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Lateral-torsional buckling analysis of wood composite I-beams with sinusoidal corrugated web
Thin-Walled Structures 119 (2017) 72–82
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Lateral-torsional buckling analysis of wood composite I-beams with sinusoidal corrugated web
MARK
⁎
Pengcheng Jiaoa, , Wassim Borchania, Sepehr Soleimania, Bradley McGrawb a b
Department of Civil and Environmental Engineering, Michigan State University, East Lansing, MI 48824, USA Department of Civil and Environmental Engineering, West Virginia University, Morgantown, WV 26505, USA
A R T I C L E I N F O
A B S T R A C T
Keywords: Critical buckling capacity Composite I-section beam Sinusoidal corrugated web Theoretical model Optimal design Lateral-torsional buckling
Buckling instability refers to a significant limit on the structures with I-section members, which causes severe decreasing of critical buckling load. Many mechanisms and techniques have been developed to increase the loading capacity of I-section structures, e.g., using composite materials or changing web geometry. This study aims at theoretically investigating the effect of geometry on the buckling capacity of wood composite I-beams with sinusoidal corrugated web. Experiments and numerical simulations are carried out to validate the theoretical predictions. The presented model is also reduced to flat web to validate with an existing study. Good agreements are observed from the validations. Parametric study is conducted to indicate the effects of web geometry on the critical buckling load. A significant increasing of buckling load (minimum 17.7%) is obtained between flat and sinusoidal web I-beams. Sensitivity study is carried out to evaluate the influences of web thickness, beam length and beam height on the loading capacity of sinusoidal I-beams. Optimal design is conducted to investigate the impact of the ratios of wavelength-to-beam length ( a ) and wave amplitude-to-flange L
width ( A ) on the critical buckling load and volume of wood corrugated I-beams. The findings in this study can be T further used as guidance for the design of composite I-beams with sinusoidal corrugated web.
1. Introduction Buckling phenomenon has been investigated over centuries since it extensively exists in various types of slender members. Many studies have been conducted to explore the applications of buckling and postbuckling response, namely, energy harvesting for self-powered sensors, damage detection for Structural Health Monitoring (SHM), structural instability of honeycomb plates in nanoscale, etc [1–15]. However, buckling-induced instability is still a severe challenge to the reliability of I-section structures in civil infrastructures. In particular, I-beams are likely to experience failures before reaching ultimate load capacities since thin-walled web tends to buckle in a pattern that combines twisting with lateral bending. This phenomenon, known as lateral-torsional buckling, severely reduces the stiffness and critical buckling capacity of I-beams [16]. The structural failure under consideration in this study is caused by lateral buckling, which is likely to be the failure mode for the slender beams with insufficiently lateral constraints. Lateral buckling typically results in the failures that critical load is much smaller than material's yield capacity. Due to initial eccentricity, buckling in I-beams mainly generates a second-order bending moment about the beams' longitudinal axes, i.e., torsional. Hence, lateral-
⁎
Corresponding author. E-mail address: [email protected] (P. Jiao).
http://dx.doi.org/10.1016/j.tws.2017.05.025 Received 7 January 2017; Received in revised form 11 May 2017; Accepted 25 May 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
torsional buckling is a common failure mode to I-beams [17]. It is of great research and practice interests to increase the buckling capacity of I-beams [18–20]. In order to increase critical buckling load, many research efforts have been dedicated to investigating the buckling response of flat web I-beams made by composite materials, e.g., wood composite, fiber-reinforced polymers (FRP), etc. Davalos and Qiao [21] and Qiao et al. [22] theoretically and experimentally investigated the lateral-distortional and lateral-torsional buckling of FRP wide-flange I-beams. Using an energy method, the authors presented theoretical models to capture the instability of FRP I-beams. Barbero and Raftoyiannis [23] examined the elastic lateral-distortional buckling of FRP I-beams. A plate theory was used to study the distortion behavior, i.e., shear effects and bending-twisting coupling, of plate cross-section. Pandey et al. [24] developed a theoretical model to investigate the lateral buckling strength of thin-walled composite I-beams. The authors used Vlasovtype linear hypothesis to evaluate the coefficients of I-beam stiffness. Hai et al. [25] examined hybrid FRP beams with respect to buckling behavior. Parametric studies were carried out to optimize the mechanical response of the presented I-beams. However, these studies have not taken into account the influence of web geometry on the
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loading capacity of I-beams. To address the research gap and enhance the resistance against the structural instability caused by buckling, many studies have been presented regarding the I-section structures with non-flat web. Elgaaly et al. [26] investigated the failure of corrugated web I-beams under uniform bending. Dubina et al. [27] experimentally studied the coldformed steel beams with trapezoidal steel web and built-up steel flanges. Vacha et al. [28] proposed experiments to examine the thermal and mechanical responses of sinusoidal corrugated web steel beams subjected to high temperatures. Aydin et al. [29] examined the behavior of corrugated web I-beams under cyclic load with respect to beamto-column connections. The authors experimentally compared two types of I-beams, i.e., conventional type with thick flat web in connection zone and alternative type that has stiffeners. Guo et al. [30] studied the in-plane mechanical response of circular steel I-section arches with sinusoidal corrugated web geometry. Thermal behavior of axially restrained corrugated web steel beams were numerically investigated by Wang et al. [31]. The authors studied the effects of different parameters, namely, load ratio, axial resistant stiffness ratio, span-depth ratio, corrugation shape of web, etc. However, these studies are proposed based on isotropic materials, such as steel, which have not considered the advantages of composites. Later on, experiments were carried out by He et al. [32] to study the bending behavior of concreteencased composite beams with corrugated web. A more general review of the mechanical performance of corrugated web composite beams had been carried out by Dayyani et al. [33]. Many theoretical models have been developed to theoretically predict the buckling behavior of composite I-beams with non-uniform web. Lee [34] developed a general geometrically nonlinear model to theoretically examine lateral buckling of composite I-beams with monosymmetric section. The author used systematic variational formulation based on the classical lamination theory. Kim et al. [35,36] developed an approach to obtain exact element stiffness matrix for composite I-beams with symmetric and arbitrary laminations. An energy method was used to solve the stability equations and force-displacement relationship in these studies. However, it is of necessity to theoretically study the lateral-torsional buckling of composite I-beams with corrugated web; especially, lack of parametric study and optimal design have been presented to guide the design of composite I-beams with sinusoidal corrugated web. This study aims at theoretically investigating the effects of geometry on the buckling capacity of wood composite I-beams with sinusoidal corrugated web. In order to achieve the research objective, the work is deployed as follows,
Fig. 1. Schematic illustration of a cantilever I-beam with sinusoidal web under a tip load.
2. Theoretical model 2.1. Problem formulation This section presents lateral-torsional buckling analysis of a cantilever I-beam with sinusoidal corrugated web under a tip load P, as illustratively displayed in Fig. 1. The deflection of the I-beam contains three components, namely, u , v and w . Fig. 2 presents the cross-section deflection of the I-beam under the external force. t , T, h , and H represent the web thickness, top and bottom flanges width, top and bottom flanges thickness, and web height, respectively. δ t , δ b , δ w , and α refer to the lateral displacements of the top and bottom flanges and web, respectively, as well as the sectional rotation of the I-beam after deflection. Since the sinusoidal I-beam consists of three segments, i.e., web, and top and bottom flanges, the superposition principle is applied in this study. For lateral-torsional buckling, the deformation due to bending is negligible [18,21]. Therefore, the displacements of web, top and bottom flanges can be written as, w ⎧uw = 0 v =0 ⎨ ww = δ w (x , z ) = δ + zα ⎩
• Section 2 develops a theoretical model to study the lateral-torsional • • •
buckling of I-beams with sinusoidal corrugated web. The superposition method is used to take into account top and bottom flanges and sinusoidal web. The Rayleigh-Ritz method is used to obtain the critical buckling load of the I-beam. Section 3 presents experimental and numerical studies to validate the theoretical predictions with respect to buckling capacity. In addition, the presented model is reduced to flat web and compared with an existing study. Good agreements are obtained. Section 4 carries out a parametric study to capture the buckling load enhancement between the flat and sinusoidal web I-beams. In addition, sensitivity ratios (Rs ) are t investigated with respect to the web thickness (t), beam length (L), and beam height (H). Section 5 proposes optimal designs. Design factors, i.e., web wavelength-to-beam length ( a ) and web wave amplitude-to-flange width L
•
( A ) ratios, are examined in terms of the buckling load capacity and T volume of I-beams. Section 6 summarizes and discusses the major findings in this study.
Fig. 2. Cross-section deformation of the I-beam.
73
(1)
Thin-Walled Structures 119 (2017) 72–82
P. Jiao et al. t
t
⎧u = −yδ , x ⎪ t v = −yα ⎨ t ⎪ w = δt = δ + ⎩
where Aij and Dij (i, j = 1, 2, 6) are the extensional and bending stiffness matrices, respectively, defined in terms of the reduced stiffness matrix Q as,
H α 2
(2) n
⎧ Aij = ∑k −1 (Qij )k (z k − z k −1) ⎨ Dij = 1 ∑n (Qij ) (z k 3 − z k −13) k −1 k 3 ⎩
and b
b
⎧u = −yδ , x ⎪ b v = −yα ⎨ b ⎪ w = δb = δ − ⎩
(9a)
2
H α 2
E11 ⎧ Q11 = E11 − v12 E12 ⎪ ⎪ Q = v12 E11 E22 12 2 E E11 − v12 22 ⎨ E11 E22 ⎪Q22 = E11 − v 2 E22 12 ⎪ ⎩Q66 = G12
(3)
where subscribes w, t, and b represent web, top and bottom flanges. 2.2. Sinusoidal web The shape function of the sinusoidal web in the x-z plate is given as,
2πx ⎞ y = A sin ⎛ ⎝ a ⎠
The sinusoidal web in the I-beam consists of two strain energy components, namely, the out-of-plane bending energy (Ubw ) and membrane strain energy (Umw ). Therefore, the total strain energy is given as [18],
(4)
where A and a are the amplitude and wavelength, respectively. The lateral displacement of the centroid and sectional rotation are given, considering only the first buckling mode, as [22],
⎧ δ = C1 ⎡1 − cos ⎪ ⎣ ⎨ α = C 1 − cos ⎡ 2 ⎪ ⎣ ⎩
( ) ( ) ⎤⎦ πx 2L
U w = Ubw + Umw
(10)
where
⎤ ⎦
w ⎧Um =
πx 2L
(9b)
⎨Ubw = ⎩
(5)
1 2 1 2
∬A [Nxw εxw + Nzw εzw + Nxzw γxzw ] dA ∬A [D11 χxw 2 +2D12 χxw χzw + D22 χzw 2 + 4D66 χxzw 2 ] dA
(11)
where C1 and C2 refers to two unknown constants. The deflected configuration of the sinusoidal web is displayed in Fig. 3. Therefore, the displacement of the web is given as,
where χxw = w ,wxx , χxzw = w ,wxz , and χzw = w ,wzz . The normal strains εi (i = x , z ) and shear strain γxz of the membrane strains can be written as,
πx ww = δ + zα = (C1 + zC2) ⎡1 − cos ⎛ ⎞ ⎤ ⎝ 2L ⎠ ⎦ ⎣
w ⎧ εx = v , z ⎪ ww εz = u,wx − R ⎨ ⎪ γxz = u,wz + v ,wx ⎩
(6)
Taking Eq. (6) into Eq. (1), the displacements of the web can be obtained. Considering the top flange as a beam element, the strain energy refers to arbitrary orthogonal coordinates, in accordance with the basic approximations of thin-plate theory, can be written as [18],
Uw =
1 2
∬A (σxw εxw + σzw εzw + τxzw γxzw ) dA
where R is the curvature radius of the web mid-surface. Substituting Eqs. (11) and (12) into Eq. (10), the total strain energy of the web can be written as,
(7)
A12 0 A22 0 0 A66 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 D11 D12 D12 D22 0 0
⎤ ⎥ εw w ⎥ ⎛ x κx ⎞ ⎥ ⎜ εzw / κ zw ⎟ 0 ⎥⎜ w w ⎟ γ κ 0 ⎥ ⎝ xz xz ⎠ ⎥ D66 ⎦
1 2
∬A ⎡⎣Nxw v,wz + Nzw (u,wx − wR ) + Nxzw (u,wz + v,wx ) ⎤⎦dsdz
+
1 2
Uw =
According to the orthotropic material property, a laminate in x-z plane consists of the membrane force, bending and twisting moments, which are given in terms of mid-surface in-plane strains and curvatures as,
⎡ A11 ⎢ A12 w w ⎛ Nx Mx ⎞ ⎢ 0 w w ⎜ Nz / Mz ⎟ = ⎢ 0 ⎜Nw Mw ⎟ ⎢ ⎝ xz xz ⎠ ⎢ 0 ⎢ 0 ⎣
(12)
w
∬A [D11 w,wxx 2+2D12 w,wxx w,wzz+D22 w,wzz 2+4D66 w,wxz 2] dsdz
(13)
where w w w w w ⎧u, x = u, z = v, x = v , z = w , zz = 0 ⎪ w2 π4 πx w , xx = 4 (C1 + zC2)2 cos2 2L 16L ⎨ ⎪ w ,wxz 2 = π 2 C22 sin2 πx 2L 4L2 ⎩
( )
( )
(8)
(14)
In order to obtain the total strain energy in Eq. (13), it is of necessity to calculate the curvilinear coordinate (s), curvature radius (R) and membrane forces. Let s be the curvilinear coordinate along the web, the curvilinear coordinate and the curvature radius are expressed as,
⎧ ds = 1 + 4 Aπ 2 cos2 2πx dx a a ⎪ ⎪ 3 2 ⎡1 + 4 Aπ cos2 2πx ⎤2 ⎨ ( a ) ( a ) ⎥⎦ ⎢ ⎪R = ⎣ Aπ 2 2πx 4 ( ) sin ( ⎪ a a ) ⎩
( )
( )
(15)
The membrane forces of the sinusoidal web consists of three comw , that refer to the in-plane normal forces in ponents, i.e., Nxw , Nzw and Nxz x and z directions, as well as in-plane shear force, respectively. Fig. 4 displays the different components of the membrane forces. The membrane forces of the web can be obtained as,
Fig. 3. Deflected configuration of the sinusoidal web.
74
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1
t t ⎧ εx = u, x + 2 (w ,tx 2 + 2 v ,tx 2) ⎪ t 1 1 ε = w ,ty + 2 (u,ty 2 + 2 v ,ty 2) ⎨ y t ⎪ γxy = w ,tx + u,ty + v ,tx v ,ty ⎩
(23)
Substituting Eq. (23) into Eq. (21), ignoring the fourth-order terms, the total strain energy of the top flange is obtained as,
Ut =
1⎡ 2⎢ ⎣
∬A Nxt ⎛⎝w,tx 2 + 12 v,tx 2 ⎞⎠ dA + ∬A Nxyt v,tx v,ty dA + A11 ∬A u,tx dA
Fig. 4. Membrane forces of deformed sinusoidal web.
+ D11
⎦ (24)
Pt
w ⎧ Nx = I w (L − x ) z ⎪ w Nz = 0 (0 < x ≤ L) ⎨ Pt h2 w ⎪ Nxz = − 2I w 4 − z 2 ⎩
(
)
where (16)
2 2
where is the moment of inertia of the web. Substituting Eqs. (15) and (16) into Eq. (13). The total strain energy of the sinusoidal web is obtained as,
16L
H
1 2
∫− H2 ∫0 2
L
( ) ( ) cos ( ) sin ( )
⎨ t 2 y 2 C22π 4 ⎪ v , xx = 16L4 ⎪ 2 2 2 ⎪ v ,txy 2 = C2 π2 4L ⎪ ⎪ (w ,tx + u,ty )2 = 0 ⎩
(17)
where
ξw =
( )
⎧ w ,tx 2 = (2C1 + HC2 2) π sin2 πx 2L 16L ⎪ ⎪ v t 2 = y 2 C22π 2 sin2 πx 2L ⎪ ,x 4L2 ⎪ 2 y2π 4 H 2 t cos2 ⎪ u, x = 4 C1 + 2 C2
Iw
Uw = ξw
∬A v,txx 2dA + A66 ∬A (w,tx + u,ty)2dA + D66 ∬A w,txy2dA⎤⎥
Aπ 2 2πx ⎞ cos2 ⎛ ⎞ dxdz [D11 w ,wxx 2+4D66 w ,wxz 2] 1 + 4 ⎛ ⎝ a ⎠ ⎝ a ⎠
( ) πx 2L
2 πx 2L πx 2L
(25)
The membrane forces of the top flange are given as,
(18)
PHh
⎧ Nxt = 2I t (L − x )(0 ≤ x ≤ L) t ⎨ N yt = Nxy = 0(0 ≤ x ≤ L) ⎩
2.3. Top flange
Substituting Eq. (26) into Eq. (24), the total strain energy of the top flange can be obtained with respect to the external force P as,
Substituting Eq. (5) into Eq. (2), the displacements of the top flange are expressed as, πy − 2L
(
) ( ) ( ) ) ( ) ⎤⎦
H C 2 2
(
U t = Pξ1t + ξ2t
πx 2L
⎧ut = C1 + sin ⎪ ⎪ t πx v = −yC2 ⎡1 − cos 2L ⎤ ⎣ ⎦ ⎨ ⎪ t H 1 − cos ⎪ w = C1 + 2 C2 ⎡ ⎣ ⎩
πx 2L
1 2
∬A (σxt εxt + σyt εyt + τxyt γxyt ) dA
T
⎧ ξ t = ∫ 2 ∫ L Hh (L − x ) w t 2 + 1 v t 2 dxdy ,x 1 − T2 0 4I t 2 ,x ⎪ ⎪ T ⎨ ξ t = 1 ∫ 2T ∫ L [A11 u t + D11 v t 2 + D66 w t 2] dxdy ,x , xx , xy 2 −2 0 ⎪ 2 ⎪ ⎩
(19)
where
)
(28)
is the moment of inertia of the top flange.
It
Taking Eq. (5) into Eq. (3), the displacements of the bottom flange are, πy
(
) ( ) ( ) ) ( )
⎧ub = − 2L C1 − H2 C2 sin πx 2L ⎪ ⎪ b πx v = −yC2 ⎡1 − cos 2L ⎤ ⎣ ⎦ ⎨ ⎪ b H πx = − − w C C 1 cos ⎡ ⎤ 1 2 ⎪ 2 2L ⎦ ⎣ ⎩
∬A Nxt εxt dA + ∬A Nxyt γxyt dA + ∬A Mxt κ xt dA + ∬A Mxyt κ xyt dA⎤⎥
⎦ (21)
(
where
(29)
Due to the symmetry of the I-beam, the total strain energy of the bottom flange is obtained in the same manner as the top flange,
t t ⎧ Nx = A11 εx t t ⎪ ⎪ Nxy = A66 γxy
⎨ Mxt = D11 κ xt ⎪ t t ⎪ Mxy = D66 κ xy ⎩
(
2.4. Bottom flange
(20)
Substituting Eq. (19) into Eq. (20), assuming the transverse resultant force and moment are negligible, the total strain energy of the top flange yields,
1 Ut = ⎡ 2⎢ ⎣
(27)
where
Considering the top flange as a beam element, the strain energy refers to arbitrary orthogonal coordinates, in accordance with the basic approximations of thin-plate theory, can be written as [21],
Ut =
(26)
Ub = (22)
1⎡ 2⎢ ⎣
∬A Nxb ⎛⎝w,bx 2 + 12 v,bx 2 ⎞⎠ dA + ∬A Nxyb v,bx v,by dA + A11 ∬A u,bx dA
+ D11
∬A v,bxx 2dA + A66 ∬A (w,bx + u,by)2dA + D66 ∬A w,bxy2dA⎤⎥
⎦ (30)
According to the compatibility of the top flange between the strains and displacements, the strains in an arbitrary location of the top flange (x-y plane) are written as,
where 75
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2 2
( ) ( ) cos ( ) sin ( )
16L
⎨ b 2 y 2 C22π 4 ⎪ v , xx = 16L4 ⎪ 2 2 2 ⎪ v ,bxy 2 = C2 π2 4L ⎪ ⎪ (w ,bx + u,by )2 = 0 ⎩
tulipifera laminated venner lumber (LVL). Fig. 5(c) presents the flange and web samples before assembly. In order to increase the bonding performance between the flanges and web, a Rilesa Rapid 1500 drilling machine was used to drill sinusoidal notch on the flanges such that the corrugated web can be assembled and bonded in the notch. The phenol formaldehyde resin was particularly used as adhesive agent to bond the members. The geometry properties of the flanges and corrugated web are given in Fig. 6. Two heights of I-beams were manufactured, i.e., 254 mm and 406 mm. Fig. 7 displays the assembled wood composite I-beam samples. Fig. 8 presents the lateral-torsional buckling experiments. The tested Ibeams were clamped at one end to achieve cantilevered boundary conditions. In particular, two steel angles were attached to a vertical steel column. The specimens were then placed between the two tightly clamped angles, along with proper lateral support. A tip load was applied at the center of the other end by a loading platform that was attached through the centroid of the cross-section by a series of chains. An initially horizontal steel hanger was placed at the loading centroid of the beam end. Three rulers were attached to the hanger. Two of the rulers were placed at the hanger ends to measure the rotation of the Ibeams, while the third ruler was placed beneath the I-beams. In this study, the wood composite is considered as a two-dimensional orthotropic material. Table 1 summarizes the material properties of the wood composite material. The critical buckling load is compared with the theoretical solution. Table 2 presents a comparison of the critical buckling load between the presented model and experimental results. Good agreements are obtained with a maximum difference of 11.54%.
( )
⎧ w ,bx 2 = (2C1 − HC2 2) π sin2 πx 2L 16L ⎪ ⎪ v b 2 = y 2 C22π 2 sin2 πx 2L ⎪ ,x 4L2 ⎪ 2 y2π 4 H 2 b cos2 ⎪ u, x = 4 C1 − 2 C2
( ) πx 2L
2 πx 2L πx 2L
(31)
The membrane forces of the bottom flange are given as,
⎧ Nxb
=−
PHh 2I b
(L − x )(0 ≤ x ≤ L)
⎨ N b = N b = 0(0 ≤ x ≤ L) xy ⎩ y
(32)
Substituting Eq. (30) into Eq. (29), leading through the same procedures, the total strain energy of the bottom flange can be gained as,
U b = −Pξ1b + ξ2b
(33)
where T
⎧ ξ b = ∫ 2 ∫ L Hh (L − x ) w b 2 + 1 v b 2 dxdy ,x 1 − T2 0 4I b 2 ,x ⎪ ⎪ T ⎨ ξ b = 1 ∫ 2T ∫ L [A11 u b + D11 v b 2 + D66 w b 2] dxdy ,x , xx , xy 2 −2 0 ⎪ 2 ⎪ ⎩
(
)
(34)
2.5. Energy method 3.2. Numerical validation The total potential energy is expressed as the summation of the strain energy and external work. According to the buckling criterion, the pre-buckling work, W, and its corresponding displacement can be ignored. Therefore, the total potential energy of the I-beam (Π) can be written as
Numerical simulations were conducted using ABAQUS to compare and validate the presented theoretical model. The FE model was created using ABAQUS 6.14 to simulate the critical load of the beam. The numerical models were built using 8-node shell elements, as shown in Fig. 9. The clamped boundary condition was imposed at one end of the beam. The critical buckling load Pcr was obtained by performing an eigenvalue analysis. Two beam lengths, L = 3.0 m and L = 3.6 m, and two heights, H = 254 mm and H = 406 mm, were considered, respectively. The material properties presented in Table 1 were used in the simulations. The numerical results of Pcr are compared with the analytical solutions, as summarized in Table 3. Satisfactory agreements are obtained between the theoretical and numerical results with a maximum difference of 5.86%.
(35)
Π = U = U w + Ut + Ub
Substituting Eqs. (17), (27), and (33) into Eq. (35), the total potential energy can be written with respect to the external force P as,
Π = P (ξ1t − ξ1b) + ξ w + ξ2t + ξ2b ξw
ξ1t ,
ξ2t , ξ1b ,
(36)
ξ2b
and are given in Eqs. (18), (28), and (34). The where , Rayleigh-Ritz method is then used to solve the constant coefficients C1 and C2 , which is written as,
∂Π = 0 (i = 1, 2) ∂Ci
3.3. Validation with an existing study
(37)
Eq. (37) leads to two linear algebraic equations. Solving the resulting eigenvalue problem, the critical buckling load, Pcr , can be obtained.
In order to validate the theoretical results with an existing study, the presented model is reduced to flat web. Assuming s → x and R → ∞ in Eq. (15), the curvilinear coordinate and the curvature radius are eliminated and, therefore, the flat web geometry is captured. Reducing the presented model to flat web I-beams, the obtained critical buckling load is compared with the existing study [22]. The comparison is presented in Table 4. It can be seen that the critical load from the current model closely matches the results from the previous study.
3. Model validations 3.1. Experimental validation Wood composite materials were used to manufacture the tested samples, as shown in Fig. 5 [37]. Fig. 5(a) displays the aluminum templates used to form the sinusoidal corrugated web. Fig. 5(b) shows the Black cherry (Prunus serotina) veneer-mill clippings that were used to make the composite beam samples. The sinusoidal web was manufactured by layering the clippings in a unidirectional mat and using a phenol formaldehyde resin under heat and pressure to bond the strands. The individual strands were aligned in the longitudinal direction between the templated and coated with a phenol formaldehyde resin (resin was 5%). The web was then compressed by a 200 tone heated platens. The flanges were made by 15-layer yellow poplar Liriodendron
4. Parametric study 4.1. Geometry property The effect of geometry property on the critical buckling load of sinusoidal I-beams is investigated in this section. Fig. 10 shows the geometries that are taken into account. In particular, the height and length of the sinusoidal I-beam are varied from 203 mm to 406 mm and 1.5 m to 5.0 m, respectively. The width and thickness of the top and bottom flanges are fixed as 58 mm and 35 mm, respectively. 76
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Fig. 5. Manufacturing of the wood composite Ibeams sinusoidal corrugated web. (a) Aluminum templates for the sinusoidal web, (b) veneer-mill clippings, and (c) flange and web before assembly.
Fig. 6. Geometry properties of the flanges and web.
buckling. The critical buckling load of sinusoidal I-beam is at least 17.7% higher than that of the flat web. Fig. 11 indicates the variation of the critical buckling loads of sinusoidal and flat web I-beams in terms of the length and height. It can be seen that the change of the web shape configuration from flat to sinusoidal critical increases the critical loading capacity. This enhancement is more significant in the I-beams with shorter length. The difference of the maximum load between the two types of I-beams is presented in Fig. 12. When the beam length is increased from 1.5 m to 5.0 m, the difference of the critical buckling load is reduced for all of the four heights. 4.3. Effects of geometry on the critical buckling load of I-beams with sinusoidal web The effects of geometry property on the critical buckling load (Pcr ) of sinusoidal web I-beams are studied. The effects are investigated with respect to three parameters, i.e., web thickness (t), height (H), and length (L). Comparing the critical buckling load of the maximum t (P t = 20 ) and the minimum t (P t = 5 ), the sensitivity ratio of thickness, Rt , is defined with respect to length and height, as expressed in Eq. (38).
Fig. 7. Assembled wood composite I-beams with sinusoidal web.
4.2. Comparison between flat and sinusoidal I-beams
Rit =
The beam height and length are varied from 203 mm to 406 mm and 1.5 m to 5 m, respectively, and the web thickness is fixed at 10 mm. Table 5 details the critical buckling load under the scenarios of both the sinusoidal and flat cases. Results show that the sinusoidal web significantly improves the beam resistance to the lateral-torsional
Pit = 20 Pit = 5
where i represents four height and length scenarios, i.e., 1) H = 203 mm, L = 1.5 m 2) H = 254 mm, L = 3.0 m 77
(38)
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Fig. 8. Experimental setup, measuring devices and loading platform.
Table 1 Material properties of the wood composite in this study. Young's modulus (GPa)
Web Flanges
Table 3 Comparison of critical buckling load between the theoretical and numerical results. Poisson's ratio
Geometry properties Height (mm)
Flange width (mm)
Length (m)
Theoretical
Numerical
Difference (%)
254 254 406 406
58 58 58 58
3.0 3.6 3.0 3.6
2.611 1.742 2.954 2.033
2.458 1.665 2.767 1.948
5.86 4.39 5.32 4.19
E1
E2
G12
G16
G26
γ12
γ12
15.58 13.38
1.24 0.44
1.05 0.42
1.02 0.46
1.02 0.46
0.39 –
0.02 –
Table 2 Comparison of the critical buckling load between the presented model and experimental results.
Critical buckling load Pcr (kN)
Table 4 Comparison of flat web I-beams between the presented model and existing study.
Geometry
Critical buckling load Pcr (kN)
Height (mm)
Theoretical
Experimental
Difference (%)
Geometry
254 406
1.742 2.033
1.541 1.819
11.54 10.52
Height (mm)
Flange width (mm)
Length (m)
Present model
Existing study [22]
Difference (%)
152 152
76 76
3.05 3.66
2.49 1.57
2.34 1.49
6.02 5.1
Critical buckling load (kN)
3) H = 305 mm, L = 3.6 m 4) H = 406 mm, L = 5.0 m In the same manner, the sensitivity ratios of length and height, i.e., RL and RH , are obtained, as summarized in Table 6. Results indicate that web thickness has a more significant effect on the critical buckling load than length, while height tends to play the least important role. The same findings are observed in Fig. 13 that the critical buckling load is most sensitive to web thickness. It is found out that the critical buckling load of sinusoidal I-beams is increased when the web thickness and height are enlarged. However, the load is dropped dramatically when the beam length is increased. 5. Optimal design of sinusoidal I-beam
Fig. 9. Lateral-torsionally buckled I-beam under tip load P.
5.1. Optimal design of web wavelength-to-beam length ratio a
The influence of the web wavelength-to-beam length ratio, RI = L , is investigated in this section. The geometry properties under 78
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Fig. 11. Comparison of critical buckling load between the sinusoidal and flat I-beams.
Fig. 10. Cross-section geometry of the sinusoidal I-beam.
consideration are summarized in Table 7. Fig. 14 displays the scenarios that are taken into account in RI design. An optimization study is conducted to maximize the critical buckling capacity while minimize the volume of the I-beam with respect to the design ratio RI . The optimization is written as,
(
⎧ Max[Pcr RI = ⎨1 < R < I ⎩8
)]
1 2
(
⎧ Min[Vol RI = ⎨1 < R < I ⎩8
a L
(39a) a L
)]
1 2
(39b)
The optimizations are conducted using the Nelder-Mead algorithm. Fig. 15 presents the optimal ratio, RI , for both Cases 1 and 2. The optimized values refer to the cross-points between the maximization of Pcr and minimization of Vol. Interestingly, the wavelength-to-beam length ratio is decreased when the beam length is increased from 1.5 m to 5 m. The results show that the most effective design ratios of critical buckling capacity-to-beam volume are obtained when RICase1 = 0.151 and RICase2 = 0.173. In addition, if I-beam length is increased, less sinusoidal wave is required to reach the designed capacity of critical buckling load.
Fig. 12. Difference between the sinusoidal and flat I-beams.
(
⎧ Max[Pcr RII = ⎨1 < R < II ⎩8
(
< RII <
)]
3 8
⎧ Min[Vol RII = ⎨1 ⎩8
A T
(40a) A T
)]
3 8
(40b)
Similarly, the Nelder-Mead algorithm is used to carry out the optimal design. Fig. 17 presents the optimal ratio, RII , for both Cases 1 and 2. The most effective design ratios of critical buckling capacity-to-beam volume are obtained as RIICase1 = 0.272 and RIICase2 = 0.329. Comparing Fig. 15 with Fig. 17, it is found out that the web wave amplitude is more significant than web wavelength in affecting the critical buckling load of sinusoidal web I-beams.
5.2. Optimal design of web wave amplitude-to-flange width ratio The influence of the web wave amplitude-to-beam flange width A ratio, RII = T , is studied. The geometry properties under consideration are summarized in Table 8. Fig. 16 displays the scenarios of RII that are taken into account in this study. Similar to Section 5.1, the optimal design is conducted with respect to the design ratio RII as,
Table 5 Comparison of critical buckling load between sinusoidal and flat I-beams with respect to different geometries. Geometry (mm)
Length L = 1.5 m
Length L = 3.0 m
Length L = 3.6 m
Length L = 5.0 m
Height
Flange width
Web thickness
Sine. (kN)
Flat (kN)
Diff. (%)
Sine. (kN)
Flat (kN)
Diff. (%)
Sine. (kN)
Flat (kN)
Diff (%)
Sine. (kN)
Flat (kN)
Diff. (%)
203 254 305 406
58 58 58 58
10 10 10 10
4.875 5.959 6.318 6.751
3.475 4.316 4.585 4.926
28.7 27.6 27.4 27.1
2.32 2.611 2.772 2.954
1.76 1.99 2.132 2.283
24.1 23.8 23.1 22.7
1.632 1.835 1.95 2.163
1.287 1.446 1.543 1.727
21.2 21.2 21.0 20.2
1.233 1.436 1.484 1.655
0.998 1.169 1.211 1.362
19.0 18.6 18.4 17.7
79
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Table 6 Effects of web thickness, length and height on critical buckling load of sinusoidal I-beams. Effect of thickness t
Effect of length L
Length (m)
Height (mm)
Rt =
1.5 3.0 3.6 5.0
203 254 305 406
16.01 15.99 16.00 16.00
Pt = 20 Pt = 5
Effect of height H
Height (mm)
Thickness (mm)
RL =
203 254 305 406
5 10 20 20
3.96 4.15 4.26 4.08
P L=5 P L = 1.5
Length (m)
Thickness (mm)
RH =
1.5 3.0 3.6 5.0
5 10 20 20
1.39 1.27 1.33 1.34
P h = 406 P h = 203
Fig. 13. Critical buckling load of sinusoidal I-beam with respect to height, length and web thickness. Table 7 Geometry properties of the cases under consideration.
Case 1 Case 2
Length (m)
Height (mm)
Flange width (mm)
Web wave amplitude (mm)
Thickness (mm)
5.0 1.5
203 406
58 58
14.5 14.5
5 20
Fig. 14. Wavelength (a) vs. beam length (L) of sinusoidal I-beams (top view).
6. Conclusions In this study, a theoretical model was developed to predict the critical buckling loading capacity of wood composite I-beams with sinusoidal corrugated web. The Rayleigh-Ritz method was used to obtain the critical load. Experiments and numerical simulations were conducted to validate the theoretical results. Reducing the presented model to flat web, the loading capacity of corrugated I-beams was also validate with an existing study. Satisfactory agreements were gained. Parametric studies were carried out to investigate the effects of geometries, e.g., web thickness, length and height, on the buckling capacity of wood composite I-beams. A significant increasing of buckling load (minimum 17.17%) was obtained between flat and sinusoidal web Ibeams. Optimal designs were carried out to investigate the impacts of the ratios of wave length-to-beam length ( a ) and wave amplitude-to-
Fig. 15. Optimal design of the ratio of RI for (a) Case 1 and (b) Case 2. Table 8 Geometry properties of the cases under consideration.
Case 1 Case 2
L
80
Length (m)
Height (mm)
Flange width (mm)
Web wavelength (mm)
Thickness (mm)
5.0 1.5
203 406
58 58
625 625
5 20
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beam height plays the least important role.
• The most effective design ratios with respect to the wave length-to-
beam length ( a ) ratio are obtained as RICase1 = 0.151 and L
•
RICase2 = 0.173. The most effective design ratios with respect to the wave amplitudeto-flange width ( A ) ratio are obtained as RIICase1 = 0.272 and
•
RIICase2 = 0.329. The design ratio is more sensitive to the wave amplitude-to-flange width ratio than wave length-to-beam length ratio.
T
Fig. 16. Wave amplitude (A) vs. flange width (T) of sinusoidal I-beams (top view).
Acknowledgement The experiments of this work were supported by the Wood Utilization Project Task No. 21 at West Virginia University. All of the financial supports as well as the donations of Weyerhaeuser Company and Universal Veneer Company are gratefully acknowledged. The authors are also grateful to Dr. An Chen and Dr. Julio F. Davalos for inspirational discussions and help. References [1] R.L. Harne, K.W. Wang, A review of the recent research on vibration energy harvesting via bistable systems, Smart Mater. Struct. 22 (2) (2013) 023001. [2] Y. Safa, T. Hocker, A validated energy approach for the post-buckling design of micro-fabricated thin film devices, Appl. Math. Model. 39 (2015) 483–499. [3] P. Jiao, W. Borchani, H. Hasni, A.H. Alavi, N. Lajnef, Post-buckling response of nonuniform cross-section bilaterally constrained beams, Mech. Res. Commun. 78 (2016) 42–50. [4] P. Jiao, W. Borchani, H. Hasni, N. Lajnef, Static and dynamic post-buckling analyses of irregularly constrained beams under the small and large deformation assumptions, Int. J. Mech. Sci. 124 (2017) 203–215. [5] A.H. Alavi, H. Hasni, P. Jiao, W. Borchani, N. Lajnef, Fatigue cracking detection in steel bridge girders through a self-powered sensing concept, J. Constr. Steel Res. 128 (2017) 19–38. [6] H. Hasni, A.H. Alavi, P. Jiao, N. Lajnef, Detection of fatigue cracking in steel bridge girders: a support vector machine approach, Arch. Civil. Mech. Eng. 17 (2017) 609–622. [7] P. Jiao, W. Borchani, A.H. Alavi, H. Hasni, N. Lajnef, An energy harvesting and damage sensing solution based on post-buckling response of non-uniform crosssection beams, Struct. Control Health Monit. (2017), http://dx.doi.org/10.1002/ stc.2052. [8] W. Borchani, P. Jiao, R. Burgueno, N. Lajnef, Control of post-buckling mode transitions using assemblies of axially-loaded bilaterally constrained beams, Int. J. Eng. Mech. (2017) In press. [9] A.H. Alavi, H. Hasni, N. Lajnef, K. Chatti, F. Faridazar, An intelligent structural damage detection approach based on self-powered wireless sensor data, Autom. Constr. 62 (2016) 24–44. [10] P. Jiao, W. Borchani, H. Hasni, N. Lajnef, A new solution of measuring thermal response of prestressed concrete bridge girders for structural health monitoring, Meas. Sci. Technol. (2017) (In press). [11] A.H. Alavi, H. Hasni, N. Lajnef, K. Chatti, Continuous health monitoring of pavement systems using smart sensing technology, Constr. Build. Mater. 114 (2016) 719–736. [12] A.H. Alavi, H. Hasni, P. Jiao, N. Lajnef, Structural health monitoring using a hybrid network of self-powered accelerometer and strain sensors, SPIE Smart Struct. NDE, Portland, OR, USA, 2017. 〈https://dx.doi.org/10.1117/12.2258633〉. [13] H. Hasni, A.H. Alavi, P. Jiao, N. Lajnef, A new method for detection of fatigue cracking in steel bridge girders using self-powered wireless sensors, SPIE Smart Struct. NDE, Portland, OR, USA, 2017. 〈https://dx.doi.org/10.1117/12.2258629〉. [14] P. Jiao, W. Borchani, H. Hasni, A.H. Alavi, N. Lajnef, An energy harvesting solution based on the post-buckling response of non-prismatic slender beams, SPIE Smart Struct. NDE, Portland, OR, USA, 2017. 〈https://dx.doi.org/10.1117/12.2268448〉. [15] K. Davami, L. Zhao, E. Lu, J. Cortes, C. Lin, D.E. Lilley, P.K. Purohit, I. Bargatin, Ultralight shape-recovering plate mechanical metamaterials, Nat. Commun., 6, p. 10019. [16] J.T. Mottram, Lateral-torsional buckling of a pultruded I-beam, Composites 32 (2) (1992) 81–92. [17] E. Ghafoori, M. Motavalli, Lateral-torsional buckling of steel I-beams retrofitted by bonded and un-bonded CFRP laminates with different pre-stress levels: experimental and numerical study, Constr. Build. Mater. 76 (2015) 194–206. [18] M. Ma, O. Hughes, Lateral distortional buckling of mono-symmetric beams under point load, J. Eng. Mech. 122 (10) (1996) 1022–1029. [19] F. Mohri, L. Azrar, M. Potier-Ferry, Lateral post-buckling analysis of thin-walled open section beams, Thin-Walled Struct. 40 (2002) 1013–1036. [20] M.A. Serna, A. Lopez, I. Puente, D.J. Yong, Equivalent uniform moment factors for lateral-torsional buckling of steel members, J. Constr. Steel Res. 62 (2006) 566–580. [21] J.F. Davalos, P. Qiao, Analytical and experimental study of lateral and distortional buckling of FRP wide-flange beams, J. Compos. Constr. 1 (4) (1997) 150–159. [22] P. Qiao, G. Zou, J.F. Davalos, Flexural-torsional buckling of fiber-reinforced plastic
Fig. 17. Optimal design of the ratio of RII for (a) Case 1 and (b) Case 2.
flange width ( A ) on the critical buckling load and volume of the T structure. The main findings can be summarized as following,
• Critical •
buckling load of composite strengthened by changing the web shape sinusoidal. Web thickness of sinusoidal corrugated nificant effect on critical buckling load
I-beams is significantly configuration from flat to I-beams has a more sigthan beam length, while 81
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and numerical investigations, Eng. Struct. 110 (2016) 105–115. [31] P. Wang, C. Liu, M. Liu, X. Wang, Numerical studies on large deflection behavior of axially restrained corrugated web steel beams at elevated temperatures, ThinWalled Struct. 98 (2016) 58–74. [32] J. He, Y. Liu, A. Chen, D. Wang, T. Yoda, Bending behavior of concrete-encased composite I-girder with corrugated steel web, Thin-Walled Struct. 74 (2014) 70–84. [33] I. Dayyani, A.D. Shaw, E.I.S. Flores, M.I. Friswell, The mechanics of composite corrugated structures: a review with applications in morphing aircraft, Compos. Struct. 133 (2015) 358–380. [34] J. Lee, Lateral buckling analysis of thin-walled laminated composite beams with monosymmetric sections, Eng. Struct. 28 (2006) 1997–2009. [35] N.I. Kim, D.K. Shin, M.Y. Kim, Exact lateral buckling analysis for thin-walled composite beam under end moment, Eng. Struct. 29 (2007) 1739–1751. [36] N.I. Kim, D.K. Shin, M.Y. Kim, Improved flexural-torsional stability analysis of thinwalled composite beam and exact stiffness matrix, Int. J. Mech. Sci. 49 (2007) 950–969. [37] B. McGraw, Recycling Veneer-Mill Residues Into Engineered Products with Improved Torsional Rigidity (Master’s thesis), West Virginia University, Morgantown, WV, USA.
composite cantilever I-beams, Compos. Struct. 60 (2003) 205–217. [23] E.J. Barbero, I.G. Raftoyiannis, Lateral and distortional buckling of pultruded Ibeams, Compos. Struct. 27 (3) (1994) 261–268. [24] M.D. Pandey, M.Z. Kabir, A.N. Sherbourne, Flexural-torsional stability of thinwalled composite I-section beams, Compos. Eng. 5 (3) (1995) 321–342. [25] N.D. Hai, H. Mutsuyoshi, S. Asamoto, T. Matsui, Structural behavior of hybrid FRP composite I-beam, Constr. Build. Mater. 24 (2010) 956–969. [26] M. Elgaaly, A. Seshadri, R. Hamilton, Bending strength of steel beams with corrugated webs, J. Struct. Eng. 123 (1997) 772–782. [27] D. Dubina, V. Ungureanu, L. Gilia, Experimental investigations of cold-formed steel beams of corrugated web and built-up section for flanges, Thin-Walled Struct. 90 (2015) 159–170. [28] J. Vacha, P. Kyzlik, I. Both, F. Wald, Beams with corrugated web at elevated temperature, experimental results, Thin-Walled Struct. 98 (2016) 19–28. [29] R. Aydin, E. Yuksel, N. Yardimci, T. Gokce, Cyclic behavior of diagonally-stiffened beam-to-column connections of corrugated-web I sections, Eng. Struct. 121 (2016) 120–135. [30] Y.L. Guo, H. Chen, Y.L. Pi, M.A. Bradford, In-plane strength of steel arches with a sinusoidal corrugated web under a full-span uniform vertical load: experimental
82
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Full length article
Lateral-torsional buckling analysis of wood composite I-beams with sinusoidal corrugated web
MARK
⁎
Pengcheng Jiaoa, , Wassim Borchania, Sepehr Soleimania, Bradley McGrawb a b
Department of Civil and Environmental Engineering, Michigan State University, East Lansing, MI 48824, USA Department of Civil and Environmental Engineering, West Virginia University, Morgantown, WV 26505, USA
A R T I C L E I N F O
A B S T R A C T
Keywords: Critical buckling capacity Composite I-section beam Sinusoidal corrugated web Theoretical model Optimal design Lateral-torsional buckling
Buckling instability refers to a significant limit on the structures with I-section members, which causes severe decreasing of critical buckling load. Many mechanisms and techniques have been developed to increase the loading capacity of I-section structures, e.g., using composite materials or changing web geometry. This study aims at theoretically investigating the effect of geometry on the buckling capacity of wood composite I-beams with sinusoidal corrugated web. Experiments and numerical simulations are carried out to validate the theoretical predictions. The presented model is also reduced to flat web to validate with an existing study. Good agreements are observed from the validations. Parametric study is conducted to indicate the effects of web geometry on the critical buckling load. A significant increasing of buckling load (minimum 17.7%) is obtained between flat and sinusoidal web I-beams. Sensitivity study is carried out to evaluate the influences of web thickness, beam length and beam height on the loading capacity of sinusoidal I-beams. Optimal design is conducted to investigate the impact of the ratios of wavelength-to-beam length ( a ) and wave amplitude-to-flange L
width ( A ) on the critical buckling load and volume of wood corrugated I-beams. The findings in this study can be T further used as guidance for the design of composite I-beams with sinusoidal corrugated web.
1. Introduction Buckling phenomenon has been investigated over centuries since it extensively exists in various types of slender members. Many studies have been conducted to explore the applications of buckling and postbuckling response, namely, energy harvesting for self-powered sensors, damage detection for Structural Health Monitoring (SHM), structural instability of honeycomb plates in nanoscale, etc [1–15]. However, buckling-induced instability is still a severe challenge to the reliability of I-section structures in civil infrastructures. In particular, I-beams are likely to experience failures before reaching ultimate load capacities since thin-walled web tends to buckle in a pattern that combines twisting with lateral bending. This phenomenon, known as lateral-torsional buckling, severely reduces the stiffness and critical buckling capacity of I-beams [16]. The structural failure under consideration in this study is caused by lateral buckling, which is likely to be the failure mode for the slender beams with insufficiently lateral constraints. Lateral buckling typically results in the failures that critical load is much smaller than material's yield capacity. Due to initial eccentricity, buckling in I-beams mainly generates a second-order bending moment about the beams' longitudinal axes, i.e., torsional. Hence, lateral-
⁎
Corresponding author. E-mail address: [email protected] (P. Jiao).
http://dx.doi.org/10.1016/j.tws.2017.05.025 Received 7 January 2017; Received in revised form 11 May 2017; Accepted 25 May 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
torsional buckling is a common failure mode to I-beams [17]. It is of great research and practice interests to increase the buckling capacity of I-beams [18–20]. In order to increase critical buckling load, many research efforts have been dedicated to investigating the buckling response of flat web I-beams made by composite materials, e.g., wood composite, fiber-reinforced polymers (FRP), etc. Davalos and Qiao [21] and Qiao et al. [22] theoretically and experimentally investigated the lateral-distortional and lateral-torsional buckling of FRP wide-flange I-beams. Using an energy method, the authors presented theoretical models to capture the instability of FRP I-beams. Barbero and Raftoyiannis [23] examined the elastic lateral-distortional buckling of FRP I-beams. A plate theory was used to study the distortion behavior, i.e., shear effects and bending-twisting coupling, of plate cross-section. Pandey et al. [24] developed a theoretical model to investigate the lateral buckling strength of thin-walled composite I-beams. The authors used Vlasovtype linear hypothesis to evaluate the coefficients of I-beam stiffness. Hai et al. [25] examined hybrid FRP beams with respect to buckling behavior. Parametric studies were carried out to optimize the mechanical response of the presented I-beams. However, these studies have not taken into account the influence of web geometry on the
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loading capacity of I-beams. To address the research gap and enhance the resistance against the structural instability caused by buckling, many studies have been presented regarding the I-section structures with non-flat web. Elgaaly et al. [26] investigated the failure of corrugated web I-beams under uniform bending. Dubina et al. [27] experimentally studied the coldformed steel beams with trapezoidal steel web and built-up steel flanges. Vacha et al. [28] proposed experiments to examine the thermal and mechanical responses of sinusoidal corrugated web steel beams subjected to high temperatures. Aydin et al. [29] examined the behavior of corrugated web I-beams under cyclic load with respect to beamto-column connections. The authors experimentally compared two types of I-beams, i.e., conventional type with thick flat web in connection zone and alternative type that has stiffeners. Guo et al. [30] studied the in-plane mechanical response of circular steel I-section arches with sinusoidal corrugated web geometry. Thermal behavior of axially restrained corrugated web steel beams were numerically investigated by Wang et al. [31]. The authors studied the effects of different parameters, namely, load ratio, axial resistant stiffness ratio, span-depth ratio, corrugation shape of web, etc. However, these studies are proposed based on isotropic materials, such as steel, which have not considered the advantages of composites. Later on, experiments were carried out by He et al. [32] to study the bending behavior of concreteencased composite beams with corrugated web. A more general review of the mechanical performance of corrugated web composite beams had been carried out by Dayyani et al. [33]. Many theoretical models have been developed to theoretically predict the buckling behavior of composite I-beams with non-uniform web. Lee [34] developed a general geometrically nonlinear model to theoretically examine lateral buckling of composite I-beams with monosymmetric section. The author used systematic variational formulation based on the classical lamination theory. Kim et al. [35,36] developed an approach to obtain exact element stiffness matrix for composite I-beams with symmetric and arbitrary laminations. An energy method was used to solve the stability equations and force-displacement relationship in these studies. However, it is of necessity to theoretically study the lateral-torsional buckling of composite I-beams with corrugated web; especially, lack of parametric study and optimal design have been presented to guide the design of composite I-beams with sinusoidal corrugated web. This study aims at theoretically investigating the effects of geometry on the buckling capacity of wood composite I-beams with sinusoidal corrugated web. In order to achieve the research objective, the work is deployed as follows,
Fig. 1. Schematic illustration of a cantilever I-beam with sinusoidal web under a tip load.
2. Theoretical model 2.1. Problem formulation This section presents lateral-torsional buckling analysis of a cantilever I-beam with sinusoidal corrugated web under a tip load P, as illustratively displayed in Fig. 1. The deflection of the I-beam contains three components, namely, u , v and w . Fig. 2 presents the cross-section deflection of the I-beam under the external force. t , T, h , and H represent the web thickness, top and bottom flanges width, top and bottom flanges thickness, and web height, respectively. δ t , δ b , δ w , and α refer to the lateral displacements of the top and bottom flanges and web, respectively, as well as the sectional rotation of the I-beam after deflection. Since the sinusoidal I-beam consists of three segments, i.e., web, and top and bottom flanges, the superposition principle is applied in this study. For lateral-torsional buckling, the deformation due to bending is negligible [18,21]. Therefore, the displacements of web, top and bottom flanges can be written as, w ⎧uw = 0 v =0 ⎨ ww = δ w (x , z ) = δ + zα ⎩
• Section 2 develops a theoretical model to study the lateral-torsional • • •
buckling of I-beams with sinusoidal corrugated web. The superposition method is used to take into account top and bottom flanges and sinusoidal web. The Rayleigh-Ritz method is used to obtain the critical buckling load of the I-beam. Section 3 presents experimental and numerical studies to validate the theoretical predictions with respect to buckling capacity. In addition, the presented model is reduced to flat web and compared with an existing study. Good agreements are obtained. Section 4 carries out a parametric study to capture the buckling load enhancement between the flat and sinusoidal web I-beams. In addition, sensitivity ratios (Rs ) are t investigated with respect to the web thickness (t), beam length (L), and beam height (H). Section 5 proposes optimal designs. Design factors, i.e., web wavelength-to-beam length ( a ) and web wave amplitude-to-flange width L
•
( A ) ratios, are examined in terms of the buckling load capacity and T volume of I-beams. Section 6 summarizes and discusses the major findings in this study.
Fig. 2. Cross-section deformation of the I-beam.
73
(1)
Thin-Walled Structures 119 (2017) 72–82
P. Jiao et al. t
t
⎧u = −yδ , x ⎪ t v = −yα ⎨ t ⎪ w = δt = δ + ⎩
where Aij and Dij (i, j = 1, 2, 6) are the extensional and bending stiffness matrices, respectively, defined in terms of the reduced stiffness matrix Q as,
H α 2
(2) n
⎧ Aij = ∑k −1 (Qij )k (z k − z k −1) ⎨ Dij = 1 ∑n (Qij ) (z k 3 − z k −13) k −1 k 3 ⎩
and b
b
⎧u = −yδ , x ⎪ b v = −yα ⎨ b ⎪ w = δb = δ − ⎩
(9a)
2
H α 2
E11 ⎧ Q11 = E11 − v12 E12 ⎪ ⎪ Q = v12 E11 E22 12 2 E E11 − v12 22 ⎨ E11 E22 ⎪Q22 = E11 − v 2 E22 12 ⎪ ⎩Q66 = G12
(3)
where subscribes w, t, and b represent web, top and bottom flanges. 2.2. Sinusoidal web The shape function of the sinusoidal web in the x-z plate is given as,
2πx ⎞ y = A sin ⎛ ⎝ a ⎠
The sinusoidal web in the I-beam consists of two strain energy components, namely, the out-of-plane bending energy (Ubw ) and membrane strain energy (Umw ). Therefore, the total strain energy is given as [18],
(4)
where A and a are the amplitude and wavelength, respectively. The lateral displacement of the centroid and sectional rotation are given, considering only the first buckling mode, as [22],
⎧ δ = C1 ⎡1 − cos ⎪ ⎣ ⎨ α = C 1 − cos ⎡ 2 ⎪ ⎣ ⎩
( ) ( ) ⎤⎦ πx 2L
U w = Ubw + Umw
(10)
where
⎤ ⎦
w ⎧Um =
πx 2L
(9b)
⎨Ubw = ⎩
(5)
1 2 1 2
∬A [Nxw εxw + Nzw εzw + Nxzw γxzw ] dA ∬A [D11 χxw 2 +2D12 χxw χzw + D22 χzw 2 + 4D66 χxzw 2 ] dA
(11)
where C1 and C2 refers to two unknown constants. The deflected configuration of the sinusoidal web is displayed in Fig. 3. Therefore, the displacement of the web is given as,
where χxw = w ,wxx , χxzw = w ,wxz , and χzw = w ,wzz . The normal strains εi (i = x , z ) and shear strain γxz of the membrane strains can be written as,
πx ww = δ + zα = (C1 + zC2) ⎡1 − cos ⎛ ⎞ ⎤ ⎝ 2L ⎠ ⎦ ⎣
w ⎧ εx = v , z ⎪ ww εz = u,wx − R ⎨ ⎪ γxz = u,wz + v ,wx ⎩
(6)
Taking Eq. (6) into Eq. (1), the displacements of the web can be obtained. Considering the top flange as a beam element, the strain energy refers to arbitrary orthogonal coordinates, in accordance with the basic approximations of thin-plate theory, can be written as [18],
Uw =
1 2
∬A (σxw εxw + σzw εzw + τxzw γxzw ) dA
where R is the curvature radius of the web mid-surface. Substituting Eqs. (11) and (12) into Eq. (10), the total strain energy of the web can be written as,
(7)
A12 0 A22 0 0 A66 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 D11 D12 D12 D22 0 0
⎤ ⎥ εw w ⎥ ⎛ x κx ⎞ ⎥ ⎜ εzw / κ zw ⎟ 0 ⎥⎜ w w ⎟ γ κ 0 ⎥ ⎝ xz xz ⎠ ⎥ D66 ⎦
1 2
∬A ⎡⎣Nxw v,wz + Nzw (u,wx − wR ) + Nxzw (u,wz + v,wx ) ⎤⎦dsdz
+
1 2
Uw =
According to the orthotropic material property, a laminate in x-z plane consists of the membrane force, bending and twisting moments, which are given in terms of mid-surface in-plane strains and curvatures as,
⎡ A11 ⎢ A12 w w ⎛ Nx Mx ⎞ ⎢ 0 w w ⎜ Nz / Mz ⎟ = ⎢ 0 ⎜Nw Mw ⎟ ⎢ ⎝ xz xz ⎠ ⎢ 0 ⎢ 0 ⎣
(12)
w
∬A [D11 w,wxx 2+2D12 w,wxx w,wzz+D22 w,wzz 2+4D66 w,wxz 2] dsdz
(13)
where w w w w w ⎧u, x = u, z = v, x = v , z = w , zz = 0 ⎪ w2 π4 πx w , xx = 4 (C1 + zC2)2 cos2 2L 16L ⎨ ⎪ w ,wxz 2 = π 2 C22 sin2 πx 2L 4L2 ⎩
( )
( )
(8)
(14)
In order to obtain the total strain energy in Eq. (13), it is of necessity to calculate the curvilinear coordinate (s), curvature radius (R) and membrane forces. Let s be the curvilinear coordinate along the web, the curvilinear coordinate and the curvature radius are expressed as,
⎧ ds = 1 + 4 Aπ 2 cos2 2πx dx a a ⎪ ⎪ 3 2 ⎡1 + 4 Aπ cos2 2πx ⎤2 ⎨ ( a ) ( a ) ⎥⎦ ⎢ ⎪R = ⎣ Aπ 2 2πx 4 ( ) sin ( ⎪ a a ) ⎩
( )
( )
(15)
The membrane forces of the sinusoidal web consists of three comw , that refer to the in-plane normal forces in ponents, i.e., Nxw , Nzw and Nxz x and z directions, as well as in-plane shear force, respectively. Fig. 4 displays the different components of the membrane forces. The membrane forces of the web can be obtained as,
Fig. 3. Deflected configuration of the sinusoidal web.
74
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1
t t ⎧ εx = u, x + 2 (w ,tx 2 + 2 v ,tx 2) ⎪ t 1 1 ε = w ,ty + 2 (u,ty 2 + 2 v ,ty 2) ⎨ y t ⎪ γxy = w ,tx + u,ty + v ,tx v ,ty ⎩
(23)
Substituting Eq. (23) into Eq. (21), ignoring the fourth-order terms, the total strain energy of the top flange is obtained as,
Ut =
1⎡ 2⎢ ⎣
∬A Nxt ⎛⎝w,tx 2 + 12 v,tx 2 ⎞⎠ dA + ∬A Nxyt v,tx v,ty dA + A11 ∬A u,tx dA
Fig. 4. Membrane forces of deformed sinusoidal web.
+ D11
⎦ (24)
Pt
w ⎧ Nx = I w (L − x ) z ⎪ w Nz = 0 (0 < x ≤ L) ⎨ Pt h2 w ⎪ Nxz = − 2I w 4 − z 2 ⎩
(
)
where (16)
2 2
where is the moment of inertia of the web. Substituting Eqs. (15) and (16) into Eq. (13). The total strain energy of the sinusoidal web is obtained as,
16L
H
1 2
∫− H2 ∫0 2
L
( ) ( ) cos ( ) sin ( )
⎨ t 2 y 2 C22π 4 ⎪ v , xx = 16L4 ⎪ 2 2 2 ⎪ v ,txy 2 = C2 π2 4L ⎪ ⎪ (w ,tx + u,ty )2 = 0 ⎩
(17)
where
ξw =
( )
⎧ w ,tx 2 = (2C1 + HC2 2) π sin2 πx 2L 16L ⎪ ⎪ v t 2 = y 2 C22π 2 sin2 πx 2L ⎪ ,x 4L2 ⎪ 2 y2π 4 H 2 t cos2 ⎪ u, x = 4 C1 + 2 C2
Iw
Uw = ξw
∬A v,txx 2dA + A66 ∬A (w,tx + u,ty)2dA + D66 ∬A w,txy2dA⎤⎥
Aπ 2 2πx ⎞ cos2 ⎛ ⎞ dxdz [D11 w ,wxx 2+4D66 w ,wxz 2] 1 + 4 ⎛ ⎝ a ⎠ ⎝ a ⎠
( ) πx 2L
2 πx 2L πx 2L
(25)
The membrane forces of the top flange are given as,
(18)
PHh
⎧ Nxt = 2I t (L − x )(0 ≤ x ≤ L) t ⎨ N yt = Nxy = 0(0 ≤ x ≤ L) ⎩
2.3. Top flange
Substituting Eq. (26) into Eq. (24), the total strain energy of the top flange can be obtained with respect to the external force P as,
Substituting Eq. (5) into Eq. (2), the displacements of the top flange are expressed as, πy − 2L
(
) ( ) ( ) ) ( ) ⎤⎦
H C 2 2
(
U t = Pξ1t + ξ2t
πx 2L
⎧ut = C1 + sin ⎪ ⎪ t πx v = −yC2 ⎡1 − cos 2L ⎤ ⎣ ⎦ ⎨ ⎪ t H 1 − cos ⎪ w = C1 + 2 C2 ⎡ ⎣ ⎩
πx 2L
1 2
∬A (σxt εxt + σyt εyt + τxyt γxyt ) dA
T
⎧ ξ t = ∫ 2 ∫ L Hh (L − x ) w t 2 + 1 v t 2 dxdy ,x 1 − T2 0 4I t 2 ,x ⎪ ⎪ T ⎨ ξ t = 1 ∫ 2T ∫ L [A11 u t + D11 v t 2 + D66 w t 2] dxdy ,x , xx , xy 2 −2 0 ⎪ 2 ⎪ ⎩
(19)
where
)
(28)
is the moment of inertia of the top flange.
It
Taking Eq. (5) into Eq. (3), the displacements of the bottom flange are, πy
(
) ( ) ( ) ) ( )
⎧ub = − 2L C1 − H2 C2 sin πx 2L ⎪ ⎪ b πx v = −yC2 ⎡1 − cos 2L ⎤ ⎣ ⎦ ⎨ ⎪ b H πx = − − w C C 1 cos ⎡ ⎤ 1 2 ⎪ 2 2L ⎦ ⎣ ⎩
∬A Nxt εxt dA + ∬A Nxyt γxyt dA + ∬A Mxt κ xt dA + ∬A Mxyt κ xyt dA⎤⎥
⎦ (21)
(
where
(29)
Due to the symmetry of the I-beam, the total strain energy of the bottom flange is obtained in the same manner as the top flange,
t t ⎧ Nx = A11 εx t t ⎪ ⎪ Nxy = A66 γxy
⎨ Mxt = D11 κ xt ⎪ t t ⎪ Mxy = D66 κ xy ⎩
(
2.4. Bottom flange
(20)
Substituting Eq. (19) into Eq. (20), assuming the transverse resultant force and moment are negligible, the total strain energy of the top flange yields,
1 Ut = ⎡ 2⎢ ⎣
(27)
where
Considering the top flange as a beam element, the strain energy refers to arbitrary orthogonal coordinates, in accordance with the basic approximations of thin-plate theory, can be written as [21],
Ut =
(26)
Ub = (22)
1⎡ 2⎢ ⎣
∬A Nxb ⎛⎝w,bx 2 + 12 v,bx 2 ⎞⎠ dA + ∬A Nxyb v,bx v,by dA + A11 ∬A u,bx dA
+ D11
∬A v,bxx 2dA + A66 ∬A (w,bx + u,by)2dA + D66 ∬A w,bxy2dA⎤⎥
⎦ (30)
According to the compatibility of the top flange between the strains and displacements, the strains in an arbitrary location of the top flange (x-y plane) are written as,
where 75
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2 2
( ) ( ) cos ( ) sin ( )
16L
⎨ b 2 y 2 C22π 4 ⎪ v , xx = 16L4 ⎪ 2 2 2 ⎪ v ,bxy 2 = C2 π2 4L ⎪ ⎪ (w ,bx + u,by )2 = 0 ⎩
tulipifera laminated venner lumber (LVL). Fig. 5(c) presents the flange and web samples before assembly. In order to increase the bonding performance between the flanges and web, a Rilesa Rapid 1500 drilling machine was used to drill sinusoidal notch on the flanges such that the corrugated web can be assembled and bonded in the notch. The phenol formaldehyde resin was particularly used as adhesive agent to bond the members. The geometry properties of the flanges and corrugated web are given in Fig. 6. Two heights of I-beams were manufactured, i.e., 254 mm and 406 mm. Fig. 7 displays the assembled wood composite I-beam samples. Fig. 8 presents the lateral-torsional buckling experiments. The tested Ibeams were clamped at one end to achieve cantilevered boundary conditions. In particular, two steel angles were attached to a vertical steel column. The specimens were then placed between the two tightly clamped angles, along with proper lateral support. A tip load was applied at the center of the other end by a loading platform that was attached through the centroid of the cross-section by a series of chains. An initially horizontal steel hanger was placed at the loading centroid of the beam end. Three rulers were attached to the hanger. Two of the rulers were placed at the hanger ends to measure the rotation of the Ibeams, while the third ruler was placed beneath the I-beams. In this study, the wood composite is considered as a two-dimensional orthotropic material. Table 1 summarizes the material properties of the wood composite material. The critical buckling load is compared with the theoretical solution. Table 2 presents a comparison of the critical buckling load between the presented model and experimental results. Good agreements are obtained with a maximum difference of 11.54%.
( )
⎧ w ,bx 2 = (2C1 − HC2 2) π sin2 πx 2L 16L ⎪ ⎪ v b 2 = y 2 C22π 2 sin2 πx 2L ⎪ ,x 4L2 ⎪ 2 y2π 4 H 2 b cos2 ⎪ u, x = 4 C1 − 2 C2
( ) πx 2L
2 πx 2L πx 2L
(31)
The membrane forces of the bottom flange are given as,
⎧ Nxb
=−
PHh 2I b
(L − x )(0 ≤ x ≤ L)
⎨ N b = N b = 0(0 ≤ x ≤ L) xy ⎩ y
(32)
Substituting Eq. (30) into Eq. (29), leading through the same procedures, the total strain energy of the bottom flange can be gained as,
U b = −Pξ1b + ξ2b
(33)
where T
⎧ ξ b = ∫ 2 ∫ L Hh (L − x ) w b 2 + 1 v b 2 dxdy ,x 1 − T2 0 4I b 2 ,x ⎪ ⎪ T ⎨ ξ b = 1 ∫ 2T ∫ L [A11 u b + D11 v b 2 + D66 w b 2] dxdy ,x , xx , xy 2 −2 0 ⎪ 2 ⎪ ⎩
(
)
(34)
2.5. Energy method 3.2. Numerical validation The total potential energy is expressed as the summation of the strain energy and external work. According to the buckling criterion, the pre-buckling work, W, and its corresponding displacement can be ignored. Therefore, the total potential energy of the I-beam (Π) can be written as
Numerical simulations were conducted using ABAQUS to compare and validate the presented theoretical model. The FE model was created using ABAQUS 6.14 to simulate the critical load of the beam. The numerical models were built using 8-node shell elements, as shown in Fig. 9. The clamped boundary condition was imposed at one end of the beam. The critical buckling load Pcr was obtained by performing an eigenvalue analysis. Two beam lengths, L = 3.0 m and L = 3.6 m, and two heights, H = 254 mm and H = 406 mm, were considered, respectively. The material properties presented in Table 1 were used in the simulations. The numerical results of Pcr are compared with the analytical solutions, as summarized in Table 3. Satisfactory agreements are obtained between the theoretical and numerical results with a maximum difference of 5.86%.
(35)
Π = U = U w + Ut + Ub
Substituting Eqs. (17), (27), and (33) into Eq. (35), the total potential energy can be written with respect to the external force P as,
Π = P (ξ1t − ξ1b) + ξ w + ξ2t + ξ2b ξw
ξ1t ,
ξ2t , ξ1b ,
(36)
ξ2b
and are given in Eqs. (18), (28), and (34). The where , Rayleigh-Ritz method is then used to solve the constant coefficients C1 and C2 , which is written as,
∂Π = 0 (i = 1, 2) ∂Ci
3.3. Validation with an existing study
(37)
Eq. (37) leads to two linear algebraic equations. Solving the resulting eigenvalue problem, the critical buckling load, Pcr , can be obtained.
In order to validate the theoretical results with an existing study, the presented model is reduced to flat web. Assuming s → x and R → ∞ in Eq. (15), the curvilinear coordinate and the curvature radius are eliminated and, therefore, the flat web geometry is captured. Reducing the presented model to flat web I-beams, the obtained critical buckling load is compared with the existing study [22]. The comparison is presented in Table 4. It can be seen that the critical load from the current model closely matches the results from the previous study.
3. Model validations 3.1. Experimental validation Wood composite materials were used to manufacture the tested samples, as shown in Fig. 5 [37]. Fig. 5(a) displays the aluminum templates used to form the sinusoidal corrugated web. Fig. 5(b) shows the Black cherry (Prunus serotina) veneer-mill clippings that were used to make the composite beam samples. The sinusoidal web was manufactured by layering the clippings in a unidirectional mat and using a phenol formaldehyde resin under heat and pressure to bond the strands. The individual strands were aligned in the longitudinal direction between the templated and coated with a phenol formaldehyde resin (resin was 5%). The web was then compressed by a 200 tone heated platens. The flanges were made by 15-layer yellow poplar Liriodendron
4. Parametric study 4.1. Geometry property The effect of geometry property on the critical buckling load of sinusoidal I-beams is investigated in this section. Fig. 10 shows the geometries that are taken into account. In particular, the height and length of the sinusoidal I-beam are varied from 203 mm to 406 mm and 1.5 m to 5.0 m, respectively. The width and thickness of the top and bottom flanges are fixed as 58 mm and 35 mm, respectively. 76
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Fig. 5. Manufacturing of the wood composite Ibeams sinusoidal corrugated web. (a) Aluminum templates for the sinusoidal web, (b) veneer-mill clippings, and (c) flange and web before assembly.
Fig. 6. Geometry properties of the flanges and web.
buckling. The critical buckling load of sinusoidal I-beam is at least 17.7% higher than that of the flat web. Fig. 11 indicates the variation of the critical buckling loads of sinusoidal and flat web I-beams in terms of the length and height. It can be seen that the change of the web shape configuration from flat to sinusoidal critical increases the critical loading capacity. This enhancement is more significant in the I-beams with shorter length. The difference of the maximum load between the two types of I-beams is presented in Fig. 12. When the beam length is increased from 1.5 m to 5.0 m, the difference of the critical buckling load is reduced for all of the four heights. 4.3. Effects of geometry on the critical buckling load of I-beams with sinusoidal web The effects of geometry property on the critical buckling load (Pcr ) of sinusoidal web I-beams are studied. The effects are investigated with respect to three parameters, i.e., web thickness (t), height (H), and length (L). Comparing the critical buckling load of the maximum t (P t = 20 ) and the minimum t (P t = 5 ), the sensitivity ratio of thickness, Rt , is defined with respect to length and height, as expressed in Eq. (38).
Fig. 7. Assembled wood composite I-beams with sinusoidal web.
4.2. Comparison between flat and sinusoidal I-beams
Rit =
The beam height and length are varied from 203 mm to 406 mm and 1.5 m to 5 m, respectively, and the web thickness is fixed at 10 mm. Table 5 details the critical buckling load under the scenarios of both the sinusoidal and flat cases. Results show that the sinusoidal web significantly improves the beam resistance to the lateral-torsional
Pit = 20 Pit = 5
where i represents four height and length scenarios, i.e., 1) H = 203 mm, L = 1.5 m 2) H = 254 mm, L = 3.0 m 77
(38)
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Fig. 8. Experimental setup, measuring devices and loading platform.
Table 1 Material properties of the wood composite in this study. Young's modulus (GPa)
Web Flanges
Table 3 Comparison of critical buckling load between the theoretical and numerical results. Poisson's ratio
Geometry properties Height (mm)
Flange width (mm)
Length (m)
Theoretical
Numerical
Difference (%)
254 254 406 406
58 58 58 58
3.0 3.6 3.0 3.6
2.611 1.742 2.954 2.033
2.458 1.665 2.767 1.948
5.86 4.39 5.32 4.19
E1
E2
G12
G16
G26
γ12
γ12
15.58 13.38
1.24 0.44
1.05 0.42
1.02 0.46
1.02 0.46
0.39 –
0.02 –
Table 2 Comparison of the critical buckling load between the presented model and experimental results.
Critical buckling load Pcr (kN)
Table 4 Comparison of flat web I-beams between the presented model and existing study.
Geometry
Critical buckling load Pcr (kN)
Height (mm)
Theoretical
Experimental
Difference (%)
Geometry
254 406
1.742 2.033
1.541 1.819
11.54 10.52
Height (mm)
Flange width (mm)
Length (m)
Present model
Existing study [22]
Difference (%)
152 152
76 76
3.05 3.66
2.49 1.57
2.34 1.49
6.02 5.1
Critical buckling load (kN)
3) H = 305 mm, L = 3.6 m 4) H = 406 mm, L = 5.0 m In the same manner, the sensitivity ratios of length and height, i.e., RL and RH , are obtained, as summarized in Table 6. Results indicate that web thickness has a more significant effect on the critical buckling load than length, while height tends to play the least important role. The same findings are observed in Fig. 13 that the critical buckling load is most sensitive to web thickness. It is found out that the critical buckling load of sinusoidal I-beams is increased when the web thickness and height are enlarged. However, the load is dropped dramatically when the beam length is increased. 5. Optimal design of sinusoidal I-beam
Fig. 9. Lateral-torsionally buckled I-beam under tip load P.
5.1. Optimal design of web wavelength-to-beam length ratio a
The influence of the web wavelength-to-beam length ratio, RI = L , is investigated in this section. The geometry properties under 78
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Fig. 11. Comparison of critical buckling load between the sinusoidal and flat I-beams.
Fig. 10. Cross-section geometry of the sinusoidal I-beam.
consideration are summarized in Table 7. Fig. 14 displays the scenarios that are taken into account in RI design. An optimization study is conducted to maximize the critical buckling capacity while minimize the volume of the I-beam with respect to the design ratio RI . The optimization is written as,
(
⎧ Max[Pcr RI = ⎨1 < R < I ⎩8
)]
1 2
(
⎧ Min[Vol RI = ⎨1 < R < I ⎩8
a L
(39a) a L
)]
1 2
(39b)
The optimizations are conducted using the Nelder-Mead algorithm. Fig. 15 presents the optimal ratio, RI , for both Cases 1 and 2. The optimized values refer to the cross-points between the maximization of Pcr and minimization of Vol. Interestingly, the wavelength-to-beam length ratio is decreased when the beam length is increased from 1.5 m to 5 m. The results show that the most effective design ratios of critical buckling capacity-to-beam volume are obtained when RICase1 = 0.151 and RICase2 = 0.173. In addition, if I-beam length is increased, less sinusoidal wave is required to reach the designed capacity of critical buckling load.
Fig. 12. Difference between the sinusoidal and flat I-beams.
(
⎧ Max[Pcr RII = ⎨1 < R < II ⎩8
(
< RII <
)]
3 8
⎧ Min[Vol RII = ⎨1 ⎩8
A T
(40a) A T
)]
3 8
(40b)
Similarly, the Nelder-Mead algorithm is used to carry out the optimal design. Fig. 17 presents the optimal ratio, RII , for both Cases 1 and 2. The most effective design ratios of critical buckling capacity-to-beam volume are obtained as RIICase1 = 0.272 and RIICase2 = 0.329. Comparing Fig. 15 with Fig. 17, it is found out that the web wave amplitude is more significant than web wavelength in affecting the critical buckling load of sinusoidal web I-beams.
5.2. Optimal design of web wave amplitude-to-flange width ratio The influence of the web wave amplitude-to-beam flange width A ratio, RII = T , is studied. The geometry properties under consideration are summarized in Table 8. Fig. 16 displays the scenarios of RII that are taken into account in this study. Similar to Section 5.1, the optimal design is conducted with respect to the design ratio RII as,
Table 5 Comparison of critical buckling load between sinusoidal and flat I-beams with respect to different geometries. Geometry (mm)
Length L = 1.5 m
Length L = 3.0 m
Length L = 3.6 m
Length L = 5.0 m
Height
Flange width
Web thickness
Sine. (kN)
Flat (kN)
Diff. (%)
Sine. (kN)
Flat (kN)
Diff. (%)
Sine. (kN)
Flat (kN)
Diff (%)
Sine. (kN)
Flat (kN)
Diff. (%)
203 254 305 406
58 58 58 58
10 10 10 10
4.875 5.959 6.318 6.751
3.475 4.316 4.585 4.926
28.7 27.6 27.4 27.1
2.32 2.611 2.772 2.954
1.76 1.99 2.132 2.283
24.1 23.8 23.1 22.7
1.632 1.835 1.95 2.163
1.287 1.446 1.543 1.727
21.2 21.2 21.0 20.2
1.233 1.436 1.484 1.655
0.998 1.169 1.211 1.362
19.0 18.6 18.4 17.7
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Table 6 Effects of web thickness, length and height on critical buckling load of sinusoidal I-beams. Effect of thickness t
Effect of length L
Length (m)
Height (mm)
Rt =
1.5 3.0 3.6 5.0
203 254 305 406
16.01 15.99 16.00 16.00
Pt = 20 Pt = 5
Effect of height H
Height (mm)
Thickness (mm)
RL =
203 254 305 406
5 10 20 20
3.96 4.15 4.26 4.08
P L=5 P L = 1.5
Length (m)
Thickness (mm)
RH =
1.5 3.0 3.6 5.0
5 10 20 20
1.39 1.27 1.33 1.34
P h = 406 P h = 203
Fig. 13. Critical buckling load of sinusoidal I-beam with respect to height, length and web thickness. Table 7 Geometry properties of the cases under consideration.
Case 1 Case 2
Length (m)
Height (mm)
Flange width (mm)
Web wave amplitude (mm)
Thickness (mm)
5.0 1.5
203 406
58 58
14.5 14.5
5 20
Fig. 14. Wavelength (a) vs. beam length (L) of sinusoidal I-beams (top view).
6. Conclusions In this study, a theoretical model was developed to predict the critical buckling loading capacity of wood composite I-beams with sinusoidal corrugated web. The Rayleigh-Ritz method was used to obtain the critical load. Experiments and numerical simulations were conducted to validate the theoretical results. Reducing the presented model to flat web, the loading capacity of corrugated I-beams was also validate with an existing study. Satisfactory agreements were gained. Parametric studies were carried out to investigate the effects of geometries, e.g., web thickness, length and height, on the buckling capacity of wood composite I-beams. A significant increasing of buckling load (minimum 17.17%) was obtained between flat and sinusoidal web Ibeams. Optimal designs were carried out to investigate the impacts of the ratios of wave length-to-beam length ( a ) and wave amplitude-to-
Fig. 15. Optimal design of the ratio of RI for (a) Case 1 and (b) Case 2. Table 8 Geometry properties of the cases under consideration.
Case 1 Case 2
L
80
Length (m)
Height (mm)
Flange width (mm)
Web wavelength (mm)
Thickness (mm)
5.0 1.5
203 406
58 58
625 625
5 20
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beam height plays the least important role.
• The most effective design ratios with respect to the wave length-to-
beam length ( a ) ratio are obtained as RICase1 = 0.151 and L
•
RICase2 = 0.173. The most effective design ratios with respect to the wave amplitudeto-flange width ( A ) ratio are obtained as RIICase1 = 0.272 and
•
RIICase2 = 0.329. The design ratio is more sensitive to the wave amplitude-to-flange width ratio than wave length-to-beam length ratio.
T
Fig. 16. Wave amplitude (A) vs. flange width (T) of sinusoidal I-beams (top view).
Acknowledgement The experiments of this work were supported by the Wood Utilization Project Task No. 21 at West Virginia University. All of the financial supports as well as the donations of Weyerhaeuser Company and Universal Veneer Company are gratefully acknowledged. The authors are also grateful to Dr. An Chen and Dr. Julio F. Davalos for inspirational discussions and help. References [1] R.L. Harne, K.W. Wang, A review of the recent research on vibration energy harvesting via bistable systems, Smart Mater. Struct. 22 (2) (2013) 023001. [2] Y. Safa, T. Hocker, A validated energy approach for the post-buckling design of micro-fabricated thin film devices, Appl. Math. Model. 39 (2015) 483–499. [3] P. Jiao, W. Borchani, H. Hasni, A.H. Alavi, N. Lajnef, Post-buckling response of nonuniform cross-section bilaterally constrained beams, Mech. Res. Commun. 78 (2016) 42–50. [4] P. Jiao, W. Borchani, H. Hasni, N. Lajnef, Static and dynamic post-buckling analyses of irregularly constrained beams under the small and large deformation assumptions, Int. J. Mech. Sci. 124 (2017) 203–215. [5] A.H. Alavi, H. Hasni, P. Jiao, W. Borchani, N. Lajnef, Fatigue cracking detection in steel bridge girders through a self-powered sensing concept, J. Constr. Steel Res. 128 (2017) 19–38. [6] H. Hasni, A.H. Alavi, P. Jiao, N. Lajnef, Detection of fatigue cracking in steel bridge girders: a support vector machine approach, Arch. Civil. Mech. Eng. 17 (2017) 609–622. [7] P. Jiao, W. Borchani, A.H. Alavi, H. Hasni, N. Lajnef, An energy harvesting and damage sensing solution based on post-buckling response of non-uniform crosssection beams, Struct. Control Health Monit. (2017), http://dx.doi.org/10.1002/ stc.2052. [8] W. Borchani, P. Jiao, R. Burgueno, N. Lajnef, Control of post-buckling mode transitions using assemblies of axially-loaded bilaterally constrained beams, Int. J. Eng. Mech. (2017) In press. [9] A.H. Alavi, H. Hasni, N. Lajnef, K. Chatti, F. Faridazar, An intelligent structural damage detection approach based on self-powered wireless sensor data, Autom. Constr. 62 (2016) 24–44. [10] P. Jiao, W. Borchani, H. Hasni, N. Lajnef, A new solution of measuring thermal response of prestressed concrete bridge girders for structural health monitoring, Meas. Sci. Technol. (2017) (In press). [11] A.H. Alavi, H. Hasni, N. Lajnef, K. Chatti, Continuous health monitoring of pavement systems using smart sensing technology, Constr. Build. Mater. 114 (2016) 719–736. [12] A.H. Alavi, H. Hasni, P. Jiao, N. Lajnef, Structural health monitoring using a hybrid network of self-powered accelerometer and strain sensors, SPIE Smart Struct. NDE, Portland, OR, USA, 2017. 〈https://dx.doi.org/10.1117/12.2258633〉. [13] H. Hasni, A.H. Alavi, P. Jiao, N. Lajnef, A new method for detection of fatigue cracking in steel bridge girders using self-powered wireless sensors, SPIE Smart Struct. NDE, Portland, OR, USA, 2017. 〈https://dx.doi.org/10.1117/12.2258629〉. [14] P. Jiao, W. Borchani, H. Hasni, A.H. Alavi, N. Lajnef, An energy harvesting solution based on the post-buckling response of non-prismatic slender beams, SPIE Smart Struct. NDE, Portland, OR, USA, 2017. 〈https://dx.doi.org/10.1117/12.2268448〉. [15] K. Davami, L. Zhao, E. Lu, J. Cortes, C. Lin, D.E. Lilley, P.K. Purohit, I. Bargatin, Ultralight shape-recovering plate mechanical metamaterials, Nat. Commun., 6, p. 10019. [16] J.T. Mottram, Lateral-torsional buckling of a pultruded I-beam, Composites 32 (2) (1992) 81–92. [17] E. Ghafoori, M. Motavalli, Lateral-torsional buckling of steel I-beams retrofitted by bonded and un-bonded CFRP laminates with different pre-stress levels: experimental and numerical study, Constr. Build. Mater. 76 (2015) 194–206. [18] M. Ma, O. Hughes, Lateral distortional buckling of mono-symmetric beams under point load, J. Eng. Mech. 122 (10) (1996) 1022–1029. [19] F. Mohri, L. Azrar, M. Potier-Ferry, Lateral post-buckling analysis of thin-walled open section beams, Thin-Walled Struct. 40 (2002) 1013–1036. [20] M.A. Serna, A. Lopez, I. Puente, D.J. Yong, Equivalent uniform moment factors for lateral-torsional buckling of steel members, J. Constr. Steel Res. 62 (2006) 566–580. [21] J.F. Davalos, P. Qiao, Analytical and experimental study of lateral and distortional buckling of FRP wide-flange beams, J. Compos. Constr. 1 (4) (1997) 150–159. [22] P. Qiao, G. Zou, J.F. Davalos, Flexural-torsional buckling of fiber-reinforced plastic
Fig. 17. Optimal design of the ratio of RII for (a) Case 1 and (b) Case 2.
flange width ( A ) on the critical buckling load and volume of the T structure. The main findings can be summarized as following,
• Critical •
buckling load of composite strengthened by changing the web shape sinusoidal. Web thickness of sinusoidal corrugated nificant effect on critical buckling load
I-beams is significantly configuration from flat to I-beams has a more sigthan beam length, while 81
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P. Jiao et al.
and numerical investigations, Eng. Struct. 110 (2016) 105–115. [31] P. Wang, C. Liu, M. Liu, X. Wang, Numerical studies on large deflection behavior of axially restrained corrugated web steel beams at elevated temperatures, ThinWalled Struct. 98 (2016) 58–74. [32] J. He, Y. Liu, A. Chen, D. Wang, T. Yoda, Bending behavior of concrete-encased composite I-girder with corrugated steel web, Thin-Walled Struct. 74 (2014) 70–84. [33] I. Dayyani, A.D. Shaw, E.I.S. Flores, M.I. Friswell, The mechanics of composite corrugated structures: a review with applications in morphing aircraft, Compos. Struct. 133 (2015) 358–380. [34] J. Lee, Lateral buckling analysis of thin-walled laminated composite beams with monosymmetric sections, Eng. Struct. 28 (2006) 1997–2009. [35] N.I. Kim, D.K. Shin, M.Y. Kim, Exact lateral buckling analysis for thin-walled composite beam under end moment, Eng. Struct. 29 (2007) 1739–1751. [36] N.I. Kim, D.K. Shin, M.Y. Kim, Improved flexural-torsional stability analysis of thinwalled composite beam and exact stiffness matrix, Int. J. Mech. Sci. 49 (2007) 950–969. [37] B. McGraw, Recycling Veneer-Mill Residues Into Engineered Products with Improved Torsional Rigidity (Master’s thesis), West Virginia University, Morgantown, WV, USA.
composite cantilever I-beams, Compos. Struct. 60 (2003) 205–217. [23] E.J. Barbero, I.G. Raftoyiannis, Lateral and distortional buckling of pultruded Ibeams, Compos. Struct. 27 (3) (1994) 261–268. [24] M.D. Pandey, M.Z. Kabir, A.N. Sherbourne, Flexural-torsional stability of thinwalled composite I-section beams, Compos. Eng. 5 (3) (1995) 321–342. [25] N.D. Hai, H. Mutsuyoshi, S. Asamoto, T. Matsui, Structural behavior of hybrid FRP composite I-beam, Constr. Build. Mater. 24 (2010) 956–969. [26] M. Elgaaly, A. Seshadri, R. Hamilton, Bending strength of steel beams with corrugated webs, J. Struct. Eng. 123 (1997) 772–782. [27] D. Dubina, V. Ungureanu, L. Gilia, Experimental investigations of cold-formed steel beams of corrugated web and built-up section for flanges, Thin-Walled Struct. 90 (2015) 159–170. [28] J. Vacha, P. Kyzlik, I. Both, F. Wald, Beams with corrugated web at elevated temperature, experimental results, Thin-Walled Struct. 98 (2016) 19–28. [29] R. Aydin, E. Yuksel, N. Yardimci, T. Gokce, Cyclic behavior of diagonally-stiffened beam-to-column connections of corrugated-web I sections, Eng. Struct. 121 (2016) 120–135. [30] Y.L. Guo, H. Chen, Y.L. Pi, M.A. Bradford, In-plane strength of steel arches with a sinusoidal corrugated web under a full-span uniform vertical load: experimental
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