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Elastic flange local buckling of I-shaped beams considering effect of web restraint
Thin-Walled Structures 105 (2016) 101–111
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Elastic flange local buckling of I-shaped beams considering effect of web restraint Kyu-Hong Han, Cheol-Ho Lee n Dept. of Architecture and Architectural Engineering, Seoul National University, Seoul 151-742, Korea
art ic l e i nf o
a b s t r a c t
Article history: Received 24 December 2015 Received in revised form 28 March 2016 Accepted 3 April 2016
The 2010 AISC flexural strength of I-shaped beams with slender flange was proposed based on Johnson (1985)’s experimental plate bucking coefficient (kc). However, our close examination revealed that most of I-beam specimens tested by Johnson to derive kc had section configuration of slender web and noncompact flange, indicating the inappropriateness of the test database itself and probable underestimation of web restraint effect in the AISC codification. In this paper, the effect of web slenderness on the elastic flange local buckling is analytically investigated using the mixed variational approach, and more accurate but still practical formula is proposed. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Flange local buckling I-shaped beams Slender flange Web slenderness Mixed variational principle
1. Introduction The use of high-strength steels in building construction increases the possibility of designing plate elements which exceed the noncompact slenderness limit ( λr ), thus often requiring the check for elastic local buckling. Local buckling of plate elements of a structural shape is interactive with adjacent elements. Several researchers investigated the effect of flange-web interaction on local buckling of plate elements of various structural shapes. For example, Johnson [1] performed an experimental program on I-shaped beams subjected to uniform moment with focusing on the effect of web slenderness on elastic flange local buckling (FLB) and provided the basis of the relevant provisions in the 2010 AISC Specification [2]. Johnson's study will be critically reviewed in the later section of this paper. Cohen [3] proposed approximate elastic buckling coefficients of a T-shaped section under uniform bending using finite difference method. Bedair [4] investigated the influence of the rotational and in-plane translational restraints on web buckling of wide flange columns. However, the majority of previous studies focused on the web local buckling. The research for elastic FLB of I-shaped beams considering web interaction is very rare and difficult to find in the literature except Johnson's research. Seif and Schafer [5] used finite strip method to derive simplified empirical expressions for various plate buckling coefficients of all the AISC hot-rolled shapes with considering elastic web-flange n
Corresponding author. E-mail address: [email protected] (C.-H. Lee).
http://dx.doi.org/10.1016/j.tws.2016.04.001 0263-8231/& 2016 Elsevier Ltd. All rights reserved.
interaction. Their elastic FLB coefficient was developed through a simple conversion of the buckling coefficient of web. However, because their derivation was based on the AISC section database only, the general applicability of their empirical formula may be limited if shapes built up in the geometric proportions different from the AISC section database are concerned. Recently, flexural tests on full-scale I-shaped beams, built up from high- and ordinary-strength steels were carried out by the authors to study the effect of flange slenderness on flexural strength and rotation capacity [6]. As shown in Fig. 1 and Table 1, large flexural overstrength in the three specimens with slender or almost slender flange was observed under moment gradient loading. The following three reasons were mentioned in our previous study to explain the large overstrength observed. First, as can be seen in Fig. 2(a), FLB does not occur at the mid-point of the beam (or at the point of maximum moment) because of the presence of the transverse stiffeners. Thus, FLB has to occur on both side of the mid-span of the beam. A numerical bifurcation buckling analysis of the slender flange specimen showed that the presence of transverse stiffeners tends to induce the antisymmetric FLB mode whose buckling strength is higher than that of the symmetric mode which is expected when transverse stiffeners are absent. Second, after the buckling, the beam can resist additional load due to the membrane action of the severely buckled flange (see Fig. 2 (a)). Third, the elastic buckling resistance of slender flange in the AISC Specification is based on the buckling strength of plates subjected to uniform stress distribution along the plate length. However, as is wellknown in the classical stability theory, plate elements subjected to linearly-varying stress distribution along the plate length under
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ratio v to be 0.3, the 2010 AISC nominal flexural strength of I-shaped sections with slender flange (Mn) is essentially the same as that based on the classical solution [Eq. (1)] as shown below:
Mn = Fcr Sx = kc
π 2E Sx 12 (1 − ν 2)(b/t f )2
⎛ 0.9E ⎞ = k c ⎜ 2 ⎟ Sx ⎝ λ ⎠
where Fcr is the critical FLB stress, Sx is the elastic section modulus, λ is the plate slenderness parameter ( λ ¼ b /t f ), tf is the flange thickness, and kc is the plate buckling coefficient for slender unstiffened elements. The plate buckling coefficient kc is given as Eq. (3) in the 2010 AISC Specification.
λ Fig. 1. Comparison of experimental and AISC nominal FLB strength.
4
kc = Table 1 Overstrength observed in slender or almost slender flange specimens [6]. Specimen designation
Section class Normalized strength Mm/Mp
Mm/Mn
SM490-S-LPD-3
Slender
0.86
1.69
HSB800-NC-LP-3
Noncompact 0.99
1.55
HSA800-S-LPD-3FHS
Slender
2.04
0.82
Failure mode
Flange local buckling Flange local buckling Flange local buckling
Note: Mm ¼ maximum experimental flexural strength; Mp ¼ plastic moment; Mn ¼ AISC nominal flexural strength.
moment gradient have higher local buckling resistance than those under uniform moment. Fig. 2(b) illustrates the effects of the presence of transverse stiffeners and the moment pattern on the elastic FLB strength of slender flange based on the numerical buckling analysis results. The authors felt that the elastic FLB provisions in the 2010 AISC Specification [2] are too conservative and that further thorough study to improve the accuracy of the relevant specification formula is warranted.
2. Review of the 2010 AISC FLB provisions An I-shaped section can be viewed as an assemblage of flat plates. The flange plate is an unstiffened element because one edge is not supported. From the classical plate theory [7], the elastic critical buckling stress under uniform axial force (Fcr) can be expressed as
⎡⎣ 0.35 ≤ k c ≤ 0.76⎤⎦
4.05 0.46
(4)
Comparison of Eqs. (3) and (4) shows that Eq. (3) in the 2010 AISC Specification is just a slight modification of Eq. (4) proposed by Johnson. The problem associated with basing Johnson's formula [Eq. (4)] in the 2010 AISC Specification is discussed in the below. Table 3 summarizes the flange and web section classification according to the 1978 and 2010 AISC Specification ([8,2]). Fig. 5 shows the web and flange slenderness distributions of Johnson's test specimens. Table 4 summarizes the flexural strength calculation with considering FLB according to the 2010 AISC Specification. It should be noted that Eq. (2) mentioned above (or Eq. F3-2 in Table 4) should be used for I-shaped beams with slender flange and compact or noncompact web. When web is slender, the bending strength reduction factor Rpg (Eq. F5-6) should be multiplied to account for web instability effect (see column 3 in Table 4). As can be seen in Table 3, the limiting values of λr for flange and web under flexural compression were relaxed somewhat during the revision from the 1978 AISC Specification [8] to
4.5
(1)
where b is the width of plate, t is the plate thickness, E is Young's modulus, v is Poisson's ratio, k is the plate buckling coefficient, and ⎡ ⎤ Et 3 D is the plate flexural rigidity ⎢D = ⎥. ⎣ 12 (1 − v 2) ⎦ In an I-shaped section, the plate buckling coefficient k represents the rotational restraint provided by the web at the flangeweb junction (see Fig. 3). The minimum value of k for a plate with one edge pinned and the other free is 0.425, and that for one edge fixed and the other free is 1.277 (see Table 2). Taking Poisson's
(3)
( h/tw )
the 2010 AISC Specification [2]; 0.56
π 2E 12 (1 − ν 2)(b/t )2
π 2D =k 2 b t
h/tw
where h is the clear distance between flanges and tw is the thickness of web. As can be seen in Eq. (2), the coefficient kc in the 2010 AISC Specification is essentially the same as the plate buckling coefficient k in Eq. (1). Note that the lower limit of kc is 0.35 (or smaller than 0.425 corresponding to one edge pinned and the other free), implying the case of “negative” web restraint. The factor kc was originally proposed by Johnson [1] to account for the interaction of flange and web local buckling. Johnson tested 19 wide flange beams having high web and high flange slenderness, and proposed the following buckling coefficient through regression of his test results (see Fig. 4).
k=
Fcr = k
(2)
E Fy
⇒ 5.7
E Fy
E Fy
⇒ 0.95
kc E FL
for flange and
for web, respectively.
It should be noted from Fig. 5 that majority of Johnson's test specimens belonged to slender web-slender flange combinations according to the 1978 AISC Specification as he intended, but now most of them fall into noncompact or compact flange and slender web combinations according to the 2010 AISC Specification. It is evident that the test database (and the resulting value of kc) is inappropriate for use within the context of the 2010 AISC Specification. In order to be consistent with the 2010 AISC Specification, the test database should be of slender flange and compact or noncompact web combination (see F3–2 in Table 4). It is not reasonable to apply the value of kc derived from the test database of
K.-H. Han, C.-H. Lee / Thin-Walled Structures 105 (2016) 101–111
SM490-S-PD-3
103
HSA800-S-PD-3-FHS
(a) Photos showing flange local buckling
Normalized Moment (M/M )
1.4 1.2
1.0 0.8 0.6 0.4 0.2 0.0
0
2
4 Normalized End Rotation (θ/θ )
6
8
(b) Effects of the presence of transverse stiffeners and moment pattern on elastic FLB Fig. 2. Behavior of specimens with slender section [6].
Beam-web junction (restraint provided by the web) Free-end Half-wave length of FLB Fig. 3. Rotational restraint provided by web at flange-web junction.
compact or noncompact flange and mostly slender web combinations to the design of slender flange and compact or noncompact web combinations. It is highly probable that the equation F3–2 in Table 4 [or Eq. (2) in the above] would underestimate the web restraint effect on the flange, leading to conservative prediction of FLB strength as observed in our previous testing (see Fig. 1 and Table 1). In the following, the mixed variational approach that can explicitly consider the web restraint effect is developed to derive more accurate but still practical FLB strength equation. The focus of this study is to
Table 2 Plate buckling coefficient (k) depending on boundary conditions. Boundary conditions of unloaded edges
Buckled shape
k
One edge simply supported, the other free
0.425
One edge fixed, the other free
1.277
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⎛ ∂ 2w ∂ 2w ⎞ Mx = − D ⎜ 2 + v 2 ⎟ ⎝ ∂x ∂y ⎠ ⎛ ∂ 2w ∂ 2w ⎞ My = − D ⎜ 2 + v 2 ⎟ ⎝ ∂y ∂x ⎠ Mxy = − D (1 − v)
Fig. 4. Regression equation proposed by Johnson.
Table 3 Flange and web section classification according to the 1978 and 2010 AISC Specification. Element Loading condition
1978 AISC
λp Flange
Flexural compression
Web
Flexural compression
a
Note: Kc =
4 h
E Fy
640/ Fy (ksi) = 3.76
λp
λr
65/ Fy (ksi) = 0.38
2010 AISC
E Fy
95/ Fy (ksi) = 0.56
0.38
E Fy
3.76
E Fy
0.95
kc E a FL
5.70
E Fy
E Fy
760/ Fy (ksi) = 4.5
λr
E Fy
; FL ¼ 0.7Fy.
tw
Q x= −
∂ ⎛ ∂ 2w ∂ 2w ⎞ ⎟ D⎜ 2 + ∂x ⎝ ∂x ∂y2 ⎠
Q y= −
∂ ⎛ ∂ 2w ∂ 2w ⎞ ⎟ D⎜ 2 + ∂y ⎝ ∂x ∂y2 ⎠
(5)
(6)
where Mx and My are the flexural moments per unit length, Mxy is the torsional moment per unit length, Q x and Q y are the transverse shear forces, and w is the transverse deflection. Applying static equilibrium condition in the z direction for all the forces shown in Fig. 6(b) and Fig. 7 gives the well-known buckling equation for plates subjected to uniform compressive force Nx.
∂ 2Mxy ∂ 2My ∂ 2Mx ∂ 2w + + = Nx 2 2 2 2 ∂ ∂ x y ∂x ∂y ∂x
(7)
It is mathematically very difficult to solve Eq. (7) directly when boundary conditions are complicated. Weighted-residual methods used in this study seek for approximate solutions through weaker formulation of the problem, or based on the weighted-integral statement of the form as shown in Eq. (8):
∫ WR
Fig. 5. Web and flange slenderness distributions of Johnson's test specimens.
∂ 2w ∂x∂y
dxdy = 0
(8)
where W is the weight function and R is the residual. In variational calculus, the weight function in the boundary expression is often termed the primary variable, and the coefficient of the weight function in the boundary expression is called the secondary variable. A mixed variational formulation is defined as one where secondary variables of the conventional formulation are also treated as dependent variables along with the primary variables [9]. This formulation can consider various boundary conditions with less difficulty and gives more accurate results than conventional formulations (like the principle of minimum total potential energy) for the force variables. Considering this advantage, the weighted-residual method with mixed variational principle is adopted for this study. The basic idea of mixed variational principle can be expressed as follows:
δΠm = δ u Πm + δN Πm = 0 propose FLB strength equation that can cover compact or noncompact web and slender flange combination because the 2010 AISC Specification lacks the technical background as discussed above. The study of the reliability of Eq. F5–7 in Table 4 (or the case where slender web is involved) is beyond the scope of this paper.
3. Mixed variational formulation for elastic FLB Fig. 6 shows the notations and the sign conventions for stress and stress resultants used in this paper. The stress resultants per unit length of a plate can be expressed as follows according to the classical thin plate theory [7]:
or δ u Πm = 0 and δN Πm = 0
(9)
where
δ u Πm =
∫ W1R1 dxdy and δN Πm = ∫ W2 R2 dxdy
(10)
where δu Πm is the virtual work done by virtual deflection, δN Πm is the virtual work done by virtual force, W1 and W2 are weight functions, and R1 and R2 are residuals. In our plate buckling problem, W1 = δw , W2 = δM , and R1 ¼ the residual of the buckling equation [Eq. (7)], and R2 ¼ the residual of the moment-deflection relations [Eq. (5)]. Mixed variational formulation to calculate elastic FLB strength of I-shaped sections is presented in the below.
K.-H. Han, C.-H. Lee / Thin-Walled Structures 105 (2016) 101–111
105
Table 4 Summary of flexural strength calculation considering FLB according to the 2010 AISC Specification. Flange
Web
C
C (1) N.A.
NC
NC (2) N.A.
Mn = Mp − (Mp − 0.7Fy Sx )(
λ − λpf λrf − λpf
) (F3–1)a
S (3) N.A.
Mn = Rpg Fcr Sxc (F5–7)a
⎛ λ − λpf ⎞ ⎟ (F4–13)a Mn = Rpc Myc − Rpc Myc − FL Sxc ⎜ ⎝ λrf − λpf ⎠
(
)
Rpg = 1 −
⎛h aw ⎜ c − 5.7 1200 + 300aw ⎝ tw
⎞
E ⎟ ≤ 1.0 Fy ⎠
(F5–6)a
⎛ λ − λpf ⎞ a Fcr = [Fy − (0.3Fy ) ⎜ ⎟ ] (F5–8) ⎝ λrf − λpf ⎠
S
Mn =
0.9Ekc Sx λ2
Mn = Rpg Fcr Sxc (F5–7)a
(F3–2)a
Rpg = 1 −
Fcr =
aw 1200 + 300aw
0.9Ekc ⎞2 ⎟ ⎟ ⎠
⎛b ⎜ f ⎜ 2t ⎝ f
⎛h ⎞ ⎜ c − 5.7 E ⎟ ≤ 1.0 (F5–6)a Fy ⎠ ⎝ tw
(F5–9)a
N. A.: the limit state of FLB does not apply a
Equation numbers in the 2010 Specification.
3.1. Weak forms
δ u Πm =
Because the condition of torsional moment Mxy is not well defined at the boundaries of a flange plate, the weak forms of plate equations [Eqs. (5) and (7)] are derived based on two force variables Mx and My, and one geometric variable w after eliminating Mxy . By using the torsional moment-deflection relation in Eq. (5), Eq. (7) can be expressed as
= −
∫Ω
(11)
⎡ Et f 3 ⎤ where Df is the flexural rigidity of the flange ⎢Df = ⎥, tf is 12 (1 − v 2) ⎦ ⎣ the flange thickness, and wf is the transverse deflection of the flange. Then, according to Eq. (9), the virtual work done by virtual deflection ( δu Πm ) should vanish.
0
⎡ 2 ∂ 4δwf ∂ 2My ∂ 2wf ∂ Mx δwf ⎢ − 2Df (1 − ν ) 2 2 + − Nx ⎢⎣ ∂x 2 ∂x ∂y ∂y 2 ∂x 2
⎤ ⎥ dxdy = 0 ⎥⎦
(12)
where Ω0 represents the total domain of the half flange. Integrating Eq. (12) by parts yields δ u Πm =
⎡ ∂ 2M ∂ 4wf ∂ 2My ⎤ ∂ 2wf x ⎥ = − Nx −⎢ − 2Df (1 − v) 2 2 + 2 2 ⎢⎣ ∂x ∂x ∂y ∂y ⎥⎦ ∂x2
∫ W1R1dxdy
⎡ ∂δw
∫Ωe ⎢⎢
f
⎣ ∂x
−
∂ 2δwf ∂ 2wf ⎤ ∂δwf ∂My ∂Mx ⎥ dxdy + 2Df (1 − ν ) + ∂x∂y ∂x∂y ⎥⎦ ∂y ∂y ∂x
⎧ ⎪
⎡
⎩
⎣
⎛
⎞
⎛ ∂ 3w ⎞⎤ ⎛ ∂My ⎞ ∂ 3wf f n ⎟⎥ ⎟ n y − Df (1 − ν ) ⎜⎜ 2 n y + 2 x⎟⎥ ∂y ⎠ ∂x∂y ⎝ ∂x ∂y ⎠⎦
∮Γe ⎨⎪ δwf ⎢⎢ ⎜⎝ ∂∂Mxx ⎟⎠ nx + ⎜⎝
⎫ ⎡ ∂ 2w ⎛ ∂δw ∂δwf ⎞ ⎤ ⎪ f f +Df (1 − ν ) ⎢ n y ⎟ ⎥ ⎬ ds nx + ⎜ ⎢⎣ ∂x∂y ⎝ ∂y ⎪ ∂x ⎠ ⎥⎦ ⎭ −
∫Ωe
∂δwf ∂x
Nx
∂wf ∂x
dxdy +
∮Γe δwf Nx
∂wf ∂x
nx ds = 0
(13)
where Γe is the boundary of the domain Ω0 , Ωe is the domain except the boundary Γe ( Ω0 = Ωe + Γe ), and nx and ny are the direction cosines of the unit normal. Similarly, according to Eq. (10), the total virtual work done by
Fig. 6. Notations and sign conventions for stresses and stress resultants.
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K.-H. Han, C.-H. Lee / Thin-Walled Structures 105 (2016) 101–111
Fig. 7. Compressive forces in bent position.
the virtual force should vanish. In this mixed variation approach, the curvature residual is computed by using the alternate form of the moment-deflection relations given in Eq. (5) as follows:
1 ∂ 2w = − (Mx − vMy ) D (1 − v2) ∂x2
δN Πm (δMy ) = = −
⇒
∂ 2w 1 = − (My − vMx ) D (1 − v2) ∂y2
(14)
∫Ω
∫ δMy R2 dxdy
⎡ ⎡ ∂ 2w ⎤⎤ 1 f ⎢ δMy ⎢ + (My − vMx ) ⎥ ⎥ dxdy = 0 2 2 ⎢ ⎥⎦ ⎦⎥ Df (1 − v ) 0 ⎢ ⎣ ∂y ⎣
⎡ ∂w ∂δM ⎤ 1 f y − δMy (My − vMx ) ⎥ dxdy 2 ⎥⎦ Df (1 − v ) ⎣ ∂y ∂y ∂wf δMy ny ds = 0 Γe ∂y
∫Ωe ⎢⎢ −
∮
(16)
Thus, for variation of Mx :
δN Πm (δMx ) =
3.2. Consideration of boundary conditions
∫ δMx R2 dxdy
= −
∫Ω
⎡ ⎤ ∂ 2wf 1 ⎢ δMx [ + (Mx − vMy )] ⎥ 2 2 ⎥⎦ ∂x Df (1 − v ) 0 ⎢ ⎣
dxdy = 0 ⎡ ∂w ∂δM ⎤ 1 f x ⎢ ⇒ − δMx (Mx − vMy ) ⎥ 2 ⎥⎦ Ωe ⎢ Df (1 − v ) ⎣ ∂x ∂x ∂wf δMx dxdy − nx ds = 0 Γe ∂x
∫
∮
(15)
The flange is rotationally restrained at the flange-web junction (y¼0) (see Fig. 8). This rotational restraint as well as other geometric and force boundary conditions along the four edges of a buckled flange should be incorporated properly in integrating the work equations given in Eqs. (13), (15), and (16). For the restrained edge ( y = 0): The following geometric and force boundary conditions should be satisfied:
wf = 0,
My = − ζ θ = − ζ
Similarly, for variation of My :
∂wf ∂y
(17)
where ζ is the web rotational stiffness (per unit length) for the half
Fig. 8. Boundary conditions of the half flange of I-shaped beams.
K.-H. Han, C.-H. Lee / Thin-Walled Structures 105 (2016) 101–111
flange of an I-shaped beam, θ is the rotation of flange-web junction (see Fig. 8b). Then, the term
( )n ∂My
in the integral expression along the
y
∂y
boundary (y¼0) in Eq. (13) should be modified as:
⎡ δwf ⎢ ⎣
( )n + ( ) ∂Mx ∂x
⎡ ⇒ δwf ⎢ ⎣
∂My
x
∂y
( )n ∂Mx ∂x
x
⎛ ∂ 3wf ny − Df (1 − ν ) ⎜ 2 ny + ⎝ ∂x ∂y
∂wf
∂wf ny ∂y
y=0
∂y
i=1
⎛ y⎞ ⎛ y ⎞5⎤ πx ⎛⎜ y ⎟⎞ ⎡ ⎢ α1 + α2 ⎜ ⎟ + ⋯ + α6 ⎜ ⎟ ⎥ = sin ⎝ b⎠ ⎝ b ⎠ ⎥⎦ a ⎝ b ⎠ ⎢⎣
∂ 3wf ∂x∂y
2
⎞⎤ nx ⎟ ⎥ ⎠⎦
y=0
9
Vy = 0
(18)
πx y i ( )] a b
and My(x, y) are assumed as a product of sine function and polynomial function as follows:
y=0
For the free edge ( y = b): The following force boundary conditions should be satisfied.
and
(20)
are the basis function of the assumed deflection. In order to satisfy the force boundary conditions ( Mx x = 0 = 0, Mx x = a = 0, and My y = b = 0), the approximate functions for Mx(x, y)
Mx (x, y) =
My = 0
i=1
πx ⎛⎜ y ⎞⎟ a ⎝ b⎠
where αi is the unknown deflection parameter, and ψi1[= sin
y=0
ny in Eq. (16) should be modified as:
1 ⇒ δMy My ny ζ
i
6
∑ αi ψi1 = ∑ αi sin
∂ 3wf
⎛ ⎛ ∂ 3wf ∂ 2wf ⎞ + ⎜ −ζ 2 ⎟ ny − Df (1 − ν ) ⎜ 2 ny + ∂y ⎠ ⎝ ⎝ ∂x ∂y
Similarly, the term
δMy
⎞⎤ nx ⎟ ⎥ ∂x∂y 2 ⎠⎦
6
wf (x, y) =
107
9
∑ βj ψ j2 = ∑ βj sin j=1
j=1
πx ⎛⎜ y ⎞⎟ a ⎝ b⎠
j
⎛ y⎞ ⎛ y ⎞8⎤ πx ⎛⎜ y ⎞⎟ ⎡ ⎢ β1 + β2 ⎜ ⎟ + ⋯ + β9 ⎜ ⎟ ⎥ = sin ⎝ ⎠ ⎝ ⎠ ⎝ b ⎠ ⎥⎦ a b ⎢⎣ b
(21)
where Vy is the effective shear force and defined as
Vy = Q y +
⎡ ∂ 3w ∂Mxy ∂ 3wf ⎤ f + (2 − ν ) 2 ⎥ = 0 = − Df ⎢ 3 ∂x ∂x ∂y ⎥⎦ ⎣⎢ ∂y
where the second term,
∂Mxy ∂x
9
My (x, y) = (19)
, represents additional shearing force
at the edge, produced by the torsional moment Mxy. Note that Eqs. (5) and (6) were used to express the effective shear force in terms of deflection in Eq. (19). It can be shown that the boundary integral term in Eq. (13) for the free edge (y = b) is expressed as follows by using the momentdeflection relations [Eq. (5)] and the effective shear force condition at the boundary [Eq. (19)]:
⎡
∮Γ δwf ⎢ e
⎣
( )n + ( )n ∂Mx ∂x
x
∂My
y
∂y
⎡ ⎛ ∂ 3wf = − ∮ δwf Df ⎢ ⎜ 3 + Γe ⎣ ⎝ ∂x
⎛ ∂ 3wf − Df (1 − ν ) ⎜ 2 ny + ⎝ ∂x ∂y
⎛ ∂ 3wf ∂ wf ⎞ n +⎜ 3 + 2⎟ x ∂x∂y ⎠ ⎝ ∂y 3
⎡ ⎤ ∂ 3wf = ∮ δwf ⎢ Df (1 − ν ) 2 ny ⎥ Γe ∂x ∂y ⎣ ⎦
⎤ ∂ wf ⎞ n ⎥ 2 ⎟ y ∂x ∂y ⎠ ⎦
∂ 3wf ∂x∂y 2
⎞⎤ nx ⎟ ⎥ ⎠⎦
y=b
3
9
∑ γj ψ j3 = ∑ γj sin j=1
j=1
⎞⎛ y ⎞ πx ⎛⎜ y − 1⎟ ⎜ ⎟ ⎠⎝ b ⎠ a ⎝b
⎞⎡ ⎛ y⎞ ⎛ y ⎞8⎤ πx ⎛⎜ y = sin − 1⎟ ⎢ γ1 + γ2 ⎜ ⎟ + ⋯ + γ9 ⎜ ⎟ ⎥ ⎠ ⎢⎣ ⎝ b⎠ ⎝ b ⎠ ⎥⎦ a ⎝b
j−1
(22)
where βj , γj are unknown parameters of Mx and My, and ⎡ ⎡ y j − 1⎤ πx y j ⎤ πx y are the basis functions ψ j2 ⎢= sin a b ⎥ , ψ j3 ⎢= sin a b − 1 b ⎣ ⎦⎥ ⎣ ⎦ of the assumed moment distribution in the x- and y directions, respectively. With assuming the approximate functions of w(x, y), Mx(x, y), and My(x, y) as above [Eqs. (20)–(22)], all the essential boundary conditions required in the mixed variational formulation are satisfied.
()
(
)( )
y=b
3.4. Derivation of web stiffness y=b
Note that the unit normal nx is zero at the boundary y ¼b. Other boundary conditions of wf y = 0 = 0 and My y = b = 0 as required by Eq. (17) and Eq. (18) are properly accounted for when we assume the approximate functions for wf(x, y), Mx(x, y) and My(x, y) in the next section.
When the flange of I-shaped beams is locally buckled with a sine shape, sinusoidal moment is induced at the flange-web junction. For simplicity and conservatism, the web is assumed simply supported along the remaining three sides as shown in Fig. 9. Note that the half of the buckled wave length is denoted as a. Then, the boundary conditions that should be satisfied are as follows:
3.3. Assumed functions We consider a six-parameter Ritz approximation [10] for w(x, y) and nine-parameter Ritz approximations for Mx(x, y) and My(x, y). The number of parameters was determined by considering the analysis time and accuracy compared to some preliminary numerical analysis. The 1st buckling mode of the flange is assumed as a sine shape in the x-direction and the half of buckling wave length is denoted as a. And we express the buckled shape in the ydirection as a polynomial form. Then, the assumed deflection function takes the form given in Eq. (20). Note that the assumed deflection function satisfies all the geometric boundary conditions; wf = 0, wf = 0, and wf y = 0 = 0. x=0
x=a
Fig. 9. Sinusoidal edge moment induced to beam web upon flange local buckling.
108
K.-H. Han, C.-H. Lee / Thin-Walled Structures 105 (2016) 101–111
x = 0 , a:
For
stiffness factor Cs needed to calculate ζ as follows.
∂ 2ww =0 ∂x2
ww = 0 ,
Cs =
2
For
y=0:
ww = 0 ,
For
y=h:
ww = 0 ,
∂ ww =0 ∂y2
⎡ ∂ 2w ∂ 2ww ⎤ πx w ⎥ = M (x) = Mf sin − Dw ⎢ +v 2 a ⎣ ∂y ∂x2 ⎦
(23)
where ww is the transverse deflection of the web, Dw is the flexural ⎡ Etw 3 ⎤ rigidity of web ⎢Dw = ⎥, Mf is the amplitude of sinusoidal ⎣ 12 (1 − v 2) ⎦ edge moment induced by flange buckling, and h is the height of web (clear distance between the top and bottom flanges). The governing differential equation of the web in this case is homogeneous and is simply
⎛ ∂ 4w ∂ 4w ∂ 4ww ⎞ w ⎟=0 Dw ⎜ + 2 2 w2 + 4 ∂x ∂y ⎝ ∂x ∂y 4 ⎠
(24)
The general solution of Eq. (24) can be expressed as the product of trigonometric and hyperbolic functions as follows (Lévy [11]; Nádai [12]):
4. Solution of mixed variational formulation and validation The mixed variational equations [Eqs. (13), (15) and (16)] with the assumed functions of w(x, y), Mx(x, y), and My(x, y) [Eqs. (20)– (22)] can be expressed in matrix form as follows.
⎡⎣ K 12⎤⎦ ⎡⎣ K 13⎤⎦ ⎤ ⎥ ⎧ {α}⎫ ⎪ ⎪ ⎡⎣ K 22⎤⎦ ⎡⎣ K 23⎤⎦⎥ ⎨ {β}⎬ = {0} ⎥⎪ ⎪ ⎡⎣ K 32⎤⎦ ⎡⎣ K 33⎤⎦⎥ ⎩ {γ} ⎭ ⎦
⎡ ⎡ K 11⎤ ⎦ ⎢⎣ ⎢ ⎡ K 21⎤ ⎦ ⎢⎣ ⎢ ⎡ K 31⎤ ⎦ ⎣⎣
∞
⎛
a
n= 1
+ Bn
nπ y nπ y nπ y + Cn sinh sinh a a a
nπ y nπy ⎞⎟ nπ x + Dn cosh sin a a ⎠ a
+∮
2 D f (1 − ν )
2 1 ∂ 2ψ 1 ∂ ψ j i dxdy ∂x ∂y ∂x ∂y
⎧ ⎛ ∂ 3ψ 1 ⎤⎫ ⎞ ⎡ ∂ 2ψ 1 ⎛ 1 ∂ 3ψ 1 ⎪ 1 ∂ψ 1 ⎞ ⎥ ⎪ j j ⎜ ∂ψ i j ⎢ i n ⎟ ⎬ ds ⎨ ψ D (1 − ν ) ⎜ n n ⎟ D 1 n ⎜ ∂x 2∂y y + ∂x∂y2 x ⎟− f ( − ν ) ⎢ ∂x∂y ⎜ ∂y x + ∂x y ⎟⎠ ⎥ ⎪ Γe ⎪ i f ⎝ ⎝ ⎦ ⎠ ⎣ ⎩
(25)
where the integration constants An, Bn, Cn and Dn should be determined from the boundary conditions of the web. Applying the boundary conditions specified in Eq. (23) to the general solution above gives the web deflection as Eq. (26):
⎡ ⎛ y⎞ ⎛ y ⎞⎤ h y 1 M (x) ⎢ cothπc sinh ⎜⎝ πc ⎠⎟ − cosh ⎝⎜ πc ⎠⎟ ⎥⎦ h h h Dw 2πc sinh πc ⎣ (26)
(30)
where Ωe
∑ ⎜⎝ An cosh nπy
(29)
Note that the factor Cs is a function of the effective aspect ratio of the web ( c = h/a ) (see Fig. 10). The half of the buckled wave length, a, is still unknown and has yet to be determined as discussed in the next section.
K ij11 = ∫
ww (x, y) =
2πc sinh2 πc (sinh πc cosh πc − πc )
⎭
∂ψ 1 ∂ψ 1 j i N dxdy + −∫ Ω e ∂x x ∂x
∂ψ 1 ∮ Γ ψi1Nx j nx ds, i, j = 1, 2, ⋯ , 6, e ∂x
⎧ ⎪
⎛ ∂ψ 2 ⎞ ⎫ j ⎟ ⎪ nx ⎬ ds i = 1, 2, ⋯ , 6; j = 1, 2, ⋯ , 9, ∂x ⎟ ⎪ ⎠ ⎭ ⎝
K ij12 = ∫
2 ∂ψ 1 ∂ψ j i dxdy − Ω e ∂x ∂x
∮ Γ ⎨ ψi1 ⎜⎜
3 ∂ψ 1 ∂ψ j i K ij13 = ∫ dxdy − Ω e ∂y ∂y
∮ Γ ⎨ ψi1 ⎜⎜
e⎪
⎩
⎧ ⎪
e⎪
⎩
⎛ ∂ψ 3 ⎞ ⎫ j ⎟ ⎪ nx ⎬ ds i = 1, 2, ⋯ , 6; j = 1, 2, ⋯ , 9, ∂y ⎟ ⎪ ⎠ ⎭ ⎝
2
ww =
where c = h/a , the effective aspect ratio of the web. The slope at the flange-web junction is obtained by differentiation of Eq. (26) as follows:
θ=
∂ww ∂y
= y=h
h −1 M (x) [πc − sinh πc cosh πc ] Dw 2πc sinh2 πc
(27)
Because web restrains “two” half flanges in I-shaped beams, ζ (¼ the web rotational stiffness per unit length for the half flange of an I-shaped beam) should be defined as follows.
ζ =
1 M (x) 1 D = Cs w 2 θ 2 h
(28)
Inserting Eq. (27) into Eq. (28) gives the non-dimensional web
Kij21 = ∫ Ω
1
∂ψ i2 ∂ψ j e
dxdy − ∮ ψi2 Γ
∂x ∂x
Kij22 = − ∫ Ω
1 2 e D f (1 − v )
v
Kij23 = ∫ Ω
2 e D f (1 − v )
Kij31 = ∫ Ω
e
1
∂ψ i3 ∂ψ j
Kij32 = ∫ Ω
2 e D f (1 − v )
∂ψ1j ∂y
ny ds i = 1, 2, ⋯, 9; j = 1, 2, ⋯, 6,
ψi3 ψ j2 dxdy i, j = 1, 2, ⋯, 9,
v 2 e D f (1 − v )
ψi3 ψ j3 dxdy i, j = 1, 2, ⋯, 9.
The order of the matrix in Eq. (30) is 24 24. The critical FLB strength Nx, min should be obtained through numerical eigenvalue analysis of Eq. (30) after incorporating the boundary conditions and ζ in integration for a problem at hand. Or, for a fixed value of ζ , a series of eigenvalue analyses should be performed to find the smallest minimum (critical) buckling compressive force Nx, min by varying the half of buckling wave length a. Once Nx, min is obtained, the FLB coefficient k min is calculated per Eq. (31) [refer to Eq. (1)].
⎡ 12 (1 − ν 2)(b/t )2N ⎤ f x, min ⎥ k min = ⎢ 2 ⎢⎣ ⎥⎦ π Et f
Fig. 10. Variation of Cs depending on the effective web aspect ratio (c).
nx dsi = 1, 2, ⋯, 9; j = 1, 2, ⋯, 6,
ψi2 ψ j3 dxdy i, j = 1, 2, ⋯, 9,
e
v
∂x
ψi2 ψ j2 dxdy i, j = 1, 2, ⋯, 9,
dxdy − ∮ ψi3 Γ
∂y ∂y
Kij33 = − ∫ Ω
∂ψ1j
e
(31)
In order to validate the accuracy of the mixed variational method (MVM) developed in this study and also derive a simplified formula, a total of thirty five I-shaped beams with slender flange and diverse web configuration covering seismic compact, compact, and noncompact web slenderness were first created (see Table 5). These sections were analyzed by using the MVM of this
K.-H. Han, C.-H. Lee / Thin-Walled Structures 105 (2016) 101–111
109
Table 5 I-shaped sections analyzed and comparison of flange local buckling coefficients. Model designation
Depth, H (mm)
Width, bf (mm)
Web thickness, tw (mm)
Flange thickness, tf (mm)
FEM AISC
k
FEM-UM-1 FEM-UM-2 FEM-UM-3 FEM-UM-4 FEM-UM-5 FEM-UM-6 FEM-UM-7 FEM-UM-8 FEM-UM-9 FEM-UM-10 FEM-UM-11 FEM-UM-12 FEM-UM-13 FEM-UM-14 FEM-UM-15 FEM-UM-16 FEM-UM-17 FEM-UM-18 FEM-UM-19 FEM-UM-20 FEM-UM-21 FEM-UM-22 FEM-UM-23 FEM-UM-24 FEM-UM-25 FEM-UM-26 FEM-UM-27 FEM-UM-28 FEM-UM-29 FEM-UM-30 FEM-UM-31 FEM-UM-32 FEM-UM-33 FEM-UM-34 FEM-UM-35
400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 753 753 753 753 753 753 753 753 753 753 911 911 911 911 911 911 911 911
650 650 650 650 650 650 650 650 650 650 650 650 650 650 200 200 200 265 265 265 265 530 530 530 530 530 530 305 305 305 305 610 610 610 610
23 (SC) 17.6 (SC) 12 (SC) 8 (SC) 20 (SC) 17.6 (SC) 15 (SC) 10 (SC) 8 (SC) 20 (SC) 17.6 (SC) 15 (SC) 12 (SC) 8 (SC) 8 (SC) 6 (C) 5 (C) 13 (SC) 11 (C) 9 (C) 7 (NC) 13 (SC) 11 (C) 9 (C) 9 (C) 7 (NC) 5 (S) 14 (C) 12 (C) 10 (C) 8 (NC) 16 (SC) 16 (SC) 12 (C) 8 (NC)
23 23 23 23 17.6 17.6 17.6 17.6 17.6 12 12 12 12 12 4 4 4 5 5 5 5 9 9 9 7 7 7 5 5 5 5 10 8 8 8
0.87 0.74 0.61 0.49 0.92 0.86 0.79 0.56 0.52 1.13 1.07 0.99 0.87 0.68 1.19 1.03 0.90 1.35 1.28 1.18 0.98 1.02 0.91 0.79 0.94 0.78 0.63 1.28 1.24 1.17 1.05 1.08 1.18 1.03 0.78
kc ⎛ ⎜= ⎝
MVM
kc/k
m (a/ b)
ζ
kmin
kmin /k
ζ
kmin
kmin /k
0.87 1.03 1.21 1.22 0.83 1.27 0.96 1.18 1.13 0.67 0.71 0.77 0.82 0.85 0.48 0.48 0.50 0.39 0.38 0.37 0.40 0.52 0.54 0.56 0.47 0.50 0.55 0.39 0.37 0.36 0.36 0.50 0.45 0.45 0.48
2.3 2.8 3.6 4.8 2.2 2.4 2.6 3.4 4.0 1.9 2.0 2.1 2.4 3.1 1.8 2.1 2.3 1.7 1.8 1.9 2.2 2.1 2.4 2.7 2.3 2.7 3.5 1.7 1.8 1.8 2.0 2.0 1.8 2.1 2.7
0.56 1.36 4.57 16.17 0.38 0.58 0.97 3.57 7.23 0.11 0.17 0.29 0.59 2.21 0.07 0.20 0.37 0.03 0.05 0.10 0.26 0.22 0.41 0.84 0.34 0.85 2.89 0.02 0.04 0.07 0.16 0.16 0.07 0.20 0.85
0.84 0.71 0.59 0.51 0.90 0.83 0.76 0.61 0.55 1.09 1.03 0.95 0.83 0.65 1.14 1.01 0.91 1.21 1.17 1.10 0.98 0.99 0.90 0.78 0.93 0.78 0.63 1.22 1.19 1.14 1.05 1.05 1.14 1.01 0.78
0.97 0.96 0.96 1.05 0.97 0.97 0.95 1.08 1.06 0.96 0.96 0.96 0.96 0.96 0.96 0.99 1.01 0.90 0.91 0.93 1.00 0.97 0.99 1.00 0.99 1.00 1.00 0.96 0.96 0.97 1.00 0.97 0.97 0.98 1.00
0.62 1.43 4.64 15.98 0.43 0.63 1.04 3.60 7.10 0.13 0.20 0.33 0.65 2.23 0.12 0.28 0.48 0.05 0.09 0.16 0.35 0.30 0.51 0.93 0.43 0.93 2.56 0.04 0.07 0.12 0.23 0.22 0.11 0.27 0.93
0.81 0.71 0.58 0.51 0.87 0.81 0.75 0.60 0.56 1.06 0.99 0.91 0.81 0.65 1.08 0.93 0.85 1.16 1.12 1.02 0.90 0.92 0.84 0.76 0.86 0.76 0.64 1.18 1.14 1.08 0.96 0.97 1.08 0.93 0.76
0.94 0.95 0.95 1.05 0.94 0.94 0.94 1.07 1.06 0.93 0.92 0.92 0.93 0.96 0.91 0.91 0.95 0.93 0.93 0.91 0.97 0.90 0.93 0.97 0.92 0.97 1.01 0.92 0.92 0.92 0.92 0.90 0.92 0.91 0.97
⎞ ⎟ h / tw ⎠
0.76 0.76 0.74 0.60 0.76 0.76 0.76 0.66 0.59 0.76 0.76 0.76 0.71 0.58 0.57 0.49 0.45 0.53 0.49 0.44 0.39 0.53 0.49 0.44 0.44 0.39 0.35 0.50 0.46 0.42 0.38 0.54 0.53 0.46 0.38
Simplified formula [per Eqs. (34)–(37)]
4
Note: i) SC ¼ seismically compact; C ¼compact; NC¼noncompact web. ii) Web sections classified with assuming the steel grade of A992 (Fy ¼ 345 MPa).
study and the commercial software ABAQUS [13]. The numerical models of I-shaped beams were modeled using the four-node shell element S4R in ABAQUS. Linear perturbation (or bifurcation buckling) analysis was conducted for the thirty-five I-shaped beams shown in Table 5. Before conducting numerical analysis, the accuracy of the finite element model was validated by comparing with several experimental results available from our previous study [6]. Fig. 2 shows that our numerically obtained FLB strength
(the case of moment gradient with stiffener in Fig. 2) is well compared with experimental strength of specimen HSA800-SLPD-3-FHS. Accurate prediction of elastic FLB strength of I-shaped beams using finite element analysis is not that difficult nowadays. The analysis results are compared in Table 5 and Fig. 11. The effect of web restraint is better evaluated in terms of the relative flexural stiffness ratio between the web and the flange. Thus, we define the non-dimensional coefficient of web restraint ( ζ ) as
ζ≡
Fig. 11. Comparison of kmin values predicted by mixed variational approach and FEM.
Df /b ζ
(32)
where ζ is the web rotational stiffness per unit length for the half flange of I-shaped beams that was already derived as Eq. (28). Theoretically, when the flange is completely fixed at the flangeweb junction, ζ should approach zero, and when the flange is free to rotate, ζ should approach infinity. As shown in Table 5 and Fig. 11, compared to ABAQUS finite element analysis results, our mixed variational method (MVM) gives much more accurate and consistent results than the AISC flange buckling coefficient [Eq. (3)] in a very wide range of ζ . Fig. 11 reveals that the variation of flange buckling coefficient, kmin, can be well described by the “single factor ζ ”.
K.-H. Han, C.-H. Lee / Thin-Walled Structures 105 (2016) 101–111
kmin values per 2010 AISC and proposed formula
110
Fig. 12. Variation of kmin depending upon ζ .
-
5. Simplified formula proposed Calculating the nominator ( Df /b) in Eq. (32) is an easy task once an I-shaped section is given. However, the effective aspect ratio of the web ( c = h/a ), which is needed to calculate ζ according to Eqs. (28) and (29), is not known until some eigenvalue solutions are available. Thus, the remaining task to derive a simplified formula is reduced to finding the reasonable estimate of the half of buckling wave length (a). Note that the factor Cs is an increasing function of the effective aspect ratio of the web ( c = h/a ) (see Fig. 10). For example, as a becomes longer, Cs becomes smaller. This in turn leads to the decrease of ζ [see Eq. (28)] and the increase of ζ , thus yielding more smaller (more conservative) flange buckling coefficient kmin (see Fig. 11). Based on the mixed variational analysis results for various beam sections, the aspect ratios of buckled flange (m ¼a/b) are also reported in Table 5. Although there are a few exceptional cases, m values are more or less stable and mostly less than 3. And the half of buckling wave length a may be conservatively taken as 3b in computing ζ according to Eqs. (28) and (29) as follows:
h h h c= = ≈ a mb 3b
k values per ABAQUS FEM Fig. 13. Comparison of kmin values per 2010 AISC and proposed formula.
Fig. 14. Bilinear approximation for kmin.
(33)
Based on the non-dimensional coefficient of web restraint ( ζ ) and regression analysis, simplified expressions for FLB coefficient (kmin) are proposed as follows (see Fig. 12) :
0.94ζ −0.072
For ζ ≤ 0.1 :
k min =
For 0.1 < ζ ≤ 4 :
k min = 0.75ζ −0.17
(35)
For 4 < ζ ≤ 16 :
k min = 0.69ζ −0.11
(36)
≤ 1.277
(34)
For ζ ≤ 4 :
k min = − 0.09 log2ζ + 0.76 ≤ 1.277
For ζ > 4 :
k min = − 0.03 log2ζ + 0.64 ≥ 0.425 (39)
(38)
Of course, Eqs. (34)–(37) and (38)–(39) in the above should be used for slender-flange I shaped beams with compact or noncompact webs.
6. Summary and conclusions The results of this analytical and numerical study on FLB of I-shaped beams with slender flange can be summarized as follows.
For ζ > 16 :
k min = 0.59ζ −0.054 ≥ 0.425
(37)
⎡ ⎤D h πc sinh πc , ζ = ⎢ (sinh πc cosh πc − πc ) ⎥ hw , and c = 3b . ζ ⎣ ⎦ The upper cap 1.277 in Eq. (34) and the lower limit 0.425 in Eq. (37) correspond to the two extreme boundary conditions, or one edge fixed and the other free, and one edge pinned and the other free, respectively. As can be seen in the last column in Table 5 and Fig. 13, the simplified expressions proposed above give much more accurate and consistent results than the AISC flange buckling coefficient [Eq. (3)] for a very wide range of ζ . More simple bilinear expressions also appear acceptable for practical purposes [see Eqs. (38)-(39) and Fig. 14]. where ζ ≡
Df / b
2
i) This study found that most of I-beam specimens tested by Johnson to derive the plate buckling coefficient (kc) had section configuration of slender (S) web and noncompact (NC) flange according to the 2010 AISC section classification, indicating the inappropriateness of the test database itself and the probable underestimation of web restraint effect in the AISC codification. ii) The mixed variational approach was used in this study to systematically investigate the effects of web slenderness on elastic FLB of I-shaped beams. The approximate equations for estimating web rotational stiffness at the flange-web junction were also proposed.
K.-H. Han, C.-H. Lee / Thin-Walled Structures 105 (2016) 101–111
iii) As this study shows, the web restraint effect should be measured in terms of the relative flexural stiffness between the web and the flange as opposed to the buckling coefficient kc in the 2010 AISC Specification which is determined only based on the web slenderness ratio, irrespective of flange flexural stiffness. iv) This study found that the variation of flange plate buckling coefficient can be well described by the single factor ζ (¼ the non-dimensional coefficient of web restraint proposed in this study). Based on this finding and the reasonable approximation of the effective aspect ratio of buckled webs, simplified expressions, more accurate and more consistent than the 2010 AISC equation, were proposed for elastic FLB strength of I-shaped beams.
Acknowledgments Support to this study by the POSCO Affiliated Research Professor Program is gratefully acknowledged.
References [1] D.L. Johnson, An investigation into the interaction of flanges and webs in
111
wide-flange shapes, in: Proceedings 1985 Annual Technical Session, 1985. [2] AISC. Specification for Structural Steel Buildings. American Institute of Steel Construction, Chicago, IL, ANSI/ASIC 360-10, 2010. [3] J.M. Cohen, Elastic buckling coefficients for long, unstiffened plates, J. Eng. Mech. 118 (12) (1992) 2491–2496. [4] O. Bedair, Stability of web plates in W-shape columns accounting for flange/ web interaction, Thin-Walled Struct. 47 (6) (2009) 768–775. [5] M. Seif, B.W. Schafer, Local buckling of structural steel shapes, J. Constr. Steel Res. 66 (10) (2010) 1232–1247. [6] C.H. Lee, K.H. Han, C.M. Uang, D.K. Kim, C.H. Park, J.H. Kim, Flexural strength and rotation capacity of I-shaped beams fabricated from 800-MPa steel, J. Struct. Eng. 139 (6) (2013) 1043–1058. [7] S.P. Timoshenko, J.M. Gere, Theory of Elastic Stability, Dover Publications, Inc, N.Y., 1961. [8] AISC, Specification for the Design, Fabrication & Erection of Structural Steel for Buildings. American Institute of Steel Construction, Inc.,1978. [9] J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, Inc, NJ, 2002. [10] W. Ritz, Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik, Journal Für Reine und Angewandte Mathematik 135 (1908) 1–61. [11] M. Levy, Sur l’équilibre élastique d’une Plaque rectangulaire, Comptes Rendus Acad. Sci. Paris 129 (1899) 535–539. [12] Á. Nádai, Die Formänderungen und die Spannungen von rechteckigen elastischen Platten, Jerry Springer, 1915. [13] ABAQUS. Abaqus user's manual 6.10. Rhode Island, USA: Dassault Systèmes Simulia Corp., 2010.
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Full length article
Elastic flange local buckling of I-shaped beams considering effect of web restraint Kyu-Hong Han, Cheol-Ho Lee n Dept. of Architecture and Architectural Engineering, Seoul National University, Seoul 151-742, Korea
art ic l e i nf o
a b s t r a c t
Article history: Received 24 December 2015 Received in revised form 28 March 2016 Accepted 3 April 2016
The 2010 AISC flexural strength of I-shaped beams with slender flange was proposed based on Johnson (1985)’s experimental plate bucking coefficient (kc). However, our close examination revealed that most of I-beam specimens tested by Johnson to derive kc had section configuration of slender web and noncompact flange, indicating the inappropriateness of the test database itself and probable underestimation of web restraint effect in the AISC codification. In this paper, the effect of web slenderness on the elastic flange local buckling is analytically investigated using the mixed variational approach, and more accurate but still practical formula is proposed. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Flange local buckling I-shaped beams Slender flange Web slenderness Mixed variational principle
1. Introduction The use of high-strength steels in building construction increases the possibility of designing plate elements which exceed the noncompact slenderness limit ( λr ), thus often requiring the check for elastic local buckling. Local buckling of plate elements of a structural shape is interactive with adjacent elements. Several researchers investigated the effect of flange-web interaction on local buckling of plate elements of various structural shapes. For example, Johnson [1] performed an experimental program on I-shaped beams subjected to uniform moment with focusing on the effect of web slenderness on elastic flange local buckling (FLB) and provided the basis of the relevant provisions in the 2010 AISC Specification [2]. Johnson's study will be critically reviewed in the later section of this paper. Cohen [3] proposed approximate elastic buckling coefficients of a T-shaped section under uniform bending using finite difference method. Bedair [4] investigated the influence of the rotational and in-plane translational restraints on web buckling of wide flange columns. However, the majority of previous studies focused on the web local buckling. The research for elastic FLB of I-shaped beams considering web interaction is very rare and difficult to find in the literature except Johnson's research. Seif and Schafer [5] used finite strip method to derive simplified empirical expressions for various plate buckling coefficients of all the AISC hot-rolled shapes with considering elastic web-flange n
Corresponding author. E-mail address: [email protected] (C.-H. Lee).
http://dx.doi.org/10.1016/j.tws.2016.04.001 0263-8231/& 2016 Elsevier Ltd. All rights reserved.
interaction. Their elastic FLB coefficient was developed through a simple conversion of the buckling coefficient of web. However, because their derivation was based on the AISC section database only, the general applicability of their empirical formula may be limited if shapes built up in the geometric proportions different from the AISC section database are concerned. Recently, flexural tests on full-scale I-shaped beams, built up from high- and ordinary-strength steels were carried out by the authors to study the effect of flange slenderness on flexural strength and rotation capacity [6]. As shown in Fig. 1 and Table 1, large flexural overstrength in the three specimens with slender or almost slender flange was observed under moment gradient loading. The following three reasons were mentioned in our previous study to explain the large overstrength observed. First, as can be seen in Fig. 2(a), FLB does not occur at the mid-point of the beam (or at the point of maximum moment) because of the presence of the transverse stiffeners. Thus, FLB has to occur on both side of the mid-span of the beam. A numerical bifurcation buckling analysis of the slender flange specimen showed that the presence of transverse stiffeners tends to induce the antisymmetric FLB mode whose buckling strength is higher than that of the symmetric mode which is expected when transverse stiffeners are absent. Second, after the buckling, the beam can resist additional load due to the membrane action of the severely buckled flange (see Fig. 2 (a)). Third, the elastic buckling resistance of slender flange in the AISC Specification is based on the buckling strength of plates subjected to uniform stress distribution along the plate length. However, as is wellknown in the classical stability theory, plate elements subjected to linearly-varying stress distribution along the plate length under
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K.-H. Han, C.-H. Lee / Thin-Walled Structures 105 (2016) 101–111
ratio v to be 0.3, the 2010 AISC nominal flexural strength of I-shaped sections with slender flange (Mn) is essentially the same as that based on the classical solution [Eq. (1)] as shown below:
Mn = Fcr Sx = kc
π 2E Sx 12 (1 − ν 2)(b/t f )2
⎛ 0.9E ⎞ = k c ⎜ 2 ⎟ Sx ⎝ λ ⎠
where Fcr is the critical FLB stress, Sx is the elastic section modulus, λ is the plate slenderness parameter ( λ ¼ b /t f ), tf is the flange thickness, and kc is the plate buckling coefficient for slender unstiffened elements. The plate buckling coefficient kc is given as Eq. (3) in the 2010 AISC Specification.
λ Fig. 1. Comparison of experimental and AISC nominal FLB strength.
4
kc = Table 1 Overstrength observed in slender or almost slender flange specimens [6]. Specimen designation
Section class Normalized strength Mm/Mp
Mm/Mn
SM490-S-LPD-3
Slender
0.86
1.69
HSB800-NC-LP-3
Noncompact 0.99
1.55
HSA800-S-LPD-3FHS
Slender
2.04
0.82
Failure mode
Flange local buckling Flange local buckling Flange local buckling
Note: Mm ¼ maximum experimental flexural strength; Mp ¼ plastic moment; Mn ¼ AISC nominal flexural strength.
moment gradient have higher local buckling resistance than those under uniform moment. Fig. 2(b) illustrates the effects of the presence of transverse stiffeners and the moment pattern on the elastic FLB strength of slender flange based on the numerical buckling analysis results. The authors felt that the elastic FLB provisions in the 2010 AISC Specification [2] are too conservative and that further thorough study to improve the accuracy of the relevant specification formula is warranted.
2. Review of the 2010 AISC FLB provisions An I-shaped section can be viewed as an assemblage of flat plates. The flange plate is an unstiffened element because one edge is not supported. From the classical plate theory [7], the elastic critical buckling stress under uniform axial force (Fcr) can be expressed as
⎡⎣ 0.35 ≤ k c ≤ 0.76⎤⎦
4.05 0.46
(4)
Comparison of Eqs. (3) and (4) shows that Eq. (3) in the 2010 AISC Specification is just a slight modification of Eq. (4) proposed by Johnson. The problem associated with basing Johnson's formula [Eq. (4)] in the 2010 AISC Specification is discussed in the below. Table 3 summarizes the flange and web section classification according to the 1978 and 2010 AISC Specification ([8,2]). Fig. 5 shows the web and flange slenderness distributions of Johnson's test specimens. Table 4 summarizes the flexural strength calculation with considering FLB according to the 2010 AISC Specification. It should be noted that Eq. (2) mentioned above (or Eq. F3-2 in Table 4) should be used for I-shaped beams with slender flange and compact or noncompact web. When web is slender, the bending strength reduction factor Rpg (Eq. F5-6) should be multiplied to account for web instability effect (see column 3 in Table 4). As can be seen in Table 3, the limiting values of λr for flange and web under flexural compression were relaxed somewhat during the revision from the 1978 AISC Specification [8] to
4.5
(1)
where b is the width of plate, t is the plate thickness, E is Young's modulus, v is Poisson's ratio, k is the plate buckling coefficient, and ⎡ ⎤ Et 3 D is the plate flexural rigidity ⎢D = ⎥. ⎣ 12 (1 − v 2) ⎦ In an I-shaped section, the plate buckling coefficient k represents the rotational restraint provided by the web at the flangeweb junction (see Fig. 3). The minimum value of k for a plate with one edge pinned and the other free is 0.425, and that for one edge fixed and the other free is 1.277 (see Table 2). Taking Poisson's
(3)
( h/tw )
the 2010 AISC Specification [2]; 0.56
π 2E 12 (1 − ν 2)(b/t )2
π 2D =k 2 b t
h/tw
where h is the clear distance between flanges and tw is the thickness of web. As can be seen in Eq. (2), the coefficient kc in the 2010 AISC Specification is essentially the same as the plate buckling coefficient k in Eq. (1). Note that the lower limit of kc is 0.35 (or smaller than 0.425 corresponding to one edge pinned and the other free), implying the case of “negative” web restraint. The factor kc was originally proposed by Johnson [1] to account for the interaction of flange and web local buckling. Johnson tested 19 wide flange beams having high web and high flange slenderness, and proposed the following buckling coefficient through regression of his test results (see Fig. 4).
k=
Fcr = k
(2)
E Fy
⇒ 5.7
E Fy
E Fy
⇒ 0.95
kc E FL
for flange and
for web, respectively.
It should be noted from Fig. 5 that majority of Johnson's test specimens belonged to slender web-slender flange combinations according to the 1978 AISC Specification as he intended, but now most of them fall into noncompact or compact flange and slender web combinations according to the 2010 AISC Specification. It is evident that the test database (and the resulting value of kc) is inappropriate for use within the context of the 2010 AISC Specification. In order to be consistent with the 2010 AISC Specification, the test database should be of slender flange and compact or noncompact web combination (see F3–2 in Table 4). It is not reasonable to apply the value of kc derived from the test database of
K.-H. Han, C.-H. Lee / Thin-Walled Structures 105 (2016) 101–111
SM490-S-PD-3
103
HSA800-S-PD-3-FHS
(a) Photos showing flange local buckling
Normalized Moment (M/M )
1.4 1.2
1.0 0.8 0.6 0.4 0.2 0.0
0
2
4 Normalized End Rotation (θ/θ )
6
8
(b) Effects of the presence of transverse stiffeners and moment pattern on elastic FLB Fig. 2. Behavior of specimens with slender section [6].
Beam-web junction (restraint provided by the web) Free-end Half-wave length of FLB Fig. 3. Rotational restraint provided by web at flange-web junction.
compact or noncompact flange and mostly slender web combinations to the design of slender flange and compact or noncompact web combinations. It is highly probable that the equation F3–2 in Table 4 [or Eq. (2) in the above] would underestimate the web restraint effect on the flange, leading to conservative prediction of FLB strength as observed in our previous testing (see Fig. 1 and Table 1). In the following, the mixed variational approach that can explicitly consider the web restraint effect is developed to derive more accurate but still practical FLB strength equation. The focus of this study is to
Table 2 Plate buckling coefficient (k) depending on boundary conditions. Boundary conditions of unloaded edges
Buckled shape
k
One edge simply supported, the other free
0.425
One edge fixed, the other free
1.277
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⎛ ∂ 2w ∂ 2w ⎞ Mx = − D ⎜ 2 + v 2 ⎟ ⎝ ∂x ∂y ⎠ ⎛ ∂ 2w ∂ 2w ⎞ My = − D ⎜ 2 + v 2 ⎟ ⎝ ∂y ∂x ⎠ Mxy = − D (1 − v)
Fig. 4. Regression equation proposed by Johnson.
Table 3 Flange and web section classification according to the 1978 and 2010 AISC Specification. Element Loading condition
1978 AISC
λp Flange
Flexural compression
Web
Flexural compression
a
Note: Kc =
4 h
E Fy
640/ Fy (ksi) = 3.76
λp
λr
65/ Fy (ksi) = 0.38
2010 AISC
E Fy
95/ Fy (ksi) = 0.56
0.38
E Fy
3.76
E Fy
0.95
kc E a FL
5.70
E Fy
E Fy
760/ Fy (ksi) = 4.5
λr
E Fy
; FL ¼ 0.7Fy.
tw
Q x= −
∂ ⎛ ∂ 2w ∂ 2w ⎞ ⎟ D⎜ 2 + ∂x ⎝ ∂x ∂y2 ⎠
Q y= −
∂ ⎛ ∂ 2w ∂ 2w ⎞ ⎟ D⎜ 2 + ∂y ⎝ ∂x ∂y2 ⎠
(5)
(6)
where Mx and My are the flexural moments per unit length, Mxy is the torsional moment per unit length, Q x and Q y are the transverse shear forces, and w is the transverse deflection. Applying static equilibrium condition in the z direction for all the forces shown in Fig. 6(b) and Fig. 7 gives the well-known buckling equation for plates subjected to uniform compressive force Nx.
∂ 2Mxy ∂ 2My ∂ 2Mx ∂ 2w + + = Nx 2 2 2 2 ∂ ∂ x y ∂x ∂y ∂x
(7)
It is mathematically very difficult to solve Eq. (7) directly when boundary conditions are complicated. Weighted-residual methods used in this study seek for approximate solutions through weaker formulation of the problem, or based on the weighted-integral statement of the form as shown in Eq. (8):
∫ WR
Fig. 5. Web and flange slenderness distributions of Johnson's test specimens.
∂ 2w ∂x∂y
dxdy = 0
(8)
where W is the weight function and R is the residual. In variational calculus, the weight function in the boundary expression is often termed the primary variable, and the coefficient of the weight function in the boundary expression is called the secondary variable. A mixed variational formulation is defined as one where secondary variables of the conventional formulation are also treated as dependent variables along with the primary variables [9]. This formulation can consider various boundary conditions with less difficulty and gives more accurate results than conventional formulations (like the principle of minimum total potential energy) for the force variables. Considering this advantage, the weighted-residual method with mixed variational principle is adopted for this study. The basic idea of mixed variational principle can be expressed as follows:
δΠm = δ u Πm + δN Πm = 0 propose FLB strength equation that can cover compact or noncompact web and slender flange combination because the 2010 AISC Specification lacks the technical background as discussed above. The study of the reliability of Eq. F5–7 in Table 4 (or the case where slender web is involved) is beyond the scope of this paper.
3. Mixed variational formulation for elastic FLB Fig. 6 shows the notations and the sign conventions for stress and stress resultants used in this paper. The stress resultants per unit length of a plate can be expressed as follows according to the classical thin plate theory [7]:
or δ u Πm = 0 and δN Πm = 0
(9)
where
δ u Πm =
∫ W1R1 dxdy and δN Πm = ∫ W2 R2 dxdy
(10)
where δu Πm is the virtual work done by virtual deflection, δN Πm is the virtual work done by virtual force, W1 and W2 are weight functions, and R1 and R2 are residuals. In our plate buckling problem, W1 = δw , W2 = δM , and R1 ¼ the residual of the buckling equation [Eq. (7)], and R2 ¼ the residual of the moment-deflection relations [Eq. (5)]. Mixed variational formulation to calculate elastic FLB strength of I-shaped sections is presented in the below.
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105
Table 4 Summary of flexural strength calculation considering FLB according to the 2010 AISC Specification. Flange
Web
C
C (1) N.A.
NC
NC (2) N.A.
Mn = Mp − (Mp − 0.7Fy Sx )(
λ − λpf λrf − λpf
) (F3–1)a
S (3) N.A.
Mn = Rpg Fcr Sxc (F5–7)a
⎛ λ − λpf ⎞ ⎟ (F4–13)a Mn = Rpc Myc − Rpc Myc − FL Sxc ⎜ ⎝ λrf − λpf ⎠
(
)
Rpg = 1 −
⎛h aw ⎜ c − 5.7 1200 + 300aw ⎝ tw
⎞
E ⎟ ≤ 1.0 Fy ⎠
(F5–6)a
⎛ λ − λpf ⎞ a Fcr = [Fy − (0.3Fy ) ⎜ ⎟ ] (F5–8) ⎝ λrf − λpf ⎠
S
Mn =
0.9Ekc Sx λ2
Mn = Rpg Fcr Sxc (F5–7)a
(F3–2)a
Rpg = 1 −
Fcr =
aw 1200 + 300aw
0.9Ekc ⎞2 ⎟ ⎟ ⎠
⎛b ⎜ f ⎜ 2t ⎝ f
⎛h ⎞ ⎜ c − 5.7 E ⎟ ≤ 1.0 (F5–6)a Fy ⎠ ⎝ tw
(F5–9)a
N. A.: the limit state of FLB does not apply a
Equation numbers in the 2010 Specification.
3.1. Weak forms
δ u Πm =
Because the condition of torsional moment Mxy is not well defined at the boundaries of a flange plate, the weak forms of plate equations [Eqs. (5) and (7)] are derived based on two force variables Mx and My, and one geometric variable w after eliminating Mxy . By using the torsional moment-deflection relation in Eq. (5), Eq. (7) can be expressed as
= −
∫Ω
(11)
⎡ Et f 3 ⎤ where Df is the flexural rigidity of the flange ⎢Df = ⎥, tf is 12 (1 − v 2) ⎦ ⎣ the flange thickness, and wf is the transverse deflection of the flange. Then, according to Eq. (9), the virtual work done by virtual deflection ( δu Πm ) should vanish.
0
⎡ 2 ∂ 4δwf ∂ 2My ∂ 2wf ∂ Mx δwf ⎢ − 2Df (1 − ν ) 2 2 + − Nx ⎢⎣ ∂x 2 ∂x ∂y ∂y 2 ∂x 2
⎤ ⎥ dxdy = 0 ⎥⎦
(12)
where Ω0 represents the total domain of the half flange. Integrating Eq. (12) by parts yields δ u Πm =
⎡ ∂ 2M ∂ 4wf ∂ 2My ⎤ ∂ 2wf x ⎥ = − Nx −⎢ − 2Df (1 − v) 2 2 + 2 2 ⎢⎣ ∂x ∂x ∂y ∂y ⎥⎦ ∂x2
∫ W1R1dxdy
⎡ ∂δw
∫Ωe ⎢⎢
f
⎣ ∂x
−
∂ 2δwf ∂ 2wf ⎤ ∂δwf ∂My ∂Mx ⎥ dxdy + 2Df (1 − ν ) + ∂x∂y ∂x∂y ⎥⎦ ∂y ∂y ∂x
⎧ ⎪
⎡
⎩
⎣
⎛
⎞
⎛ ∂ 3w ⎞⎤ ⎛ ∂My ⎞ ∂ 3wf f n ⎟⎥ ⎟ n y − Df (1 − ν ) ⎜⎜ 2 n y + 2 x⎟⎥ ∂y ⎠ ∂x∂y ⎝ ∂x ∂y ⎠⎦
∮Γe ⎨⎪ δwf ⎢⎢ ⎜⎝ ∂∂Mxx ⎟⎠ nx + ⎜⎝
⎫ ⎡ ∂ 2w ⎛ ∂δw ∂δwf ⎞ ⎤ ⎪ f f +Df (1 − ν ) ⎢ n y ⎟ ⎥ ⎬ ds nx + ⎜ ⎢⎣ ∂x∂y ⎝ ∂y ⎪ ∂x ⎠ ⎥⎦ ⎭ −
∫Ωe
∂δwf ∂x
Nx
∂wf ∂x
dxdy +
∮Γe δwf Nx
∂wf ∂x
nx ds = 0
(13)
where Γe is the boundary of the domain Ω0 , Ωe is the domain except the boundary Γe ( Ω0 = Ωe + Γe ), and nx and ny are the direction cosines of the unit normal. Similarly, according to Eq. (10), the total virtual work done by
Fig. 6. Notations and sign conventions for stresses and stress resultants.
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K.-H. Han, C.-H. Lee / Thin-Walled Structures 105 (2016) 101–111
Fig. 7. Compressive forces in bent position.
the virtual force should vanish. In this mixed variation approach, the curvature residual is computed by using the alternate form of the moment-deflection relations given in Eq. (5) as follows:
1 ∂ 2w = − (Mx − vMy ) D (1 − v2) ∂x2
δN Πm (δMy ) = = −
⇒
∂ 2w 1 = − (My − vMx ) D (1 − v2) ∂y2
(14)
∫Ω
∫ δMy R2 dxdy
⎡ ⎡ ∂ 2w ⎤⎤ 1 f ⎢ δMy ⎢ + (My − vMx ) ⎥ ⎥ dxdy = 0 2 2 ⎢ ⎥⎦ ⎦⎥ Df (1 − v ) 0 ⎢ ⎣ ∂y ⎣
⎡ ∂w ∂δM ⎤ 1 f y − δMy (My − vMx ) ⎥ dxdy 2 ⎥⎦ Df (1 − v ) ⎣ ∂y ∂y ∂wf δMy ny ds = 0 Γe ∂y
∫Ωe ⎢⎢ −
∮
(16)
Thus, for variation of Mx :
δN Πm (δMx ) =
3.2. Consideration of boundary conditions
∫ δMx R2 dxdy
= −
∫Ω
⎡ ⎤ ∂ 2wf 1 ⎢ δMx [ + (Mx − vMy )] ⎥ 2 2 ⎥⎦ ∂x Df (1 − v ) 0 ⎢ ⎣
dxdy = 0 ⎡ ∂w ∂δM ⎤ 1 f x ⎢ ⇒ − δMx (Mx − vMy ) ⎥ 2 ⎥⎦ Ωe ⎢ Df (1 − v ) ⎣ ∂x ∂x ∂wf δMx dxdy − nx ds = 0 Γe ∂x
∫
∮
(15)
The flange is rotationally restrained at the flange-web junction (y¼0) (see Fig. 8). This rotational restraint as well as other geometric and force boundary conditions along the four edges of a buckled flange should be incorporated properly in integrating the work equations given in Eqs. (13), (15), and (16). For the restrained edge ( y = 0): The following geometric and force boundary conditions should be satisfied:
wf = 0,
My = − ζ θ = − ζ
Similarly, for variation of My :
∂wf ∂y
(17)
where ζ is the web rotational stiffness (per unit length) for the half
Fig. 8. Boundary conditions of the half flange of I-shaped beams.
K.-H. Han, C.-H. Lee / Thin-Walled Structures 105 (2016) 101–111
flange of an I-shaped beam, θ is the rotation of flange-web junction (see Fig. 8b). Then, the term
( )n ∂My
in the integral expression along the
y
∂y
boundary (y¼0) in Eq. (13) should be modified as:
⎡ δwf ⎢ ⎣
( )n + ( ) ∂Mx ∂x
⎡ ⇒ δwf ⎢ ⎣
∂My
x
∂y
( )n ∂Mx ∂x
x
⎛ ∂ 3wf ny − Df (1 − ν ) ⎜ 2 ny + ⎝ ∂x ∂y
∂wf
∂wf ny ∂y
y=0
∂y
i=1
⎛ y⎞ ⎛ y ⎞5⎤ πx ⎛⎜ y ⎟⎞ ⎡ ⎢ α1 + α2 ⎜ ⎟ + ⋯ + α6 ⎜ ⎟ ⎥ = sin ⎝ b⎠ ⎝ b ⎠ ⎥⎦ a ⎝ b ⎠ ⎢⎣
∂ 3wf ∂x∂y
2
⎞⎤ nx ⎟ ⎥ ⎠⎦
y=0
9
Vy = 0
(18)
πx y i ( )] a b
and My(x, y) are assumed as a product of sine function and polynomial function as follows:
y=0
For the free edge ( y = b): The following force boundary conditions should be satisfied.
and
(20)
are the basis function of the assumed deflection. In order to satisfy the force boundary conditions ( Mx x = 0 = 0, Mx x = a = 0, and My y = b = 0), the approximate functions for Mx(x, y)
Mx (x, y) =
My = 0
i=1
πx ⎛⎜ y ⎞⎟ a ⎝ b⎠
where αi is the unknown deflection parameter, and ψi1[= sin
y=0
ny in Eq. (16) should be modified as:
1 ⇒ δMy My ny ζ
i
6
∑ αi ψi1 = ∑ αi sin
∂ 3wf
⎛ ⎛ ∂ 3wf ∂ 2wf ⎞ + ⎜ −ζ 2 ⎟ ny − Df (1 − ν ) ⎜ 2 ny + ∂y ⎠ ⎝ ⎝ ∂x ∂y
Similarly, the term
δMy
⎞⎤ nx ⎟ ⎥ ∂x∂y 2 ⎠⎦
6
wf (x, y) =
107
9
∑ βj ψ j2 = ∑ βj sin j=1
j=1
πx ⎛⎜ y ⎞⎟ a ⎝ b⎠
j
⎛ y⎞ ⎛ y ⎞8⎤ πx ⎛⎜ y ⎞⎟ ⎡ ⎢ β1 + β2 ⎜ ⎟ + ⋯ + β9 ⎜ ⎟ ⎥ = sin ⎝ ⎠ ⎝ ⎠ ⎝ b ⎠ ⎥⎦ a b ⎢⎣ b
(21)
where Vy is the effective shear force and defined as
Vy = Q y +
⎡ ∂ 3w ∂Mxy ∂ 3wf ⎤ f + (2 − ν ) 2 ⎥ = 0 = − Df ⎢ 3 ∂x ∂x ∂y ⎥⎦ ⎣⎢ ∂y
where the second term,
∂Mxy ∂x
9
My (x, y) = (19)
, represents additional shearing force
at the edge, produced by the torsional moment Mxy. Note that Eqs. (5) and (6) were used to express the effective shear force in terms of deflection in Eq. (19). It can be shown that the boundary integral term in Eq. (13) for the free edge (y = b) is expressed as follows by using the momentdeflection relations [Eq. (5)] and the effective shear force condition at the boundary [Eq. (19)]:
⎡
∮Γ δwf ⎢ e
⎣
( )n + ( )n ∂Mx ∂x
x
∂My
y
∂y
⎡ ⎛ ∂ 3wf = − ∮ δwf Df ⎢ ⎜ 3 + Γe ⎣ ⎝ ∂x
⎛ ∂ 3wf − Df (1 − ν ) ⎜ 2 ny + ⎝ ∂x ∂y
⎛ ∂ 3wf ∂ wf ⎞ n +⎜ 3 + 2⎟ x ∂x∂y ⎠ ⎝ ∂y 3
⎡ ⎤ ∂ 3wf = ∮ δwf ⎢ Df (1 − ν ) 2 ny ⎥ Γe ∂x ∂y ⎣ ⎦
⎤ ∂ wf ⎞ n ⎥ 2 ⎟ y ∂x ∂y ⎠ ⎦
∂ 3wf ∂x∂y 2
⎞⎤ nx ⎟ ⎥ ⎠⎦
y=b
3
9
∑ γj ψ j3 = ∑ γj sin j=1
j=1
⎞⎛ y ⎞ πx ⎛⎜ y − 1⎟ ⎜ ⎟ ⎠⎝ b ⎠ a ⎝b
⎞⎡ ⎛ y⎞ ⎛ y ⎞8⎤ πx ⎛⎜ y = sin − 1⎟ ⎢ γ1 + γ2 ⎜ ⎟ + ⋯ + γ9 ⎜ ⎟ ⎥ ⎠ ⎢⎣ ⎝ b⎠ ⎝ b ⎠ ⎥⎦ a ⎝b
j−1
(22)
where βj , γj are unknown parameters of Mx and My, and ⎡ ⎡ y j − 1⎤ πx y j ⎤ πx y are the basis functions ψ j2 ⎢= sin a b ⎥ , ψ j3 ⎢= sin a b − 1 b ⎣ ⎦⎥ ⎣ ⎦ of the assumed moment distribution in the x- and y directions, respectively. With assuming the approximate functions of w(x, y), Mx(x, y), and My(x, y) as above [Eqs. (20)–(22)], all the essential boundary conditions required in the mixed variational formulation are satisfied.
()
(
)( )
y=b
3.4. Derivation of web stiffness y=b
Note that the unit normal nx is zero at the boundary y ¼b. Other boundary conditions of wf y = 0 = 0 and My y = b = 0 as required by Eq. (17) and Eq. (18) are properly accounted for when we assume the approximate functions for wf(x, y), Mx(x, y) and My(x, y) in the next section.
When the flange of I-shaped beams is locally buckled with a sine shape, sinusoidal moment is induced at the flange-web junction. For simplicity and conservatism, the web is assumed simply supported along the remaining three sides as shown in Fig. 9. Note that the half of the buckled wave length is denoted as a. Then, the boundary conditions that should be satisfied are as follows:
3.3. Assumed functions We consider a six-parameter Ritz approximation [10] for w(x, y) and nine-parameter Ritz approximations for Mx(x, y) and My(x, y). The number of parameters was determined by considering the analysis time and accuracy compared to some preliminary numerical analysis. The 1st buckling mode of the flange is assumed as a sine shape in the x-direction and the half of buckling wave length is denoted as a. And we express the buckled shape in the ydirection as a polynomial form. Then, the assumed deflection function takes the form given in Eq. (20). Note that the assumed deflection function satisfies all the geometric boundary conditions; wf = 0, wf = 0, and wf y = 0 = 0. x=0
x=a
Fig. 9. Sinusoidal edge moment induced to beam web upon flange local buckling.
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x = 0 , a:
For
stiffness factor Cs needed to calculate ζ as follows.
∂ 2ww =0 ∂x2
ww = 0 ,
Cs =
2
For
y=0:
ww = 0 ,
For
y=h:
ww = 0 ,
∂ ww =0 ∂y2
⎡ ∂ 2w ∂ 2ww ⎤ πx w ⎥ = M (x) = Mf sin − Dw ⎢ +v 2 a ⎣ ∂y ∂x2 ⎦
(23)
where ww is the transverse deflection of the web, Dw is the flexural ⎡ Etw 3 ⎤ rigidity of web ⎢Dw = ⎥, Mf is the amplitude of sinusoidal ⎣ 12 (1 − v 2) ⎦ edge moment induced by flange buckling, and h is the height of web (clear distance between the top and bottom flanges). The governing differential equation of the web in this case is homogeneous and is simply
⎛ ∂ 4w ∂ 4w ∂ 4ww ⎞ w ⎟=0 Dw ⎜ + 2 2 w2 + 4 ∂x ∂y ⎝ ∂x ∂y 4 ⎠
(24)
The general solution of Eq. (24) can be expressed as the product of trigonometric and hyperbolic functions as follows (Lévy [11]; Nádai [12]):
4. Solution of mixed variational formulation and validation The mixed variational equations [Eqs. (13), (15) and (16)] with the assumed functions of w(x, y), Mx(x, y), and My(x, y) [Eqs. (20)– (22)] can be expressed in matrix form as follows.
⎡⎣ K 12⎤⎦ ⎡⎣ K 13⎤⎦ ⎤ ⎥ ⎧ {α}⎫ ⎪ ⎪ ⎡⎣ K 22⎤⎦ ⎡⎣ K 23⎤⎦⎥ ⎨ {β}⎬ = {0} ⎥⎪ ⎪ ⎡⎣ K 32⎤⎦ ⎡⎣ K 33⎤⎦⎥ ⎩ {γ} ⎭ ⎦
⎡ ⎡ K 11⎤ ⎦ ⎢⎣ ⎢ ⎡ K 21⎤ ⎦ ⎢⎣ ⎢ ⎡ K 31⎤ ⎦ ⎣⎣
∞
⎛
a
n= 1
+ Bn
nπ y nπ y nπ y + Cn sinh sinh a a a
nπ y nπy ⎞⎟ nπ x + Dn cosh sin a a ⎠ a
+∮
2 D f (1 − ν )
2 1 ∂ 2ψ 1 ∂ ψ j i dxdy ∂x ∂y ∂x ∂y
⎧ ⎛ ∂ 3ψ 1 ⎤⎫ ⎞ ⎡ ∂ 2ψ 1 ⎛ 1 ∂ 3ψ 1 ⎪ 1 ∂ψ 1 ⎞ ⎥ ⎪ j j ⎜ ∂ψ i j ⎢ i n ⎟ ⎬ ds ⎨ ψ D (1 − ν ) ⎜ n n ⎟ D 1 n ⎜ ∂x 2∂y y + ∂x∂y2 x ⎟− f ( − ν ) ⎢ ∂x∂y ⎜ ∂y x + ∂x y ⎟⎠ ⎥ ⎪ Γe ⎪ i f ⎝ ⎝ ⎦ ⎠ ⎣ ⎩
(25)
where the integration constants An, Bn, Cn and Dn should be determined from the boundary conditions of the web. Applying the boundary conditions specified in Eq. (23) to the general solution above gives the web deflection as Eq. (26):
⎡ ⎛ y⎞ ⎛ y ⎞⎤ h y 1 M (x) ⎢ cothπc sinh ⎜⎝ πc ⎠⎟ − cosh ⎝⎜ πc ⎠⎟ ⎥⎦ h h h Dw 2πc sinh πc ⎣ (26)
(30)
where Ωe
∑ ⎜⎝ An cosh nπy
(29)
Note that the factor Cs is a function of the effective aspect ratio of the web ( c = h/a ) (see Fig. 10). The half of the buckled wave length, a, is still unknown and has yet to be determined as discussed in the next section.
K ij11 = ∫
ww (x, y) =
2πc sinh2 πc (sinh πc cosh πc − πc )
⎭
∂ψ 1 ∂ψ 1 j i N dxdy + −∫ Ω e ∂x x ∂x
∂ψ 1 ∮ Γ ψi1Nx j nx ds, i, j = 1, 2, ⋯ , 6, e ∂x
⎧ ⎪
⎛ ∂ψ 2 ⎞ ⎫ j ⎟ ⎪ nx ⎬ ds i = 1, 2, ⋯ , 6; j = 1, 2, ⋯ , 9, ∂x ⎟ ⎪ ⎠ ⎭ ⎝
K ij12 = ∫
2 ∂ψ 1 ∂ψ j i dxdy − Ω e ∂x ∂x
∮ Γ ⎨ ψi1 ⎜⎜
3 ∂ψ 1 ∂ψ j i K ij13 = ∫ dxdy − Ω e ∂y ∂y
∮ Γ ⎨ ψi1 ⎜⎜
e⎪
⎩
⎧ ⎪
e⎪
⎩
⎛ ∂ψ 3 ⎞ ⎫ j ⎟ ⎪ nx ⎬ ds i = 1, 2, ⋯ , 6; j = 1, 2, ⋯ , 9, ∂y ⎟ ⎪ ⎠ ⎭ ⎝
2
ww =
where c = h/a , the effective aspect ratio of the web. The slope at the flange-web junction is obtained by differentiation of Eq. (26) as follows:
θ=
∂ww ∂y
= y=h
h −1 M (x) [πc − sinh πc cosh πc ] Dw 2πc sinh2 πc
(27)
Because web restrains “two” half flanges in I-shaped beams, ζ (¼ the web rotational stiffness per unit length for the half flange of an I-shaped beam) should be defined as follows.
ζ =
1 M (x) 1 D = Cs w 2 θ 2 h
(28)
Inserting Eq. (27) into Eq. (28) gives the non-dimensional web
Kij21 = ∫ Ω
1
∂ψ i2 ∂ψ j e
dxdy − ∮ ψi2 Γ
∂x ∂x
Kij22 = − ∫ Ω
1 2 e D f (1 − v )
v
Kij23 = ∫ Ω
2 e D f (1 − v )
Kij31 = ∫ Ω
e
1
∂ψ i3 ∂ψ j
Kij32 = ∫ Ω
2 e D f (1 − v )
∂ψ1j ∂y
ny ds i = 1, 2, ⋯, 9; j = 1, 2, ⋯, 6,
ψi3 ψ j2 dxdy i, j = 1, 2, ⋯, 9,
v 2 e D f (1 − v )
ψi3 ψ j3 dxdy i, j = 1, 2, ⋯, 9.
The order of the matrix in Eq. (30) is 24 24. The critical FLB strength Nx, min should be obtained through numerical eigenvalue analysis of Eq. (30) after incorporating the boundary conditions and ζ in integration for a problem at hand. Or, for a fixed value of ζ , a series of eigenvalue analyses should be performed to find the smallest minimum (critical) buckling compressive force Nx, min by varying the half of buckling wave length a. Once Nx, min is obtained, the FLB coefficient k min is calculated per Eq. (31) [refer to Eq. (1)].
⎡ 12 (1 − ν 2)(b/t )2N ⎤ f x, min ⎥ k min = ⎢ 2 ⎢⎣ ⎥⎦ π Et f
Fig. 10. Variation of Cs depending on the effective web aspect ratio (c).
nx dsi = 1, 2, ⋯, 9; j = 1, 2, ⋯, 6,
ψi2 ψ j3 dxdy i, j = 1, 2, ⋯, 9,
e
v
∂x
ψi2 ψ j2 dxdy i, j = 1, 2, ⋯, 9,
dxdy − ∮ ψi3 Γ
∂y ∂y
Kij33 = − ∫ Ω
∂ψ1j
e
(31)
In order to validate the accuracy of the mixed variational method (MVM) developed in this study and also derive a simplified formula, a total of thirty five I-shaped beams with slender flange and diverse web configuration covering seismic compact, compact, and noncompact web slenderness were first created (see Table 5). These sections were analyzed by using the MVM of this
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Table 5 I-shaped sections analyzed and comparison of flange local buckling coefficients. Model designation
Depth, H (mm)
Width, bf (mm)
Web thickness, tw (mm)
Flange thickness, tf (mm)
FEM AISC
k
FEM-UM-1 FEM-UM-2 FEM-UM-3 FEM-UM-4 FEM-UM-5 FEM-UM-6 FEM-UM-7 FEM-UM-8 FEM-UM-9 FEM-UM-10 FEM-UM-11 FEM-UM-12 FEM-UM-13 FEM-UM-14 FEM-UM-15 FEM-UM-16 FEM-UM-17 FEM-UM-18 FEM-UM-19 FEM-UM-20 FEM-UM-21 FEM-UM-22 FEM-UM-23 FEM-UM-24 FEM-UM-25 FEM-UM-26 FEM-UM-27 FEM-UM-28 FEM-UM-29 FEM-UM-30 FEM-UM-31 FEM-UM-32 FEM-UM-33 FEM-UM-34 FEM-UM-35
400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 753 753 753 753 753 753 753 753 753 753 911 911 911 911 911 911 911 911
650 650 650 650 650 650 650 650 650 650 650 650 650 650 200 200 200 265 265 265 265 530 530 530 530 530 530 305 305 305 305 610 610 610 610
23 (SC) 17.6 (SC) 12 (SC) 8 (SC) 20 (SC) 17.6 (SC) 15 (SC) 10 (SC) 8 (SC) 20 (SC) 17.6 (SC) 15 (SC) 12 (SC) 8 (SC) 8 (SC) 6 (C) 5 (C) 13 (SC) 11 (C) 9 (C) 7 (NC) 13 (SC) 11 (C) 9 (C) 9 (C) 7 (NC) 5 (S) 14 (C) 12 (C) 10 (C) 8 (NC) 16 (SC) 16 (SC) 12 (C) 8 (NC)
23 23 23 23 17.6 17.6 17.6 17.6 17.6 12 12 12 12 12 4 4 4 5 5 5 5 9 9 9 7 7 7 5 5 5 5 10 8 8 8
0.87 0.74 0.61 0.49 0.92 0.86 0.79 0.56 0.52 1.13 1.07 0.99 0.87 0.68 1.19 1.03 0.90 1.35 1.28 1.18 0.98 1.02 0.91 0.79 0.94 0.78 0.63 1.28 1.24 1.17 1.05 1.08 1.18 1.03 0.78
kc ⎛ ⎜= ⎝
MVM
kc/k
m (a/ b)
ζ
kmin
kmin /k
ζ
kmin
kmin /k
0.87 1.03 1.21 1.22 0.83 1.27 0.96 1.18 1.13 0.67 0.71 0.77 0.82 0.85 0.48 0.48 0.50 0.39 0.38 0.37 0.40 0.52 0.54 0.56 0.47 0.50 0.55 0.39 0.37 0.36 0.36 0.50 0.45 0.45 0.48
2.3 2.8 3.6 4.8 2.2 2.4 2.6 3.4 4.0 1.9 2.0 2.1 2.4 3.1 1.8 2.1 2.3 1.7 1.8 1.9 2.2 2.1 2.4 2.7 2.3 2.7 3.5 1.7 1.8 1.8 2.0 2.0 1.8 2.1 2.7
0.56 1.36 4.57 16.17 0.38 0.58 0.97 3.57 7.23 0.11 0.17 0.29 0.59 2.21 0.07 0.20 0.37 0.03 0.05 0.10 0.26 0.22 0.41 0.84 0.34 0.85 2.89 0.02 0.04 0.07 0.16 0.16 0.07 0.20 0.85
0.84 0.71 0.59 0.51 0.90 0.83 0.76 0.61 0.55 1.09 1.03 0.95 0.83 0.65 1.14 1.01 0.91 1.21 1.17 1.10 0.98 0.99 0.90 0.78 0.93 0.78 0.63 1.22 1.19 1.14 1.05 1.05 1.14 1.01 0.78
0.97 0.96 0.96 1.05 0.97 0.97 0.95 1.08 1.06 0.96 0.96 0.96 0.96 0.96 0.96 0.99 1.01 0.90 0.91 0.93 1.00 0.97 0.99 1.00 0.99 1.00 1.00 0.96 0.96 0.97 1.00 0.97 0.97 0.98 1.00
0.62 1.43 4.64 15.98 0.43 0.63 1.04 3.60 7.10 0.13 0.20 0.33 0.65 2.23 0.12 0.28 0.48 0.05 0.09 0.16 0.35 0.30 0.51 0.93 0.43 0.93 2.56 0.04 0.07 0.12 0.23 0.22 0.11 0.27 0.93
0.81 0.71 0.58 0.51 0.87 0.81 0.75 0.60 0.56 1.06 0.99 0.91 0.81 0.65 1.08 0.93 0.85 1.16 1.12 1.02 0.90 0.92 0.84 0.76 0.86 0.76 0.64 1.18 1.14 1.08 0.96 0.97 1.08 0.93 0.76
0.94 0.95 0.95 1.05 0.94 0.94 0.94 1.07 1.06 0.93 0.92 0.92 0.93 0.96 0.91 0.91 0.95 0.93 0.93 0.91 0.97 0.90 0.93 0.97 0.92 0.97 1.01 0.92 0.92 0.92 0.92 0.90 0.92 0.91 0.97
⎞ ⎟ h / tw ⎠
0.76 0.76 0.74 0.60 0.76 0.76 0.76 0.66 0.59 0.76 0.76 0.76 0.71 0.58 0.57 0.49 0.45 0.53 0.49 0.44 0.39 0.53 0.49 0.44 0.44 0.39 0.35 0.50 0.46 0.42 0.38 0.54 0.53 0.46 0.38
Simplified formula [per Eqs. (34)–(37)]
4
Note: i) SC ¼ seismically compact; C ¼compact; NC¼noncompact web. ii) Web sections classified with assuming the steel grade of A992 (Fy ¼ 345 MPa).
study and the commercial software ABAQUS [13]. The numerical models of I-shaped beams were modeled using the four-node shell element S4R in ABAQUS. Linear perturbation (or bifurcation buckling) analysis was conducted for the thirty-five I-shaped beams shown in Table 5. Before conducting numerical analysis, the accuracy of the finite element model was validated by comparing with several experimental results available from our previous study [6]. Fig. 2 shows that our numerically obtained FLB strength
(the case of moment gradient with stiffener in Fig. 2) is well compared with experimental strength of specimen HSA800-SLPD-3-FHS. Accurate prediction of elastic FLB strength of I-shaped beams using finite element analysis is not that difficult nowadays. The analysis results are compared in Table 5 and Fig. 11. The effect of web restraint is better evaluated in terms of the relative flexural stiffness ratio between the web and the flange. Thus, we define the non-dimensional coefficient of web restraint ( ζ ) as
ζ≡
Fig. 11. Comparison of kmin values predicted by mixed variational approach and FEM.
Df /b ζ
(32)
where ζ is the web rotational stiffness per unit length for the half flange of I-shaped beams that was already derived as Eq. (28). Theoretically, when the flange is completely fixed at the flangeweb junction, ζ should approach zero, and when the flange is free to rotate, ζ should approach infinity. As shown in Table 5 and Fig. 11, compared to ABAQUS finite element analysis results, our mixed variational method (MVM) gives much more accurate and consistent results than the AISC flange buckling coefficient [Eq. (3)] in a very wide range of ζ . Fig. 11 reveals that the variation of flange buckling coefficient, kmin, can be well described by the “single factor ζ ”.
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kmin values per 2010 AISC and proposed formula
110
Fig. 12. Variation of kmin depending upon ζ .
-
5. Simplified formula proposed Calculating the nominator ( Df /b) in Eq. (32) is an easy task once an I-shaped section is given. However, the effective aspect ratio of the web ( c = h/a ), which is needed to calculate ζ according to Eqs. (28) and (29), is not known until some eigenvalue solutions are available. Thus, the remaining task to derive a simplified formula is reduced to finding the reasonable estimate of the half of buckling wave length (a). Note that the factor Cs is an increasing function of the effective aspect ratio of the web ( c = h/a ) (see Fig. 10). For example, as a becomes longer, Cs becomes smaller. This in turn leads to the decrease of ζ [see Eq. (28)] and the increase of ζ , thus yielding more smaller (more conservative) flange buckling coefficient kmin (see Fig. 11). Based on the mixed variational analysis results for various beam sections, the aspect ratios of buckled flange (m ¼a/b) are also reported in Table 5. Although there are a few exceptional cases, m values are more or less stable and mostly less than 3. And the half of buckling wave length a may be conservatively taken as 3b in computing ζ according to Eqs. (28) and (29) as follows:
h h h c= = ≈ a mb 3b
k values per ABAQUS FEM Fig. 13. Comparison of kmin values per 2010 AISC and proposed formula.
Fig. 14. Bilinear approximation for kmin.
(33)
Based on the non-dimensional coefficient of web restraint ( ζ ) and regression analysis, simplified expressions for FLB coefficient (kmin) are proposed as follows (see Fig. 12) :
0.94ζ −0.072
For ζ ≤ 0.1 :
k min =
For 0.1 < ζ ≤ 4 :
k min = 0.75ζ −0.17
(35)
For 4 < ζ ≤ 16 :
k min = 0.69ζ −0.11
(36)
≤ 1.277
(34)
For ζ ≤ 4 :
k min = − 0.09 log2ζ + 0.76 ≤ 1.277
For ζ > 4 :
k min = − 0.03 log2ζ + 0.64 ≥ 0.425 (39)
(38)
Of course, Eqs. (34)–(37) and (38)–(39) in the above should be used for slender-flange I shaped beams with compact or noncompact webs.
6. Summary and conclusions The results of this analytical and numerical study on FLB of I-shaped beams with slender flange can be summarized as follows.
For ζ > 16 :
k min = 0.59ζ −0.054 ≥ 0.425
(37)
⎡ ⎤D h πc sinh πc , ζ = ⎢ (sinh πc cosh πc − πc ) ⎥ hw , and c = 3b . ζ ⎣ ⎦ The upper cap 1.277 in Eq. (34) and the lower limit 0.425 in Eq. (37) correspond to the two extreme boundary conditions, or one edge fixed and the other free, and one edge pinned and the other free, respectively. As can be seen in the last column in Table 5 and Fig. 13, the simplified expressions proposed above give much more accurate and consistent results than the AISC flange buckling coefficient [Eq. (3)] for a very wide range of ζ . More simple bilinear expressions also appear acceptable for practical purposes [see Eqs. (38)-(39) and Fig. 14]. where ζ ≡
Df / b
2
i) This study found that most of I-beam specimens tested by Johnson to derive the plate buckling coefficient (kc) had section configuration of slender (S) web and noncompact (NC) flange according to the 2010 AISC section classification, indicating the inappropriateness of the test database itself and the probable underestimation of web restraint effect in the AISC codification. ii) The mixed variational approach was used in this study to systematically investigate the effects of web slenderness on elastic FLB of I-shaped beams. The approximate equations for estimating web rotational stiffness at the flange-web junction were also proposed.
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iii) As this study shows, the web restraint effect should be measured in terms of the relative flexural stiffness between the web and the flange as opposed to the buckling coefficient kc in the 2010 AISC Specification which is determined only based on the web slenderness ratio, irrespective of flange flexural stiffness. iv) This study found that the variation of flange plate buckling coefficient can be well described by the single factor ζ (¼ the non-dimensional coefficient of web restraint proposed in this study). Based on this finding and the reasonable approximation of the effective aspect ratio of buckled webs, simplified expressions, more accurate and more consistent than the 2010 AISC equation, were proposed for elastic FLB strength of I-shaped beams.
Acknowledgments Support to this study by the POSCO Affiliated Research Professor Program is gratefully acknowledged.
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wide-flange shapes, in: Proceedings 1985 Annual Technical Session, 1985. [2] AISC. Specification for Structural Steel Buildings. American Institute of Steel Construction, Chicago, IL, ANSI/ASIC 360-10, 2010. [3] J.M. Cohen, Elastic buckling coefficients for long, unstiffened plates, J. Eng. Mech. 118 (12) (1992) 2491–2496. [4] O. Bedair, Stability of web plates in W-shape columns accounting for flange/ web interaction, Thin-Walled Struct. 47 (6) (2009) 768–775. [5] M. Seif, B.W. Schafer, Local buckling of structural steel shapes, J. Constr. Steel Res. 66 (10) (2010) 1232–1247. [6] C.H. Lee, K.H. Han, C.M. Uang, D.K. Kim, C.H. Park, J.H. Kim, Flexural strength and rotation capacity of I-shaped beams fabricated from 800-MPa steel, J. Struct. Eng. 139 (6) (2013) 1043–1058. [7] S.P. Timoshenko, J.M. Gere, Theory of Elastic Stability, Dover Publications, Inc, N.Y., 1961. [8] AISC, Specification for the Design, Fabrication & Erection of Structural Steel for Buildings. American Institute of Steel Construction, Inc.,1978. [9] J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, Inc, NJ, 2002. [10] W. Ritz, Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik, Journal Für Reine und Angewandte Mathematik 135 (1908) 1–61. [11] M. Levy, Sur l’équilibre élastique d’une Plaque rectangulaire, Comptes Rendus Acad. Sci. Paris 129 (1899) 535–539. [12] Á. Nádai, Die Formänderungen und die Spannungen von rechteckigen elastischen Platten, Jerry Springer, 1915. [13] ABAQUS. Abaqus user's manual 6.10. Rhode Island, USA: Dassault Systèmes Simulia Corp., 2010.