design of steel plate girders with very thin webs …
TRANSCRIPT
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, [email protected]
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