CC BY
13Magazine of Civil Engineering. 2021. 101(1). Article No. 10101
Magazine of Civil Engineering issn
2712-8172
journal homepage: http://engstroy.spbstu.ru/
DOI: 10.34910/MCE.101.1
Cold-formed steel joints with partial warping restraint
I.M. Selyantsev*a, A. Tusninb
a Ltd"Non-state expertise of the Pskov region", Pskov, Russia b Moscow State University of Civil Engineering, Moscow, Russia * E-mail: [email protected]
Keywords: cold-formed structures, warping tests, semi-rigid joints, partial restraints, warping factor, warping restraint
Abstract. The article investigates the influence of joints on the warping torsion of cold-formed steel bars. The modern warping torsion theory suggests cold-formed steel bars to be simply supported or fixed at the ends. Simple support provides zero warping restraint. Fixed support provides fool warping restraint at the joint of the bar. In real constructions cold-formed steel joints are partial warping restrained. Not considering the partial restraint of deplanations by real joints leads to an incorrect assessment of the twist angles and the stress state of thin-walled steel bars in warping torsion. This article deals with an experimental and analytical investigation of warping torsion of cold-formed steel bars with bolted joints. Considered 142C16, 142C20, 262C23 and 262C29 sections. Five types of joints considered: a wall and both flanges of the bar end sections are fixed; the upper and lower flanges are fixed; the wall is fixed; the wall and the lower flange are fixed; the lower flange is fixed. First, analytical expressions for twist angles and bimoments for warping torsion for bars with partial warping restraints obtained. Analytical results are compared with the results of the warping torsion experiment conducted at Moscow State University of Civil Engineering. The cold-formed steel specification is shown to be a poor predictor for the twist angle and bimoment value of twisting members. The warping factor coefficient is recommended for the estimation of the degree of the joint warping constraint. Experimental values of warping factors for different joint types are obtained. The influence of partial warping restraints and cross-section deformation on the work of the tested cold-formed steel bars are evaluated.
1. Introduction
The technical theory of torsion of thin-walled rods of an open profile was created in the 30s of the 20th century in [1]. One of the main assumptions used in its development is the hypothesis of a rigid contour. It is assumed that the cross-section contour maintains its shape when the bar is twisted. Torsion leads not only to cross-section rotation about the center of twist but at the same time the points of the section undergo different displacements along the longitudinal axis. These displacements, called deplanations, lead to warping of cross-sections of a thin-walled bar in the torsion. Because of warping restraint, additional sectorial normal stresses arise in the bar. The basic theory of thin-walled members with open cross-sections was developed by Vlasov [1]. To determine the twist angles and bimoments in the rod, Vlasov proposed the differential equation:
eV _ k 2q = m(z)_M (1)
EJm
where 0 is the angle of twist of the rod; m(z) is the intensity of external distributed twisting moments; b (z) is the derivative with respect to z of the intensity of external distributed bimoments; Jm is the warping constant of the cross-section; Jd is the torsion constant of the cross-section; E is the modulus of elasticity;
G is the shear modulus; k = Gjd is the elastic flexural-torsional characteristic of the thin-walled rod.
\Eja
Selyantsev, I.M., Tusnin, A. Cold-formed steel joints with partial warping restraint. Magazine of Civil Engineering. 2021. 101(1). Article No. 10101. DOI: 10.34910/MCE. 101.1
I This work is licensed under a CC BY-NC 4.0
Equations of the elastic line of twist angles and bimoments for different types of load arrangements are presented in tabular form by Bychkov [2]. Solutions presented in [1] and [2] are valid for rods subjected to zero or completely warping restraints in the absence of a rod twisting in the supported sections.
A number of research studies have been carried out over the past decades to determine the influence of restraints on the work of cold-formed steel bars. A significant part of the studies is devoted to the study of the carrying capacity of cold-formed thin-walled steel bars, working in bending, compression, and torsion. In the studies reviewed issues of strength, overall and local stability. In [3-8], the results of theoretical and numerical studies of the strength and stability of cold-formed bars are presented. Solved problems of determining forces and displacements using rod and shell finite elements, propose theoretical solutions for determining bimoments taking into account bending moments, consider the problem of distortional buckling. The stress-strain state of cold-formed steel bars under torsion was studied experimentally [9] and theoretically [10]. In [9], considering the experimental data obtained, an expression for the bearing capacity of an eccentrically loaded C-profile bar was obtained. In [10], three methods are compared, the theory of calculating thin-walled constructions [1], the method of representing constrained torsion by bending with torsion, and the method of representing torsion by bending the shelves of a thin-walled rod by a pair of forces. The performed work confirmed the effectiveness of the theoretical and numerical methods used in the calculation of thin-walled structures experiencing bending, compression, and torsion.
Extensive experience has been gained in experimental studies of thin-walled systems [11-25]. The articles show the results of tests for bending and compression of C and Z shaped profiles, solid, perforated, with simple and complex edge stiffeners. The obtained data on strength and stability were compared with theoretical and numerical solutions. Based on the tests carried out in [12], an expression was given to evaluate the carrying capacity of thin-walled Z-profiles in biaxial bending. In [15] the method of selecting the effective width of the section elements for calculating the effective section properties was justified. In [16] the method for determining the effective width of the elements sections and method of determining the bearing capacity of the rod. The experience of experimental studies can be successfully adapted to solve the problems of the operation of cold-formed steel bars with various joints.
The Consideration of Saint-Venant's principle of using the Vlasov theory [1] is the work of [26]. It shows the influence of boundary conditions on the operation of thin-walled rods. In [27], the effect of conjunction flexibility on the critical load of cold-formed C-shaped profiles was studied. In addition to analytical data, tests of the considered structures are given.
Experimental and theoretical studies of the behavior of trusses made of thin-walled members were carried out in [28]. Taking into account the research carried out, a technique for strengthening the upper chord of the truss has been proposed, and an eaves joint has been developed.
The results of numerical and experimental studies of frame structures made of cold-formed steel members are presented in [29-31]. In [32], the results of experimental studies of bolted joints of cold-formed steel trusses are given. Bolted joints are considered semi-rigid to bend. The authors proposed a technique for numerical modeling of bolted joints, which is based on previous studies [33] and [34]. In [35, 36], experimental and numerical studies of structures made of thin-walled rods were carried out, and the rotation stiffness of semi-rigid nodal joints was determined. In [36], recommendations were given on numerical modeling, taking into account the initial imperfections and features of bolted joints.
The results of cyclic tests of the cold-formed frame joints and the method for determining the bearing capacity of bolted joints are presented in [37]. The refined method for calculating the bolted connections of thin-walled elements [38] is based on the results of experimental studies. Authors give recommendations on the determining the carrying capacity of bolted joints. Experimental studies of bolted connections for cold-formed frames were carried out in [39].
The common feature for these studies is that the joints supporting the rods are "idealized", considered to be zero or completely warping restraint. The degree of restriction of deplanations with real joints is not considered. The effect of the degree of deplanation restriction is not taken into account. Also, in addition to [1] and [2], little attention is paid to the separately constrained torsion of the rod. Thus, despite extensive studies of the operation of thin-walled structures, the issues of accurately determining the forces and deformations in such structures, taking into account their actual joints, are of definite practical and scientific interest.
This article discusses the warping torsion of cold-formed steel bars with different bolted joint types. The Autors carried out a series of warping torsion tests of cold-formed C-shaped profiles of different lengths and sizes under different joint conditions. The Experimental results were compared with analytical solutions. The main goal of the study is to provide a method for bolted joints warping stiffness calculation. This method will be used to specify the influence of partial warping restraints on the twist angles and bimoments value of cold-formed steel bars. According to the warping torsion experimental results, a part of the torsion angles of the cold-formed bar, obtained by deforming the contour of the cross-section, will be estimated. The range
of the flexural-torsional characteristic kl will be determined beyond which the cross-section contour deformation can be neglected.
2. Methods
Practically used joints of cold-formed steel bars impose partial warping restraint in accordance with complete twist restraint. They do not provide zero or complete deplanations restriction at supported sections. Considering the joints to be zero or completely warping restraint leads to significant errors in determining the twist angles and the bimoments. Experimental and theoretical studies have been carried out at Moscow State University of Civil Engineering aimed at identifying the features of the behavior of cold-formed steel bars under various boundary conditions.
A series of warping torsion tests were conducted. A total of 40 cold-formed C-shaped bars of four sections 142C16, 142C20, 262C23, and 262C20 under different joint conditions were tested. Steel grade of cold-formed steel sections is S450GD EN 1036:2015. The elastic modulus is E = 210000 N/mm2, the shear modulus is G ~ 81000 N/mm2, the yield stress is fy = 450 N/mm2. The lengths, dimensions of the cross-sections and the values of the flexural-torsional characteristics kl of the tested bars are given in Table 1. Warping torsion test arrangements are given in Fig. 1 and Fig. 2. the Bars were fixed at the ends. Torque was applied to the middle of the span of the bar. At the distance of 260 mm from the central section of the bar four LVDTs were located (see Fig. 1). Two on the upper flange and two on the lower flange. Torque moment values are listed in Table 2. The bars ends were bolted to a rigid fixed support structure. Five types of joints were considered
1. fixed wall and both flanges;
2. fixed upper and lower flange;
3. fixed wall;
4. fixed wall and a bottom flange;
5. fixed bottom flange.
Table 1. Section geometrical characteristics.
Profile h (mm) b (mm) c (mm) t (mm)
142C16 142 60 13 1.6
142C20 142 60 13 2.0
262C23 262 65 13 2.3
262C29 262 65 13 2.9
where: h is the wall height. b is the flange width. c is the flange stiffeners height. t is the wall thickness.
..........Joint Type 1 Joint Type 2 Joint Type 3
Joint section
Bolted to a rigid construction segment of the joint section (see Fig.2l
Figure 1. Warping torsion test arrangement. types of joints.
Table 2. Torque moment values.
Profile l (mm) kl Mt (kN mm)
Type1 Type2 Type3 Type4 Type5
142C16 1955 0.864 146.6 146.8 859.6 703.0 20.0
142C20 1955 1.094 156.7 146.7 933.0 104.2 31.1
262C23 1955 0.698 229.1 229.1 205.9 229.2 49.2
262C29 1955 0.895 377.1 377.1 357.3 377.2 54.1
142C16 3940 1.706 380.9 337.9 298.7 360.0 17.8
142C20 3940 2.160 419.4 419.5 250.9 419.4 21.7
262C23 3940 1.379 133.8 133.8 113.8 133.9 54.5
262C29 3940 1.767 145.8 145.8 146.1 145.8 62.2
where: l is the length of the bar, kl is the dimensionless flexural-torsional characteristic of the bar,
kl = I
V
GJt
EJ o
. Jt is the St. Venan torsion constant of the bar cross section. Jm is the warping constant of
the bar cross section. Torque moment Mt value is calculated as product of numbers of a load applied at the end of a loading device console to a distance from load application point to an axis of rotation of the loading device (see Fig. 2). Type 1, 2, 3, 4, 5 indicates joint types (see Fig. 1).
Figure 2. Warping torsion of cold-formed C-shaped steel bar. Joint type 3.
Table 3. Experimental angles of twist dexp. The ratios of experimental and theoretical values of angles of twist.
Profile l (mm)
142C16 142C20 262C23 262C29 142C16 142C20 262C23 262C29
1955 1955 1955 1955 3940 3940 3940 3940
kl
0.864 1.094 0.698 0.895 1.706 2.160 1.379 1.767
dexp (grad)
dexp / d complete
Type Type Type Typ Type Type Type Type Type Type 123 e 4 512345
3.44 2.74 1.06 1.43 4.39 3.55 2.77 2.38
3.17 2.95 1.07 1.45 4.48 3.55 2.80 2.43
2.69 2.05 1.85 1.87 4.50 3.05 3.65 3.33
2.02 2.11 1.26 1.57 4.37
3.55 3.11
2.56
1.15 1.27 2.22 1.47 4.49 4.27 4.03 3.25
1.72 1.45 2.06 2.07 1.09 0.98 1.24 1.22
1.58 1.66 2.08 2.10 1.26 0.98 1.26 1.25
2.29 1.81 4.01 2.86 1.43 1.41 1.93 1.71
2.11 1.68 2.45 2.27 1.15 0.98 1.39 1.32
4.22 3.38 20.9 14.9 2.38 2.29 4.44 3.92
Table 3 employs the following notations: 6exp is experimental angles of twist at the central section of the bar. dcomptete is theoretical angles of twist calculated for bars at the central section of the bar with joints providing complete warping restriction (23).
Table 4. Experimental bimoment values B0J,exp. The ratios of experimental and theoretical values of bimoments.
Profile
l (mm)
Bm,exp / 104 (kN mm2)
/ B m,complete
Type Type Type Typ Type Type Type Type Type Type
1 2 3 e 4 5 1 2 3 4 5
142C16 1955 0.864 1.39 1.53 1.34 0.86 0.19 0.85 0.93 1.40 1.10 0.84
142C20 1955 1.094 1.79 1.85 1.71 1.55 0.46 0.99 1.09 1.58 1.28 1.28
262C23 1955 0.698 2.93 3.02 4.36 3.64 1.19 1.09 1.12 1.80 1.35 2.05
262C29 1955 0.895 4.88 4.98 7.02 5.52 1.22 1.11 1.13 1.68 1.25 1.93
142C16 3940 1.706 1.17 1.03 1.07 1.16 0.78 0.92 0.91 1.07 0.96 1.31
142C20 3940 2.160 1.37 1.39 1.32 1.47 0.97 1.01 1.02 1.62 1.08 1.38
262C23 3940 1.379 4.99 5.01 5.42 5.31 3.11 1.08 1.08 1.38 1.15 1.65
262C29 3940 1.767 5.13 5.22 6.18 5.34 3.43 1.05 1.07 1.26 1.09 1.64
kl
where: Bm,exp is experimental bimoment value. Bm,compiete is theoretical bimoment value calculated for bars with joints providing complete warping restriction (24).
As a result, the angles of twist dexp in the central section of the bar, Table 3, and bimoments Bm,exp in the LVDTs location section, Table 4, were determined. A significant difference between the experimental and theoretical results calculated for joints with zero and complete warping restriction was found, Table 3, Table 4. To identify the causes of this mismatch, the effects of contour and form of cross-section deformation on the supports and in the place where the load was applied were studied. As a result, theoretical expressions were obtained for twisting angles and bimoments for bars with partial torsional and warping restraints. Theoretical expressions for the twisting angles and the bimoments described below. The level of joint warping restraint was determined on the reference bimoment perceived by the joint.
The theoretical solution to the effect of the influence of partial torsional and warping restraints on the behavior of cold-formed steel bars is based on Vlasov theory [1]. Consider the case when, on the supports (due to the flexibility of the joint), the twist angle and the deplanation are not equal to zero. In this case, on the supports, the twist angle 0 is proportional to the level of torque moment on the joint section Me, and the degree of deplanation S is proportional to the level of bimoment on the joint section Bmo- In this case, the relationship between internal forces and deformations for twist angles and deplanations on the supports can be written in the form:
001 = keiM 0i
002 = ke2M 02; £01 = ksiBm0i 802 = ks2 Ba 02,
where ke and ks are the twist and deplanation flexibility of the joint, respectively. Index 0 represents the cross-section of the bar, and indices 1 and 2 represent the beginning and the end of the bar.
The solution of Eq. (1) can be written as:
0 = A ■ sh kz + B ■ ch kz + C ■ z + D + f (z) (3)
where f(z) is the particular solution of Eq (3). Instead of ch kz, sh kz, z and 1 in Eq. (3), we introduce the partial integrals yi(z), ^2(z), ty3(z), ty4(z) which are linear combinations of the first:
y/1 = a1sh kz + a2ch kz + a3z + a4; = b1sh kz + b2ch kz + b3z + b4; = c1sh kz + c2ch kz + c3z + c4; y/4 = d 1sh kz + d2ch kz + d3z + d4,
Then the solution of Eq. (1) can be written as:
0 = + B^2+C + D f (z), (5)
The constants A, B, C, and D are evaluated from the boundary conditions for the bar with partial torsional and warping restraints:
at z = 0:
at z = l:
0=keiM01 and Q ' = -
0 = k02M 02 and Q = -
BqQ1 EJ a
Bw02 EJ a
(6)
To simplify further calculations, we take the partial integrals of the Eq. (5) so that they satisfy the following conditions:
^(0) -1; (0) - 0; Vx{l) - 0; ¥1 (l) - 0; » » » »
^2(0) - 0; ¥l (0) -1; ¥l(l) - 0; ¥l (l) - 0; » » » »
¥3(0) - 0; ¥3 (0) - 0; ^(l) -1; w3 (l) - 0; » » » »
¥4(0) - 0; ¥4 (0) - 0; ¥a(1 ) - 0; ¥4 (l) -1 From these conditions we find:
(7)
wi =1 - y ;
w = k2 1-1+
sh kl ■ ch kz - ch kl ■ sh kz
sh kl
(8)
w3=y ;
¥4
1 sh kz
k2 sh kl k2 l
Consequently,
.(, z ^ B ( z , sh kl ■ ch kz - ch kl ■ sh kz \ _z D ( sh kz z \ .
d = AI 1 — l + —I --1 +-1 + C-+ ---1 + f (z).
V l ) k V l sh kl ) l k shkl l ) JyJ
Choose the particular solution of Eq. (3) f(z) such that:
(9)
f (0) - 0; f (0) - 0; f (0) - 0; f (0) - 0. (10) Using the boundary conditions Eq. (6) for Eq. (9), using Eq. (10), we find four non-zero constants A, B, C and D:
A = koM01; B = - a C = k02M02 - /(l); D = - Ba02 - f '(l).
(11)
EJ w EJ w
Substituting Eq. (11) into Eq. (9), we obtain the equation of the elastic line of the twist angles for a bar with partial torsional and warping restraints:
Q z) .-/(l ) z + if ( z - f| ) + / ( z) + k01M 01(1 -1 ]
z Bmrn ( z , , , chkl , , | Bam ( shkz z +k02M02---£aI— 1 + ch kz--sh kz |-Ba02 1
(12)
l EJak211
sh kl
EJ ak2 shkl l.
Upon double differentiating the Eq. (12) with the respect to the z and multiplying it by -EJm, we obtain the equation of bimoments for a bar with partial torsional and warping restraints:
f (l) !» /•' V \ (11 ch kl , . ^ sh kz
B a = EJ a^TJ sh kz - EJ af (z) + Ba01 ch kz--- sh kz + B«02"—
sh kl V sh kl j sh kl
<
z
It can be seen that when ke1 = ke2 = 0 and Ba01 = Ba02 = 0 , Eq. (12) will be transferred to the equation of the elastic line of the twist angles for the bar, with a complete twisting and zero warping restraints.
e( z) = - f (l )-+
z , f (l )
k
2
V
z sh kz l sh kl
+ f ( z).
(14)
From the same boundary conditions, Eq. (13) will be transferred to the equation of the bimoments for the bar, with a complete twisting and zero warping restraints.
f (l )
Ba(z) = EJv^-jsh kz -EJaf (z). sh kl
(15)
Similarly, taking the twist angles on the supports equal to zero ke1 = ke2 = 0 and assuming that the
warping deformations are completely restrained, that is, Bmoi and Bm02 are equal to the support bimoments for a bar with a complete warping restraints, the Eq. (12) will be transformed into an equation of the elastic line of the twist angles for a bar with a complete warping and torsion restraints.
0( z) = - f (l )
ch k(l - z) - ch kz + kz ■ sh kl - ch kl +1 kl ■ sh kl - 2ch kl + 2
' „ kl ■ ch kz - kz ■ ch kl + sh kl - sh kz - sh k(l - z) - k(l - z) x
-f (l)-------- + f (z).
k(kl ■ sh kl - 2ch kl + 2) Eq. (13), becomes the equation of bimoments for a bar with similar boundary conditions.
(16)
B®( z ) = EJ mk 2f (l )
ch k (l - z ) - ch kz kl ■ sh kl - 2ch kl + 2
, „' kl ■ ch kz - sh kz - sh k(l - z) j .
+EJ Jkf(l)-„,„„,„ „ ' - EJa f (z).
(17)
kl ■ sh kl - 2ch kl + 2
Substituting into Eq. (12) and Eq. (13) the expressions for the particular solutionf(z), one can obtain the values of the twist angles and the bimoments for different load arrangement.
3. Results and Discussion
Using the proposed theoretical solutions, the influence of distortion on the behavior of the tested cold-formed steel bars was evaluated. Particular solution f(z) tested bars with torque applied to the center section of the bar can be written as:
f ( z) = ■
M
k 3EJ a
whereMis the applied torque moment. The values of particular solutionsf(z) for different load arrangement are listed in [2]. Substituting Eq. (18) into Eq. (12) and Eq. (13), taking into account that the right and left ends of the bar are fixed equally, kei=ke2 and ksi=ks2, we obtain the expressions for the twist angles and the bending-twisting bimoment for tested partially torsion and warping restrained bars with a torque applied in the central section:
sh k
l
z--I-k
V 2
l
z--
V2
(18)
f
0( z) =
M
2k 3 EJ c
\
kz -
sh kz
ch
kl 2 ;
BaQ
k 2 EJ a
sh kz sh kl
(1 - ch kl)-1 + ch kz
+ keM o,
M sh kz
Ba( z) = 2k T^ + Ba0
ch — 2
chkz + sh kz (1 - ch kl ) sh kl
(19)
(20)
where Bmo is the bimoment on the joint section; M0 is the torque moment on the joint section.
When Bmo = 0 and, ke = 0 Eq. (19) and Eq. (20) will be transferred to the equations for the bar, with a complete twisting and zero warping restraints.
0( z) =
M
2k EJ a
kz —
sh kz
ch
kl 2 ;
Ba( z) =
M sh kz
2k , kl ' ch — 2
(21)
(22)
If Ba0 =
M
1 — ch
kl
2k j kl sh — 2
2
— and ke=0, Eq. (19) and Eq. (20) will be transferred to the equations for the
bar, with a complete twisting and complete warping restraints.
0( z ) = ■
M
kz 7 kl 9 kz , kz , k(l - z)
— sh--sh 2--sh — sh-
2 2 2 2 2
k EJ
a
kl sh —
2
(23)
Ba(z) =
M
ch kz — ch k
l
— z
2k
sh
M_ 2
(24)
Let us compare the experimental values of the twist angles and bimoments with the theoretical values obtained for the bar with complete twisting and warping restraints. Calculations of the theoretical values of the twist angles and bimoments for bars with complete twisting and warping restraints are carried out using Eq (23) and Eq. (24). To obtain the values of the twist angles 0 and the bimoments Bm for bars with partial restraints, using Eq. (19) and Eq. (20), it is required to know the value of the bimonent on the joint section Bmo corresponding to the type joint.
Let us agree that the cross-section of the bar during the torque can be deformed in two ways - the cross-section contour can be deformed out of a plane, which leads to warping displacements. Cross-section in this case does not remain planar after deformation. And the cross-section form can be deformed in-plane without violating the flat section hypothesis. Warping deformations are considered when calculating the twist angles and bimoments according to the Eq. (19) and Eq. (20). In-plane cross-section deformations can be estimated from the results of the experiment.
During the experiment, stresses were measured at four points of the cross-section, located at 260 mm from the center of the bar. Placing the strain gauges at some distance from the point of application of the load minimizes the effect of cross-section in-plane deformation on the value of the measured bimoment. The value of the bimoment Bmo on the joint section is obtained from the processing of experimental data. Bmo was calculated using the experimental value of the bimoment Bm on the strain gauges section, using Eq. (20).
The degree of constraint by the joint of the warping deformations, so-called deplanations, can be described by the warping factor coefficient:
KS =
Ba0 , partial
(25)
Bm0 , complete
where Ks is the warping factor; Bmo,partiai is the bimoment on the joint section for the bar with partial warping restraint; Bmo,compiete is the bimoment on the joint section for the bar with complete warping restraint.
A decrease in the joint bimoment is proportional to the warping factor coefficient Ks, which leads to an increase in the value of the bimoment in the span of the bar. The values of the warping factor Ks, for examined joint types, are given in Table 5.
Table 5. Mean values of the warping factors.
Joint type
Warping factor
type 1
type 2
type 3
type 4
type 5
Ks
BqjQpartial / Bo)Q,complete
0.98 1.03
0.96 1.04
0.7 1.24
0.83 1.11
0.55 1.35
From Table 5, only two of the examined joint types, type 1 and 2, can be considered as complete warping restraint. And none of the examined joints can be considered as zero warping restraint. So, when only one flange is fixed, the bimoment on the joint is about half of the bimoment for the joint with complete warping restraint.
Using Eq. (25) for the warping factor Ks, Eq. (19) and Eq. (20) for bimoment and twist angles can be written as:
f
0( z) =
M
2k 3 EJ c
kz —
sh kz
, kl ch —
2 ;
K sBWQ , complete
k 2 EJ a
sh kz sh kl
(1 — chkl ) — 1 + ch kz
M sh kz
Ba(z) ~ - i K sB a0,complete
ch— 2
ch kz + sh kz (1 — ch kl ) sh kl
+ kdM o, (26)
(27)
Considering the real joint conditions with the help of the proposed method leads to a more accurate, in comparison with the traditional method [1, 2], the assessment of the stress-strain state with warping torsion of cold-formed steel bars.
It should be noted that the question of determining the warping factor Ks requires additional studies. When processing the experiment, the values of Ks determined from the experimental results were used, which leads to the complete coincidence of the experimental values of the bimoment Bm with the values obtained from Eq. (27). In the absence of a sufficient amount of experimental data, the Ks values for different joint types can be preliminarily taken from Table 5.
Considering the experimental and theoretical studies carried out, the influence of the cross-section in-plane deformation on the behavior of the tested cold-formed bars was evaluated. The experimental angle of twist was estimated by the rotation of the loading device Fig. 2. The total twist angle is found as the sum of three values, the theoretical value taking into account the additional warping deformation of the bar on the joints Eq. (26), the additional angle of rotation of the loading device due to the in-plane deformation of the bar cross-section at the point of application of the load and the additional angle of twist due to in-plane bar cross-section deformation on the joints.
Comparing the value of the obtained angle of twist of the bar according to Eq. (26) with the experimental value dexp, it is possible to calculate the part of the angle of twist 0X obtained by in-plane cross-section deformation of the cold-formed bar at the joints and in the place of load application. The obtained values of the 0/dexp relations for the tested bars with different types of joints are listed in Table 6.
Table 6. Qy/Qexp-
6J6t
X/0exp
Profile
mm
kl
type 1 type 2 type 3 type 4 type 5
262C23 1955 0.684 0.43 0.42 0.41 0.37 0.88
142C16 1955 0.876 0.43 0.35 0.27 0.37 0.73
262C29 1955 0.895 0.43 0.41 0.32 0.39 0.85
142C20 1955 1.072 0.26 0.27 0.13 0.14 0.54
262C23 3940 1.379 0.10 0.10 0.06 0.09 0.46
142C16 3940 1.766 0.17 0.19 0.10 0.18 0.26
262C29 3940 1.933 0.16 0.15 0.13 0.13 0.40
142C20 3940 2.161 0.00 0.03 0.11 0.05 0.20
l
Where 0exp is the experimental value of the bar twist angle; 0X = 0exp - 0 is the twist angle due to inplane cross-section deformation; 0 is the twist angle of the bar with partial warping restraints.
From Table 6 it can be seen that for joint types 1, 2, 3, and 4, depending on the flexural-torsional characteristic kl, up to 43 % of the angle of twist of a cold-formed bar, occurs due to in-plane cross-section deformation at the load application point and on the joint sections. For joint type 5, the contribution of the in-plane cross-section deformation to the twist angle of the cold-formed bar can be up to 88 %.
For joint types 1, 2, 3, and 4, with kl > 2, the magnitude of the twist angles of the cold-formed bar can be determined by the Eq. (26) with sufficient accuracy for practical use. The joint number 5 requires additional research, but it can be said that taking into account the partial warping restraint of the joint allows using Eq. (26) with a value of kl > 3. When joint types are 1, 2, 3, 4 and kl < 2, as well as when joint type 5 and kl < 3, the cross-section in-plane deformation cannot be neglected.
4. Conclusions
Based on the research conducted, it was established:
1. The degree of warping restriction with real joint types can be described by the value of the bimoment on the joint section Bmo. When Bmo=0, the joint is zero warping restrained. When Bmo=M/2k(1-ch kl/2)sh kl/2 the joint is complete warping restrained.
2. Not considering the partial warping restriction by the joint leads to a significant error in the determination of the twist angles of the bar and bimoments. generally, the magnitudes of the twist angles 0 and the bimoments Bm with warping torsion of a thin-walled cold-formed steel bar can be determined by Eq. (12) and Eq. (13). In the case of a load in the form of a torque applied to the central section of the bar according to Eq. (26) and Eq. (27). To consider the type of joint, it is possible to use experimentally obtained values of the warping factor Ks according to Table 5.
3. According to Eq. (13), additional twisting angles on the bar joints due to deformation of the bar ends or insufficient rigidity of the joint assembly design do not affect the values of bimoments along the bar length.
4. An indicator of the deformability of the cross-sectional contour is the value of the flexural-torsional characteristic of the bar, kl. The smaller it is, the greater the influence of the contour deformations on the operation of the bar during torsion.
5. For cases of fastening a cold-formed bar along two flanges and a wall, along a flange and a wall, along two flanges, as well as for fastening along a wall, the in-plane cross-section deformation can be neglected for kl > 2. The case of fastening a cold-formed bar along one flange requires additional study. Previously, we can say that in this case, the in-plane cross-section deformations can be neglected when kl > 3.
References
1. Vlasov, V.Z. Tonkostennyye uprugiye sterzhni [Thin-walled elastic rods]. Moscow: Fizmatgiz,1959. 574 p. (rus)
2. Bychkov, D.V. Stroitelnaya mekhanika sterzhnevykh tonkostennykh konstruktsiy [Structural mechanics of thin-walled bar structures]. Moscow: Gosstroyizdat, 1962. 475 p. (rus)
3. Erkmen, R.E. Bridging multi-scale approach to consider the effects of local deformations in the analysis of thin-walled members. Computational Mechanics. 2013. No. 52. Pp. 65-79.
4. Keerthan, P., Mahendran, M. Numerical modeling of litesteel beams subject to shear. Journal of Structural Engineering. 2011. No. 137(12). Pp. 1428-1439.
5. Chen, J.K., Li, L.Y. Distortional buckling of cold-formed steel sections subjected to uniformly distributed transverse loading. International Journal of Structural Stability and Dynamics. 2010. No. 10(5). Pp. 1017-1030.
6. Schafer, B.W., Sarawit, A., Pekoz, T. Complex edge stiffeners for thin-walled members. Journal of Structural Engineering. 2006. No. 2. Pp. 212-226.
7. Nazmeeva, T.V., Vatin, N.I. Numerical investigations of notched C-profile compressed members with initial imperfections. Magazine of Civil Engineering. 2016. No. 62(2). Pp. 92-101.
8. Pavlenko, A.D., Rybakov, V.A., Pikht, A.V., Mikhailov, E.S. Non-uniform torsion of thin-walled open-section multi-span beams. Magazine of Civil Engineering. 2016. No. 67(7). Pp. 55-69.
9. Put, B.M., Pi, Y.L., Trahair, N.S. Bending and torsion of cold-formed channel beams. Journal of Structural Engineering. 2002. No. 5. Pp. 540-546.
10. Melcher, J., Karmazlnova, M. On problems of torsion analysis of steel members with open cross section. Procedia Engineering. 2012. Pp. 262-267.
11. Ghosn, A.A. Deflection of nested cold-formed steel Z-section beams. Journal of Structural Engineering. 2002. No. 11. Pp. 1423-1428.
12. Put, B.M., Pi, Y.L., Trahair, N.S. Biaxial bending of cold-formed Z-beams. Journal of Structural Engineering. 1999. No. 11. Pp. 1284-1290.
13. Garifullin, M., Bronzova, M., Sinelnikov, A., Vatin, N. Buckling analysis of cold-formed c-shaped columns with new type of perforation. Proceedings of the International Conference on Engineering Sciences and Technologies. ESaT. 2015. Pp. 63-68.
14. Vatin, N.I., Nazmeeva, T., Guslinscky, R. Problems of cold-bent notched C-shaped profile members. Advanced Materials Research. 2014. 941-944. Pp. 1871-1875.
15. Yu, C., Schafer, B.W. Local buckling tests on cold-formed steel beams. Proceedings of the International Specialty Conference on Cold-Formed Steel Structures: Recent Research and Developments in Cold-Formed Steel Design and Construction. 2002. No. 1. Pp. 1596-1606.
16. Zhou, X.H., Shi, Y. Flexural strength evaluation for cold-formed steel lip-reinforced built-up I-beams. Advances in Structural Engineering. 2011. No. 14(4). Pp. 597-611.
17. Simic, D. Modelling of load transfer to laterally restrained thin-walled beams with open cross-section. Transactions of Famena. 2010. Pp. 47-56.
18. Pham, C.H., Hancock, G.J. Experimental investigation of high strength cold-formed C -sections in combined bending and shear. Journal of Structural Engineering. 2010. No. 10. Pp. 866-878.
19. Young, B., Hancock, G.J. Cold-formed steel channels subjected to concentrated bearing load. Journal of Structural Engineering.
2003. No. 8. Pp. 1003-1010.
20. Ghosn, A.A., Sinno, R.R. Load capacity of nested, laterally braced, cold-formed steel Z-section beams. Journal of Structural Engineering. 1996. No. 8. Pp. 968-971.
21. Pham, C.H., Hancock G.J. Direct strength design of cold-formed purlins. Journal of Structural Engineering. 2009. No. 3. Pp. 229-238.
22. Penava, D.S., Radic, A., Ilijas, T. Elastic stability analysis of thin-walled C- And Z-section beams without lateral restraints. Transactions of Famena. 2014. Pp. 41-52.
23. Yakovleva, Ye.L., Atavin, I.V., Kazakova, Yu.D., Maksudov, I.Kh. Strength characteristics of thin-walled elements. Construction of Unique Buildings and Structures. 2017. No. 12(63). DOI: 10.18720/CUBS.63.7 (rus)
24. Vatin, N., Sinelnikov, A., Garifullin, M., Trubina, D. Simulation of cold-formed steel beams in global and distortional buckling. Applied Mechanics and Materials. 2014. 633-634. Rr. 1037-1041.
25. Vatin, N.I., Havula, J., Martikainen, L., Sinelnikov, A.S., Orlova, A.V., Salamakhin, S.V. Thin-walled cross-sections and their joints: Tests and FEM-Modelling. Advanced Materials Research. 2014. 945-949. Pp. 1211-1215.
26. Goldshteyn, Yu.B. Printsip Sen-Venana i stesnennoye krucheniye tonkostennykh sterzhney otkrytogo profilya [Saint-Venant principle and constrained torsion of thin-walled open profile rods]. Izvestiya vuzov. Investitsii. Stroitelstvo. Nedvizhimost. 2016. No. 4(19) S. 75-83. (rus)
27. Rybakov, V., Molchanova, N., Laptev, V., Suslova, A., Sivokhin, A. The effect of conjunction flexibility on the local stability of steel thin-walled slab beams. Proceedings of MATEC Web of Conferences. 2016.
28. Wood, J.V., Dawe, J.L. Full-scale test behavior of cold-formed steel roof trusses. Journal of Structural Engineering. 2006. No. 4. Pp. 616-623.
29. Chung, K.F., Ho, H.C., Wang, A.J., Yu, W.K. Advances in analysis and design of cold-formed steel structures. Advances in Structural Engineering. 2009. No. 11(6) Pp. 615-632.
30. Wrzesien, A., Lim, J.B.P., Nethercot, D.A. Optimum joint detail for a general cold-formed steel portal frame. Advances in Structural Engineering. 2012. No. 15(9). Pp. 1623-1639.
31. Lim, J.B.P., Nethercot, D.A. Finite element idealization of a cold-formed steel portal frame. Journal of Structural Engineering.
2004. No. 1. Pp. 78-94.
32. Zaharia, R., Dubina, D. Stiffness of joints in bolted connected cold-formed steel trusses. Journal of Constructional Steel Research. 2006. No. 62. Pp. 240-249.
33. Toma, A.W., Stark, Jan W.B. Connections in cold-formed sections and steel sheets. Proceedings of International Specialty Conference on Cold-Formed Steel Structures. 1978. No. 5. Pp. 951-987.
34. Zadanfarrokh, F., Bryan, E.R. Testing and design of bolted connections in cold formed steel sections. Proceedings of Eleventh International Specialty Conference on Cold-Formed Steel Structures. 1992. No. 3. Pp. 625-662.
35. Sngle, K.K., Bajoria, K.M., Talicotti, R.S. Stability and dynamic analysis of cold-formed storage rack structures with semirigid connections. International Journal of Structural Stability and Dynamics. 2011. No. 11(6). Pp. 1059-1088.
36. Dunai, L., Jakab, G. Stability behavior and design of nonconventional cold-formed steel structures - research review. International Journal of Structural Stability and Dynamics. 2011. No. 11(5). Pp. 903-927.
37. Uang, C.M., Sato, A., Hong, J.K., Wood, K. Cyclic testing and modeling of cold-formed steel special bolted moment frame connections. Journal of Structural Engineering. 2010. No. 8. Pp. 953-960.
38. Bolandim, E.A., Beck, A.T., Malite, M. Bolted connections in cold-formed steel: reliability analysis for rupture in net section. Journal of Structural Engineering. 2013. No. 5. Pp. 748-756.
39. Kwon, Y.B., Chung, H.S., Kim, G.D. Development of cold-formed steel portal frames with PRY sections. Advances in Structural Engineering. 2008. No. 11(6). Pp. 633-649.
Contacts:
Ilya Selyantsev, [email protected] Alexandr Tusnin, [email protected]
© Selyantsev, I.M.,Tusnin, A., 2021
