^{1}

^{1}

^{1}

^{2}

^{3}

Thin-walled member is structurally superior to a construction member. However, by reason of complexity in structure the stress and the deformation to yield the cross section are complicated. Specially, in case thin-walled members, such as thin-walled channel section columns, which are subjected to compressive force, these members produce the local buckling, distortional buckling and overall buckling. A number of experimental and theoretical investigations subjected to axial compressive force are generated for thin-walled channel section columns with triangle-shaped folded groove by Hancock [1] and with complex edge stiffeners and web stiffeners by Wang [2]. In case thin-walled channel section column with folded groove which is subjected to axial compressive force, it is cleared that the buckling mode shapes are ordinarily generated for local buckling mode shape of plate-panel composing cross section of member in short member aspect ratio and overall buckling mode shape as column and distortional buckling mode shape interacting between local buckling and overall buckling similarly normal thin-walled member. It is cleared analytically and experimentally that buckling strength and critical strength of thin-walled channel section column with folded groove can increase sharply in comparison with that of normal thin-walled member composing only plate-panel. In this paper a new cross section of shell-shaped curved groove [3] was proposed instead of the thin-walled lipped channel section column with triangle- and rectangle-shaped folded grooves used ordinarily, and therefore the comparison and the examination of buckling strength and buckling behavior were generated in the case of preparing triangle-shaped folded and shell-shaped curved grooves to web and flange of thin-walled channel section column. And then in order to investigate the buckling behavior on the thin-walled channel section column with folded and curved grooves, exact buckling strength and the buckling mode shapes are generated by using the transfer matrix method. The analytical local distortional and overall elastic buckling loads of thin-walled channel section column with folded and curved grooves can be obtained simultaneously by use of the transfer matrix method. Furthermore, a technique to estimate the buckling mode shapes of these members is also shown.

Thin-walled member is structurally superior to a construction member. However by reason of complexity in structure the stress and the deformation to yield the cross section are complicated.

Specially, in case thin-walled members, such as thin-walled channel section columns, which are subjected to compressive force, the members with folded grooves produce the local buckling, distortional buckling and overall buckling.

In recent year, thin-walled channel section columns with folded grooves fabricated from cold-reduced high strength steel plate have been used by advance of manufacturing technique. However, a design of thin-walled member with folded grooves subjected to axial force is generated by obtained strength during assumed bending test as use of deck-plate. And experimental and theoretical investigations are extremely insufficient for various complicated buckling behavior above-mentioned. Especially, in the case of thin-walled member with folded and curved grooves subjected to axial force, the examination of most effective shape to cross section is present state to be not almost investigated.

A number of experimental and theoretical investigations subjected to axial compressive force are performed for thin-walled channel section column with triangle-shaped folded groove by Hancock [^{ }and Wang [

It is cleared analytically and experimentally that the buckling strength and the critical strength of thin-walled channel section column with folded groove can increase sharply in comparison with normal thin-walled member composing only plate-panels.

Further, on the position, the number and the size of groove-cross section and the effect exerting buckling behavior and buckling strength are examined. Then, it is clear that more large stiffened effect is obtained by establishing groove-cross section not only web but also flanges.

As regards the investigation of the shape of folded groove, the comparison and the examination on the effect exerting bending strength of deck-plate during the shape of folded groove are reported in the case of using deck-plate the thin-walled channel section column with triangle- and rectangle-shaped grooves.

This study aims to propose a new cross section of shell-shaped curved groove [

And then in order to investigate the buckling behavior on the thin-walled channel section column with folded and curved grooves, exact buckling strength and the buckling mode shapes are generated by using the transfer matrix method. In analysis the transfer equations are proposed by considering the compatibility and the equilibrium conditions between plate-panel and groove. The analytical local, distortional and overall elastic buckling loads of thin-walled channel section column with folded and curved grooves can be obtained simultaneously by use of the transfer matrix method. Furthermore, a technique to estimate the buckling mode shapes of these members is also shown.

From the equilibrium equations of forces for the shell-panel subjected to in-plane compressive force (

by the sides x = 0 and x = L of the shell-pate into the partial differential equations, the following ordinary differential equations

d Z S r d φ = Z S • = [ w ∗ S • , φ ∗ S φ • , M ∗ S φ • , V ∗ S φ • , v ∗ S • , u ∗ S • , N ∗ S φ • , N ∗ S φ x • ] (1)

referenced to the variable φ are only obtained.

( w ∗ S φ ∗ S φ M ∗ S φ V ∗ S φ v ∗ S u ∗ S N ∗ S φ N ∗ S φ x ) • = ( 0 − α 0 0 1 r 0 0 0 − α K 21 K 22 0 α K 0 K 22 0 0 − I 21 r I 22 − α 2 K 0 r I 22 0 0 4 α K 33 K 0 0 α 0 0 0 0 A 41 0 α K 12 K 22 0 0 0 1 r 0 − 1 r 0 0 0 0 α I 21 I 22 α 3 K 0 I 22 0 0 2 α 2 K 33 r I 33 0 0 − α 0 0 α 3 K 0 I 33 0 0 0 − 1 r 0 0 0 α 0 0 0 0 0 A 86 − α I 12 I 22 0 ) ( w ∗ S φ ∗ S φ M ∗ S φ V ∗ S φ v ∗ S u ∗ S N ∗ S φ N ∗ S φ x ) (2)

or

d Z S r d φ = Z S • = A S ⋅ Z S (3)

Here, α ≡ m π a

I 11 = I 22 = E t 1 − μ 2 , I 21 = I 12 = μ E t 1 − μ 2 , I 33 = E t 2 ( 1 + μ )

K 11 = K 22 = E t 3 24 ( 1 − μ 2 ) , K 21 = K 12 = μ E t 3 24 ( 1 − μ 2 ) , K 33 = K 34 = K 43 = K 44 = E t 3 24 ( 1 + μ )

A 41 = α K 0 ( K 11 − K 12 K 21 K 22 − π 2 k K 0 α 2 b 2 ) ， A 86 = − 1 α K 0 ( I 12 I 21 I 22 − I 11 )

Integrating Equation (2), the field transfer matrix F S is obtained as follows

Z S = exp ( A S r φ ) ⋅ Z 0 = F S ⋅ Z 0 (4)

where

exp ( A S φ ) = I + ( A S φ ) + 1 2 ! ( A S φ ) 2 + 1 3 ! ( A S φ ) 3 + ⋯ (5)

I is the unit matrix.

During the equilibrium equations of forces for the plate-panel subjected to in-plane compressive force and considering the relations between strains and deformations for plate-panel (

d Z P d y = Z P • = [ w ∗ P • , φ ∗ P y • , M ∗ P y • , V ∗ P y • , v ∗ P • , u ∗ P • , N ∗ P y • , N ∗ P y x • ] T (6)

Referred to the variable y are obtained only:

( w ∗ P φ ∗ P y M ∗ P y V ∗ P y v ∗ P u ∗ P N ∗ P y N ∗ P y x ) • = ( 0 − α 0 0 1 r 0 0 0 − α K 21 K 22 0 α K 0 K 22 0 0 0 0 0 0 4 α K 33 K 0 0 α 0 0 0 0 A 41 0 α K 12 K 22 0 0 0 0 0 0 0 0 0 0 α I 21 I 22 α 3 K 0 I 22 0 0 0 0 0 − α 0 0 α 3 K 0 I 33 0 0 0 0 0 0 0 α 0 0 0 0 0 A 86 − α I 12 I 22 0 ) ( w ∗ P φ ∗ P y M ∗ P y V ∗ P y v ∗ P u ∗ P N ∗ P y N ∗ P y x ) (7)

or

d Z P d y = Z P • = A P ⋅ Z P (8)

Integrating Equation (7), the field transfer matrix F P is obtained as follows:

Z P = exp ( A P y ) ⋅ Z 0 = F P ⋅ Z 0 (9)

where

exp ( A P y ) = I + ( A P y ) + 1 2 ! ( A P y ) 2 + 1 3 ! ( A P y ) 3 + ⋯ (10)

I is the unit matrix.

Here, by considering r ≅ ∞ and y = r φ , the partial differential equations of plate-panel Equation (7) is obtained from the partial differential equations Equation (2) of shell-panel.

As shown in

( w ∗ φ ∗ y M ∗ y V ∗ y v ∗ u ∗ N ∗ y N ∗ y x ) L = ( cos θ 0 0 0 − sin θ 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 cos θ 0 0 − sin θ 0 sin θ 0 0 0 cos θ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 sin θ 0 0 cos θ 0 0 0 0 0 0 0 0 1 ) ( w ∗ φ ∗ y M ∗ y V ∗ y v ∗ u ∗ N ∗ y N ∗ y x ) R (11)

or

Z ∗ i L = P i ⋅ Z ∗ i R (12)

where, P_{i} is the field point matrix relating the state vectors between two consecutive panels.

Performing the transfer procedure from section 0 to 10 on lipped channel column with shell-shaped grooves in the web and the flanges, the relation between the initial state vector, Z 0 , and that at section 10, Z 10 is described as follows (

Z 10 = F P 10 ⋅ P 9 ⋅ F P 9 ⋅ P 8 ⋅ F S 8 ⋅ P 7 ⋅ F P 7 ⋅ P 6 ⋅ F P 6 ⋅ P 5 ⋅ F S 5 ⋅ P 4 ⋅ F P 4 ⋅ P 3 ⋅ F P 3 ⋅ P 2 ⋅ F S 2 ⋅ P 1 ⋅ F P 1 ⋅ P 0 ⋅ F P 0 ⋅ Z 0 (13)

where, F p and F S are the field transfer matrices, and P is the point transfer matrix. Similarly, the transfer procedure is performed on the lipped channel column with triangle-shaped grooves in the web and the flanges (

Z 13 = F P 13 ⋅ P 12 ⋅ F P 12 ⋅ P 11 ⋅ F P 11 ⋅ P 10 ⋅ F P 10 ⋅ P 9 ⋅ F P 9 ⋅ P 8 ⋅ F P 8 ⋅ P 7 ⋅ F P 7 ⋅ P 6 ⋅ F P 6 ⋅ P 5 ⋅ F P 5 ⋅ P 4 ⋅ F P 4 ⋅ P 3 ⋅ F P 3 ⋅ P 2 ⋅ F P 2 ⋅ P 1 ⋅ F P 1 ⋅ P 0 ⋅ F P 0 ⋅ Z 0 (14)

For the purpose of grasping fundamental mechanical behavior on cross section of folded and curved grooves the buckling strength and the buckling mode shape of the thin-walled channel section columns with stiffeners subjected to axial compressive force are investigated by use of the transfer matrix method.

The transfer matrix method is analytical method to perform analysis multiplying the field transfer matrix derived by the governing equations for plate- or shell-panel composing the thin-walled member and the transform coordinates matrix relating the vectors between plate-panels composing the thin-walled members, and owing to finish construction of transfer matrices a time the structural analysis is consequential to multiplication of matrices only and is very automatically advanced to be not necessary intervention of mechanical principle during analysis [

of variables notably.

Although the present method is naturally a solution procedure for one-dimensional problems, this method is extended to two-dimensional problems by introducing the trigonometric series into the governing equations of problems. Further this transfer matrix method is expanded for the thin-walled member by introducing the field transfer matrix and the transform coordinates matrix relating the vectors between plate-panels composing the thin-walled member. In present paper the buckling analysis is generated by introducing the field transfer matrix and the transform coordinates matrix on lipped channel section columns with folded and curved grooves.

In

In

specimens | t (mm) | b (mm) | h (mm) | S1 (mm) | S2 (mm) | L (mm) |
---|---|---|---|---|---|---|

A | 1.3 | 130 | 130 | - | - | 5, 10, 15, 20 |

B-shell | 1.3 | 130 | 130 | 10 | 20 | 5, 10, 15, 20 |

B-triangle | 1.3 | 130 | 130 | 10 | 20 | 5, 10, 15, 20 |

C-shell | 1.3 | 130 | 130 | 10 | 20 | 5, 10, 15, 20 |

C-triangle | 1.3 | 130 | 130 | 10 | 20 | 5, 10, 15, 20 |

D-shell | 1.3 | 130 | 130 | 10 | 20 | 5, 10, 15, 20 |

D-triangle | 1.3 | 130 | 130 | 5, 10, 15, 20 | 10, 20, 30, 40 | 5, 10, 15, 20 |

E-shell | 0.42 | 120 | 90 | 10 | 20 | 12 |

E-triangle | 0.42 | 120 | 90 | 10 | 20 | 12 |

small member aspect ratio (a/b). However, before sifting over to overall buckling the distortion buckling is produced, and after that the buckling behavior is sifting over to overall buckling. In the case of L = 5 mm local and overall buckling behavior is produced failing of buckling strength in comparison with that of lip length L = 15 mm and 20 mm.

In

In

In

with that with triangle-shaped folded grooves. As compared with the local and distortional buckling strength of

In

In

In

In

In

This analytical model with triangle-shaped folded grooves is same the model (Model E) calculated by Hancock. In

by Hancock and Yang during finite strip method are simultaneously shown in local and distortional regions, and then the buckling coefficients of normal lipped channel section column without stiffener is together shown as comparison. As shown in

comparison with that on lipped channel column with triangle-shaped folded grooves in the web and the flanges.

In this chapter a new cross section of shell-shaped curved groove was proposed instead of the thin-walled lipped channel section column with triangle-and rectangle-shaped folded grooves used ordinarily, and therefore the comparison and the examination of buckling strength and buckling behavior were generated in the case of preparing triangle-shaped folded and shell-shaped curved grooves to the web and the flanges of thin-walled channel section column. As compared with the buckling strength stiffening the web, that stiffening the flanges is naturally appreciated to be increasing.

Regarding to the buckling strength, that of the member with shell-shaped curved groove is a little rising in comparison with the member with triangle shaped folded groove.

Therefore, it is considered that the use of the shell-shaped curved groove is possible sufficiently.

Then, it is considered that there are sufficient possibilities when using this thin-walled lipped channel section with shell-shaped curved grooves as compression.

The authors declare no conflicts of interest regarding the publication of this paper.

Hoshide, K., Ohga, M., Chun, P.-j., Shigematsu, T. and Kawamura, S. (2018) Strength of Thin-Walled Lipped Channel Section Columns with Shell-Shaped Curved Groov. Open Journal of Civil Engineering, 8, 508-523. https://doi.org/10.4236/ojce.2018.84036