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DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE
EGYPTIAN CODE OF PRACTICE: A PROPOSED SIMPLE EQUATION
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DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
Al Azhar University Engineering Journal
Vol. 7, No. 5, December 2004, 1043-1063
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS
ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
Ezzeldin Yazeed Sayed-Ahmed
Ain Shams University, Faculty of Engineering, Structural Engineering Dept., Cairo, Egypt.
(on leave to University of Qatar, Civil Engineering Dept., P.O. Box 2713, Doha, Qatar)
‫ وغير المثبتة جانبيا بتقنية اإلجيادات المسموح بيا أو بتقنية معامالت الحمل‬I ‫يتطمب تصميم الكمرات المعدنية بشكل‬
‫والمقاومة إستخدام معادالت متعددة تعتمد عمى رتبة "إنضغاطية" قطاع الكمرة وعمى طول الكمرة غير المثبت جانبيا وعمى‬
‫ وباإلضافة إلى ذلك فإن اإلنبعاج المحمي وانبعاج المي‬.‫الخصائص اليندسية لقطاع الكمرة وعمى إجياد الخضوع لمحديد‬
‫ ويعتمد اإلنبعاج المحمي عمى النسبة بين عرض‬،‫اإلنحنائي الجانبي يؤثران أيضا عمى مقاومة ىذه الكمرات لإلنحناء‬
‫الفالنجة الحر إلى تخانتيا والنسبة بين إرتفاع العصب إلى تخانتو وتعرف ىاتان النسبتان أيضا إنضغاطية أو رتبة قطاع‬
‫ وعمى الجانب األخر فإن الطول الجانبي الحر من الكمرة يؤثر في قيمة العزم الحرج الذي يسبب إنبعاج المي‬.‫الكمرة‬
‫ وتعرف معظم مواصفات التصميم ثالثة نطاقات لتصميم الكمرات المعدنية غير المثبتة جانبيا حيث‬،‫اإلنحنائي الجانبي‬
‫تتأثر مق اومة الكمرة في النطاق األول بإنبعاج المي اإلنحنائي الجانبي المرن وفي الثاني بإنبعاج المي اإلنحنائي الجانبي‬
‫ ويقدم ىذا البحث معادلة واحدة لتعريف‬.‫المدن بينما التتأثر ىذه المقاومة بإنبعاج المي اإلنحنائي الجانبي في النطاق الثالث‬
‫ وغير المثبتة جانبيا حيث تأخذ ىذه المعادلة في اإلعتبار‬I ‫إجيادات اإلنحناء المسموح بيا لتصميم الكمرات المعدنية بشكل‬
‫ وقد تمت مقارنة المعادلة المقترحة مع طريقة التصميم المتبعة في المواصفات المصرية لتشييد‬،‫كل العوامل السابق ذكرىا‬
‫المنشآت المعدنية والكباري – طريقة اإلجيادات المسموح بيا – ويوصي البحث بإستخدام ىذه المعادلة كبديل لممعادالت‬
.‫المستخدمة حاليا في ىذه المواصفات‬
ABSTRACT
Design of laterally unsupported steel I-section beams according to ASD (Allowable Stress Design) and
LRFD (Load and Resistance Factor Design) techniques requires the use of multiple equations. These
equations depend on the section compactness, the laterally unsupported length of the beam, the
geometric properties of the cross section and the yield strength of the steel. Furthermore, local
buckling and lateral torsion-flexure buckling significantly affect the behaviour of steel I-section beams.
Flange outstand-to-thickness and web height-to-thickness ratios define the I-section compactness and
hence control the local buckling behaviour. The laterally unsupported length of the beam on the other
hand, affects the critical moment initiating lateral flexure-torsion buckling. According to most codes of
practice, three distinct zones are established for the behaviour of laterally unsupported steel beams; the
moment resistance or the allowable bending stress is defined by a different equation in each zone. The
first zone defines the elastic lateral torsion-flexure buckling behaviour while the second defines the
1043
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
elasto-plastic lateral torsion-flexure buckling behaviour. The third zone is not affected by lateral
buckling and failure is controlled by steel yielding. The ECPSCB-ASD (Egyptian Code of Practice for
Steel Construction and Bridges – Allowable Stress Design) even treats the warping and the torsion
resistances of the beam’s cross section separately. In this paper, a single equation which defines the
allowable bending stress for laterally unsupported steel I-section beams is proposed to cover all these
zones. The proposed equation results are compared to those obtained using the design provisions of
the ECPSCB (ASD).
The equation is proposed to replace the discontinuous definitions currently adopted by the ECPSCB
(ASD) defining the allowable bending stress for laterally unsupported steel I-section beams.
Key words: Allowable Bending Stress, Lateral Torsion-Flexure Buckling, Local
Buckling, Moment Resistance, Plastic Moment, Yield Moment, Steel I-beams.
1. INTRODUCTION
In Load and Resistance Factor Design (LRFD) technique, the factored nominal resistance of the
structural member is selected so that it equals or exceeds the factored load effects. The load and
resistance factors are derived such that the probability of failure of the whole structure would be under
an acceptable limit. On the other hand, the Allowable Stress Design (ASD) technique is based on a
different concept; no load factors are applied and service loads are considered.
The stresses resulting from the service loads acting in each member must not exceed the permissible
(allowable) stress for the considered loading effect. The allowable stress is the ultimate strength of this
member divided by a certain factor of safety. In both design techniques, all the possible modes of
failure must be considered in design of any member.
I-sections steel beams may fail in different failure modes (Figure 1): local buckling of the web or the
compression flange, lateral torsion-flexure buckling of the section and/or steel yielding [1-4]. Local
buckling is mainly affected by the flange outstand-to-thickness and the web-to-thickness ratios. These
two ratios define the “compactness” of the steel I-sections. Codes of practice [e.g. 5-7] place limits on
these ratios such that the critical stress initiating local buckling would not be reached before the yield
stress is reached at selected locations.
AUEJ, 7, 5,2004
1044
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
y
y
z
z
Lateral torsion-flexure buckling
x
Local flange buckling
Local web buckling
Figure 1. Local and lateral torsion-flexure buckling of a steel I-section.
Lateral torsion-flexure buckling represents another major design consideration for steel I-beams. It is
affected by the unsupported length of the I-section’s compression flange, the type of loading, the
geometric properties of the beam’s cross section and the boundary conditions of the beam. When a
steel I-beam is subjected to flexure about its axis of greatest flexural rigidity with insufficient lateral
bracing, out-of-plane bending and twisting occur as the applied load reaches its critical value. At this
critical value of the load, in-plane bending deformation ceases to be a stable configuration for the
beam and lateral buckling takes place. For design purposes, the critical moment causing lateral
torsion-flexure buckling for a simply supported beam subjected to uniform bending is determined.
Then, it is modified using an “equivalent moment factor” which depends on the type of loading. Steel
I-section beams have two components resisting lateral buckling: warping restraint resistance and
torsion resistance. The total resistance of the beam to lateral buckling is obtained by a vector
summation of the two components [4,6,7]. However, the ECPSCB (ASD) specifications [5] treat the
two components separately and consider the larger of the two components to be the resistance of the
beam to the lateral buckling.
Codes of practice often deal with design of steel I-beams by first classifying the section compactness
(class). Then, the allowable bending stress in case of adopting the ASD technique (or the beam
moment resistance in case of using LRFD technique) is evaluated taking into account the lateral
torsion-flexure buckling behaviour of the beam.
1045
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DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
2. CRITICAL MOMENT INITIATING LATERAL BUCKLING
Lateral torsion-flexure buckling has traditionally been approached by assuming that the beam’s crosssection does not distort during buckling [2]. The beam is also assumed to be geometrically perfect and
undergoes only a small deflection when subject to in-plane bending.
Referring to Figure 2, for a simply supported I-beam subjected to two equal and opposite end moments,
the out-of-plane equilibrium condition is [2,4]:
ECw
d 4
d 2 M o2

GJ

 0
dx 4
dx 2 EI y
(1)
y
Mo
z
C.L.
Mo
C.L.
x
y
w
z

Figure 2. Lateral torsion-flexure buckling of a steel I-section beam.
where Mo is the applied end moment, Iy is the cross-section’s second moment of area about the y-y
axis, J is the torsional constant, Cw is the warping constant of the section,  is the angle of twist, E is
Young’s modulus and G is the shear modulus. The torsional and warping constants of an I-section are
defined by:
n
J 
 bi t i3
i 1
3
CW 
I f h2
(2)
2
where b is the larger dimension, t is the smaller dimension of the plates making up the section, h is
distance between the centroid of the compression and the tension flanges and If is the second moment
of area of the compression flange about the y-y axis.
2.1 Critical Moment of I-Sections with Two Axes of Symmetry
Solution of Equation 1 yields the critical moment for a simply supported beam subjected to two equal
and opposite end moments which is:
AUEJ, 7, 5,2004
1046
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
M ocr 

L
EI y GJ ( 1  Wr2 )
WR 

L
ECw
GJ
(3)
where WR represents the warping restraint contribution to the beam’s resistance. The first term under
the root of Equation 3 represent the torsion resistance of the beam’s section while the second one
represents the resistance of the section to warping.
For beams subjected to other types of loading, the effect of the moment gradient on the critical
moment can be accounted for by the use of an “equivalent moment factor” Cb [8,9]. This concept has
been adopted in design by most codes of practice. The critical moment is generalized to be:
M cr  Cb M ocr  Cb
 E 
EI y GJ  
 I y Cw
L
 L 

2
(4)
For beams subjected to unequal end moments (MA and MB), the equivalent moment factor may be
given by:
Cb  1.75  1.05( M A / M B )  0.3( M A / M B )2
 2.3
(5)
where MA is the smaller moment and the ratio MA/MB is positive for beams bent in double curvature
and negative for beams bent in single curvature. AISC-LRFD [6] defines another general equation for
the equivalent moment factor:
Cb 
12.5 M max
3 M 1  4 M 2  3 M 3  2.5 M max
(6)
where M1, M2, M3 are the absolute values of the moments at the quarter point, midpoint and threequarter point of the beam, respectively and Mmax is the maximum moment acting on the beam.
2.2 Critical Moment for I-Sections with One Axis of Symmetry
Steel I-sections with a single axis of symmetry are used frequently in steel construction (e.g. for crane
track girders). The critical moment initiating lateral torsion flexure buckling for steel I-section beams
with one axis of symmetry is:

M cr  Cb M ocr

2
 EI y 
  EI y 

 E 
 
 Cb
EI y GJ  
 I y C w  

L

2
 L 

L

L


(7)

t w y 23 
t y 3 
1  
2
  y1  I y1  b1t1 y12  w 1 
  yo 
 y 2  I y 2  b2 t 2 y 2 

2 I z  
4 
4 

where yo is the distance between the centre of gravity of the section and its shear centre (Figure 3) and
the subscripts 1 and 2 refers to the upper and the lower flanges respectively.
1047
AUEJ, 7, 5,2004
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
y
b1 x t1
Shear centre
y1
yo
z
z
C.G.
y2
hw x tw
b2 x t2
y
Figure 3. Steel I-section with one axis of symmetry.
3. LOCAL BUCKLING OF STEEL I-BEAMS
When, a simply supported steel I-beam web is subjected to flexure about its major axis of inertia, the
stress distribution along the cross section changes with increasing applied moment as shown in Figure
4. The yield moment My and the plastic moment Mp of the cross section are defined by:
M y  S x FY
M P  Z x Fy
(8)
where Sx and Zx are the elastic and plastic section modulii around the major axis of inertia and Fy is the
yield strength of the steel.
The mid-span deflection of the beam is plotted versus the maximum moment in Figure 5. The upper
curve in this figure represents the ideal response of a beam suffering no local buckling of its web or
flanges. In this case, the cross section can develop the fully plastic moment Mp and has a high rotation
capacity: this is classified as a Class 1 compact section [7]. The second class of sections can also
develop the fully plastic moment but has less rotation capacity (Figure 5) and is known as a Class 2
compact section [7]. The ECPSCB (ASD) [5] and the AISC-LRFD [6] defines these two classes as
one class: compact section class.
For an I-section which has a slender flange or web, local buckling may take place in the flange or the
web before the fully plastic moment is developed. If local buckling occurs after developing the yield
moment, then the cross section is classified as a Class 3 non-compact section [7]; the AISC-LRD [6]
and the ECPSCB (ASD) [5] define this class as non-compact sections. On the other hand, if the
section cannot even develop the yield moment and suffers local buckling before reaching My, then it is
classified as a Class 4 or slender section [5-7]. The capacity of a slender section is based on the
mechanical properties of a virtual “effective” section which is determined based on the effective width
concept [10]: the elastic section modulus Sx of Equation 8 is replaced by the effective section elastic
modulus Se [11].
AUEJ, 7, 5,2004
1048
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
F<Fy
bfl x tfl
Fy
-
Fy
Fy
-
-
-
hw x tw
+
bfl x tfl
F<Fy
M<My
+
+
+
Fy
M=My
Fy
My<M<Mp
Fy
M=Mp
“Yield moment”
“Plastic moment”
Figure 4. Stress distribution across the I-section beam subject to increasing flexure.
M
M
M
MP
My
Class 3
Non-compact Section
Class 1
Class 2
Compact Section
F< Fy
Class 4
Fy
Fy
Fy
Slender Section
F<Fy
Fy
Fy
Fy
Class 1
Class 3
Class 2
Non-compact
Compact
Section
Section
Class 4
Slender
Section

Figure 5. Effect of local buckling on the moment capacity of steel I-section beams and
classification of cross section.
As mentioned earlier, plate component elements of an I-section (flanges and web) may buckle locally.
For plates in compression, the basic elastic buckling equation is:
1049
AUEJ, 7, 5,2004
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
Fcr  k
 2E
12( 1  2 )( b / t )2
(9)
where Fcr is the critical stress initiating buckling, b/t is the width to thickness ratio of the plate,  is
Poisson’s ratio, E is the elasticity modulus and k is the buckling coefficient which depends on the type
of stress, the plate end conditions and the plate length-to-width ratio. Figure 6 lists values for the
buckling coefficient k for different plate end conditions and shows an idealization for both the flanges
and the web of an I-section.
Fixed
Fixed
Simply supported
kmin=6.97
kmin =5.42
kmin =4.00
Fixed
Simply supported
Simply supported
Fixed
Simply supported
Simply supported
kmin =1.277
kmin =0.425
Free
kmin =23.9
Free
bfl
Simply supported
Web stiffened edge
bfl/2
tfl
bfl/2
Free edge
hw
Flange
outstand
Flange stiffened edge
tw
hw
Web
Flange stiffened edge
Figure 6. Buckling coefficients of long plates for different end and loading conditions
(above) and an idealization for both the flanges and the web of an I-section (below).
AUEJ, 7, 5,2004
1050
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
Codes of practice [e.g. 5-7] prevent local buckle to occur before the extreme fibre of the beam’s cross
section reaches the yield stress. Hence, satisfying this limit and substituting in Equation 9 for E and 
result-in:
Fcr 
180762k
( b / t )2
( MPa )  Fy
Fcr 
1843k
( b / t )2
(
ton
)  Fy
cm 2
(10)
where Fy is the yield strength of the steel. Thus, the b/t limit of a non-compact section is defined by:
180762k
Fy ( MPa )
b

t
b

t
1843 k
Fy ( ton / cm 2 )
(11)
For I-section flanges, b/t represents the flange outstand-to-thickness ratio (bfl/2tfl). The actual flange
outstand should be marginally less than this value by the sum of half the web thickness and the flangeto-web weld size (welded I-sections) or the flange-to-web fillet (rolled I-sections). The buckling
coefficient k may be taken as 0.425 simulating a plate which is supported from three sides (Figure 6).
To account for the effect of the residual stress, the flange outstand-to-thickness ratio may be reduced
by about 30% which results-in:
b fl
2t fl
b fl
194

2t fl
Fy ( MPa )

1921
(12a)
Fy ( ton / cm 2 )
On the other hand, for an I-section web, b/t of Equation 11 represents the web height-to-thickness ratio
(hw/tw) and the buckling coefficient k is taken as 23.9 (Figure 6) which results-in:
hw

tw
hw

tw
2078
Fy ( MPa )
65600
(12b)
Fy ( ton / cm 2 )
The flange outstand-to-thickness and the web height-to-thickness ratios defining the class of I-section
beams subjected to flexure according to the CAN/CSA-S16 specifications [7] are given by:
Class 1 :
Class 2 :
Class 3 :
b fl
2t fl
b fl
2t fl
b fl
2t fl

145

170

200
hw 1100

tw
Fy
Fy
hw 1700

tw
Fy
Fy
(13a)
hw 1900

tw
Fy
Fy
where bfl/2 is considered to be the flange outstand and Fy is the yield strength of the steel in MPa. On
the other hand, according to the AISC-LRFD [6], the above classification becomes:
Compact sections :
Non - compact sections :
b fl
2t fl
b fl
2t fl
  p  0.38
E
Fy
E
 r  0.83
Fy  Fr
1051
hw
E
  p  3.76
tw
Fy
(13b)
hw
E
 r  5.70
tw
Fy
AUEJ, 7, 5,2004
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
where Fr is the residual stress in the section. Both Class 1 and Class 2 sections of the CAN/CSA-S16
specifications [7] are considered as one class according to the AISC-LRFD specifications; the compact
section class [6]. The AISC-LRFD specification uses (Fy – Fr) to account for a residual stress of 69
MPa for hot-rolled steel sections and 114 MPa for built-up welded I-sections.
The flange outstand-to-thickness and the web height-to-thickness ratios defining the I-section class for
beams according to the ECPSCB (ASD) [5] are:
Compact section :
rolled :
Welded :
b fl
2t fl
b fl
2t fl

16.9

15.3
hw 127

tw
Fy
Fy
hw 127

tw
Fy
Fy
(13c)
Non - compactsections : rolled :
Welded :
b fl
2t fl
b fl
2t fl

23

21
hw 190

tw
Fy
Fy
hw 190

tw
Fy
Fy
with Fy being the yield strength of the steel in ton/cm2.
4. MOMENT RESISTANCE OF STEEL I-SECTION BEAMS
The flexural capacity of steel I-section beams is significantly affected by local and lateral torsionflexure buckling. For example, according to the CAN/CSA-S16 specifications [7], the section class is
first defined based on the flange-to-thickness and the web-to-thickness ratios. For Class 1 and Class 2
sections, the flexural capacity follows the behaviour conceptually shown in Figure 7a while for Class 3
sections, it follows the behaviour shown in Figure 7b.
AUEJ, 7, 5,2004
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DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
Elastic
plastic
buckling
Mr
Mr
Elastic
plastic
buckling
Elastic
buckling
Mp
Elastic
buckling
My
2/3 Mp
Lp
Lr Unsupported length
Lp
(a)
Zone 1
Zone 2
Zone 3
Zone 1
Zone 2
Zone 3
2/3 My
Lr
Unsupported length
(b)
Figure 7. Moment resistance of I-beams for compact (left) and for non-compact sections (right).
Three distinct zones are classified in Figure 7: Zone 1 is controlled by elastic lateral torsion-flexure
buckling, Zone 2 is controlled by elastic-plastic lateral torsion-flexure buckling and Zone 3 is not
affected by this buckling mode at all. Thus, according to the behaviour shown in this figure, the
moment resistance of steel I-section beams may be defined using the following equations:
Zone 1 : M r   M cr

M 
Zone 2 : M r  1.15 M 1  0.28

M cr 

Zone 3 : M r   M
for M cr 
2
M
3
for M cr 
2
M
3
(14)
where M = Mp for compact sections and M = My for non-compact sections. In Equation 14, Mcr is the
elastic critical moment causing lateral buckling and is determined using Equation 5, Mp and My are the
plastic and the yield moment respectively, and  is a material resistance factor ( = 0.9). The elastic
lateral-torsional buckling zone (Zone 1 in Figure 7) is terminated at 2/3 Mp or 2/3 My, depending on
the section class, to account for the effect of the residual stresses which may reach 1/3 Fy.
On the other hand, according to the AISC-LRFD [6] specifications, local buckling behaviour is
investigated using Equation 13b: the flange outstand-to-thickness and the web height-to-thickness
ratios  are compared to p and r. The moment resistance of an I-section beam is first determined by
considering only the local buckling behaviour of the flange and the web as follows:

M r   M p  M p  M y



   


p
r
p




 M p
(15)

M y  Fy  Fr S x
1053
AUEJ, 7, 5,2004
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
where Mp is the plastic moment of the section (Equation 8) and My is the moment initiating yield at the
extreme fibres of the section considering a residual stress Fr equals to 69 MPa in case of hot rolled Isections and 114 MPa in case of welded I-sections. If both the flange outstand-to-thickness and the
web height-to-thickness ratios are less than p, the moment resistance would be based on Mp. On the
other hand, if of either the web or the flange is equal to r, the moment resistance would be based on
My. In both cases, a resistance factor = 0.9 is adopted.
The lateral-torsional buckling in zone 2 is then considered separately where the moment resistance is
calculated using:

M r   M p  M p  M y



u

r
 L p 

 L p 
  LL
(16)
where Lu is the laterally unsupported length of the beam. Lr and Lp which are conceptually shown in
Figure 7 are defined for I-sections by:
L p  1.76 ry
X1 

E
Fy
Lr 
Fy  Fr
1  1  X 2 ( Fy  Fr )2
C S 
X2  4 w  x 
I y  GJ 
EG J A
2
Sx
ry X 1
(17)
2
where ry is the radius of gyration of the I-section about the weak axis, Cw is the warping constant, Sx is
the section modulus about the strong axis, J is the torsional constant, A is the cross section area, E is
Young’s modulus and G is the shear modulus.
The moments resistance considering local buckling of the flange and of the web (Equation 15) and
lateral torsion flexure buckling of the I-section (Equation 16) are determined separately; then the
lowest value of the three moments are considered as the moment resistance of the cross section.
5. ALLOWABLE BENDING STRESS DEFINED BY THE ECPSCB (ASD)
SPECIFICATIONS
The allowable bending stress for laterally unsupported I-section beams is determined according to the
ECPSCB (ASD) provisions [5] as follows (in ton – cm units):
If L 
20b fl
If L 
20b fl
Fy
Fy
and L 
or L 
1380A fl
hw Fy
1380A fl
hw Fy
Cb :
Cb :
Fc  Fcb
Fc  Greater of Fltb1 or Fltb 2
(18a)
(18b)
where, L is the laterally unsupported length, bfl and Afl are the compression flange width and area
respectively and Fy is the yield strength of the steel. The stresses Fcb, Fltb1 and Fltb2 are defined by:
Fcb  0.64Fy
For Compact Sections
(19a)
 0.58Fy
AUEJ, 7, 5,2004
For Non  Compact Section
1054
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
Fltb1 
800
C  0.58Fy
Lhw b
Af
(19b)
Fltb 2  0.58Fy
For
C
L
 84 b
rt
Fy
2
L
  Fy
r
 ( 0.64   t  5
)Fy  0.58Fy
1.176x10 Cb

For 84
12000
Cb  0.58Fy
2
L
 
 rt 
For
Cb L
C
  188 b
Fy rt
Fy
(19c)
C
L
 188 b
rt
Fy
where rt is the radius of gyration of a section composed of the compression flange and 1/6 the web
area. The ECPSCB (ASD) provisions [5] do not differentiate between the behaviour of I-sections with
single and two axes of symmetry as this effect is conceptually considered through rt.
The stresses Fltb1 and Fltb2 defined according to the ECPSCB (ASD) provisions [5] represent the two
components resisting lateral buckling: the warping restraint resistance and torsion resistance
respectively. Instead of taking the vector summation of these two components to obtain the section
resistance to lateral buckling, the ECPSCB (ASD) defines the greater of the two components as the
beam resistance; a conservative approach.
6. PROPOSED EQUATION FOR THE ALLOWABLE BENDING STRESS OF IBEAMS
Equations 18 and 19 estimate the allowable bending stress for steel I-section beams in a multiple and
discontinuous form. Here, an alternative simple design equation is proposed for the allowable bending
stress of laterally unsupported steel I-section beams. The proposed equation is reached using an
approach similar to the one proposed by Loov [12] and adopted by CAN/CSA-S16 specifications [7]
for axially loaded columns. The new equation takes the form:
If L 
20b fl
Fy
and L 
1380A fl
hw Fy
Cb :
Fc  0.64  Fy
 0.58  Fy
For Compact Sections
For Non  Compact Section
  0.58  F
y
Fc  0.58  Fy 1  
  Fltb

Otherwise :




n





(20a)
1
n
where Fltb is the greater of Fltb1 or Fltb2 defined by:
Fltb1 
800
 Lhw 


 Af 


 Cb
Fltb 2 
12000
L
 
 rt 
2
 Cb
1055
(20b)
AUEJ, 7, 5,2004
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
with no upper limit on Fltb1 or Fltb2. The greater value of Fltb1 or Fltb2 is chosen as opposed to the vector
summation of the two values in order to match the procedures adopted by ECPSCB (ASD). The
exponent n in Equation 20a is taken 3.5 (will be discussed in the following section).
The predictions of the allowable bending stress obtained using the proposed equation, are compared to
the currently used procedures adopted by the ECPSCB (ASD) provisions [5]. Compact and noncompact I-sections are considered in this comparison: the web height-to-thickness and flange outstandto-thickness ratios are chosen to be on the limiting edges of compact and non-compact section classes.
Steel grades St52 and St37 with 3.6 ton/cm2 (353 MPa) and 2.4 ton/cm2 (235 MPa) yield strengths
respectively, are adopted in the analysis. The results of this investigation are plotted in Figures 8 to 11.
The maximum and minimum percentages of difference between the ECPSCB (ASD) allowable stress
prediction and that of the proposed equation are also shown in these figures.
The comparisons shown in Figures 8 to11 reveal an excellent behaviour of the proposed equation. It is
also evident from these figures that the maximum and minimum percentages of difference between the
ECPSCB (ASD) allowable stress predictions and those of the proposed equation are in the order of
±4%. This percentage is significantly affected by the exponent n of Equation 20. The investigation
performed on the relation between the exponent n of Equation 20 and the percentage of difference is
briefly shown in Figure 12. It is evident form this investigation that n of 3.5 minimizes the percentage
difference between the ECPSCB (ASD) allowable stress predictions and the ones obtained by the
proposed equation.
AUEJ, 7, 5,2004
1056
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
3
ECPSCB (ASD)
Proposed Equation
Lumax
Allowable bending stress (t/cm2)
2.5
Fcb
2
Class  1 Compact Section
tf  12 mm bf  193 mm
1.5
tw  6 mm
1
Fy  3.6
0
ton
2
cm
hw
 66.833
tw
0.5
0
2
hw  401 mm
bf
2tf
 8.042
4
6
Unsupported length (m)
8
10
10
% of diff between proposed and Code Eqs
Class  1 Compact Section
4.4%
5
0
-3.9%
5
tf  12 mm bf  193 mm
tw  6 mm
10
0
2
Fy  3.6
hw  401 mm
4
6
Unsupported length (m)
8
ton
2
cm
10
Figure 8. Comparison between results of the proposed allowable bending stress equation and
the procedures adopted by ECPSCB (ASD) for compact I-section beams (Steel St52).
7. REGRESSION ANALYSIS ON THE PROPOSED ALLOWABLE BENDING
STRESS EQUATION RESULTS
A regression analysis has been performed on the results of the proposed allowable bending stress
equation to evaluate the coefficient of determination R2 and to verify the chosen value for the exponent
n. The coefficient of determination is given by [13]:
1057
AUEJ, 7, 5,2004
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
2.5
ECPSCB (ASD)
Proposed Equation
Allowable bending stress (t/cm2)
Lumax
2
Fcb
Class  2
Non-compact Section
1.5
Fy  3.6
ton
1
tw  6 mm
hw  600 mm
hw
 100
tw
0.5
0
2
cm
tf  12 mm bf  265 mm
0
2
bf
 11.042
2tf
4
6
Unsupported length (m)
8
10
10
% of diff between proposed and Code Eqs
Class  2 Non-compact Section
4.4%
5
0
-3.9%
5
tf  12 mm bf  265 mm
tw  6 mm
10
0
2
hw  600 mm
4
6
Unsupported length (m)
Fy  3.6
8
ton
2
cm
10
Figure 9. Comparison between results of the proposed allowable bending stress equation and
the procedures adopted by ECPSCB (ASD) for non-compact I-section beams (Steel St52).
R2 
St  S r
St
(21a)
where St is the total sum of the squares of the residuals between the allowable stress obtained using the
ECPSCB (ASD) provisions and their mean value. On the other hand, Sr is sum of the squares of the
AUEJ, 7, 5,2004
1058
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
residuals between the allowable stresses obtained using the code provisions and those obtained using
the proposed equation. Thus, St and Sr may be given by:

S t   Fccode  Fccode


2
S r   Fccode  Fcproposed eq
(21b)

2
Lumax
Allowable bending stress (t/cm2)
2
ECPSCB (ASD)
Proposed Equation
Fcb
1.5
bf  237 mm
tf  12 mm
1
hw  491 mm
tw  6 mm
0.5
0
Class  1
Compact Section
0
2
Fy  2.4
ton
2
cm
hw
 81.833
tw
bf
 9.875
2tf
4
6
Unsupported length (m)
8
10
10
% of diff between proposed and Code Eqs
Class  1 Compact Section
4.4%
5
0
n  3.5
5
-3.9%
tf  12 mm bf  237 mm
tw  6 mm
10
0
2
hw  491 mm
Fy  2.4
4
6
Unsupported length (m)
ton
2
cm
8
10
Figure 10. Comparison between results of the proposed allowable bending stress equation
and the procedures adopted by ECPSCB (ASD) for compact I-section beams (Steel St37).
1059
AUEJ, 7, 5,2004
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
Lumax
Allowable bending stress (t/cm2)
1.5
Fcb
Class  2
Non-compact Section
1
Fy  2.4
ton
2
cm
tf  12 mm bf  325 mm
0.5
0
tw  6 mm hw  735 mm
hw
bf
 122.5
 13.542
tw
2tf
0
2
4
6
8
10
Unsupported length (m)
12
14
16
14
16
10
% of diff between proposed and Code Eqs
Class  2
Non-compact Section
4.4%
5
0
n  3.5
5
-3.9%
tf  12 mm bf  325 mm
tw  6 mm
10
0
2
hw  735 mm
4
Fy  2.4
6
8
10
Unsupported length (m)
ton
2
cm
12
Figure 11. Comparison between results of the proposed allowable bending stress equation and
the procedures adopted by ECPSCB (ASD) for non-compact I-section beams (Steel St37).
For a perfect fit, Sr should be zero and R2 would be 1.0, signifying that the proposed equation 100
percent satisfies the code of practice procedures.
AUEJ, 7, 5,2004
1060
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
Figure 13 shows the coefficient of correlation between results of the proposed equation and the
ECPSCB (ASD) predictions for the allowable bending stress. The figure reveals that a value of 3.5 for
the n coefficient of Equation 20 yields a coefficient of correlation that is greater than 99%.
10
% of diff between proposed and Code Eqs
8.4%
Compact Section
Fy=3.6 t/cm2
n=4.5
4.4%
5
n=3.5
0
-1.9%
-3.9%
5
n=2.5
-9.1%
10
0
2
4
6
Unsupported length (m)
8
10
8
10
10
% of diff between proposed and Code Eqs
Non-compact Section
Fy=3.6 t/cm2
8.4%
n=4.5
4.4%
5
n=3.5
0
-1.9%
-3.9%
5
n=2.5
-9.1%
10
0
2
4
6
Unsupported length (m)
Figure 12. Percentages of difference between the ECPSCB (ASD) allowable stress
predictions and those of the proposed equation for different vales of the exponent n of
Equation 20.
1061
AUEJ, 7, 5,2004
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
8. CONCLUSIONS
Design of laterally unsupported steel I-beams currently requires the use of multiple equations which
are controlled by local and lateral torsion-flexure buckling behaviour. According to ECPSCB (ASD),
the allowable bending stress for steel I-beams depend on the section compactness, the laterally
unsupported length of the beam, the properties of the cross section and the yield strength of the steel.
1.0
Coefficient of determination R2
Compact Section
bfl /2tfl = 8
hw/tw = 66
0.95
1.0
Non-compact Section
bfl /2tfl = 11
hw/tw = 100
0.95
Fy = 3.6 ton/cm2
1
1.5
2
2.5
Coefficient n
3
3.5
4
Coefficient of determination R2
1.0
Compact Section
bfl /2tfl = 8
hw/tw = 66
0.95
1.0
Non-compact Section
bfl /2tfl = 11
hw/tw = 100
0.95
Fy = 2.4 ton/cm2
1
1.5
2
2.5
Coefficient n
3
3.5
4
Figure 13. Correlation coefficient between the proposed equation and ECPSCB (ASD)
provisions
AUEJ, 7, 5,2004
1062
DESIGN OF LATERALLY UNSUPPORTED STEEL I-BEAMS ACCORDING TO THE EGYPTIAN CODE OF PRACTICE: A
PROPOSED SIMPLE EQUATION
Furthermore, since flange outstand-to-thickness and web height-to-thickness ratios governs the local
buckling behaviour of an I-section, they significantly affect the allowable bending stress predictions.
The laterally unsupported length of the beam on the other hand, affects the critical moment initiating
lateral flexure-torsion buckling and thus, it also significantly affects the allowable bending stress.
According to most codes of practice (e.g. ECBSCB–ASD), distinct zones are specified for the beam’s
behaviour; each has an equation for defining the allowable bending stress.
To simplify the design procedures, a simple equation is proposed for the allowable bending stress of a
laterally unsupported steel I-section beam. The proposed equation considers all the parameters
considered by the ECPSCB (ASD). Results obtained using the proposed equation are compared to
those obtained using the design equations of ECPSCB (ASD) and showed a very good fit to the
ECPSCB (ASD) provisions. A regression analysis performed on the allowable bending stress obtained
using the proposed equation and the predictions of allowable stress obtained by adopting the ECPSCB
(ASD) provisions reveals a coefficient of correlation that is higher than 99%.
REFERENCES
1. Kulak G.L., Gilmore, M.I. 1998. Limit states design in structural steel. 6th edition, Canadian
Institute of Steel Construction, Ontario, Canada.
2. Galambos, T. V. 1998. A guide to Stability design criteria for metal structures. 5 th edition, John
Wiley & Sons, Inc., NY, USA.
3. Salmon, C.G., Johnson, J.E. 1996. Steel structures: design and behavior. 4th edition, Printice Hall,
NJ, USA.
4. Chen, W.F., and Lui, E. M. Structural stability: theory and implementation. Elsevier Science
Publishing Co., Inc. N.Y., USA. 1987.
5. Ministry of Housing, Utilities and Urban Communities; Housing and Building Research Centre.
2001. Egyptian Code of Practice for Steel Construction and Bridges (Allowable Stress Design).
Ministerial Degree No. 279 - 2001.
6. American Institute of Steel Construction (AISC). 2000. Load and Resistance Factor Design.
American Institute of Steel Construction, Illinois, USA.
7. Canadian Standard Association, CAN/CSA-S16-01. 2001. Limit states design of steel structures.
Canadian standard association, Ontario, Canada.
8. Salvadori, M. G. 1955. Lateral buckling of I-beams. ASCE Transaction. Vol. 120, pp. 1165-1177.
9. Sayed-Ahmed, E.Y. 2004. Lateral buckling of steel I-beams: a numerical investigation and
proposed equivalent moment factor equations. Al-Azhar University Engineering Journal (AUEJ),
Al-Azhar University, Faculty of Engineering Vol. 7, No. 1, pp. 111-123.
10. Yu, W. Cold-formed steel design, 3rd edition. John Wiley & Sons, NY, USA, 2000.
11. AISI – American Iron and Steel Institute - North American Specifications for the Design of ColdFormed Steel Structural Members (NAS 2001), Washington, USA. 2001.
12. Loov, R. 1996. A Simple equation for axially loaded steel column design curves. Canadian Journal
of Civil Engineering, Vol. 23, pp. 272-276.
13. Chapra, S.C. and Raymond P.C. 1998. Numerical methods for engineers, 3rd ed., McGraw-Hill,
USA.
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