- Email: [email protected]
Estimation of maximum torsional moment for multicorner tubes
Thin-Walled Structures 112 (2017) 66–77
Contents lists available at ScienceDirect
Thin–Walled Structures journal homepage: www.elsevier.com/locate/tws
Estimation of maximum torsional moment for multicorner tubes a
b
Siti Aisyah Mohd Zaifuddin , Dai-Heng Chen , Kuniharu Ushijima a b c
c,⁎
MARK
Department of Mechanical Engineering, Graduate School of Engineering, Tokyo University of Science, Niijuku 6-3-1, Katsushika-ku, Tokyo 1258585, Japan Faculty of Civil Engineering & Mechanics, Jiangsu University, 301 Xuefu Road, Jingkou, Zhenjiang 212013, China Department of Mechanical Engineering, Faculty of Engineering, Tokyo University of Science, Niijuku 6-3-1, Katsushika-ku, Tokyo 1258585, Japan
A R T I C L E I N F O
A B S T R A C T
Keywords: Elastic buckling Plastic yielding Plastic flattening Multicorner tubes Torsional collapse
In this paper, the estimation of maximum torsional moment for multicorner tubes under torsional loading was investigated using nonlinear FE analysis. The effects of tube geometries and strain-hardening coefficient on the torsional behaviour were discussed. The maximum torsional moment was due to the occurrence of sectional collapse of the tubes, and the mechanism of sectional collapse could be classified into three physical phenomena, elastic buckling, plastic yielding and plastic flattening of the cross section. Moreover, based on our numerical results, analytical solution for estimating the maximum torsional moment for each phenomenon was proposed.
1. Introduction Thin-walled circular and multicorner tubes have been widely used in many structural applications as lightweight structural components. Once the external load is applied to the tubes, the external work is dissipated by the elastic and plastic strain energy due to axial tension or compression, bending and torsion. Over the decades, the application of such tubes as impact energy absorbers for vehicles has been investigated by many researchers [1–6]. For example, Alexander [1] proposed his theoretical model to estimate the mean crushing load of circular tubes subjected to axial compressive load. Also, Abramowicz and Wierzbicki [2] developed their theoretical analysis to evaluate the plastic resistance of rectangular and multicorner tubes under axial compression. When the tubes with moderate length are subjected to axial load, the first peak load which associates with local plastic buckling are found in the early stage of deformation, and the following load fructuates due to the wrinkle's folding behaviour. The absorbed energy can be estimated by the load-displacement curve after plastic buckling took place. Therefore, the estimation of average compressive load under such load fructuation is significant for the energy absorption capacity for tubes. On the contrary, when the tubes are subjected to bending or torsional load, the reaction bending or torsional moment increases and reaches a peak, and then drops dramatically and never recover due to the occurrence of sectional collapse. Therefore, the estimation of maximum bending or torsional moment for tubes is important to estimate its load-carrying capacity. The elastic and plastic collapse behaviour of thin-walled tubes under torsion have also been published by many researchers [6–11]. ⁎
For example, Murray [6] investigated the maximum torsional load for thin-walled square tubes, and proposed a solution to distinguish the load caused by elastic buckling with that caused by plastic yielding. In Murray's research, the effect of strain-hardening was neglected [6]. Also, Chen and Wierzbicki [9], Chen et al. [10] and Chen [11] studied the torsional collapse of thin-walled square and circular columns analytically and numerically. They mainly discussed the collapse behaviour of square column under large plastic rotation. In their study, they proposed three-phase of collapse mechanism, namely, pre-buckling, post-buckling and collapse-spreading, and developed their solution to evaluate the torsional moment versus rotation curve for each phase using energy method. Also, they extended their solution for rectangular and hexagonal thin-walled columns, and compared analytical and numerical results. Their analytical results agreed fairly with numerical results, however, there were still large differences between these results, especially the maximum torsional moment. This implies that their solution is limited to be applied in square tubes, and that other modified analytical development is still needed for estimating the torsional moment of multicorner tubes. Basically, the maximum torsional moment and deformation behaviour for tubes depend strongly on the tube geometries and the amount of strain-hardening. The estimation technique should be constructed by observing the variation of stress states and deformation behaviour in details. The main purpose of our study is to evaluate the maximum torsional moment for multicorner tubes by using nonlinear FE analysis. The appearance of maximum torsional moment is due to the occurrence of sectional collapse of the tubes, and the mechanism of sectional collapse could be classified into three physical phenomena, elastic buckling, plastic yielding and plastic flattening of the cross-section. Based on the
Corresponding author. Tel.: +81 (0) 358761332; fax: +81 (0) 358761332. E-mail address: [email protected] (K. Ushijima).
http://dx.doi.org/10.1016/j.tws.2016.12.005 Received 6 April 2016; Received in revised form 17 November 2016; Accepted 12 December 2016 0263-8231/ © 2016 Elsevier Ltd. All rights reserved.
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
each direction. Here, the wall's width b and the tube's length L are set to 50 mm and 200–250 mm respectively, and the thickness t varying from 0.35 to 4.0 mm. The tube's length L may affect the torsional response of tubes, but our FE results confirmed that when the length-to-width ratio L / b is larger than 4.0, the effect can be negligible. In this paper, the estimation of maximum torsional moment of square tubes is mainly investigated analytically and numerically, while the validation of the proposed theoretical equations on the multicorner tubes is summarized in Section 3.4.
Table 1 List of material parameters. Young's modulus E [GPa] Poisson's ratio Yield strass σY [MPa] Hardening coefficient Eh [MPa]
72.4 0.3 72.4 E /100 ≤ Eh ≤ E /10
investigation of stress variation and cross-sectional distortion in our FE analysis, the analytical solutions for estimating the maximum torsional moment for each phenomenon are proposed.
3. Results and discussion It is well known that for the torsional response of hollow straight tubes, the sectional collapse would occur when the end rotation reaches a certain angle. Once the sectional collapse occurs, the torsional moment reaches a maximum and drops considerably. The occurrence of sectional collapse has two reasons:
2. Method of numerical analysis In our study, the commercial finite element analysis software, MSC.Marc, is used to demonstrate the torsional deformation behaviour of multicorner tubes. The multicorner tubes are assumed to be isotropic, homogeneous elastoplastic material, and obey the bilinear hardening curve as shown in Eq. (1) where Eh shows the strain-hardening coefficient.
⎛ σ ⎞ σ = σY + Eh ⎜ε − Y ⎟ . ⎝ E⎠
• •
the shear stress distributed over the cross-section reaches a limit the large amount of plastic deformation, namely, plastic flattening deformation arises
The former includes two physical phenomena, namely, the occurrence of elastic buckling and plastic yielding, and these phenomena can be observed in relatively thin-walled tubes. In the following, the torsional moment due to elastic buckling and plastic yielding are written by Mcole and Mcolp respectively. On the contrary, as for the latter case, the walls of the tubes move inward and the applied moment drops significantly even when the shear stress still increases after the collapse. This phenomenon can be observed in relatively thick-walled tubes. Here the torsional moment due to plastic flattening is written by Mcolf. Fig. 2(a)–(c) show examples of the torsional moment MT versus torsional angle θ (MT-θ) diagram of square tubes having different tube's thickness-to-width ratio t / b and different strain-hardening coefficient Eh / E . Also in these figures, two kinds of stress variation are plotted by dotted and dashed lines. Here, τA represents the shear stress at point ‘A’ where the maximum deflection takes place, and τave represents the average shear stress at the same position x = xA . It can be found in these
(1)
Values of the material properties applied in our FE model are as shown in Table 1 unless it is stated. As for the finite element mesh, the 2-dimensional shell element is used for the thin-walled tubes with 5 mm×5 mm mesh size. also in our calculation, the mises’ yield criterion and the updated lagrangian method are used for formulating the nonlinear problem, and the newton-raphson numerical method is applied for finding the root effectively. The boundary condition and tube's geometry of our FE model are explained as follows. All rotations and displacements except for the movement in the axial direction Ux is fixed at one end of the tube, and the torsional rotation θ is applied at the other end of the tube via an attached rigid plate. Fig. 1(a) shows the schematic of boundary condition, and Fig. 1(b) portrays three types of cross-sectional shape of multicorner tubes discussed in our paper. The longitudinal coordinate sets x, while y and z denote the coordinate of the cross-section in
Fig. 1. Multicorner tubes subjected to torsional loading. (a) Hollow straight tube under torsional loading, (b) cross-section of multicorner tubes.
67
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 2. MT-θ curves of square tubes with different thickness-to-width ratio t /b and hardening coefficient Eh /E . (a) t /b = 7.0 × 10−3, Eh /E = 0.01. (b) t /b = 0.02 , Eh /E = 0.01. (c) t /b = 0.02 , Eh /E = 0.07 .
figures that the characteristic of MT-θ diagram is affected strongly on the thickness-to-width ratio t / b and the strain-hardening coefficient Eh / E . Finally, the maximum torsional moment for tubes can be settled by the competition of the above three moments, Mcole, Mcolp and Mcolf and written in the following equation: f p e Mcol = Minimum [Mcol , Mcol , Mcol ].
(2)
The torsional moment Mcol may depend on tube's geometries and material's strain-hardening properties. In our study, the torsional response of tubes is examined by changing the tube thickness t and strain-hardening coefficient Eh / E systematically, and the theoretical e and analytical solutions for the estimation of Mcol , Mcolp and Mcolf are discussed in the next section. 3.1. Torsional moment due to elastic buckling and elastic collapse When the tube's thickness-to-width ratio t / b is sufficiently small, the sectional collapse caused by elastic buckling can be observed in the tube. Fig. 2(a) shows the MT-θ diagram for a square tube with the ratio t / b =7.0 × 10−3. The distribution of shear stress over a cross-section at four points ‘B’, ‘C’, ‘D’ and ‘E’ in Fig. 2(a) is shown in Fig. 3, where the normalized distance of ‘A’ along the tube xA / L =0.55. Here, the parameter s represents the distance over a cross-section from one corner. It can be found that a uniform shear stress distributed along the width until the angle θ=7.0 × 10−3 rad (at ‘C’ in Fig. 2(a)). Then, the shear stress at the center of the wall (s / b =0.5) drops when the angle θ exceeds 7.0 × 10−3 rad although the shear stress is smaller than the yield value τY(=σY / 3 =41.8 MPa). This is because the elastic buckling
Fig. 3. Stress distribution for a square tube with t /b = 7.0 × 10−3 and Eh /E = 0.01 at the most deformed cross-section ( xA /L = 0.55).
occurs at the center of the wall (s / b =0.5) when the angle θ is equal to 7.0 × 10−3 rad. However, the torsional moment still increases until θ=9.0 × 10−3 rad (at ‘D’ in Fig. 2(a)), as the elements near four corners in a cross-section can sustain the shear stress to some extent after the
68
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
elastic buckling, which is called the effect of “effective width” proposed by Karman et al. [12]. When the angle θ exceeds 9.0 × 10−3 rad, the elements near corners would also yield, and the torsional moment drops dramatically. In Fig. 2(a), the variation of average shear stress τave over the cross-section is also plotted by a dashed curve. It is found that the angle θ where the average shear stress reaches its maximum point coincides with that where the torsional moment becomes a maximum. From the above results, it can be concluded that the elastic buckling and elastic collapse can be distinguished by the variation of shear stress. In the following, the analytical solution of torsional moments for each phenomenon is explained. The analysis of torsional buckling for a thin-walled square tube can be regarded as an elastic buckling problem of a rectangular plate subjected to shear loading. There are some papers and books published regarding the ultimate strength of thin-walled plates under shear loading. For example, Vilnay and Burt [13] studied the effective width of plates made of aluminium under shear loading. As mentioned above, the effective width is firstly proposed by Karman et al. [12] in order to estimate the maximum compressive stress of a plate larger than the elastic buckling stress. Also, the theoretical result of elastic buckling stress τbuce for a plate under shear loading has already been reported in Timoshenko and Goodier's book [14] as follows: e = τbuc
⎛ t ⎞2 kπ 2E ⎜ ⎟ . 2 12(1 − ν ) ⎝ b ⎠
cross-section. Next, the modification on the estimation of torsional moment due to elastic buckling will be discussed. As explained above, the buckling coefficient k in Eq. (3) depends on the length l of a plate. Based on the inclination angle shown in Fig. 4, the length l can be regarded as the length of a wrinkle which gives l=b/tan 35°. Therefore, the value of the parameter k becomes 7.31, which is obtained from Eq. (4). Finally, the estimation of the shear stress and the torsional moment at elastic buckling can be modified and shown in the following equations: e = 0.61 × τbuc
e = 1.22 × Mbuc
Here, the parameter k represents the buckling coefficient depending on the length-to-width ratio, l / b of a plate and the edge boundary condition. As for a simply-supported rectangular plate, the parameter k can be approached by
(4)
e Mcol p Mcol
0.35 42.18 30.01 41.02
0.40 59.86 44.81 61.24
=
e τbuc
τY
1−
+
e τbuc τY
3 . · 2 1 + α2
(9)
p Mcol = 2τY b 2t .
(10) p e Mcol / Mcol
Fig. 5 shows a variation of as a function of normalized e shear stress τY / τbuc for square, hexagonal and octagonal tubes. Analytical results obtained by Eq. (9) and proposed by Vilnay and Burt [13] are also shown in the figure by solid and dashed lines, respectively. It can be concluded from Fig. 5 that the maximum torsional moment due to elastic collapse can be estimated sufficiently from Eqs. (7), (9) and (10) regardless of the shape of cross-section.
(5)
0.30 29.18 18.90 25.83
(8)
The torsional moment at plastic yielding can be written as
3.2. Torsional moment due to plastic yielding When the tube becomes thicker, the torsional behaviour is mainly dominated by the plastic deformation. Fig. 2(b) displays the MT-θ response of a square tube with t / b =0.02 and with relatively small strain-hardening coefficient Eh / E =0.01. The corresponding shear stress distribution at point ‘F’, ‘G’, ‘H’ and ‘I’ in Fig. 2(b) is shown in Fig. 6(a). Here, the normalized distance along the tube xA / L is 0.275 where the maximum deformation can be observed. Also, the torsional response of a square tube with relatively large strain-hardening coefficient Eh / E =0.07 are shown in Fig. 2(c) and Fig. 6(b) where the normalized distance along the tube xA / L is 0.225. As can be found in Fig. 2(b) and (c), the torsional angle where the average shear stress reaches a peak is almost equal to that of the maximum torsional moment. This behaviour is similar to the elastic collapse as explained in the previous section. However, it is found in Fig. 2(b) and Fig. 6(a) that when a square tube has relatively thick wall but with small strain-hardening coefficient, the average torsional shear stress τave increases monotonically and reaches the yielding value, τY without the appearance of elastic buckling or
Table 2 Comparison of Mbuce for a plate under shear load obtained from FEM and predictions (Eq. (5)). 0.25 19.80 10.94 14.95
e τbuc τY
Here, the parameter α represents the length-to-width ratio L / b of the plate. Applying this equation to the estimation of the torsional moment due to elastic collapse gives
Table 2 summarizes torsional moment at the initiation of elastic buckling of square tubes for b=50 mm and varying the thickness t from 0.25 to 0.45 mm obtained by FE analysis and Eq. (5). It can be found in Table 2 that there are large differences between them. In the following, the modification on the estimation of the elastic buckling moment Mbuce based on the deformation behaviour of multicorner tubes will be carried out. Comparisons of deformed shape at the initiation of elastic buckling for the case of tube's thickness t=0.25 mm, 0.35 mm and 0.45 mm are given in Fig. 4(a)–(c) respectively. In these figures, the contour bands represent the in-plane shear stress distribution on the square tube. From these figures, it can be found that the inclination angle of a wrinkle observed in each tube is almost the same regardless of the tube's thickness t, and the angle can be measured as about 35°. In addition, our FE results confirmed that this angle is also independent of the shape of
Thickness t [mm] FEM [N m] Eq. (5) [N m] Eq. (7) [N m]
(7)
Vu 3 = + · . VY τY 2 1 + α2
where l denotes the length of the plate. White et al. [8] also investigated the elastic torsional buckling of a simply-supported plate, and they concluded that the lowest value of τbuce would occur for the lowest value of k, simply when the normalized parameter l / b approaches infinity. Thus, they regarded the value of k as 5.35, and concluded that the torsional moment of square tube at the occurrence of elastic buckling can be estimated by the following equations: e Mbuc
π 2E 3 t . (1 − ν 2 )
1−
e τbuc
⎞2
π 2E 3 = 0.89 × t . (1 − ν 2 )
(6)
The modified results of torsional moment caused by elastic buckling are also shown in Table 2, which agree well with FE results. Next, the estimation of the maximum torsional moment due to elastic collapse will be discussed. Basler [15] investigated analytically the strength of plate girders with flanges subjected to shear loading, and derived the ratio of the ultimate shear load Vu and the plastic shear load VY as a function of the buckling shear stress τbuce and the plastic yielding shear stress τY as follows:
(3)
⎛b k = 5.35 + 4 ⎜ ⎟ , ⎝l⎠
2 π 2E ⎛ t ⎞ ⎜ ⎟ , 2 (1 − ν ) ⎝ b ⎠
0.45 80.48 63.80 87.19
69
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 4. Comparison of wrinkle's angle for square tubes with different thickness t. (a) t=0.25 mm, (b) t=0.35 mm, (c) t=0.45 mm.
70
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 5. Variation of Mcole/ Mcolp for thin-walled multicorner tubes (t /b < 0.01).
Fig. 7. MT − θ curve of a square tube with t /b = 0.06 and Eh /E =0.1.
elastic collapse. After that, the stress decreases rapidly due to the occurrence of plastic yielding. On the contrary, as can be observed in Fig. 2(c), when the strain-hardening coefficient is relatively large, the average shear stress τave increases gradually until its peak, but its maximum value is larger than the yield stress τY. Then, the average shear stress τave decreases gently although the torsional angle increases. It can be understood by comparing both stress results shown in Fig. 6(a) and (b) that when the strain-hardening is sufficiently large, the stress distribution does not drop and keeps high even after the occurrence of the maximum torsional moment. This implies that the maximum torsional moment for tubes with larger strain-hardening material is not mainly due to the degradation of stress value, but due to the inward deflection of a cross-section. In this paper, this deformation behaviour is called as flattening, and the estimation of maximum moment due to flattening will be discussed in the next section. Here, as for the case of t / b =0.02 and Eh / E =0.07, the square tube collapses in transition phase from plastic yielding to plastic collapse (flattening), thus, this deformation behaviour can be regarded as mixed-type collapsing. As for the perfectly-rigid plastic material, Mahendran and Murray [7] proposed a formulation of torsional moment due to plastic yielding, Mcolp as shown in Eq. (10). In Eq. (10), the effect of strain-hardening coefficient is being ignored. The analytical result obtained by Eq. (10) is also shown in Fig. 2(b) and (c) by a solid line. It is found from Fig. 2(c) that as for thick-walled square tubes with relatively large strain-
hardening, the maximum torsional moment cannot be predicted effectively by Eq. (10). 3.3. Maximum torsional moment due to plastic collapse (flattening) Fig. 7 represents MT-θ response of a thick-walled square tube with t / b =0.06 and large strain-hardening coefficient Eh / E =0.1. Also in Fig. 7, the variation of average shear stress at xA / L =0.5 is shown by a dashed curve. It is found that although the torsional moment increases and reaches a peak at θ=0.82 rad, the average shear stress keeps on increasing until θ=1.1 rad. In this case, Fig. 7 proved that the average shear stress τave is not the main parameter that affect the drop in torsional moment. Instead, it is anticipated that the large sectional deflection could be the reason to the occurrence of maximum torsional moment. Thus, in this section, the deformation behaviour of the crosssection of square tubes will be investigated. Fig. 8(a) and (b) plot the deformed cross-section at maximum torsional moment for two cases of hardening coefficient. Here, the relative rotational movement is removed in both figures whereas θc is defined as the critical angle where the maximum torsional moment takes place. As can be seen in Fig. 8(a), the cross-section of square tube deformed slightly when the material has small hardening coefficient (Eh / E =0.01). On the contrary, when the square tube has large hardening coefficient (Eh / E =0.1), the large amount of flattening deformation in which the tube walls move inward can be observed in Fig. 8(b).
Fig. 6. Variations of shear stress observed in two cases of square tubes with different hardening coefficient Eh /E at the most deformed cross-section. (a) Eh /E =0.01, (b)Eh /E =0.07.
71
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 8. The deformed cross-section of square tubes at peak torsional moment with different thickness-to-width ratio t /b and hardening coefficient Eh /E . (a) t /b = 0.05, Eh /E = 0.01, (b) t /b = 0.05 , Eh /E = 0.1.
Such movement results in the degradation of the torsional moment. Therefore, it can be concluded that for a thicker tube made by larger strain-hardening material, the formulation of flattening movement is important to estimate the maximum torsional moment. W. Chen [11] has proposed analytical equations to describe the deformation of square tubes for two phases under torsional loading. In his study, it is approximated that in the pre-buckling phase, the walls of the tube become spiral surfaces, and the displacements of the upper b b b flange (− 2 ≤ z ≤ 2 , y = 2 ) for x-, y- and z-directions can be written in the following equations:
⎧Ux = 0 ⎪ ⎪ b xθ xθ b ⎨Uy = 2 cos( L ) − z sin( L ) − 2 ⎪ xθ b xθ ⎪Uz = z cos( ) + sin( ) − z. ⎩ L 2 L
(11)
Also, in the post-buckling phase, since the walls would deform inward as shown in Fig. 8, it is predicted that the displacements can be expressed by the following equations:
⎧Ux = 0 ⎪ ⎪ b zπ xθ zπ xθ b ⎨Uy = ( 2 − A cos b )cos L − (z − δ sin b )sin L − 2 ⎪ ⎪Uz = (z − δ sin zπ )cos xθ + ( b − A cos zπ − δ )sin xθ − z, ⎩ b L 2 b L
Fig. 9. Comparisons of deformed shape of upper flange for pre- and post-buckling phases obtained by FE analysis and theoretical results proposed by Chen [11].
(12)
where δ and A are two parameters which depend on torsional angle θ, and can be approximated by
⎧ ⎪ (1 − δ =⎨ b ⎪ (1 − ⎩
2 2
)· L · π
x θ
2 2
)(1 − L )· π ( 2 ≤ x ≤ L ),
W. Chen fixed the x-axis displacement while free axial displacement is set in this study. As can be observed in Fig. 8, the maximum displacement occurs at the center of the wall. Thus, as the displacement at the mentioned point cannot be predicted by using W. Chen's theory, another new model to evaluate the maximum torsional moment due to flattening is proposed as follows. Fig. 12 shows a schematic of simplified flattening deformation applied in our theoretical analysis. There are some assumptions in our theoretical model. The deformed shape is composed of four identical semi-arcs having the same radius ρ and angle ϕ. Here, the parameter Δ is the deflection at the center of each plate. Fig. 13 plots the Δ/ b – θ curve for square tubes on double logarithmic axes. As can be seen in Fig. 13, the gradient of Δ/ b – θ response keeps almost constant regardless of the relative thickness t / b and the amount of torsional angle θ. Based on this result, the following approximation of Δ is proposed in our study.
L
(0 ≤ x ≤ 2 ) x
⎛ δ⎞ A = 0.24 ⎜1 − e−40 b ⎟ . ⎠ ⎝ b
θ
L
(13)
(14)
Comparisons of in-plane distortion of the upper flange at θ = θc =0.675 rad obtained by Eqs. (11)–(14) and FE results are shown in Fig. 9. It is observed that the displacement at both corners of the upper flange for numerical and analytical results fit reasonably with each other. However, there are big difference between FE results and Eqs. (11)–(14) for the displacement at the center of the upper flange. Detailed examination on the displacement of the edge (namely point P) and middle point (namely point Q) of the upper flange is shown in Figs. 10 and 11 respectively. Fig. 10 plots the displacement and torsional angle response at one corner of the square tube. The FE result for displacement Uy at point P agrees well with both Eqs. (11) and (12) (see Fig. 10(a)) but a big difference between FE results and Eq. (12) for displacement Uz can be observed in Fig. 10(b). On the other hand, as can be seen in Fig. 11, the numerical and analytical results for the displacement in both y and z directions at point Q do not fit with each other. This error might be due to the different boundary conditions as
⎛ b ⎞m Δ = C ⎜ ⎟ θ m. ⎝t⎠ b
(15)
Here, C and m are parameters that can be approximated from FE results. For a square tube, the value of parameters C and m in Eq. (15) can be obtained as 3.61 × 10−4 and 2.0, respectively. Fig. 14 shows variations of deflection Δ obtained from Eq. (15) and 72
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 10. The displacement at corner (point P) of square tube. (a) Displacement Uy, (b) Displacement Uz.
FE analysis as the angle θ changes for three cases of thickness-to-width ratio t / b . It can be seen that both results agree well regardless of torsional angle θ and tube thickness t. In the following, the prediction of maximum torsional moment based on deflection Δ obtained by Eq. (15) will be presented. Firstly, let eα be the distance of tangent line from the point O (see Fig. 12) until the arc at angle α, it can be calculated by following equation: (16)
eα = (ρ + d )cos α − ρ .
Secondly, assuming that the overall length of the cross-section keeps constant, the parameters ρ and ϕ can be presented as function of Δ as follows:
⎧ ρϕ = b ⎨ ⎩ ρ (1 − cos ϕ /2) = Δ.
(17)
Fig. 12. Schematic of deformed shape due to flattening in our theoretical model.
Thirdly, the shear strain is assumed to be distributed uniformly over the cross-section, and is written by the following equation:
Finally, the torsional moment can be calculated by
θ γ=d . L
MT = n
ϕ /2
(18)
∫−ϕ/2 τ·eα ·tρdα.
(20)
Based on the uniaxial tensile stress-strain behaviour described by Eq. (1), the shear stress τ and shear strain γ relationship can be derived by
Here, n is the number of sides, and for square tube, n is equal to 4. The approximation of torsional moment can be simplified as follows:
⎧Gγ (γ < τY / G ) ⎪ τ = ⎨ 3τY (E − Eh ) + EEh γ ⎪ 3(E − E ) + 2(1 + ν ) E (γ ≥ τY / G ). ⎩ h h
MT = nτρ [2(ρ + d )sin ϕ /2 − b] t .
(21)
The comparison MT − θ curves for square tubes obtained by FE analysis and Eq. (21) is shown in Fig. 15. The solid line plots FE results while the
(19)
Fig. 11. The displacement at center (point Q) of square tube. (a) Displacement Uy, (b) Displacement Uz.
73
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 16. Classification of collapse mode of square tube. Fig. 13. Variation of deflection Δ corresponding with torsional angle.
Fig. 16 concludes the FE results based on their collapse mode varies with thickness-to-width ratio, t / b and strain-hardening coefficient, Eh / E for square tubes. It is observed that the occurrence of elastic collapse takes place when t / b< 0.01, which is independent of the strainhardening coefficient. As the thickness-to-width ratio, t / b exceeds 0.01, the torsional response is mainly dominated by plastic deformation, and the consideration of plastic flattening becomes significant to estimate the maximum torsional moment. For example, as for the small hardening coefficient (Eh / E =0.01), the square tubes undergone only elastic collapse (t / b< 0.01) and plastic yielding (t / b> 0.01). However, when the strain-hardening coefficient is relatively large (Eh / E > 0.01), the square tubes also undergone mixed-type collapse and flattening in addition to elastic collapse and plastic yielding. The mixed-type collapse and flattening can be occurred easily in square tube with thicker wall and larger strain-hardening coefficient. 3.4. Approximation of maximum torsional moment of multicorner tubes As explained in Section 3.1, the elastic buckling and collapse are observed in multicorner tubes when the tube's thickness-to-width ratio t / b is sufficiently small (t / b<0.01). Therefore, Eq. (9) is effective for estimating the maximum torsional moment for thin-walled tubes. As for the thick-walled tubes having small strain-hardening coefficient (Eh / E ≤ 0.01), the maximum torsional moment is mainly dominated by the plastic yielding of the constituent rectangular plate. Based on Eq. (10), the torsional moment of multicorner tubes due to plastic yielding can be written by the following equation:
Fig. 14. Estimation of deflection Δ for three types of square tubes with large hardening coefficient.
p = Mcol
ntb 2τY π . 2 tan n
(22)
Table 3 shows values of maximum torsional moment of multicorner tubes obtained by FE analysis (Eh / E =0.01, and t / b =0.02) and Eq. (22). It can be seen that both results fit well with each other. Moreover, as explained in Section 3.3, the deflection of the tube Δ is the main parameter in determining the torsional moment of thickwalled tube with large strain-hardening coefficient. Therefore, the deformed cross-section of multicorner tubes must also be investigated before checking the validation of Eq. (21). Fig. 17(a) and (b) illustrate the deformed shape of hexagonal and Fig. 15. Comparison of MT − θ curve between FE results and analytical prediction for relatively thick-walled square tubes with large hardening coefficient.
Table 3 Prediction of MTp for multicorner tubes with Eh /E =0.01, and t /b =0.02.
dashed line is the analytical results derived from Eq. (21). As can be seen, the torsional moment derived from Eq. (21) agrees fairly with FE results. 74
Section type
Square
Hexagon
Octagon
FEM [N m] Eq. (22) [N m]
214.0 209.0
570.5 543.0
1011.0 1003.2
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 17. The most deformed cross-section of multicorner tubes. (a) Hexagonal tube, (b) octagonal tube.
Fig. 19. Comparison of MT − θ curves between FE results and analytical prediction Eq. (21) for hexagonal and octagonal tubes with large hardening coefficient. (a) Hexagonal tube, (b) octagonal tube.
Fig. 18. Comparison of Δ/b – θ curves between FE results and analytical prediction Eq. (15) for Hexagonal and octagonal tubes with large hardening coefficient. (a) hexagonal tube, (b) octagonal tube.
slanted sides do not deflect inward and keep the straight line. The Δ/ b -θ response varying in t / b for hexagonal and octagonal tubes are shown in Fig. 18. Here, the values of parameters C and m in Eq. (15) for each model are written in the figures. It is observed that the prediction of Δ obtained by Eq. (21) agrees fairly well with FE results and gives better assumption as the thickness t increases.
octagonal tubes corresponding at certain torsional angle. As can be seen in Fig. 17(a), all side of tubes moved inward symmetrically, and the biggest deflection Δ occurs at the center of each wall. On the other hand, as for octagonal tube (see Fig. 17(b)), the deformation behaviour is slightly different from square and hexagonal tubes, in which the 75
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 20. Prediction of maximum torsional moment MT , max of multicorner tubes with various thickness-to-width ratio t /b and hardening coefficient Eh /E . (a) Square tube, (b) hexagonal tube, (c) octagonal tube.
atically by using FE analysis. Also, based on our numerical results, estimation procedure of maximum torsional moment is proposed. From our study, the following conclusion can be obtained.
Fig. 19(a) and (b) conclude the comparison of MT-θ response between FE results and prediction by Eq. (21) for hexagonal and octagonal tubes. It is found that Eq. (21) predicts slightly bigger value of maximum torsional moment than FE results for almost all models. In addition, the critical torsional angle θc is also larger than FE results. However, as the ratio t / b increases, the error between FE results and predicted value becomes small. This implies that as the ratio t / b increases, the deformation behaviour is mainly dominated by the flattening deformation. Thus, since Eq. (15) agrees well with deflection Δ from FE results, Eq. (21) also makes better prediction of torsional moment for thick-walled multicorner tubes. In short, Eqs. (15) and (21) are pronounced for relatively thick-walled multicorner tubes. Fig. 20(a)–(c) summarize the validation of proposed equations by comparing FE results and analytical results. The straight line represents p = τY . In addition, Eq. (9) while dotted line shows Eq. (22) with τbuc dashed line displays analytical results of Eq. (21). It is observed in Fig. 20, the proposed equations can be used to estimate the maximum torsional moment for square, hexagonal and octagonal tubes. From Fig. 20, it can be concluded that Eqs. (9), (21) and (22) can approximate the elastic collapse, plastic flattening and plastic yielding respectively, regardless of the geometrical shapes.
1. When the torsional load is applied on thin-walled multicorner tubes, the deformation behaviour can be divided into three modes; elastic collapse, plastic yielding and plastic collapse (flattening). These deformation modes are influenced by the geometrical properties and strain-hardening coefficient of the multicorner tubes. 2. When the tube's thickness-to-width ratio t / b is sufficiently small, the maximum torsional moment takes place due to the occurrence of elastic collapse. The maximum moment can be estimated by developing the conventional equation of elastic collapse of plate girders with flange. In this case, the length of the plate l must be set to the wrinkle's length. 3. When the tube's thickness-to-width ratio t / b is large but the material's strain-hardening coefficient is small, the maximum torsional moment is dominated by the occurrence of plastic yielding. In other words, when the shear stress distribution in a cross section reaches the yield stress, the torsional moment also approaches its peak. Therefore, the maximum moment can be estimated by using the yield stress τy. 4. When the tube's thickness-to-width ratio t / b and the strain-hardening coefficient are large, the large amount of distortion, namely, the flattening deformation can be observed before the moment reaches the peak. In this case, the estimation of flattening deformation is significant to evaluate the maximum torsional moment. Based on our
4. Conclusion In this paper, the estimation of maximum torsional moment for multicorner tubes subjected to torsional loading is investigated system76
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
[7] M. Mahendran, N. Murray, Ultimate load behaviour of box-columns under combined loading of axial compression and torsion, Thin-Walled Struct. 9 (1990) 91–120. [8] G. White, R. Grzebieta, N. Murray, Maximum strength of square thin-walled sections subjected to combined loading of torsion and bending, Int. J. Impact Eng. 13 (2) (1993) 203–214. [9] W. Chen, T. Wierzbicki, Torsional collapse of thin-walled prismatic columns, ThinWalled Struct. 36 (3) (2000) 181–196. [10] W. Chen, T. Wierzbicki, O. Breuner, K. Kristiansen, Torsional crushing of foam-filled thin-walled square columns, Int. J. Mech. Sci. 43 (10) (2001) 2297–2317. [11] W. Chen, Plastic Resistance of Thin-walled Prismatic Tubes Under Large Twisting Rotations, Massachusetts Institute of Technology, Department of Ocean Engineering; and, (S.M.)–Massachusetts Institute of Technology, Department of Mechanical Engineering, U.S.A., 2000. [12] T. Karman, E. Sechler, L. Donnell, The strength of thin plates in compression, Trans. ASME 54 (2) (1932) 53–57. [13] O. Vilnay, C. Burt, The shear effective width of aluminium plates, Thin-Walled Struct. 6 (1988) 119–128. [14] S. Timoshenko, J. Goodier, Theory of Elasticity, McGraw Hill, U.S.A, 1951. [15] K. Basler, Strengh of plate girders in shear, Fritz Eng. Lab. Rep. 251 (20) (1960) 1–57.
FE analysis, a deformation model is proposed, and the maximum torsional moment of multicorner tubes can be approximated successfully from the analytical equations. References [1] J. Alexander, An approximate analysis of the collapse of thin cylindrical shells under axial loading, Q. J. Mech. Appl. Mech. 13 (1960) 10–15. [2] W. Abramowicz, T. Wierzbicki, Axial crushing of multicorner sheet metal columns, J. Appl. Mech. 54 (1) (1989) 113–120. [3] A. Mamalis, D. Manolakos, A. Baldoukas, Energy dissipation and associated failure modes when axially loading polygonal thin-walled cylinder, Thin-Walled Struct. 12 (1) (1991) 17–34. [4] A. Mamalis, D. Manolakos, M. Loannidis, P. Kostazos, C. Dimitriou, Finite element simulation of the axial collapse of metallic thin-walled tubes with octagonal crosssection, Thin-Walled Struct. 41 (10) (2003) 891–900. [5] D. Chen, K. Ushijima, Estimation of the initial peak load for circular tubes subjected to axial impact, Thin-Walled Struct. 49 (7) (2011) 889–898. [6] N. Murray, Introduction to The Theory of Thin-Walled Structures, Oxford Engineering Science Series, Clarendon Press, U.K, 1984.
77
Contents lists available at ScienceDirect
Thin–Walled Structures journal homepage: www.elsevier.com/locate/tws
Estimation of maximum torsional moment for multicorner tubes a
b
Siti Aisyah Mohd Zaifuddin , Dai-Heng Chen , Kuniharu Ushijima a b c
c,⁎
MARK
Department of Mechanical Engineering, Graduate School of Engineering, Tokyo University of Science, Niijuku 6-3-1, Katsushika-ku, Tokyo 1258585, Japan Faculty of Civil Engineering & Mechanics, Jiangsu University, 301 Xuefu Road, Jingkou, Zhenjiang 212013, China Department of Mechanical Engineering, Faculty of Engineering, Tokyo University of Science, Niijuku 6-3-1, Katsushika-ku, Tokyo 1258585, Japan
A R T I C L E I N F O
A B S T R A C T
Keywords: Elastic buckling Plastic yielding Plastic flattening Multicorner tubes Torsional collapse
In this paper, the estimation of maximum torsional moment for multicorner tubes under torsional loading was investigated using nonlinear FE analysis. The effects of tube geometries and strain-hardening coefficient on the torsional behaviour were discussed. The maximum torsional moment was due to the occurrence of sectional collapse of the tubes, and the mechanism of sectional collapse could be classified into three physical phenomena, elastic buckling, plastic yielding and plastic flattening of the cross section. Moreover, based on our numerical results, analytical solution for estimating the maximum torsional moment for each phenomenon was proposed.
1. Introduction Thin-walled circular and multicorner tubes have been widely used in many structural applications as lightweight structural components. Once the external load is applied to the tubes, the external work is dissipated by the elastic and plastic strain energy due to axial tension or compression, bending and torsion. Over the decades, the application of such tubes as impact energy absorbers for vehicles has been investigated by many researchers [1–6]. For example, Alexander [1] proposed his theoretical model to estimate the mean crushing load of circular tubes subjected to axial compressive load. Also, Abramowicz and Wierzbicki [2] developed their theoretical analysis to evaluate the plastic resistance of rectangular and multicorner tubes under axial compression. When the tubes with moderate length are subjected to axial load, the first peak load which associates with local plastic buckling are found in the early stage of deformation, and the following load fructuates due to the wrinkle's folding behaviour. The absorbed energy can be estimated by the load-displacement curve after plastic buckling took place. Therefore, the estimation of average compressive load under such load fructuation is significant for the energy absorption capacity for tubes. On the contrary, when the tubes are subjected to bending or torsional load, the reaction bending or torsional moment increases and reaches a peak, and then drops dramatically and never recover due to the occurrence of sectional collapse. Therefore, the estimation of maximum bending or torsional moment for tubes is important to estimate its load-carrying capacity. The elastic and plastic collapse behaviour of thin-walled tubes under torsion have also been published by many researchers [6–11]. ⁎
For example, Murray [6] investigated the maximum torsional load for thin-walled square tubes, and proposed a solution to distinguish the load caused by elastic buckling with that caused by plastic yielding. In Murray's research, the effect of strain-hardening was neglected [6]. Also, Chen and Wierzbicki [9], Chen et al. [10] and Chen [11] studied the torsional collapse of thin-walled square and circular columns analytically and numerically. They mainly discussed the collapse behaviour of square column under large plastic rotation. In their study, they proposed three-phase of collapse mechanism, namely, pre-buckling, post-buckling and collapse-spreading, and developed their solution to evaluate the torsional moment versus rotation curve for each phase using energy method. Also, they extended their solution for rectangular and hexagonal thin-walled columns, and compared analytical and numerical results. Their analytical results agreed fairly with numerical results, however, there were still large differences between these results, especially the maximum torsional moment. This implies that their solution is limited to be applied in square tubes, and that other modified analytical development is still needed for estimating the torsional moment of multicorner tubes. Basically, the maximum torsional moment and deformation behaviour for tubes depend strongly on the tube geometries and the amount of strain-hardening. The estimation technique should be constructed by observing the variation of stress states and deformation behaviour in details. The main purpose of our study is to evaluate the maximum torsional moment for multicorner tubes by using nonlinear FE analysis. The appearance of maximum torsional moment is due to the occurrence of sectional collapse of the tubes, and the mechanism of sectional collapse could be classified into three physical phenomena, elastic buckling, plastic yielding and plastic flattening of the cross-section. Based on the
Corresponding author. Tel.: +81 (0) 358761332; fax: +81 (0) 358761332. E-mail address: [email protected] (K. Ushijima).
http://dx.doi.org/10.1016/j.tws.2016.12.005 Received 6 April 2016; Received in revised form 17 November 2016; Accepted 12 December 2016 0263-8231/ © 2016 Elsevier Ltd. All rights reserved.
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
each direction. Here, the wall's width b and the tube's length L are set to 50 mm and 200–250 mm respectively, and the thickness t varying from 0.35 to 4.0 mm. The tube's length L may affect the torsional response of tubes, but our FE results confirmed that when the length-to-width ratio L / b is larger than 4.0, the effect can be negligible. In this paper, the estimation of maximum torsional moment of square tubes is mainly investigated analytically and numerically, while the validation of the proposed theoretical equations on the multicorner tubes is summarized in Section 3.4.
Table 1 List of material parameters. Young's modulus E [GPa] Poisson's ratio Yield strass σY [MPa] Hardening coefficient Eh [MPa]
72.4 0.3 72.4 E /100 ≤ Eh ≤ E /10
investigation of stress variation and cross-sectional distortion in our FE analysis, the analytical solutions for estimating the maximum torsional moment for each phenomenon are proposed.
3. Results and discussion It is well known that for the torsional response of hollow straight tubes, the sectional collapse would occur when the end rotation reaches a certain angle. Once the sectional collapse occurs, the torsional moment reaches a maximum and drops considerably. The occurrence of sectional collapse has two reasons:
2. Method of numerical analysis In our study, the commercial finite element analysis software, MSC.Marc, is used to demonstrate the torsional deformation behaviour of multicorner tubes. The multicorner tubes are assumed to be isotropic, homogeneous elastoplastic material, and obey the bilinear hardening curve as shown in Eq. (1) where Eh shows the strain-hardening coefficient.
⎛ σ ⎞ σ = σY + Eh ⎜ε − Y ⎟ . ⎝ E⎠
• •
the shear stress distributed over the cross-section reaches a limit the large amount of plastic deformation, namely, plastic flattening deformation arises
The former includes two physical phenomena, namely, the occurrence of elastic buckling and plastic yielding, and these phenomena can be observed in relatively thin-walled tubes. In the following, the torsional moment due to elastic buckling and plastic yielding are written by Mcole and Mcolp respectively. On the contrary, as for the latter case, the walls of the tubes move inward and the applied moment drops significantly even when the shear stress still increases after the collapse. This phenomenon can be observed in relatively thick-walled tubes. Here the torsional moment due to plastic flattening is written by Mcolf. Fig. 2(a)–(c) show examples of the torsional moment MT versus torsional angle θ (MT-θ) diagram of square tubes having different tube's thickness-to-width ratio t / b and different strain-hardening coefficient Eh / E . Also in these figures, two kinds of stress variation are plotted by dotted and dashed lines. Here, τA represents the shear stress at point ‘A’ where the maximum deflection takes place, and τave represents the average shear stress at the same position x = xA . It can be found in these
(1)
Values of the material properties applied in our FE model are as shown in Table 1 unless it is stated. As for the finite element mesh, the 2-dimensional shell element is used for the thin-walled tubes with 5 mm×5 mm mesh size. also in our calculation, the mises’ yield criterion and the updated lagrangian method are used for formulating the nonlinear problem, and the newton-raphson numerical method is applied for finding the root effectively. The boundary condition and tube's geometry of our FE model are explained as follows. All rotations and displacements except for the movement in the axial direction Ux is fixed at one end of the tube, and the torsional rotation θ is applied at the other end of the tube via an attached rigid plate. Fig. 1(a) shows the schematic of boundary condition, and Fig. 1(b) portrays three types of cross-sectional shape of multicorner tubes discussed in our paper. The longitudinal coordinate sets x, while y and z denote the coordinate of the cross-section in
Fig. 1. Multicorner tubes subjected to torsional loading. (a) Hollow straight tube under torsional loading, (b) cross-section of multicorner tubes.
67
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 2. MT-θ curves of square tubes with different thickness-to-width ratio t /b and hardening coefficient Eh /E . (a) t /b = 7.0 × 10−3, Eh /E = 0.01. (b) t /b = 0.02 , Eh /E = 0.01. (c) t /b = 0.02 , Eh /E = 0.07 .
figures that the characteristic of MT-θ diagram is affected strongly on the thickness-to-width ratio t / b and the strain-hardening coefficient Eh / E . Finally, the maximum torsional moment for tubes can be settled by the competition of the above three moments, Mcole, Mcolp and Mcolf and written in the following equation: f p e Mcol = Minimum [Mcol , Mcol , Mcol ].
(2)
The torsional moment Mcol may depend on tube's geometries and material's strain-hardening properties. In our study, the torsional response of tubes is examined by changing the tube thickness t and strain-hardening coefficient Eh / E systematically, and the theoretical e and analytical solutions for the estimation of Mcol , Mcolp and Mcolf are discussed in the next section. 3.1. Torsional moment due to elastic buckling and elastic collapse When the tube's thickness-to-width ratio t / b is sufficiently small, the sectional collapse caused by elastic buckling can be observed in the tube. Fig. 2(a) shows the MT-θ diagram for a square tube with the ratio t / b =7.0 × 10−3. The distribution of shear stress over a cross-section at four points ‘B’, ‘C’, ‘D’ and ‘E’ in Fig. 2(a) is shown in Fig. 3, where the normalized distance of ‘A’ along the tube xA / L =0.55. Here, the parameter s represents the distance over a cross-section from one corner. It can be found that a uniform shear stress distributed along the width until the angle θ=7.0 × 10−3 rad (at ‘C’ in Fig. 2(a)). Then, the shear stress at the center of the wall (s / b =0.5) drops when the angle θ exceeds 7.0 × 10−3 rad although the shear stress is smaller than the yield value τY(=σY / 3 =41.8 MPa). This is because the elastic buckling
Fig. 3. Stress distribution for a square tube with t /b = 7.0 × 10−3 and Eh /E = 0.01 at the most deformed cross-section ( xA /L = 0.55).
occurs at the center of the wall (s / b =0.5) when the angle θ is equal to 7.0 × 10−3 rad. However, the torsional moment still increases until θ=9.0 × 10−3 rad (at ‘D’ in Fig. 2(a)), as the elements near four corners in a cross-section can sustain the shear stress to some extent after the
68
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
elastic buckling, which is called the effect of “effective width” proposed by Karman et al. [12]. When the angle θ exceeds 9.0 × 10−3 rad, the elements near corners would also yield, and the torsional moment drops dramatically. In Fig. 2(a), the variation of average shear stress τave over the cross-section is also plotted by a dashed curve. It is found that the angle θ where the average shear stress reaches its maximum point coincides with that where the torsional moment becomes a maximum. From the above results, it can be concluded that the elastic buckling and elastic collapse can be distinguished by the variation of shear stress. In the following, the analytical solution of torsional moments for each phenomenon is explained. The analysis of torsional buckling for a thin-walled square tube can be regarded as an elastic buckling problem of a rectangular plate subjected to shear loading. There are some papers and books published regarding the ultimate strength of thin-walled plates under shear loading. For example, Vilnay and Burt [13] studied the effective width of plates made of aluminium under shear loading. As mentioned above, the effective width is firstly proposed by Karman et al. [12] in order to estimate the maximum compressive stress of a plate larger than the elastic buckling stress. Also, the theoretical result of elastic buckling stress τbuce for a plate under shear loading has already been reported in Timoshenko and Goodier's book [14] as follows: e = τbuc
⎛ t ⎞2 kπ 2E ⎜ ⎟ . 2 12(1 − ν ) ⎝ b ⎠
cross-section. Next, the modification on the estimation of torsional moment due to elastic buckling will be discussed. As explained above, the buckling coefficient k in Eq. (3) depends on the length l of a plate. Based on the inclination angle shown in Fig. 4, the length l can be regarded as the length of a wrinkle which gives l=b/tan 35°. Therefore, the value of the parameter k becomes 7.31, which is obtained from Eq. (4). Finally, the estimation of the shear stress and the torsional moment at elastic buckling can be modified and shown in the following equations: e = 0.61 × τbuc
e = 1.22 × Mbuc
Here, the parameter k represents the buckling coefficient depending on the length-to-width ratio, l / b of a plate and the edge boundary condition. As for a simply-supported rectangular plate, the parameter k can be approached by
(4)
e Mcol p Mcol
0.35 42.18 30.01 41.02
0.40 59.86 44.81 61.24
=
e τbuc
τY
1−
+
e τbuc τY
3 . · 2 1 + α2
(9)
p Mcol = 2τY b 2t .
(10) p e Mcol / Mcol
Fig. 5 shows a variation of as a function of normalized e shear stress τY / τbuc for square, hexagonal and octagonal tubes. Analytical results obtained by Eq. (9) and proposed by Vilnay and Burt [13] are also shown in the figure by solid and dashed lines, respectively. It can be concluded from Fig. 5 that the maximum torsional moment due to elastic collapse can be estimated sufficiently from Eqs. (7), (9) and (10) regardless of the shape of cross-section.
(5)
0.30 29.18 18.90 25.83
(8)
The torsional moment at plastic yielding can be written as
3.2. Torsional moment due to plastic yielding When the tube becomes thicker, the torsional behaviour is mainly dominated by the plastic deformation. Fig. 2(b) displays the MT-θ response of a square tube with t / b =0.02 and with relatively small strain-hardening coefficient Eh / E =0.01. The corresponding shear stress distribution at point ‘F’, ‘G’, ‘H’ and ‘I’ in Fig. 2(b) is shown in Fig. 6(a). Here, the normalized distance along the tube xA / L is 0.275 where the maximum deformation can be observed. Also, the torsional response of a square tube with relatively large strain-hardening coefficient Eh / E =0.07 are shown in Fig. 2(c) and Fig. 6(b) where the normalized distance along the tube xA / L is 0.225. As can be found in Fig. 2(b) and (c), the torsional angle where the average shear stress reaches a peak is almost equal to that of the maximum torsional moment. This behaviour is similar to the elastic collapse as explained in the previous section. However, it is found in Fig. 2(b) and Fig. 6(a) that when a square tube has relatively thick wall but with small strain-hardening coefficient, the average torsional shear stress τave increases monotonically and reaches the yielding value, τY without the appearance of elastic buckling or
Table 2 Comparison of Mbuce for a plate under shear load obtained from FEM and predictions (Eq. (5)). 0.25 19.80 10.94 14.95
e τbuc τY
Here, the parameter α represents the length-to-width ratio L / b of the plate. Applying this equation to the estimation of the torsional moment due to elastic collapse gives
Table 2 summarizes torsional moment at the initiation of elastic buckling of square tubes for b=50 mm and varying the thickness t from 0.25 to 0.45 mm obtained by FE analysis and Eq. (5). It can be found in Table 2 that there are large differences between them. In the following, the modification on the estimation of the elastic buckling moment Mbuce based on the deformation behaviour of multicorner tubes will be carried out. Comparisons of deformed shape at the initiation of elastic buckling for the case of tube's thickness t=0.25 mm, 0.35 mm and 0.45 mm are given in Fig. 4(a)–(c) respectively. In these figures, the contour bands represent the in-plane shear stress distribution on the square tube. From these figures, it can be found that the inclination angle of a wrinkle observed in each tube is almost the same regardless of the tube's thickness t, and the angle can be measured as about 35°. In addition, our FE results confirmed that this angle is also independent of the shape of
Thickness t [mm] FEM [N m] Eq. (5) [N m] Eq. (7) [N m]
(7)
Vu 3 = + · . VY τY 2 1 + α2
where l denotes the length of the plate. White et al. [8] also investigated the elastic torsional buckling of a simply-supported plate, and they concluded that the lowest value of τbuce would occur for the lowest value of k, simply when the normalized parameter l / b approaches infinity. Thus, they regarded the value of k as 5.35, and concluded that the torsional moment of square tube at the occurrence of elastic buckling can be estimated by the following equations: e Mbuc
π 2E 3 t . (1 − ν 2 )
1−
e τbuc
⎞2
π 2E 3 = 0.89 × t . (1 − ν 2 )
(6)
The modified results of torsional moment caused by elastic buckling are also shown in Table 2, which agree well with FE results. Next, the estimation of the maximum torsional moment due to elastic collapse will be discussed. Basler [15] investigated analytically the strength of plate girders with flanges subjected to shear loading, and derived the ratio of the ultimate shear load Vu and the plastic shear load VY as a function of the buckling shear stress τbuce and the plastic yielding shear stress τY as follows:
(3)
⎛b k = 5.35 + 4 ⎜ ⎟ , ⎝l⎠
2 π 2E ⎛ t ⎞ ⎜ ⎟ , 2 (1 − ν ) ⎝ b ⎠
0.45 80.48 63.80 87.19
69
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 4. Comparison of wrinkle's angle for square tubes with different thickness t. (a) t=0.25 mm, (b) t=0.35 mm, (c) t=0.45 mm.
70
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 5. Variation of Mcole/ Mcolp for thin-walled multicorner tubes (t /b < 0.01).
Fig. 7. MT − θ curve of a square tube with t /b = 0.06 and Eh /E =0.1.
elastic collapse. After that, the stress decreases rapidly due to the occurrence of plastic yielding. On the contrary, as can be observed in Fig. 2(c), when the strain-hardening coefficient is relatively large, the average shear stress τave increases gradually until its peak, but its maximum value is larger than the yield stress τY. Then, the average shear stress τave decreases gently although the torsional angle increases. It can be understood by comparing both stress results shown in Fig. 6(a) and (b) that when the strain-hardening is sufficiently large, the stress distribution does not drop and keeps high even after the occurrence of the maximum torsional moment. This implies that the maximum torsional moment for tubes with larger strain-hardening material is not mainly due to the degradation of stress value, but due to the inward deflection of a cross-section. In this paper, this deformation behaviour is called as flattening, and the estimation of maximum moment due to flattening will be discussed in the next section. Here, as for the case of t / b =0.02 and Eh / E =0.07, the square tube collapses in transition phase from plastic yielding to plastic collapse (flattening), thus, this deformation behaviour can be regarded as mixed-type collapsing. As for the perfectly-rigid plastic material, Mahendran and Murray [7] proposed a formulation of torsional moment due to plastic yielding, Mcolp as shown in Eq. (10). In Eq. (10), the effect of strain-hardening coefficient is being ignored. The analytical result obtained by Eq. (10) is also shown in Fig. 2(b) and (c) by a solid line. It is found from Fig. 2(c) that as for thick-walled square tubes with relatively large strain-
hardening, the maximum torsional moment cannot be predicted effectively by Eq. (10). 3.3. Maximum torsional moment due to plastic collapse (flattening) Fig. 7 represents MT-θ response of a thick-walled square tube with t / b =0.06 and large strain-hardening coefficient Eh / E =0.1. Also in Fig. 7, the variation of average shear stress at xA / L =0.5 is shown by a dashed curve. It is found that although the torsional moment increases and reaches a peak at θ=0.82 rad, the average shear stress keeps on increasing until θ=1.1 rad. In this case, Fig. 7 proved that the average shear stress τave is not the main parameter that affect the drop in torsional moment. Instead, it is anticipated that the large sectional deflection could be the reason to the occurrence of maximum torsional moment. Thus, in this section, the deformation behaviour of the crosssection of square tubes will be investigated. Fig. 8(a) and (b) plot the deformed cross-section at maximum torsional moment for two cases of hardening coefficient. Here, the relative rotational movement is removed in both figures whereas θc is defined as the critical angle where the maximum torsional moment takes place. As can be seen in Fig. 8(a), the cross-section of square tube deformed slightly when the material has small hardening coefficient (Eh / E =0.01). On the contrary, when the square tube has large hardening coefficient (Eh / E =0.1), the large amount of flattening deformation in which the tube walls move inward can be observed in Fig. 8(b).
Fig. 6. Variations of shear stress observed in two cases of square tubes with different hardening coefficient Eh /E at the most deformed cross-section. (a) Eh /E =0.01, (b)Eh /E =0.07.
71
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 8. The deformed cross-section of square tubes at peak torsional moment with different thickness-to-width ratio t /b and hardening coefficient Eh /E . (a) t /b = 0.05, Eh /E = 0.01, (b) t /b = 0.05 , Eh /E = 0.1.
Such movement results in the degradation of the torsional moment. Therefore, it can be concluded that for a thicker tube made by larger strain-hardening material, the formulation of flattening movement is important to estimate the maximum torsional moment. W. Chen [11] has proposed analytical equations to describe the deformation of square tubes for two phases under torsional loading. In his study, it is approximated that in the pre-buckling phase, the walls of the tube become spiral surfaces, and the displacements of the upper b b b flange (− 2 ≤ z ≤ 2 , y = 2 ) for x-, y- and z-directions can be written in the following equations:
⎧Ux = 0 ⎪ ⎪ b xθ xθ b ⎨Uy = 2 cos( L ) − z sin( L ) − 2 ⎪ xθ b xθ ⎪Uz = z cos( ) + sin( ) − z. ⎩ L 2 L
(11)
Also, in the post-buckling phase, since the walls would deform inward as shown in Fig. 8, it is predicted that the displacements can be expressed by the following equations:
⎧Ux = 0 ⎪ ⎪ b zπ xθ zπ xθ b ⎨Uy = ( 2 − A cos b )cos L − (z − δ sin b )sin L − 2 ⎪ ⎪Uz = (z − δ sin zπ )cos xθ + ( b − A cos zπ − δ )sin xθ − z, ⎩ b L 2 b L
Fig. 9. Comparisons of deformed shape of upper flange for pre- and post-buckling phases obtained by FE analysis and theoretical results proposed by Chen [11].
(12)
where δ and A are two parameters which depend on torsional angle θ, and can be approximated by
⎧ ⎪ (1 − δ =⎨ b ⎪ (1 − ⎩
2 2
)· L · π
x θ
2 2
)(1 − L )· π ( 2 ≤ x ≤ L ),
W. Chen fixed the x-axis displacement while free axial displacement is set in this study. As can be observed in Fig. 8, the maximum displacement occurs at the center of the wall. Thus, as the displacement at the mentioned point cannot be predicted by using W. Chen's theory, another new model to evaluate the maximum torsional moment due to flattening is proposed as follows. Fig. 12 shows a schematic of simplified flattening deformation applied in our theoretical analysis. There are some assumptions in our theoretical model. The deformed shape is composed of four identical semi-arcs having the same radius ρ and angle ϕ. Here, the parameter Δ is the deflection at the center of each plate. Fig. 13 plots the Δ/ b – θ curve for square tubes on double logarithmic axes. As can be seen in Fig. 13, the gradient of Δ/ b – θ response keeps almost constant regardless of the relative thickness t / b and the amount of torsional angle θ. Based on this result, the following approximation of Δ is proposed in our study.
L
(0 ≤ x ≤ 2 ) x
⎛ δ⎞ A = 0.24 ⎜1 − e−40 b ⎟ . ⎠ ⎝ b
θ
L
(13)
(14)
Comparisons of in-plane distortion of the upper flange at θ = θc =0.675 rad obtained by Eqs. (11)–(14) and FE results are shown in Fig. 9. It is observed that the displacement at both corners of the upper flange for numerical and analytical results fit reasonably with each other. However, there are big difference between FE results and Eqs. (11)–(14) for the displacement at the center of the upper flange. Detailed examination on the displacement of the edge (namely point P) and middle point (namely point Q) of the upper flange is shown in Figs. 10 and 11 respectively. Fig. 10 plots the displacement and torsional angle response at one corner of the square tube. The FE result for displacement Uy at point P agrees well with both Eqs. (11) and (12) (see Fig. 10(a)) but a big difference between FE results and Eq. (12) for displacement Uz can be observed in Fig. 10(b). On the other hand, as can be seen in Fig. 11, the numerical and analytical results for the displacement in both y and z directions at point Q do not fit with each other. This error might be due to the different boundary conditions as
⎛ b ⎞m Δ = C ⎜ ⎟ θ m. ⎝t⎠ b
(15)
Here, C and m are parameters that can be approximated from FE results. For a square tube, the value of parameters C and m in Eq. (15) can be obtained as 3.61 × 10−4 and 2.0, respectively. Fig. 14 shows variations of deflection Δ obtained from Eq. (15) and 72
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 10. The displacement at corner (point P) of square tube. (a) Displacement Uy, (b) Displacement Uz.
FE analysis as the angle θ changes for three cases of thickness-to-width ratio t / b . It can be seen that both results agree well regardless of torsional angle θ and tube thickness t. In the following, the prediction of maximum torsional moment based on deflection Δ obtained by Eq. (15) will be presented. Firstly, let eα be the distance of tangent line from the point O (see Fig. 12) until the arc at angle α, it can be calculated by following equation: (16)
eα = (ρ + d )cos α − ρ .
Secondly, assuming that the overall length of the cross-section keeps constant, the parameters ρ and ϕ can be presented as function of Δ as follows:
⎧ ρϕ = b ⎨ ⎩ ρ (1 − cos ϕ /2) = Δ.
(17)
Fig. 12. Schematic of deformed shape due to flattening in our theoretical model.
Thirdly, the shear strain is assumed to be distributed uniformly over the cross-section, and is written by the following equation:
Finally, the torsional moment can be calculated by
θ γ=d . L
MT = n
ϕ /2
(18)
∫−ϕ/2 τ·eα ·tρdα.
(20)
Based on the uniaxial tensile stress-strain behaviour described by Eq. (1), the shear stress τ and shear strain γ relationship can be derived by
Here, n is the number of sides, and for square tube, n is equal to 4. The approximation of torsional moment can be simplified as follows:
⎧Gγ (γ < τY / G ) ⎪ τ = ⎨ 3τY (E − Eh ) + EEh γ ⎪ 3(E − E ) + 2(1 + ν ) E (γ ≥ τY / G ). ⎩ h h
MT = nτρ [2(ρ + d )sin ϕ /2 − b] t .
(21)
The comparison MT − θ curves for square tubes obtained by FE analysis and Eq. (21) is shown in Fig. 15. The solid line plots FE results while the
(19)
Fig. 11. The displacement at center (point Q) of square tube. (a) Displacement Uy, (b) Displacement Uz.
73
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 16. Classification of collapse mode of square tube. Fig. 13. Variation of deflection Δ corresponding with torsional angle.
Fig. 16 concludes the FE results based on their collapse mode varies with thickness-to-width ratio, t / b and strain-hardening coefficient, Eh / E for square tubes. It is observed that the occurrence of elastic collapse takes place when t / b< 0.01, which is independent of the strainhardening coefficient. As the thickness-to-width ratio, t / b exceeds 0.01, the torsional response is mainly dominated by plastic deformation, and the consideration of plastic flattening becomes significant to estimate the maximum torsional moment. For example, as for the small hardening coefficient (Eh / E =0.01), the square tubes undergone only elastic collapse (t / b< 0.01) and plastic yielding (t / b> 0.01). However, when the strain-hardening coefficient is relatively large (Eh / E > 0.01), the square tubes also undergone mixed-type collapse and flattening in addition to elastic collapse and plastic yielding. The mixed-type collapse and flattening can be occurred easily in square tube with thicker wall and larger strain-hardening coefficient. 3.4. Approximation of maximum torsional moment of multicorner tubes As explained in Section 3.1, the elastic buckling and collapse are observed in multicorner tubes when the tube's thickness-to-width ratio t / b is sufficiently small (t / b<0.01). Therefore, Eq. (9) is effective for estimating the maximum torsional moment for thin-walled tubes. As for the thick-walled tubes having small strain-hardening coefficient (Eh / E ≤ 0.01), the maximum torsional moment is mainly dominated by the plastic yielding of the constituent rectangular plate. Based on Eq. (10), the torsional moment of multicorner tubes due to plastic yielding can be written by the following equation:
Fig. 14. Estimation of deflection Δ for three types of square tubes with large hardening coefficient.
p = Mcol
ntb 2τY π . 2 tan n
(22)
Table 3 shows values of maximum torsional moment of multicorner tubes obtained by FE analysis (Eh / E =0.01, and t / b =0.02) and Eq. (22). It can be seen that both results fit well with each other. Moreover, as explained in Section 3.3, the deflection of the tube Δ is the main parameter in determining the torsional moment of thickwalled tube with large strain-hardening coefficient. Therefore, the deformed cross-section of multicorner tubes must also be investigated before checking the validation of Eq. (21). Fig. 17(a) and (b) illustrate the deformed shape of hexagonal and Fig. 15. Comparison of MT − θ curve between FE results and analytical prediction for relatively thick-walled square tubes with large hardening coefficient.
Table 3 Prediction of MTp for multicorner tubes with Eh /E =0.01, and t /b =0.02.
dashed line is the analytical results derived from Eq. (21). As can be seen, the torsional moment derived from Eq. (21) agrees fairly with FE results. 74
Section type
Square
Hexagon
Octagon
FEM [N m] Eq. (22) [N m]
214.0 209.0
570.5 543.0
1011.0 1003.2
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 17. The most deformed cross-section of multicorner tubes. (a) Hexagonal tube, (b) octagonal tube.
Fig. 19. Comparison of MT − θ curves between FE results and analytical prediction Eq. (21) for hexagonal and octagonal tubes with large hardening coefficient. (a) Hexagonal tube, (b) octagonal tube.
Fig. 18. Comparison of Δ/b – θ curves between FE results and analytical prediction Eq. (15) for Hexagonal and octagonal tubes with large hardening coefficient. (a) hexagonal tube, (b) octagonal tube.
slanted sides do not deflect inward and keep the straight line. The Δ/ b -θ response varying in t / b for hexagonal and octagonal tubes are shown in Fig. 18. Here, the values of parameters C and m in Eq. (15) for each model are written in the figures. It is observed that the prediction of Δ obtained by Eq. (21) agrees fairly well with FE results and gives better assumption as the thickness t increases.
octagonal tubes corresponding at certain torsional angle. As can be seen in Fig. 17(a), all side of tubes moved inward symmetrically, and the biggest deflection Δ occurs at the center of each wall. On the other hand, as for octagonal tube (see Fig. 17(b)), the deformation behaviour is slightly different from square and hexagonal tubes, in which the 75
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
Fig. 20. Prediction of maximum torsional moment MT , max of multicorner tubes with various thickness-to-width ratio t /b and hardening coefficient Eh /E . (a) Square tube, (b) hexagonal tube, (c) octagonal tube.
atically by using FE analysis. Also, based on our numerical results, estimation procedure of maximum torsional moment is proposed. From our study, the following conclusion can be obtained.
Fig. 19(a) and (b) conclude the comparison of MT-θ response between FE results and prediction by Eq. (21) for hexagonal and octagonal tubes. It is found that Eq. (21) predicts slightly bigger value of maximum torsional moment than FE results for almost all models. In addition, the critical torsional angle θc is also larger than FE results. However, as the ratio t / b increases, the error between FE results and predicted value becomes small. This implies that as the ratio t / b increases, the deformation behaviour is mainly dominated by the flattening deformation. Thus, since Eq. (15) agrees well with deflection Δ from FE results, Eq. (21) also makes better prediction of torsional moment for thick-walled multicorner tubes. In short, Eqs. (15) and (21) are pronounced for relatively thick-walled multicorner tubes. Fig. 20(a)–(c) summarize the validation of proposed equations by comparing FE results and analytical results. The straight line represents p = τY . In addition, Eq. (9) while dotted line shows Eq. (22) with τbuc dashed line displays analytical results of Eq. (21). It is observed in Fig. 20, the proposed equations can be used to estimate the maximum torsional moment for square, hexagonal and octagonal tubes. From Fig. 20, it can be concluded that Eqs. (9), (21) and (22) can approximate the elastic collapse, plastic flattening and plastic yielding respectively, regardless of the geometrical shapes.
1. When the torsional load is applied on thin-walled multicorner tubes, the deformation behaviour can be divided into three modes; elastic collapse, plastic yielding and plastic collapse (flattening). These deformation modes are influenced by the geometrical properties and strain-hardening coefficient of the multicorner tubes. 2. When the tube's thickness-to-width ratio t / b is sufficiently small, the maximum torsional moment takes place due to the occurrence of elastic collapse. The maximum moment can be estimated by developing the conventional equation of elastic collapse of plate girders with flange. In this case, the length of the plate l must be set to the wrinkle's length. 3. When the tube's thickness-to-width ratio t / b is large but the material's strain-hardening coefficient is small, the maximum torsional moment is dominated by the occurrence of plastic yielding. In other words, when the shear stress distribution in a cross section reaches the yield stress, the torsional moment also approaches its peak. Therefore, the maximum moment can be estimated by using the yield stress τy. 4. When the tube's thickness-to-width ratio t / b and the strain-hardening coefficient are large, the large amount of distortion, namely, the flattening deformation can be observed before the moment reaches the peak. In this case, the estimation of flattening deformation is significant to evaluate the maximum torsional moment. Based on our
4. Conclusion In this paper, the estimation of maximum torsional moment for multicorner tubes subjected to torsional loading is investigated system76
Thin-Walled Structures 112 (2017) 66–77
S. Aisyah Mohd Zaifuddin et al.
[7] M. Mahendran, N. Murray, Ultimate load behaviour of box-columns under combined loading of axial compression and torsion, Thin-Walled Struct. 9 (1990) 91–120. [8] G. White, R. Grzebieta, N. Murray, Maximum strength of square thin-walled sections subjected to combined loading of torsion and bending, Int. J. Impact Eng. 13 (2) (1993) 203–214. [9] W. Chen, T. Wierzbicki, Torsional collapse of thin-walled prismatic columns, ThinWalled Struct. 36 (3) (2000) 181–196. [10] W. Chen, T. Wierzbicki, O. Breuner, K. Kristiansen, Torsional crushing of foam-filled thin-walled square columns, Int. J. Mech. Sci. 43 (10) (2001) 2297–2317. [11] W. Chen, Plastic Resistance of Thin-walled Prismatic Tubes Under Large Twisting Rotations, Massachusetts Institute of Technology, Department of Ocean Engineering; and, (S.M.)–Massachusetts Institute of Technology, Department of Mechanical Engineering, U.S.A., 2000. [12] T. Karman, E. Sechler, L. Donnell, The strength of thin plates in compression, Trans. ASME 54 (2) (1932) 53–57. [13] O. Vilnay, C. Burt, The shear effective width of aluminium plates, Thin-Walled Struct. 6 (1988) 119–128. [14] S. Timoshenko, J. Goodier, Theory of Elasticity, McGraw Hill, U.S.A, 1951. [15] K. Basler, Strengh of plate girders in shear, Fritz Eng. Lab. Rep. 251 (20) (1960) 1–57.
FE analysis, a deformation model is proposed, and the maximum torsional moment of multicorner tubes can be approximated successfully from the analytical equations. References [1] J. Alexander, An approximate analysis of the collapse of thin cylindrical shells under axial loading, Q. J. Mech. Appl. Mech. 13 (1960) 10–15. [2] W. Abramowicz, T. Wierzbicki, Axial crushing of multicorner sheet metal columns, J. Appl. Mech. 54 (1) (1989) 113–120. [3] A. Mamalis, D. Manolakos, A. Baldoukas, Energy dissipation and associated failure modes when axially loading polygonal thin-walled cylinder, Thin-Walled Struct. 12 (1) (1991) 17–34. [4] A. Mamalis, D. Manolakos, M. Loannidis, P. Kostazos, C. Dimitriou, Finite element simulation of the axial collapse of metallic thin-walled tubes with octagonal crosssection, Thin-Walled Struct. 41 (10) (2003) 891–900. [5] D. Chen, K. Ushijima, Estimation of the initial peak load for circular tubes subjected to axial impact, Thin-Walled Struct. 49 (7) (2011) 889–898. [6] N. Murray, Introduction to The Theory of Thin-Walled Structures, Oxford Engineering Science Series, Clarendon Press, U.K, 1984.
77