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International Journal of Civil Engineering and Technology (IJCIET)

Volume 10, Issue 01, January 2019, pp. 891–903, Article ID: IJCIET_10_01_082

Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=10&IType=1

ISSN Print: 0976-6308 and ISSN Online: 0976-6316

©IAEME Publication Scopus Indexed

DEVELOPMENT OF EFFICIENCY BASED

STANDARDS FOR OPTIMUM DESIGN OF

STIFFENED PLATE GIRDERS

Priya A Jacob

Associate Professor, North Malabar Institute of Technology, Kerala, India

Research Scholar, Karunya Institute of Technology and Sciences, Coimbatore, India

Justin S

Chief Engg. Manager (Civil), EDRC – Buildings & Factories,

L&T Construction, Chennai, India

R Mercy Shanthi

Associate Professor, Dept. of Civil Engineering,

Karunya Institute of Technology & Sciences, Coimbatore, India

ABSTRACT

In recent years, stiffened plate girders have been used extensively for long spans

due to its high flexural rigidity and buckling resistance. While designing, the amount

of costly steel used in a girder can be reduced by adopting optimum dimensions for

web depth, web thickness, flange thickness, flange breadth and spacing of stiffeners.

Such design can finally result in an economical design. Indian standards have code

provisions which can be used for design of stiffened plate girders. However, relatively

little attention has been devoted to developing efficiency based standards in the design

of plate girders. These efficiency based standards in the form of design charts can

help a designer in the economical design of stiffened plate girders considering

strength and serviceability conditions. Herein, relationships between the design

variables are developed using genetic algorithm (GA) based optimization formulation.

Both transversely stiffened and corrugated web plate girders are considered. The

relationships are further used to develop design charts which can be useful for design

engineers. The design charts are developed considering strength and serviceability

conditions as specified by the IS 800:2007.

Key words: Optimum design, Stiffened plate girder, Corrugated web, Design charts

Cite this Article: Priya A Jacob, Justin S, R Mercy Shanthi, Development of

Efficiency Based Standards for Optimum Design of Stiffened Plate Girders,

International Journal of Civil Engineering and Technology (IJCIET) 10(1), 2019, pp.

891–903.

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=1

Development of Efficiency Based Standards for Optimum Design of Stiffened Plate Girders


1. INTRODUCTION

Plate girders are built-up flexural members made up from separate structural steel plates

which are either welded or bolted together to form the horizontal flanges and vertical web of

the girder. A plate girder has enormous flexural strength and the resistance to bending and

shear can be increased by increasing the distance between the flanges. Web buckling can be

prevented by providing stiffeners. Use of stiffeners and thin corrugated steel plates in web of

plate girder can help in achieving out-of-plane stiffness and buckling resistance in place of a

thick web plate and thus save girder production cost. Span lengths can be increased due to the

use of stiffened plate girders because of high strength to weight ratio. Due to their many

favourable properties, these plate girders are widely used in many fields of application. For

the last 25 years, there have been a remarkable use of corrugated plate girders and many

theoretical studies and experimental investigations are carried out on corrugated plate girders.

Recently in India many manufacturers started investing on cutting edge technology, to

develop corrugated plate girders to be used as main frames of single-storey steel buildings.

Figure-1 a) Stiffened Plate b) Corrugated Plate girder

Optimization of stiffened girders is necessary to take complete advantage of many

favourable properties. Majority of the investigations on stiffened plate girders mainly focused

on analytical and experimental work. Few of the significant numerical and experimental

investigations include the study of interactive shear behaviour of trapezoidally corrugated

webs by Jogwon Yi et al. [6], study of influence of geometric parameters on patch load

resistance in corrugated webs. Performance of sinusoidal corrugated plate girders when used

as main frames of single-storey steel buildings were studied by Hartmut et al. [5]. However

insufficient investigations are carried out on the optimum design of plate girders. Kuan Chen

Fu. et al. [7] used genetic algorithm (GA) with elitism in the optimum design of plate girders.

Fatima Zohra Chalal [4] developed a formulation using Generalised Reduced Gradient

method for optimum design of continuous girder with variable depth. Optimum cost design of

steel box girder by varying plate thickness was carried out by Do Dai Thang et al. [3]. They

developed design guidelines which can be useful to the designer in the first stage of the

designing procedure. Furthermore Shon S D et al. [9] and Sudeok Shon et al. [12] conducted

analytical studies on optimum structural design of sinusoidal corrugated web beam using real-

valued genetic algorithm. Shon et al. [9, 10] recently researched on the optimum design of

thin corrugated web for shear buckling using genetic algorithm.

Many algorithms based on natural phenomena have been developed since 1970s. In

particular, the genetic algorithm (GA) has often been used in the optimization problems of

civil engineering structures. GA performs effectively in performing global search. The main

purpose of this work is to identify relationship of design variables and establish design

guidelines for the optimum design of stiffened plate girders. Plate girders with transverse

stiffeners and corrugated web stiffeners are used in this work. GA based algorithm has been

Priya A Jacob, Justin S, R Mercy Shanthi


used to perform optimization. These guidelines can be useful for structural engineers during

the first stage of design.

2. OPTIMUM DESIGN OF PLATE GIRDERS

Optimum design of plate girders is governed by flexural strength, shear strength,

serviceability and overall girder weight. Therefore optimum design of plate girders can be

formulated as a weight minimization problem keeping in view the flexural strength, shear

strength and serviceability aspects as suggested by the design codes. To increase the buckling

strength plate girder web, they are usually reinforced with transverse stiffeners or provided

with thin corrugated webs. Optimum design of such plate girders hence helps a designer to

also optimize the parameters influencing the weight and strength. This can finally result in an

economical design which satisfies strength and serviceability conditions as specified by the

design codes.

2.1. Objective Function

The common engineering objective function involves minimization of overall cost of

manufacturing of a structure or minimization of its overall weight or maximization of its

strength. Objective function is represented in terms of the design variables and other problem

parameters. In this work the objective function used is weight minimization. Usually objective

function in a case of unconstrained optimization is f(x). The modified objective function Ф

(x) in case of constraint optimization is written as

Ф (x) =f(x) {1+KC}

where parameter K has to be judiciously selected depending on the required influence of a

violation individual in the next generation and C is the constraint coefficient.

2.1.1. Transversely Stiffened Plate Girder (TSPG)

Use of transverse stiffeners in plate girders help in safeguarding the web against local

buckling failure. A bench mark problem is identified here for carrying out the formulation

using GA. The objective function, design variables and constraints are identified.

Objective function of weight minimization in TSPG is,

( ) [( ) ( )]

(1)

where

L : Span of girder ρ : Density of steel

d : Depth of web n : Number of stiffeners / folds

tw : Thickness of web bs : Breadth of stiffener

bf : Breadth of flange ts : Thickness of stiffener

tf : Thickness of flange

Figure 2 Transversely stiffened plate girder



2.1.2. Corrugated Web Plate Girder (CWPG)

Corrugated plate girders afford a significant weight reduction compared with welded I

section. Due to developments in automatic fabrication process, thin corrugated webs are

possible and this has helped engineers to design optimal structures.

The objective function, design variables and constraints are identified.

Objective function of weight minimization in CWPG is,

( ) ∑ ( ) (2)

subject to ( ) ( )

where ( ) {

}

Figure 3 Corrugated web plate girder

2.2. Constraints

Constraints represent some functional relationship among the design variables and other

design parameters which satisfy certain resource limitations and physical phenomena. GA is

ideally appropriate for unconstrained optimization problems. The problem used here is a

constraint optimization problem; hence it is necessary to convert the same by using exterior or

interior penalty functions. GA performs the search in parallel using populations of points in

the given search space. Traditional transformations using penalty or barrier functions are not

appropriate for this genetic algorithm. Hence, a formulation based on the violation of

normalized constraints is generally adopted. It is found to work very well for these classes of

problems. The constraints and design checks are applied as per IS 800:2007. The constraints

used in this work are,

a) Stress constraints

b) Dimensional constraints

c) Serviceability constraints

2.2.1. Stress constraints

Stress constraints consist of shear stress, flexural stress and buckling stress. In a plate girder,

shear stress is zero at the extreme fibres, increases to a high value at flange-web intersection

and attains maximum value at the neutral axis. The nature of flexural stress distribution

indicates that flanges carry most of the bending stress. Hence from the comparison of the

shear and flexural stress distributions in a plate girder, it is observed that flanges carry a major

portion of the flexural load, whereas the web carries most of the shear load.

The following requirements shall be satisfied according to IS 800: 2007in design of plate

girders.

The applied bending moment shall not exceed the moment capacity of the plate girder.

The applied shear force shall not exceed the shear buckling resistance of the plate girder.



Shear buckling resistance of web depends on two factors.

a) Depth to web thickness

b) Spacing of intermediate web stiffeners

Resistance to shear buckling shall be verified as specified, when

> 67 ε for stiffened web.

IS 800:2007 specifies minimum web thickness requirement from a

serviceability point of view with regard to provision of stiffeners. The web

thickness should satisfy the following requirements with respect to

serviceability:

When only transverse stiffeners are provided

≤ 200 εw when 3d ≥ c ≥ d

≤ 200 εw

when 0.74 d ≤ c < d

≤ 270 εw when c < d

web shall be considered as unstiffened when c > 3d

In order to avoid buckling of the compression flange into the web, the web

thickness shall satisfy the following:

When transverse stiffeners are provided and

≤ 345 εf

2 when c>l.5d

≤ 345 εf when c < 1.5d

where

c : panel width

εw : √

fyw yield stress of the web

εf : √

fyf yield stress of compression flange

2.2.2. Dimensional constraints

Dimensional constraints are used to control the proportions and size of the design variables.

The ratio between equivalent span to the full depth of plate girder is chosen such that the ratio

does not exceed 25.

For webs with only transverse stiffeners, the maximum

ratio of the web is kept below 400

in order to meet serviceability and compression flange buckling requirements.

The flange rigidity ratio should be such that the flange is either plastic or compact or semi-

compact. Hence

ratio is limited to 13.6.

The panel aspect ratio

is limited to 3 beyond which the web panel is considered as

unstiffened.



Due to headroom constraint and web buckling susceptibility, the depth of web cannot be

increased infinitely. Hence deflection limits recommended by IS 800:2007 are adopted as

constraints to check the most adverse but realistic combination of service loads.

2.2.3. Serviceability constraints

Serviceability limit state includes deflection limit and vibration limit.

Deflection limit according to Table 6 of IS 800: 2007 is adopted as:

Allowable deflection ≤ Span/300

Vibration limit according to ANNEX C of IS 800:2007 is adopted as:

Natural floor frequency

√ where

= Natural floor frequency in Hz, = Maximum deflection in mm

Natural floor frequency less then 5Hz is avoided.

The stress, dimensional and serviceability constraints in normalized form is given by

≤0

≤0

≤0 (3)

where

σc : Stress in the member σa : Allowable stress

: Calculated dimension : Allowable dimension

: Calculated deflection : Allowable deflection

Violation coefficient C is computed as,

Ci=g(x) if g(x) > 0

Ci=0 if g(x) <= 0

C= ∑ , where n is the number of constraints.

g i(x) =

g i(x) =

gi(x)=

(4)

Now the modified function Ф(x) is written as

Ф(x) = f(x) {1+KC}. (5)

2.3. Design variables

The design variables used in this work are depth of the web (d), thickness of the web (tw),

thickness of the flange (tf), breadth of the flange (bf) and panel width (c).

3. PROBLEM STATEMENT

To investigate the performance of GA on the optimum design of plate girder, a simply

supported beam example with transverse web as well as corrugated web were analysed with

program developed using MATLAB. Optimization was performed using the equations

mentioned in section 2.

The adopted model in the problem has a span of 24m and uniform load of w = 100kN/m is

applied. The elastic modulus E, Poisson ratio μ and yield stress fy are 210GPa, 0.3 and

250MPa. The values for the variables are set to vary from a lower bound to an upper bound.

The range of variation is selected such that there is no constraint violation in the final

optimum solution.



A set of results obtained for TSPG at an intermediate iteration process is shown in table 1.

This table also gives the decoded values of design variables and the plate girder weight after

the GA process.

Table 1 Design variables, weight of the plate girder and fitness value

bf (mm) tf (mm) d(mm) tw (mm) Weight (ton) Fitness

value

340 18 1100 12 4.6 8.83E-15

410 18 1400 6 2.9 7.76 E-15

390 18 1400 6 3.7 9.50 E-15

320 22 1100 12 3.8 1.26E-14

320 25 1250 10 3.6 1.05E-14

330 20 1400 12 3.5 8.11 E-15

380 20 1400 10 5.1 1.30E-14

320 22 1250 10 4.7 6.14E-15

380 20 1400 14 4.1 1.16E-14

310 22 1250 7 3.6 1.28E-14

Figure 4 Weight of the plate girder vs number of generations

After the convergence has occurred, a graph is plotted between weight of the plate girder

and number of generations (Figure 4). It is evident from the graph that convergence occurred

at 215 generations. Thus the obtained graph gives the optimum weight after convergence. For

evaluating the effectiveness of GA in arriving at an optimum design of plate girder, the GA

results are compared with conventional design according to IS 800:2007. These results are

shown in Table 2.

Table 2 Comparative analysis of plate girder weight using GA and conventional design

Span (m) 20 30

Loading (kN/m) GA(ton) Conventional

Design(ton) GA(ton)

Conventional

Design(ton)

10 1.66 2.55 2.84 3.74

15 1.84 2.83 3.14 4.13

20 2.18 3.35 3.27 4.30

25 2.31 3.55 3.49 4.59

30 2.43 3.74 3.62 4.76

35 2.65 4.08 3.74 4.92

40 2.81 4.32 3.88 5.11



A comparison between optimized weight from GA and IS 800: 2007 for varying span is

also carried out to validate the GA results.

Figure 5 Comparisons between optimized weight from GA and IS 800: 2007

From table 2 and figure 5, it can be concluded that optimized weights of the plate girder

obtained through GA are less in comparison with conventional design. Thus GA has

convincing possibilities of finding optimal solution to a problem influenced by complex

parameters.

In order to understand the independent contribution by each design variable in attaining

the optimum weight, another study using GA is carried out. Of all the design variables, the

variables which influences the optimum weight of the girder to an appreciable extend is

identified. Figure 6 shows the influence of each design variables and their ratios on optimum

weight.

Figure 6 Influence of each design variables on optimum weight

The results show that varying depth (d) and thickness (tw) of web influences more in

achieving economical weight of a plate girder. Therefore the ratio between d and tw, denoted

as web slenderness ratio (d/tw) is identified as the most influencing parameter in optimum

design of stiffened plate girder. The relation between spacing of stiffeners or folds (c) and

depth of web (d) denoted as panel aspect ratio (c/d) is the another influencing parameter.

Flange rigidity ratio which is the relation between breadth (bf) and thickness (tf) of flange is



also found to influence the design to an appreciable extend. Varying all the design variables

also influences, but this can lead to an occurrence of more noisy population in GA.

4. DESIGN CHARTS

Design charts are developed from the optimization study to outline the behaviour of these

influential design parameters on economical design of plate girders. These charts can be

utilized by structural engineers in designing economical plate girders. Design variables

common to transversely stiffened plate girder (TSPG) and corrugated web plate girder

(CWPG) are used. Design charts are developed for optimizing the following ratios:

a) Web slenderness ratio (d/tw)

b) Panel aspect ratio (c/d)

c) Flange rigidity ratio (b/tf)

4.1. Web slenderness ratio (d/tw)

Weight of the web for a TSPG is about 25%-28% of the overall weight of the girder. Since the

flanges provide most of the flexural strength, most of the steel must be concentrated in flanges

and as far as possible away from the neutral axis of the girder which consequently results in

deep, thin web. A study of relation between depth of web and thickness of flange plate is

carried out to understand the behavioural pattern.

Figure 7 Variation of web depth with flange thickness

It is observed that an increase in the web depth of the plate girder causes a decrease in the

flange plate thickness. Hence it can be understood that a deeper web reduces the flange plate

thickness thereby reducing the weight of the flanges. Flanges of a plate girder carry a major

portion of the flexural load and flange area method is often used for a quick estimation of trail

sections. Using this method, the moment resisting capacity of a plate girder is determined

from the product of depth of the girder and flange area. This relationship shown in figure 7 is

realistic since the moment capacity is dependent on depth and flange area of the girder.

According to Fatima Zohra Chalal [4], results demonstrated from the study of optimum

design of continuous plate girder with variable depth show a similar behaviour.

The depth of the plate girder for which the area of steel used is minimum, will have

minimum weight and is called optimum depth. But as the web depth increases, the thickness

of the web should also be increased to prevent lateral buckling of the web and compression

flange buckling into the web. Hence an optimum web depth to web thickness ratio need to be

identified which can minimize the total weight of stiffened girders and provide the required

buckling resistance. This result of this study is illustrated in figure 8.



Figure 8 Influence of web slenderness ratio

The results from the optimization studies indicate that web slenderness ratio between 175

and 200 is appropriate when panel aspect ratio is between 1 and 3 and web slenderness ratio

between 260 and 270 is appropriate when panel aspect ratio is less than 1 in case of TSPG. In

case of CWPG used in buildings, web slenderness ratio about 500 and more are used during

conventional design. From the present study it is observed that above web slenderness ratio of

325, the weight increases marginally. Such behaviour can be due to vertical flange buckling

for higher web slenderness ratios. For higher ratios, corrugated webs may not have adequate

strength to support flanges vertically.

4.2. Panel aspect ratio (c/d)

Web stiffeners play a critical role in achieving the ultimate capacity of the girder, even though

they contribute only about 5%-7% of the total girder weight. Higher values of aspect ratios

lead to less number of web stiffeners. But this does not reduce the overall girder weight due to

increase in flange dimensions to meet the serviceability requirements. Also as the number of

stiffeners reduces, the girder starts to undergo shear buckling. Hence the panel aspect ratio

needs to be optimized which can minimize the girder weight and prevent shear buckling. In

conventional design of stiffened plate girders, panel aspect ratio in the range of 1.2 – 1.6 is

usually chosen. Panel aspect ratio from 0.05 to 3 is adopted for this work. Beyond aspect ratio

of 3, the web panel is considered as unstiffened [IS 800: 2007, Clause 8.6.1.1.1]. In case of

CWPG, panel aspect ratio of 0.05 to 1.5 is considered.

Figure 9 Influence of panel aspect ratio



From this study of influence of panel aspect ratio on optimum weight (figure 9), it is

observed that for TSPG, the minimum panel aspect ratio of 1.1 gives an optimum weight of

the girder. Above panel aspect ratio of 1.1, the percentage reduction in optimum weight is

negligible. In a CWPG, failure in shear by local and global buckling has been usually seen.

Local buckling is predominant in course corrugations and global buckling is predominant in

dense corrugations. Considering these buckling modes within the permissible limits, optimum

girder weight is achieved for aspect ratio of 1. The percentage reduction in optimum weight is

negligible for higher values of aspect ratios above 1. Sedky et al. [11] in their work had

reported that lower values of panel aspect ratios yield higher load carrying capacities. Hence

to achieve high strength to weight ratios, these lower values of panel aspect ratios are suitable.

4.3. Flange rigidity ratio (b/tf)

Flanges are designed from the consideration of strength and rigidity. For a non-composite

plate girder, the ratio of width of the flange plate and the depth of the section is usually

adopted as 0.3 in conventional designs. Weight of the flanges of a stiffened plate girder is

55%-60% of the total girder weight. When the applied load is increased, the failure mode of a

plate girder will depend largely on the web slenderness ratio. Hence, in this work another

study is carried out to study the influence of web slenderness ratio and flange rigidity ratio on

optimum girder weight. This study is done for TSPG.

Flange rigidity ratio is represented as b/tf, where

in case of TSPG.

Figure 10 Influence of web slenderness ratio and flange rigidity ratio on optimum weight

From this study (figure 10) it is concluded that for flange rigidity ratio over 8, the weight

saving increases with increase in web slenderness ratio. Shahabian et al. [8] had earlier

reported that weight savings can be achieved for a flange rigidity ratio over 14. Thus these

observations prove to be more economical.

The moment resistance capacity of a plate girder depends on the product of breadth and

thickness of flange. Increasing the flange thickness may cause a reduction in design strength

of flanges; hence during design the flange width must be increased to provide the required

design moment resistance. Also during design the flange rigidity ratio is selected such that the

flange is plastic, compact or semi-compact to avoid local buckling before reaching the yield

stress. Considering all the above mentioned constraints, another study is carried out to

identify an optimum flange rigidity ratio which can minimize the total girder weight and



satisfy the moment constraint. The results of this study are illustrated in figures 11a and 11 b.

This study was carried out for various web slenderness ratios ranging from 220 – 270. As

weight savings can be achieved for flange rigidity ratio above 8, the lower limit for flange

rigidity ratio is taken as 8 and upper limit as 14 which are to avoid local buckling of flanges.

Figure 11 a) & b) Influence of flange rigidity ratio on optimum flange area

In this study the breadth and thickness of flanges which are sufficient to provide the

required design moment resistance are considered. It was identified that optimum flange areas

can be obtained for a flange rigidity ratio up to 13.5.

In case of CWPG, there is larger outstand at one side where web is parallel to the axis of

girder and smaller outstand on the other side. Optimum flange rigidity ratio in case of CWPG

can be identified by considering either larger or smaller outstand or average outstands. Since

IS 800:2007 does not provide guidelines for design of corrugated web plate girders, studies on

flange rigidity ratio is limited to only TSPG in this paper.

5. CONCLUSIONS

Design variables and constraints imposed by the design codes are important factors that

influence a structural design. Optimal designs of steel structures are even more complex due

to discrete sizes of steel sections in market. This paper has presented the use of Genetic

Algorithm, a robust optimization technique in solving such complex problems. Weight of

plate girder obtained using IS 800: 2007 when compared with that of GA is found to be

overestimated. Among all the design variables used for weight optimization, web slenderness

ratio (d/tw) is identified as the most influential parameter on optimum girder weight. In order

to aid a designer in effective design of stiffened plate girders both transversely stiffened and

corrugated web type, design charts were developed. These charts outline the behaviour of

influential parameters on economical design of plate girders. A study of interrelationship

between ratios of design variables reveals that for flange rigidity ratio over 8, the weight

saving increases with increase in web slenderness ratio. Suitable range of values for web

slenderness ratio, panel aspect ratio and flange rigidity ratio are attained from the design

charts for corrugated web plate girder and transversely stiffened plate girder. Thus use of GA

in optimum design of stiffened plate girders can lead to cost-effective and structurally

practicable design. The outcomes of the work are very promising as it may open a new era for

the accurate and effective use of transversely stiffened plate girders as well as corrugated web

plate girders in buildings.



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[14] IS 800-2007, Code of Practice for General Construction in Steel.

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