JOURNAL OF ENGINEERING AND APPLIED SCIENCE, VOL. 67, NO. 3, JUNE 2020, PP. 683-702

FACULTY OF ENGINEERING, CAIRO UNIVERSITY

DESIGN OF STEEL PLATE GIRDERS WITH VERY

THIN WEBS UNDER PURE BENDING

M. M. E. ABDEL-GHAFFAR1 AND A. M. ABOUSDIRA2

ABSTRACT

Unlike web-depth, web-thickness has a minor effect on the bending capacity of

plate girders. Therefore, the optimum way to build a plate girder is by reducing web-

thickness and adding to flange area, which resists bending as well as lateral and local-

buckling. However, web slenderness has an upper-limit in all design codes to prevent

vertical induced buckling of the compression flange. This upper-limit was based on only

one laboratory test performed in the 1960’s by Basler et al. This single test is believed

to be not enough to limit most design codes worldwide. For more accurate specification

of this limit, laboratory tests on ten girders with web slenderness ratios varying from

400 to 800 under pure bending were tested at Delft University. The tested girders were

loaded up to failure. Their failure modes and capacity loads were recorded. In this work,

non-linear finite element analysis was performed using ANSYS as a good Direct

Analysis (DA) tool to simulate Delft University plate-girder-tests. The results of these

tests are discussed versus the results of the finite element analysis and compared with

theoretical capacities given by international codes and other researchers. A new equation

is proposed to include the effect of web slenderness on the bending resistance of plate

girders with very thin webs.

KEYWORDS: Plate-girder, buckling, direct-analysis, advanced-design, very-thin-

webs, steel, bending-capacity.

1. INTRODUCTION

The main goal of structural engineering is to design safe and economic structures.

Application of this concept on plate girders it is found that before 1960 the bending

capacity of a plate girder was limited by the onset of local buckling of its web. The post-

buckling behavior of plate girder was investigated by many scientists [1], but they could

not exactly define its carrying capacity in bending until five full-scale-girders were

tested under pure bending [2-3]. Each girder was subjected to two tests. The plate girders

dimensions were chosen to present the different parameters affecting the bending

1 Associate Professor, Structural Engineering Department, Faculty of Engineering, Cairo University, Giza, Egypt. 2 Structural Design Engineer, VINCI Construction Co., Cairo, Egypt, a.abousdira@gmail.com

M. M. E. ABDEL-GHAFFAR AND A. M. ABOUSDIRA

684

capacity. These laboratory tests were used to determine the web contribution in the

bending capacity, the maximum web slenderness, and local and lateral buckling of the

compression flange. In 2016, a statical and statistical analysis was done based on these

tests [4] to present a new equation for the plastic instability of plate girders in bending

with sufficient accuracy for use in design.

To prevent vertical induced buckling, Basler [2-3] gave a maximum limit for the

web slenderness. This limit was based on the failure of only one test girder (G4-T2).

Because only one test is a very limited number, in 2014 additional ten tests with different

dimensions and different web slenderness ratios were conducted in Stevin II Laboratory

of the Delft University of Technology by Abspoel et al. [5]. These laboratory tests were

used to determine the mode of failure of the plate girders with very slender unstiffened

webs. In these tests, it was observed that yielding of the compression flange occurred

first, followed by vertical induced buckling during unloading.

In this research, non-linear finite element analysis was performed using ANSYS

[6] to simulate Delft University plate girder tests and the results are verified. Theoretical

capacities of these girders are also calculated by several international design codes and

research theories. All these results are discussed and compared. A new equation is then

proposed to take the effect of web slenderness on bending resistance of plate girders

with very thin unstiffened webs; where: [ℎ

𝑡𝑤>

0.42𝐸

𝐹𝑦].

2. ULTIMATE BENDING CAPACITY

Several researchers and standards specify formulas for the ultimate bending

resistance of plate girders. Commonly used formulas will be discussed in following

sections.

2.1. Basler’s Theory

Basler gave a distribution of stresses over the web by assuming that the effective

depth in the compression part of the web is equal to thirty times the web thickness, as

shown in Fig. 1. Based on this distribution, Basler proposed Eq. (1) for the ultimate

bending capacity of plate girders.

DESIGN OF STEEL PLATE GIRDERS WITH VERY ….

685

Effective cross section Stress distribution

Fig. 1. Post-buckling stress distribution of the web according to Basler et al. [2-3].

2.2. Herzog’s Theory

Herzog used the method of reducing stresses by reducing the web strength of the

compression part of the girder from Fyw to Fyw/2 , as shown in Fig. 2.

Cross-section Reduced Stress distribution

Fig. 2. Post-buckling stress distribution of the web according to Herzog [7].

Herzog proposed Eq. (2) for the ultimate bending capacity of plate girders as follows:

𝑀𝑢= 𝐾1𝐾2𝐾3 𝑀𝑢𝑜 (2)

𝑀𝑢

𝑀𝑒𝑙

= 1 − 0.0005 𝐴𝑤

𝐴𝑓

[ 𝛽𝑤 − 5.7 √𝐸

𝐹𝑦𝑤

]

(1)

M. M. E. ABDEL-GHAFFAR AND A. M. ABOUSDIRA

686

2.3. Theory of Veljkovic and Johansson

Veljkovic and Johansson [8] presented the following equation for a hybrid plate

girder with slender web based on the stress distribution given in EN 1993-1-1 [9], as

shown in Fig. 3.

Effective cross section Stress distribution

Fig. 3. Post-buckling stress distribution [9].

If the effect of hybrid plate girder is not taken into account, Eq. (3) will determine

the web slenderness effect.

𝑀𝑢

𝑀𝑒𝑙

= [ 1 − 0.1 𝐴𝑤

𝐴𝑓

(1 − 124 𝜀 𝑡𝑤

ℎ𝑤

)] (3)

2.4. AISC Equation

AISC 360-16 standard presented Eq. (4) which is based on Basler’s theory to take

into account the web slenderness effect [10].

𝑀𝑢

𝑀𝑒𝑙

= 1 − ρ

1200 + 300 ρ (

ℎ𝑐

𝑡𝑤

− 5.7 √𝐸

𝑓𝑦

)

(4)

2.5. ECP Equation

ECP 205-LRFD standard proposed Eq. (5) to take into account the web

slenderness effect [11].

DESIGN OF STEEL PLATE GIRDERS WITH VERY ….

687

𝑀𝑢

𝑀𝑒𝑙

= 1 − ρ

1200 + 300 ρ [

ℎ𝑐

𝑡𝑤

− 222

√𝐹𝑐𝑟

]

(5)

Where, 𝐹𝑐𝑟 is in ton /𝑐𝑚2.

2.6. Maximum Web Slenderness

The maximum web slenderness given in AISC360-16 and ECP205-LRFD is

based on the limit given by Basler and his team. This limit is based on the failure of

girder G4-T2 with web slenderness 388 [2]. Dimensions of this girder are as shown in

Fig. 4. Failure of that girder occurred by lateral-buckling of the compression flange.

After the ultimate load was reached the girder collapsed with an explosive sound and a

plastic hinge was formed, as shown in Fig. 5.

Fig. 4. Dimensions of panels of plate girder G4-T2 [2].

The maximum web slenderness based on Basler’s assumption is given in Eq. (6).

𝛽𝑚𝑎𝑥 = √ 𝜋2𝐸2

24(1 − 𝜇2) .

𝐴𝑤

𝐴𝑓

1

𝐹𝑦 [ 𝐹𝑦 + 𝜎𝑟 ]

(6)

M. M. E. ABDEL-GHAFFAR AND A. M. ABOUSDIRA

688

(a) G4-T1 (b) G4-T2

Fig. 5. Failure mode of girder G4-T1 and G4-T2 [2].

Few remarks may be made on the maximum web slenderness limit given by

Basler:

G4-T1 has the same web slenderness as girder G4-T2, but G4-T1 failed by lateral

buckling of the top flange as shown in Fig.5a.

For G4-T2, according to Basler: “when a yield line concentration appeared in the

top flange over the tested panel, an attempt was made to stop the straining, but the

compression flange pushed into the web of this panel”. It is obvious that yielding of

the compression flange occurred followed by vertical induced buckling.

Basler assumed that the web is simply supported by both flanges, therefore, he took

the web-buckling-length equal to web height; but this is not true. The buckling

length lies between 0.5ℎ𝑤 (if web is assumed fixed by welding to flanges) and ℎ𝑤

(if web is assumed hinged).

Basler assumed that residual stresses in the compression flange are equal to half the

value of the yield stress 𝜎𝑟 = 𝐹𝑦 / 2; while it is not a specific value.

Basler assumed that the web behaves like a column, as he neglected the effect of

longitudinal stress, which is much more than vertical induced normal stress by the

flange buckling. Therefore, checking the column buckling of the web must be

replaced by checking of plate buckling and should take into account the actual

stresses. This would increase the maximum web slenderness limitation.

DESIGN OF STEEL PLATE GIRDERS WITH VERY ….

689

Based on the previous remarks ten girder specimens were tested by Abspoel et

al. [5] to identify the maximum web slenderness limit. The nominal web slenderness for

the tested girders varied from 400 to 800.

3. LABORATORY TESTS OF ABSPOEL

Four-point load tests on ten girder specimens were conducted with nominal

thickness of 1 mm for the tested web panels. The nominal height of the webs varied

from 400 to 800mm. The span of all girders was 6 m and the tested panel length was

3m. There were 4 transverse stiffeners at end supports and point-loads. The dimensions

of top and bottom flanges were varied to take the influence of the area-ratio of web-to-

flange into account. Therefore, the flanges nominal dimensions were 50x4, 80x5, and

100x4 mm. The geometry of a typical test girder is presented in Fig. 6.

Fig. 6. Geometry of typical tested girder [5].

The dimensions of the tested plate girders are shown in Table 1. The table also

shows the web slenderness of the plate girders and the maximum web slenderness based

on Basler. It is noticed that all test specimens exceed the maximum web slenderness

𝛽𝑚𝑎𝑥 except test girder G400x50. Therefore, the expected collapse mode for these

girders is “flange induced buckling”.

The actual yield stress of the plates used to build the plate girders is illustrated in

Table 2. This yield stress is measured at stress 𝜎0.20 in the tensile test.

M. M. E. ABDEL-GHAFFAR AND A. M. ABOUSDIRA

690

Table 1. Dimensions of tested girders, web slenderness, and area ratios [5].

Girder ℎ𝑤 mm

𝑡𝑤 mm

𝑡𝑤.𝑒𝑝

mm

𝑏𝑡𝑓

mm

𝑡𝑡𝑓

mm

𝑏𝑏𝑓

mm

𝑡𝑏𝑓

mm

ℎ𝑤

𝑡𝑤

𝛽𝑀𝑎𝑥

Basler

𝐴𝑤

𝐴𝑓

400x50 400.0 1.0 4.0 49.7 4.4 49.8 4.3 396.1 443.8 1.87

400x80 (1) 399.3 1.0 4.0 80.0 5.4 80.0 5.3 399.3 343.2 0.92

400x80 (2) 399.8 1.02 4.07 80.10 5.57 79.80 5.53 392.0 347.8 0.91

400x100 399.8 1.0 4.1 80.1 5.6 79.8 5.5 392.0 347.8 0.88

600x50 601.6 1.02 3.99 49.60 4.48 49.90 4.47 589.8 583.5 2.76

600x80 400.1 0.9 4.1 98.7 4.3 98.9 4.4 430.2 315.1 1.32

600x100 601.6 1.0 4.0 49.6 4.5 49.9 4.5 589.8 583.5 1.36

800x50 600.2 1.0 4.0 79.9 5.5 79.9 5.7 618.7 402.3 3.51

800x80 600.1 1.0 4.0 99.1 4.3 98.7 4.3 618.7 394.0 1.75

800x100 801.0 1.0 4.0 50.2 4.4 49.5 4.4 825.8 662.1 1.91

Table 2. Yield strength of the elements of the tested girders [5].

Girder

400x50

400x80

(1)

400x80

(2)

400x100

600x50

600x80

600x100

800x50

800x80

800x100

Top flange 355 322 316 343 328 329 341 326 320 339

Bottom flange 319 331 315 342 309 314 344 310 317 350

Web of tested panel 288 284 278 208 240 287 286 292 296 290

3.1. Properties of the Tested Girders

Based on the material properties and geometry of the tested plate-girders as

shown in Tables 1 and 2, Table 3 shows some properties of girders cross section in the

test panel area including elastic, plastic, critical moment of resistance, and the bending

moment due to flanges only. The elastic-limit bending moment 𝑀𝑒𝑙 is calculated based

on the web initial yielding.

Table 3. Theoretical properties of tested girders.

Girder 𝑀𝑓𝑙 𝑀𝑒𝑙 𝑀𝑝𝑙 𝑀𝑐𝑟. hinged 𝑀𝑐𝑟. fixed 𝑀𝑡𝑒𝑠𝑡

400x50 27.81 36.00 40.89 3.26 5.40 32.5

400x80(1) 56.29 63.31 67.44 5.59 9.26 53.5

400x80(2) 56.35 64.23 67.69 6.01 9.96 62.0

400x100 58.74 66.84 66.66 4.79 7.93 58.2

600x50 41.77 60.12 64.95 2.53 4.20 50.2

600x80 86.02 102.19 111.69 3.85 6.39 88.5

600x100 88.03 107.01 113.10 3.72 6.16 90.6

800x50 54.01 86.42 101.74 1.89 3.13 65.3

800x80 114.47 146.38 161.02 3.14 5.21 115.0

800x100 114.13 149.20 161.92 3.12 5.17 114.7

DESIGN OF STEEL PLATE GIRDERS WITH VERY ….

691

4. FEM CALCULATIONS

4.1. Model

4.1.1. General

To simulate the theoretical bending moment capacity of Delft plate girders, three

dimensional (3D) finite element models are conducted using the finite element software

ANSYS [12]. The element type used is Shell 181 with four nodes with six degrees of

freedom at each: three rotations and three translations. Shell 181 is suitable for analyzing

thin shell structures. It is also well-suited for large rotation, and/or large strain nonlinear

applications [6]. Therefore, it is suitable to be used in the nonlinear analysis of plate

girders.

4.1.2. Geometry of the FEM

Using ANSYS, only half of the plate girder is modeled by the program as shown

in Fig. 6, due to the full symmetry about mid span.

4.1.3. Material properties used in FEM

The material properties used in the FEM-model are shown in Table 2. The

modulus of elasticity E= 210000 MPa and Poisson’s ratio μ = 0.3. Residual stresses

over web and flanges are based on the Dutch Code [13]; as shown in

Fig. 7.

Web Flanges

Fig. 7. Residual stresses in flanges and web [13].

M. M. E. ABDEL-GHAFFAR AND A. M. ABOUSDIRA

692

Unfortunately, ANSYS accepts only uniform residual stresses. Therefore, a self-

equilibrating re-distribution is used for the residual stresses, as shown in Fig. 8 [12].

Web Flanges

Fig. 8. Residual stresses distribution used in ANSYS [12].

4.1.4. Results of the FEM

Table 4 shows the load capacity and deflection of the plate girders using FEM

[12].

Table 4. Load capacity and deflection of FEM [12].

Tested girder

400x50

400x80

(1)

400x80

(2)

400x100

600x50

600x80

600x100

800x50

800x80

800x100

Load Capacity, kN 44.9 76.4 80.3 76.4 68.5 120.7 115.0 99.0 163.8 152.5

Deflection, mm 26.9 26.4 27.0 27.0 16.2 16.8 16.7 13.2 12.3 12.0

5. COMPARISON BETWEEN TEST AND FEM RESULTS

The results of the laboratory tests [5] and the FEM results [12] are given in Table

5 and in

Fig. 9. It is noticed that Abousdira FEM [12] is closer to test results than

Abspoel's FEM except for girders 400x50, 600x50, and 800x50. The ratio between

Abousdira FEM results and test results vary from 95.17% (girder 600x100) to 113.62%

(girder 800x50). It is also noticed that the stiffness of the FEM is larger than the actual

DESIGN OF STEEL PLATE GIRDERS WITH VERY ….

693

stiffness of the tested girders, which is caused by the influence of the residual stresses

of the scaled test girders with webs of 1 mm only and the influence of the rather big

geometrical imperfections.

Table5. Summary of capacity loads and deflections of delft girders [5, 12].

Test girder Experimental [5] Abspoel FEM results [5] Abousdira FEM results [12]

Capacity Defl. Capacity Defl. FEM/test % Capacity Defl. FEM/test %

400x50 43.36 27.14 42.96 32.70 99.08 44.86 26.94 103.46

400x80(1) 71.35 32.00 78.70 28.09 110.30 76.406 26.38 107.09

400x80(2) 82.73 41.00 80 29.00 96.70 80.336 27.03 97.11

400x100 77.59 37.00 80.54 26.50 103.80 76.38 26.97 98.44

600x50 66.89 28.00 65.74 19.15 98.28 68.49 16.235 102.39

600x80 118.02 22.50 122.95 21.50 104.18 120.65 16.75 102.23

600x100 120.82 23.65 113.20 15.60 93.69 114.98 16.69 95.17

800x50 87.09 21.50 89.07 17.1 102.27 98.95 13.15 113.62

800x80 153.28 19.10 164.40 21.00 107.25 163.75 12.27 106.83

800x100 152.96 18.00 141.50 10.70 92.51 152.49 12.01 99.69

Fig. 9. Comparison between Experimental and FEM Load-Capacity (kN).

5.1. Collapse Modes

As listed in Table 1, nine out of ten plate girders that were tested by Abspoel had

web slenderness ratios more than the maximum web slenderness allowed by Basler.

Therefore, flange induced buckling was the expected collapse mode. Abspoel described

that the failure mode of those girders was mainly due to yielding of the compression

flange and the vertical induced buckling in the unloading part of the load-displacement

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

160.0

180.0

400x50 400x80(1) 400x80(2) 400x100 600x50 600x80 600x100 800x50 800x80 800x100

Cap

acit

y [k

N]

Tested Girders

Experimental Abspoel FEM Abousdira FEM

M. M. E. ABDEL-GHAFFAR AND A. M. ABOUSDIRA

694

curve (similar to Basler description for G2-T2). Figure 10 shows the failure mode of

girders 400x50 and 400x80(1) in laboratory [5]. Figure 11 shows the failure mode of

the same girders in the FEM [12].

Girder 400x50 Girder 400x80(1)

Fig. 10. Failure mode pictures for girders 400x50 and 400x80(1) from Lab. [5].

Girder 400x50 Girder 400x80(1)

Fig. 11. Failure mode for girders 400x50 and 400x80(1) from FEM [12].

5.2. Comparison between Test Results and Theoretical Resistance

Tables 6 and 7 together with Fig. 12 show a comparison between test results and

theoretical resistances of the tested plate girders. Theoretical resistances are based on

Basler, Herzog, Veljkovic, AISC360-16, and ECP205-LRFD. Note that maximum web

slenderness limits in design codes are not listed. From the above comparison, the

following can be pointed out:

The reduction factors related to web slenderness given by AISC360 and ECP205-

LRFD which are based on Basler’s theory are very conservative. This reduction

DESIGN OF STEEL PLATE GIRDERS WITH VERY ….

695

increases significantly by increasing both web slenderness and the web-to-flange area

ratio.

For girders with lower web slenderness, the capacities given by theories and codes

are closer to the test capacity.

Bending capacities calculated by Veljkovic are closer to the tests capacities than other

theories, in spite of overestimating 6 out of 10 tests.

Table 6. Reduction factor (𝑀𝑢/𝑀𝑒𝑙) due to web slenderness for theories and codes.

Tested girder

40

0x

50

40

0x

80

(1)

40

0x

80

(2)

40

0x

10

0

60

0x

50

60

0x

80

60

0x

10

0

80

0x

50

80

0x

80

80

0x

10

0

Basler 0.76 0.88 0.89 0.87 0.38 0.69 0.67 - 0.42 0.37

Herzog 0.97 0.97 0.97 0.96 0.88 0.86 0.86 0.76 0.76 0.77

Veljkovic 0.86 0.93 0.93 0.93 0.77 0.89 0.89 0.69 0.85 0.83

AISC 360-16 0.73 0.84 0.85 0.83 0.39 0.61 0.60 - 0.32 0.29

ECP205-LRFD 0.71 0.83 0.83 0.82 0.36 0.59 0.59 - 0.30 0.28

Table 7. Comparison between test results and theoretical results.

Test

girder

Test Basler Herzog Veljkovic AISC360-16 ECP205-

LRFD

Cap

acit

y,

kN

Cap

acit

y,

kN

Bas

ler/

tes

t %

Cap

acit

y,

kN

Her

zog/

test

%

Cap

acit

y,

kN

Vel

jkovic

/ t

est

%

Cap

acit

y,

kN

AIS

C/

test

%

Cap

acit

y,

kN

EC

P/

test

%

400x50 43.4 40.7 93.8 39.2 90.3 40.8 94.0 38.9 89.7 37.3 86.0

400x80(1) 71.4 74.3 104.2 74.9 104.9 78.3 110.0 71.2 99.7 69.8 97.8

400x80(2) 82.7 76.3 92.3 75.3 91.1 80.0 97.0 72.9 88.1 71.5 86.5

400x100 77.6 62.9 81.1 63.8 82.3 82.9 107.0 67.6 87.1 58.3 75.1

600x50 66.9 32.8 49.1 52.8 78.9 62.0 93.0 33.6 50.2 30.1 45.0

600x80 118.0 98.1 83.1 102.5 86.8 119.3 101.0 86.9 73.7 84.0 71.2

600x100 120.8 77.7 64.3 87.3 72.2 124.2 103.0 77.8 64.4 67.7 56.0

800x50 87.1 0.0 0.0 60.6 69.5 79.6 91.0 0.0 0.0 0.0 0.0

800x80 153.3 82.0 53.5 121.8 79.4 164.6 107.0 63.5 41.4 59.0 38.5

800x100 153.0 59.0 38.6 103.6 67.7 165.2 108.0 52.8 34.5 44.7 29.3

M. M. E. ABDEL-GHAFFAR AND A. M. ABOUSDIRA

696

Fig. 12. Comparison between experimental and theoretical capacities.

6. PROPOSED EQUATION

Based on the previous discussions and results of tests and FEM, Abousdira [12]

presented the following equations (8-12) to calculate bending capacity of plate girders

with very thin webs. The same post-buckling stress distribution given in Fig. 2 is

adopted; and a doubly-symmetric non-hybrid cross-section is assumed.

𝑀𝑢 = 𝜂𝑠 𝑀𝑢𝑜 (8)

𝑀𝑢𝑜 can be calculated as:

𝑀𝑢𝑜 = 𝐴𝑓 . 𝑓𝑦𝑓. (ℎ𝑤 + 𝑡𝑓) +

1

9. 𝐴𝑤 . ℎ𝑤 . 𝑓𝑦𝑤

(9)

While the elastic moment M𝑒𝑙 can be calculated as:

Mel = 𝐴𝑓 . 𝑓𝑦𝑓. (ℎ𝑤 + 𝑡𝑓) +

1

6. 𝐴𝑤 . ℎ𝑤 . 𝑓𝑦𝑤

(10)

By assuming that (ℎ𝑤 + 𝑡𝑓) ≃ ℎ𝑤 , 𝑓𝑦𝑓= 𝑓𝑦𝑤 (non-hybrid section) and, hence:

𝑀𝑢𝑜

𝑀𝑒𝑙

=𝐴𝑓 . 𝑓𝑦𝑓

. ℎ𝑤 +19

. 𝐴𝑤 . ℎ𝑤 . 𝑓𝑦𝑤

𝐴𝑓 . 𝑓𝑦𝑓. ℎ𝑤 +

16

. 𝐴𝑤 . ℎ𝑤 . 𝑓𝑦𝑤

=

1 +19

.𝐴𝑤

𝐴𝑓

1 +16

.𝐴𝑤

𝐴𝑓

(11)

Therefore, the bending capacity can be calculated as:

0

20

40

60

80

100

120

140

160

180

400x50 400x80(1) 400x80(2) 400x100 600x50 600x80 600x100 800x50 800x80 800x100

Cap

acit

y [k

N]

Tested girders

Experimental Basler Herzog AISC 360 ECP-205 VeljecovicVeljkovic

DESIGN OF STEEL PLATE GIRDERS WITH VERY ….

697

𝑀𝑢

𝑀𝑒𝑙

= (1.17 − √βw

100 )

1 + 𝐴𝑤

9 𝐴𝑓

1 + 𝐴𝑤

6 𝐴𝑓

(12)

6.1. Comparison between the Proposed Equation and Test Results

Figure 13 and Table 8 show a comparison between test results [5] and theoretical

resistances of the tested plate girders predicted by the proposed Eq. (12) above.

Fig. 13. Test capacity vs. theoretical capacity based on the proposed equation.

Table 8. Comparison between test capacity [5] and theoretical capacity [12].

Tested girder

400x50

400x80

(1)

400x80

(2)

400x100

600x50

600x80

600x100

800x50

800x80

800x100

Test, kN 43.4 71.4 82.7 77.6 66.9 118.0 120.8 87.1 153.3 153.0

Abousdira, kN 39.1 74.8 75.2 77.2 55.9 109.7 112.0 70.6 141.3 144.0

Ratio % 90.2 104.9 90.9 99.5 83.6 92.9 92.7 81.1 92.2 94.1

6.2. The Proposed Equation versus Other Theories and Design Codes

Figures 14 a to d show the relation between web slenderness and the reduction

factor in bending capacity with different values of web-to-flange areas ( 𝐴𝑤

𝐴𝑓 ) according

to common theories and codes.

0

20

40

60

80

100

120

140

160

180

400x50 400x80(1) 400x80(2) 400x100 600x50 600x80 600x100 800x50 800x80 800x100

Cap

acit

y [K

N]

Tested Girders

Test Abousdira Proposed Equation

M. M. E. ABDEL-GHAFFAR AND A. M. ABOUSDIRA

698

Fig. 14 a. Bending capacity vs. web slenderness for ( 𝐴𝑤

𝐴𝑓 ) = 0.50.

Fig. 14 b. Bending capacity vs. web slenderness for ( 𝐴𝑤

𝐴𝑓 ) = 1.0.

DESIGN OF STEEL PLATE GIRDERS WITH VERY ….

699

Fig. 14 c. Bending capacity vs. web slenderness for ( 𝐴𝑤

𝐴𝑓 ) = 1.5.

Fig. 14 d. Bending capacity vs. web slenderness for ( 𝐴𝑤

𝐴𝑓 ) = 2.0.

M. M. E. ABDEL-GHAFFAR AND A. M. ABOUSDIRA

700

7. CONCLUSIONS

From the previous discussion, the following conclusions may be drawn:

1. For very thin webs, yielding of the compression flange occurs first; then the web

bends out of its plane. Afterwards, the compression flange presses into the web

causing “flange induced buckling”.

2. The maximum permitted web slenderness 𝛽𝑚𝑎𝑥 shall be raised to 800, which is more

than twice of the limit permitted by most design codes.

3. Direct analysis (DA) is highly recommended for design of plate-girders with very

thin webs. Therefore, DA shall be allowed-for in the new Egyptian Design Code.

4. The reduction factor related to very thin webs given by AISC360 and by ECP205-

LRFD is very conservative. This factor decreases significantly by increasing both

web slenderness and the ratio of web-to-flange areas.

5. Bending capacities calculated based on Veljkovic [8] theory are close to the test

capacities. However, his theory gives resistance more than the experimental results

in few cases (max. of 8% more in 6 of the 10 tests).

6. The reduction factor given by our proposed equation is suitable for plate girders with

both slender and very slender webs.

7. Bending capacities calculated based on our proposed equation are closer to the test

capacities (max. of 19% less in 9 out of 10 tests; and max. of 5% more in one test

only) if compared with other theories and design codes.

DECLARATION OF CONFLICT OF INTERESTS

The authors have declared no conflict of interests.

REFERENCES

1. Abu-hamd, M. and Elmahdy, G., “The Effective Width of Slender Plate Elements

in Plate Girders”, Journal of Engineering and Applied Science, Vol. 50, No. 2, pp.

259–278, 2003.

2. Basler, K., and Thürlimann, B., “Strength of Plate Girders in Bending”,

Bethlehem, Pennsylvania: Fritz Engineering Laboratory, Lehigh University, 1960.

3. Basler, K., Yen, B., Muller, J., and Thürlimann, B., “Web Buckling Tests on

Welded Plate Girders, Part 2: Tests on Plate Girders Subjected to Bending”,

Bethlehem, Pennsylvania: Fritz Engineering Laboratory, Lehigh University,1960.

DESIGN OF STEEL PLATE GIRDERS WITH VERY ….

701

4. Hanna, M. S., and Ghaffar, M. A., “Buckling and Post Buckling of Plate Girders

in Bending”, Welding Research Counsil Inc., WRC Bulletin 557, 2016.

5. Abspoel, R., and Bijlaard, F., “Optimization of Plate Girders”, Steel Construction,

Vol. 7, pp. 116-125, 2014.

6. ANSYS Finite Element Analysis Software “User’s Guide”, V16.0, 2015.

7. Herzog, M., “Die Traglast Versteifter, Dunwandiger Blechträger”, Der

Bauingenieur, No. 48, 1958.

8. Veljkovic, M., Johansson, B., “Design of Hybrid Steel Girders”, Journal of

Constructional Steel Research,Vol. 60, pp. 535-547, 2004.

9. EN 1993-1-1, “Eurocode 3 : Design of Steel Structures - Part 1-1: General Rules

and Rules for Buildings”, 2006.

10. AISC 360-16. "Specification for Structural Steel Buildings”, American National

Standard, 2016.

11. ECP 205-LRFD, “Egyptian Code of Practice for Steel Construction (LRFD)”,

Egypt: Housing and Building National Research Center, 2008.

12. Abousdira, A., “Design of Steel Plate Girders With Very Thin Webs Under Pure

Bending”, M. Sc. thesis, Faculty of Engineering, Cairo University, 2019.

13. NEN6771, “Technische Grondslagen Voor Bouwconstructies F- TGB 1990 -

Staalconstructies-Stabiliteit (in Dutch)”, 2002.

LIST OF SYMBOLS

Symbol Discription Unit

𝐴𝑓 Area of flange 𝑚𝑚2

𝐴𝑤 Area of web 𝑚𝑚2 DA Direct Analysis -

E Steel modulus of elasticity MPa

𝐹𝑦 Yield strength MPa

𝐹𝑐𝑟 Critical stress of the compression flange according to ECP205 MPa

𝐹𝑦𝑓 Yield strength of flange MPa

𝐹𝑦𝑤 Yield strength of web MPa

𝐹𝐸𝑀 Finite Element Method -

𝐾1 The effect of local buckling of the compression flange [7] -

𝐾2 The effect of lateral buckling of the compression flange [7] -

𝐾3 The effect of web slenderness [7] -

𝑀𝑢 Ultimate bending moment resistance kN.m

Muo The effective bending moment resistance according to stress

distribution (Fig.2) kN.m

𝑀𝑐𝑟 Critical elastic bending moment kN.m

𝑀𝑒𝑙 Elastic bending moment resistance (My) kN.m

𝑀𝑓𝑙 Bending moment resistance of a cross-section consisting of the

flanges only kN.m

𝑀𝑝𝑙 Plastic bending moment resistance kN.m

M. M. E. ABDEL-GHAFFAR AND A. M. ABOUSDIRA

702

𝑏𝑏𝑓 Bottom flange width mm

𝑏𝑡𝑓 Top flange width mm

ℎ Depth of the cross-section mm

ℎ𝑤 Web depth between the flanges mm

ℎ𝑐 Twice the compression part of the web mm

𝑡𝑏𝑓 Bottom flange thickness mm

𝑡𝑡𝑓 Top flange thickness mm

𝑡𝑤 Web thickness mm

𝑡𝑤. 𝑒𝑝 Web thickness of an end panel of a plate girder mm

𝑡𝑤. 𝑡𝑝 Web thickness of a test panel of a plate girder Mm

𝛽𝑤 Web slenderness [ℎ𝑤

𝑡𝑤] -

𝛽𝑚𝑎𝑥 Maximum web slenderness [ℎ𝑤

𝑡𝑤]max -

𝜇 Poisson’s ratio in elastic stage -

𝜌 Ratio between areas of web and compressive flange (Aw/Af) -

𝜂𝑠 Reduction factor due to web slenderness (1.17-√𝛽𝑤

100 ≤ 1) -

𝜎𝑟 Residual stress MPa

𝜀 Factor according to Eurocode (235

𝐹𝑦)

0.5

-

تصميم الكمرات اللوحية الحديدية ذات العصب النحيف جدا تحت تأثير عزوم انحناء خالصة

م تحميل تلنسبه عمق العصب للكمرات الحديدية الى سمكه حيث لحد الأقصىلدراسة يقدم البحث ائج تم عرض ومناقشة نت، والهبوط الكمرات حتى الانهيار وتسجيل شكل الانهيار وقيمة الحمل الأقصى

النظرية والمقاومة القصوى ا بنتائج طريقة العناصر المحدودة ميمة والحديثة وعلاقتهدالتجارب المعملية الق ةمقاوم ىاقتراح معادلة جديدة لمقدار التخفيض فو عن طريق بعض النظريات والمواصفات العالمية ةالمعطا

مح به أكثر مما تس ب النحيف جدااعصالعزوم المتعلق بنحافة العصب للكمرات اللوحية الحديدية ذات الأ .الأكواد الحالية