirical

180

equation is little conservative in predicting the ultimate load for specimens in

flexural buckling range of slenderness ratio. The modified IS 456 short

column equation is conservative in predicting the ultimate load of specimens

in the torsional-flexural buckling range. All the three equations predict the

ultimate load in an equally good and consistent manner for specimens in the

terminal range of torsional-flexural buckling failure. The best standard

deviation value is possessed by the IS 456 modified short column equation

and the best mean and coefficient of correlation is possessed by the modified

ACI – 318 concrete wall equation.

Table 7.8 (Puc/Pue) Ratio Predicted by Proposed Equations - Trapezoidal

Sl.

No

Sp

ecim

en

(Puc/Pue) Ratio

Predicted by Proposed

equations

Sp

ecim

en (Puc/Pue) Ratio Predicted

by Proposed equations

Empiri

cal

IS

2000

ACI

1989

Empiri

cal IS 2000

ACI

1989

1. T30 - 1.1 0.901 1.074 1.107 T25 - 1.1 0.709 1.000 0.927

2. T30 - 1.2 0.933 1.000 1.056 T25 - 1.2 0.738 0.901 0.866

3. T30 - 1.3 0.955 0.941 1.007 T25 - 1.3 0.898 0.955 0.960

4. T30 - 1.4 0.947 0.855 0.935 T25 - 1.4 0.863 0.828 0.848

5. T30 - 2.1 0.901 1.099 1.134 T25 - 2.1 0.662 0.946 0.869

6. T30 - 2.2 0.902 1.000 1.053 T25 - 2.2 0.723 0.901 0.872

7. T30 - 2.3 0.929 0.937 1.012 T25 - 2.3 0.774 0.852 0.858

8. T30 - 2.4 0.951 0.895 0.974 T25 - 2.4 0.895 0.883 0.920

9. T30 - 3.1 0.859 1.169 1.218 T25 - 3.1 0.510 0.885 0.813

10. T30 - 3.2 0.881 1.054 1.137 T25 - 3.2 0.604 0.847 0.847

11. T30 - 3.3 0.903 0.984 1.075 T25 - 3.3 0.720 0.864 0.915

12. T30 - 3.4 0.915 0.910 1.005 T25 - 3.4 0.837 0.886 0.951

Mean 0.830 0.944 0.973

Standard Deviation 0.121 0.088 0.109

Coefficient of Correlation 0.992 0.963 0.966

181

Figure 7.10 Ultimate Load Predicted by Proposed Equations – C Series

Specimens

Figure 7.11 Ultimate Load Predicted by Proposed Equations – T Series

Specimens

182

7.7 CONCLUDING REMARKS

In this chapter, based on regression analysis of test data using PSO

technique, slenderness and buckling factors for IS 456 and ACI 318 design

equations are found. The above equations are modified without disturbing

their popular basic form. Their suitability to the type of members tested is

found using the test data and a series of graphs and tables are used to illustrate

the same. The slenderness factor and buckling factor for C and T series

specimens for the above modified equations are odd with more decimals,

making them difficult to use and remember. This has been made good by

rounding the above factors to simple form, so as to get 90 to 95% design

curve for the open section members tested. The buckling coefficients for IS

456 and ACI 318 original unmodified expression are also calculated and

necessary graphs are made available for finding the buckling coefficient

directly from the graph. Finally the compatibility of proposed empirical

equations to C and T series specimens is compared with that of modified IS

and ACI equations. It is found that the empirical equation is nominally

conservative but reliable compared to the modified IS and ACI equations.