CC BY
27Magazine of Civil Engineering. 2022. 110(2). Article No. 11009
Magazine of Civil Engineering
journal homepage: http://engstroy.spbstu.ru/
ISSN 2712-8172
DOI: 10.34910/MCE.110.9
The behaviour of thin-walled beam with restrained torsion
V.V. Galishnikovaa , T.H. Gebreb*
a National Research Moscow State Civil Engineering University, Moscow, Russia b Peoples' Friendship University of Russia, Moscow, Russia *E-mail: tesfaldethg@gmail. com
Keywords: thin-walled structures, restrained torsion, section properties, angle of twist, open section, closed sections, non-uniform warping, torsional stress
Abstract. In this paper, the behaviour of thin-walled sections of a bar with restrained torsion is studied. Neglecting these warping behaviours may generate significant errors, particularly for open profile torsion or shear bending of short beams. The governing equation for non-uniform torsion is used to study the behaviour of the beam with restrained torsion. The variation of the primary torsional moment, secondary torsional moments and warping moments for different value of characteristic number for torsion are presented graphically. Finally, the behaviour and comparison of all torsional moment components with three different thin-walled sections are illustrated by presenting and discussing their results. The section properties, displacements, rotations, stresses and their distribution within the span are compared based on the required value of characteristic number for torsion. It is found that for all thin-walled sections, the characteristic number for torsion is the key criteria for the study of the behaviour of thin-walled sections of a bar with restrained torsion.
The behaviour of a bar with restrained torsion differs significantly from stretching and bending moreover their mathematical formulations varies. In analyses of thin-walled structures subjected to torsion, the effect of warping must be considered as the axial stresses mainly occur at the points of action of concentrated torsion moments (except for free ends) and at sections with warping restraints. The most profitable and effective way to construction of prefabricated structures is to use the system of light steel thin-walled structures, and also thermal insulation, facing and vapor sealing [1]. Normally, thin-walled sections do not behave according to the law of the plane sections employed by Euler-Bernoulli-Navier however the general theory of thin-walled section developed by Vlasov [2]. If warping is not restrained, the applied twisting moment is entirely carried by uniform torsion as a result the shear contribution to the deformation energy can be considered small enough [3-4]. Torsion leads not only to cross-section rotation about the centre of twist but at the same time the points of the section undergo different displacements along the longitudinal axis [5]. For warping restrained, the member develops additional shearing stresses, as well as normal stresses as warping stresses are not to be ignored [6-7]. The resulting stresses are primarily shear stresses (uniform torsion) or a combination of shear and longitudinal stresses (non-uniform torsion). The distribution of these stresses through the thin-wall section depends to a large extent on the cross sectional geometry and specifically whether it is open or closed sections [8]. In order to include warping shear stresses in the global equilibrium of the bar, that is to account for the secondary torsional moment deformation effect, an additional kinematical component (along with the angle of twist) is generally required, increasing the difficulty of the problem at hand [9].
Remarkable growth has been made since the 1945s in classical theories of torsion of thin-walled beams members with the consideration of warping, but it was limited to symmetric cross sections due to its complexity. There are many studies on non-uniform torsion with or without consideration of warping effects
Galishnikova, V.V., Gebre, T.H. The behaviour of thin-walled beam with restrained torsion. Magazine of Civil
Engineering. 2022. 110(2). Article No. 11009. DOI: 10.34910/MCE.110.9
© Galishnikova, V.V., Gebre, T.H., 2022. Published by Peter the Great St. Petersburg Polytechnic University.
This work is licensed under a CC BY-NC 4.0
1. Introduction
as theories and also as analysis using commercial FEM codes [8-16]. Similarly, researchers worldwide have extensively used fibre-reinforced polymer (FRP) strengthening materials to enhance the combined load strengths of reinforced concrete (RC) beams [18-23]. It has various well-known advantages such as high strength to weight ratio, high corrosive resistance, and easy-to-apply character [24]. The effect of span length of cantilever RC beams under pure torsion had been studied using a non-linear finite element analysis [25]. The element stiffness matrix and load vectors are derived using the primary and secondary warping functions [17-18]. Warping effects occur mainly at the points of action of the concentrated torsional moments (except for free beam ends) and at sections with free-warping restrictions [19-20]. The warping effect was included through an additional degree of freedom at each nodal point in the form of the first derivative of the angle of twist of the cross-section of the beam [10, 21]. Meanwhile, the torsional analysis of thin walled sections including shear has been studied in Ref. [15, 22-26]. The analysis of uniform torsion shows that the warping of bars depends on the shape of their section. A new torsion element of thin-walled beams including shear deformation which accounts for the warping deformation and shear deformation due to restrained torsion is developed [27-28]. Prismatic bars of solid and hollow sections or thin-walled section with a single interior vertex (for example an angle, a tee or a cross) do not warp but all other sections will experience warping of the cross section, depending on the geometry of the cross section [38]. Neglecting these warping stresses may generate significant errors specially for open profile torsion or shear-bending of short beams, and the situation may be even more critical for composite beams [6-7]. Warping-based stresses and deformations in closed sections, however, are assumed to be insignificant and have been therefore neglected [39]. Reasonably strong warping is expected to have effect in open than closed cross-sections and its significance is anticipated to be restricted to open cross-sections. Restraining the warping deformation causes the twist rate to be zero at the point of restraint and this causes a local effective torsional stiffening that affects the global torsional response of the beam [40]. The difference between the displacements and stresses due to uniform and non-uniform torsion is most pronounced for thin-walled sections. Warping of a thin-walled structures due to torsion stands for all the longitudinal displacements caused by the torsional rotation of the member cross-section around its shear centre. Thin-walled beam, because of its specific properties, has a unique internal force factor - bi-moment, which in certain cases can cause large normal stresses in cross-section [41].
In this paper, the behaviour of thin-walled sections of a bar with restrained torsion using the governing equation for non-uniform torsion is considered. This study is extensively applied to bars of closed and open sections of thin-walled steel cross section subjected to concentrated torsional loading and to the most
general torsional boundary conditions. The variation of primary torsional moment (MTp), secondary torsional moments ( Mts ) and warping moments (Mro) for different value of characteristic number for
torsion (9) are presented graphically. Finally, the behavior and comparison of all torsional moment
components with three different thin-walled sections are illustrated by presenting and discussion their results.
2. Methods
2.1. Defining Governing Equations
The governing equation for non-uniform torsion is used to study the behaviour of a bar with restrained torsion and it is derived for bars with thin-walled sections with local coordinate systems yj, y2, y3 as shown in Fig. 1 [31].
support
Figure 1. Torsion of prismatic bar.
When the bar is subjected to the arbitrarily distributed twisting moment MT and its angle of twist is defined by the following governing equation (2.1) with boundary value problem [13, 18, 34, 42].
dM■
T
+ mT = 0,
dyi
C d4Pi d2Pi (21)
dyi dyi
where Pi is angle of twist, E, G are elastic constant, Cro is warping constant, J is torsion constant, mT is twisting load per unit length of bar.
If a bar undergoes non-uniform torsion, the total applied twisting moment MT is resisted by an internal twisting moment that consists of two components as given in equation (2.2). The primary internal moment is the twisting moment Mtp due to uniform torsion (St Venant torsion). The secondary internal
moment is the twisting moment Mts is due to warping restraint and the twisting moments are expressed
in terms of the angle of rotation Pi and the properties of the shape of the section. The applied torque is
resisted by a combination of the uniform and warping torques and it is given with its differential equation of non-uniform torsion.
Mt = Mtp + Mts = GJ^- ECm ^ (2.2a)
dyi dyi
where Mt is total twisting moment, Mtp is primary twisting moment due to uniform torsion, Mts is secontary twisting moment due to warping restraint.
For an element with constant Mt, the warping torsion Mts generates a generalized force called
Bimoment ( Mro) as follows:
Mro=-ECro ^ (2.3b)
dyi
The total shear stress o^t due to torsion is the sum of the primary shear stress o^p due to the
primary twisting moment for open section and the secondary shear stress Oi2S due to the secondary twisting moment for warping restraints.
o12T =0i2p + °i2s- (2.3)
Consider the homogeneous form of the governing equation (2) and let the characteristic number for torsion of a bar be defined as follows:
EC^ ^-rj^P. = 0,
dyi dyi
d^-i^ = 0, (2.4)
dy4 Va J dy2
e := a[rr
EC
where 9 is characteristic number for torsion, a is length of the bar.
The general solution for the homogeneous equation (2.4) is satisfied by the following assumed twisting angle function Pj (yj) and it yields to the exact solutions of the angle of twists, its derivatives, twisting moments, and bimoments of a node. They are given by
0Vi 0Vi Vi
P1 = c, sinh—— + c2 cosh —— + c3— + c4. (2.5)
a a a
Based on the derivatives of equation (2.5) the following equations are obtained:
^ = c11 -|cosh^ + c2 | -|sinh^ + c3L, (2.6)
dy1 V a J a V a J a a
d2p1 fef . u6y1 fef u6y1
—f = c11 - I sinh —— + c2l - I cosh——, (2.7)
dy2 V a J a V a J a
d3p1 feV , ey1 feV . , ey1
—^ = c1l — I cosh —— + c2l -I sinh ——, (2.8)
dy- V a J a V a J a
di=c, Í^Ysinh^ïI+c2 r^14cosh^xL. (2.9)
dy4 V a J a V a J a
The free coefficients c, to c4 are determined so that the boundary conditions at the ends of the bar are satisfied. If the vertex y1 = 0 is fixed and a twisting moment MT is applied at vertex y1 = a which is free to warp. Solving equation (2.5) with the boundary conditions and the free coefficients are given below:
^^ y a c c c
c3 =——, c, =—-, c2 = —tanhe, c4 =—-tanhe.
3 gj 1 e e 4 e
These coefficients are substituted into expression (2.5) and the angle of twist is given below:
Pi =
M
gj e
f f 6Vi ^ tanh e cosh——1 -1
<*1 isinh-VL nil
V a a Jy
(2.10)
V v a 7
Based on equation (2.10), the following equations of MTp, MTs and Mro for a node are obtained.
MTP = MT + MT | tanh 0 sinh ^ - cosh ^ |, (2.11)
V a a 7
MTS =-MT | tanh 0 sinh 0yi - cosh |, (2.12)
V a a )
Mro=-
MTef , _ , ey, . , ey,1 T tanh ecosh^1 — sinh -
a a y
(2.13)
2.2. Numerical case studies
In this study, the variation of Mtp, Mts and Mro with different value of 9 is investigated. Different
section types are considered as it is given in Table 5 and the section properties, displacements, rotations, stresses are to be compared for all cases in addition their distribution with in the span is compared based on their values of 9. Tables 1 to 3 show comparatively for different value of 9 along the span of the beam, the initial value of 9 is 1 and the maximum is 10. The variation of Mtp, Mts and Mro for different value
of 9 are presented. The results are tabulated below to bars of closed and/or open sections of thin walled steel cross section subjected to different torsional loading and to the most general torsional boundary conditions. In Table 1, the values are obtained based on equation (2.11) as it is expressed in a dimensionless form for diverse values of 9 as shown below.
T
a
Table 1. Variation of Mtp for different value of 0.
Mtp / Mt
yi/a 9 = 1 9 = 2.5 9 = 5 9 = 7.5 9 = 10
0 1.00 1.00 1.00 1.00 1.00
0.2 0.87 0.61 0.37 0.22 0.14
0.4 0.77 0.38 0.14 0.05 0.02
0.6 0.70 0.25 0.05 0.01 0.00
0.8 0.66 0.18 0.02 0.00 0.00
1 0.65 0.16 0.01 0.00 0.00
Similarly, In Table 2 and Tables 3-4, the values are obtained based on equations (2.12) and (2.13) respectively and they are expressed in a dimensionless form for different values of 9 as shown below.
Table 2. Variation of Mts for different value of 0.
Mts / Mt
yi/a 9 = 1 9 = 2.5 9 = 5 9 = 7.5 9 = 10
0 0.00 0.00 0.00 0.00 0.00
0.2 0.13 0.39 0.63 0.78 0.86
0.4 0.23 0.62 0.86 0.95 0.98
0.6 0.30 0.75 0.95 0.99 1.00
0.8 0.34 0.82 0.98 1.00 1.00
1 0.35 0.84 0.99 1.00 1.00
Table 3. Variation of M® for different value of 0 with 0/a = 2 .
_Mm / Mt_
yi/a
9 = 1 9 = 2.5 9 = 5 9 = 7.5 9 = 10
0 -0.38 -0.49 -0.50 -0.50 -0.50
0.2 -0.29 -0.30 -0.18 -0.11 -0.07
0.4 -0.21 -0.17 -0.07 -0.02 -0.01
0.6 -0.13 -0.10 -0.02 -0.01 0.00
0.8 -0.07 -0.04 -0.01 0.00 0.00
1 0.00 0.00 0.00 0.00 0.00
Table 4. Variation of Mm for different value of 0 with 0/a = 4 .
_Mm / Mt_
yi/a
9 = 1 9 = 2.5 9 = 5 9 = 7.5 9 = 10
0 -0.19 -0.25 -0.25 -0.25 -0.25
0.2 -0.14 -0.15 -0.09 -0.06 -0.03
0.4 -0.10 -0.09 -0.03 -0.01 0.00
0.6 -0.07 -0.05 -0.01 0.00 0.00
0.8 -0.03 -0.02 0.00 0.00 0.00
1 0.00 0.00 0.00 0.00 0.00
According to equation (2.5), with the solution function for Pi (yi) and replacing the integration
constants with the deformations, the following stuffiness relation is obtained. Considering the non-uniform torsion, the warping effect is contained within through an additional degree of freedom at each nodal point of the beam.
Kt — "
kT1 kT2 kT3 kT4
ECro kT2 kT6 kT7 kT8
a3 kT3 kT7 kT11 kT12
kT4 kT8 kT12 kT16
KT1 — KT11 — S *0sinh 0, KT6 — KT16 — S * (cosh 0-
sinh 0
sinh 0
KT2 — KT4 — S*(cosh0-1)*a, KT8 — S*| |*a
) *a
2
2
S—
2 ^
02 Q
Q — 2(1 -cosh0) + 0sinh0, kt3 — -K^, ^7 — kt12 —-kt2
Different section types are given in Table 5 which are the I section, rectangular hollow section, and channel section. Considering the prismatic bar of Fig. 1 as it is fixed at vertex yj = 0 and subjected to a
twisting moment of MT without warping restraint at vertex yj = a. The section properties, displacements, rotations, stresses are to be compared for all cases and compared their distribution within the span based on the required value of 9 .
Table 5. Different section types.
Section type
h(mm) 400 400 400
f(mm) 180 180 180
tf(mm) 11 11 11
tw(mm) 8 8 8
E — 200*106 kN/m2 MT — 1.0 kNm G — 77*106 kN/m2
3. Results and Discussions
3.1. The variation of Mtp, Mts and Mro for different value of 9
The behavior and variation of Mtp and Mts for different value of 9 which ranges from 1 to 10 are presented graphically in Fig. 2. Considering a point nearly to the support at y1 — 0 for Fig. 1, the total
torsional moment is largely carried by the primary twisting moment, whereas the secondary twisting moment is small. At points in the span, the torsion is carried partly as St. Venant torsion (i.e. by the St. Venant shear stresses) and partly as warping torsion (i.e. by the shear stresses caused by the restraint of warping. For 9=Z both torsion mechanisms contribute to Mt throughout the beam in both cases but with the increasing
of the value of 9 the influence of the twisting moment role to the Mt varies as shown in the Fig. 2. In view of Fig. 2 (a) with increasing of 9 value of MTp reduces fast and the torsional moment is dominated by St. Venant in a major part of the beam. Similarly, in Fig. 2(b), with the increasing value of 9, the St. Venant moment is growing, and the torsional moment Mt is dominated by St. Venant in a major part of the beam in both cases.
0
?
v V
\ \ v X \
v. Vs
! Vi.
\ \ NT ---«■■■-J
---«0 U
- -Mi b)
-----0-1 '
"uMJ M --0-1
„■0L4
S fl
£ m""w J« 1»
fl r i" ■ 1! ' / / / / _ if J"
ft J ■x f ■d ' j* y
!; -
■ e-w
-Û i
friî -A L
ih i>.[i
I U U U M 1 Fj1'* y,,'»
Figure 2. Variation of Mtp and Mts for different value of 9.
Similarly, the behavior and variation of warping moment (Mro ) for different value of 9 which ranges
from 1 to 10 are presented graphically in Fig. 3 As the characteristics number for torsion (9) is an indicator of how quickly the effect of warping restraint dissipates because the warping moment varies based on the restrained conditions of the beam. The variation of Mro in the Fig. 3 displays the greater values at the fixed support where the warping is prevented and induces the largest normal stresses. At the free end the beam can warp freely as a result the normal stresses is zero, thus Mro (a) = 0. The two graphs below are plotted
for different value of 9/a (i.e. 9/a = 2 and 4) as it is for general case. The distribution of Mro varies
differently in both cases with varying values of 9 and thus the normal warping stress differs along the beam based on the position of restrained torsion and section type.
o
-0.1
-0.2
s
S -0.3
0.4
-0.5
/
a i, / t t f
Hi i,/
if if/ y y
For (Va =2 - ■ - 9-7.5
-----9=5
......... &-Z5 gH
-9-1 C
s
" -0.15
-0.2
-0.25
j ty ' ' . / / —ij. - ..
> U * / / t 9
tif t /
If / a /
' /
For 01* = 4
— —9=10 -""9=5 -9=1
.........9-2.5
---9-7.5
0 9.2 0.4 0.6 0.8 1 0 0.2 0.4 (1.6 OlS 1
yi/a y,/a
Figure 3. Variation of Mm for different value of 0.
3.2. Comparison of Mt components with three different thin walled sections
The section properties, displacements, rotations, stresses are to be compared for all cases and compared their distribution with the span based on the required value of 0 .
Case I: Refereeing Table 5, for an I- sections, the values of Cm, J, 0 and Pj (a) are computed
below:
Cro = 1 tfb3h2 = 0.4277* 10-6m6, J = 1(2ft/ + htj) = 0.2280*10-6m4,
9Isec. = 1812,
P1 = -0.1257 sinh 1.812 + 0.119208 cosh 1.812 + 0.2278 - 0.119208, P1 = 0.109 radians = 6.25 degrees.
The angle of twist for uniform torsion of an I-section of the bar is:
aMT 4.0
P1
.uniform
GJ
77 *106 • .228 *10-
— 0.228 radians — 13.06 degrees
The variation of the uniform, non-uniform angle of twist, Mtp, Mts and Mt on the axis of the bar are shown in Fig. 4. The rotation at vertex y1 — a due to non-uniform torsion is around 50 percent of the
rotation for uniform torsion of an I-section. Both Mtp and Mts contribute to Mt through the span of the beam and the torsional stresses are due to St Venant shear stresses and the restraint of warping. The applied twisting moment is resisted entirely by the secondary twisting moment at support ( y1 — 0 ) and
entirely by the primary twisting moment at y1 — a. According to the studies on [18, 27], our results for the
variation of the uniform, non-uniform angle of twist, Mtp,Mts and Mro on the axis are compared for
further verifications and the results are some. The distribution of the total moment between uniform and non-uniform torsion at intermediate points is shown in Fig. 4.
Figure 4. The normalised graphs of Pi, Mtp, Mts and Mt of restrained I-beam section.
Furthermore, the values of Pj of equation 2.5 and its derivatives are plotted in Fig. 5 to show their comparisons and behaviours along the length of the member. The behaviour of the flanges can be observed in the graph for Pj and its first derivatives (P'). They show the variation in twist to which St. Venant shear
strains and stresses are proportional to the primary torsional moments. The curvature of the flanges is also associated with the graph Pi" and it is proportional to the warping moment in one flange. A combined graph for P' and P!" is shown in the Fig. 5(b) so P' is representing for the primary twisting moment and Pj
belongs to the secondary twisting moments. The graph of Pj belongs to the characteristics of the rate of change of curvature and it is proportional to the warping shear force in the flange in addition to the warping torsional moment. Based on the Fig. 5(b) Pj contributes 68 % and P" contributes 32 % of the total torsional moment.
Figure 5. Decomposition of total twisting moment for an I-beam.
For an I-section, the longitudinal stress an at yj = 0 due to non-uniform torsion and the warping function are computed as below:
© = ± Ifh = ± ¡8.0 * j0-3 m2, 4
an = -© MTtanh e[ =± 0.0980 *j06 kN/m2
c© lej
The primary shear stress at the free end (yj = a) is computed as given below and it varies linearly over the thickness of the walls of the section.
Mt tf 2
flange: Tmax = ±-yX = ±48.2 N/mm2,
Mt t 1
web: Tmax = ± T w = ±35.j N/mm2
The secondary shear stress at node yj = 0 is computed by determining the static moments of the
warping function and it is constant over the thickness of the walls of the section. The contributions of the walls to the static moment of the warping function are computed based on the shear flow at the external vertices. The shear stress due to warping restraint is:
F 20 83 o o
a^ = -© = -^1 = -j893.6 kN/m2 = -L9N/mm2 j2s tf 0.0U
The distributions of the stress an , warping function (©) and secondary shear stresses an over the section due to the non-uniform torsion are shown in Fig. 6.
Figure 6. Distribution of aii, © and ai2p (N/mm2).
Case II: Similarly, for rectangular hollow cross-section, which is given in Table 5, the values for C©, J, e and Pj (a) are computed below:
C© = 2©2 (f tf + htw ) = 0.2875 * j0j2 mm6 ,
2f2 h2 tw tf ftw + htf
j — „ twtf — 0.1562 *109 mm4
0Rec.sec. — 57.85
P1 — 5.749 *10-6 • (- sinh 57.85 + cosh 57.85 -1.0) + 0.3326*10-3 P1 — 0.327*10-3 radians — 0.0187 degrees
The angle of twist for uniform torsion of the rectangular section of the bar is:
aMT 4 0 -3
Piunifbrm — - — -^-t — 0.333 *10 radians
1 GJ 77*106 • 0.1562*10-3
The variation of the uniform and non-uniform angle of twist (restrained torsion) on the axis of rectangular hollow cross-section of the bar are shown in Fig. 7. For rectangular hollow cross-section as the value of 9 is large and the angle of twist is very small in its magnitude. The angle of twist at points nearer to the support has significant magnitude as shown in the Fig. 7 and suddenly drops to a small value at the free end. The rotation at vertex yj = a due to uniform and non-uniform torsion is almost the same and its magnitude is negligible comparing with an open section.
Figure 7. The normalised graphs of Pi of restrained rectangular hollow cross-section.
The longitudinal stress on at yj = 0 due to non-uniform torsion of rectangular hollow section is given below and the distribution of this stress over the section is shown in Fig. 8:
U. U — ©0 = — f h-
0 4 u + 1
h tf -in
with u :=--f- © = ±©0 = ±9.124*10 3 m2
t f 0
M
on = -©
— tanhef-l = + 2.2 N/mm2
cm le J
The primary shear stress is null at y1 = a and it is computed with expressions shown below. It is constant over the thickness of the walls of the section.
fl-nge: ai2p =
M
2f htf
= ±0.63 N/mm
2
web : o
M
12p
2fhtw
T = ±0.86 N/mm2
The secondary shear stresses (012 ) at the support ( yi = 0) depend on the Static moments ( Sb)
of the warping function. The increments of the static moment for the walls are given by the expression below:
180 400 „„^ n = -+- = 66.36
11
8
->-3
S0 = - 9124*10" • (0.182 - 0.402) =-2.92*10-6 m4 0 6 • 66.36
wall AB : S(1) = S0 -1 ©0 f tf = -2.92*10-6 - 4.516*10-6 = -7.44 *10-6m4
wall BC : S(2) = S0 +1 ©0ftw = - 2.92 *10-6 + 7.299 *10-6 = 4.38 *10-6m4
Based on the above solutions and static moment of the section, the variations of secondary shear stress over the section are shown in Fig. 8:
Mt s,
a12s = -
t C
10 S© = -3.4783*106 S© kN/m2
© 0.2875 *10-6 t t
The distributions of the stress an , warping function (©) and secondary shear stresses a^ over
the section due to the non-uniform torsion are shown in Fig. 8.
Figure 8. Stress (aii) (a), Static moments (Sra), secondary shear stress (ai2s) (N/mm2). Case III: Likewise, by considering the channel section of Table 5, the values for C©, J, 9 and P1 (a) are calculated as given below:
f - tw | *h2*tf
C© =
12
2htw + 31 fItf
htw + 6lfItf
= 0.580*10-6 m6,
J = 1(2f tf3 + htw3) = 0.218*10-6 m4 9 = 1.52
Pj = -0.157 sinh1.52 + 0.142 cosh1.52 + 0.238 - 0.142 Pj = 0.083radians = 4.75 degrees
The angle of twist for uniform torsion of the rectangular section bar is:
P1
.uniform
aMT
"gT
4.0
77 *106 x 0.218 *10-3
= 0.24 radians = 13.75 degrees
The variation of the uniform and non-uniform angle of twist (restrained torsion) on the axis of the channel section of a bar is shown in Fig. 9. The rotation at vertex y1 = a due to non-uniform torsion is 40 percent of the rotation for uniform torsion of the channel section.
Figure 9. The normalised graphs of P1 of restrained Chunnel section.
For channel section, the longitudinal stress (an) at yj = 0 due to non-uniform torsion and the warping function are computed as below:
®a = ^*(f -tw -e) = 21.4*10
-3m2
qb =-e * —f =-13.8 * 10-3m2
M
a11A = mA —xtanh0| - | = 124.6 N/mm2
C
m
M
a
a11B = - m^^^tanh0| ^ | = -80.3 N/mm2
a
The values and distribution of the warping function at the external vertices and longitudinal stress are given in Fig. 8: Correspondingly the primary shear stress in the flange and web plates at support is zero and at node y1 = a is computed as follow. It varies linearly over the thickness of the walls of the section.
flange: web:
MT tf T = +—LX = ±
lmax — j —
MT tw T = +-J—w = +-
lmax — —
0 011 kN/m2 = ±50.0 N/mm2
0.218 *10-6
0 008 kN/m2 4 ±36.4 N/mm2
^-6
J 0.218 *10-
The secondary shear stress is null at section and at node y1 = 0 is computed by determining the static moments of the warping function.
S0 = 0.5tf (f - 0.5tw)h * (0.5f - 0.25tw - e) = 7.36*10-6 m4
wall AB: S(2) = 1h tf [(f -0.5tw)-e]2 = 12.6* 10-6m4
wall BC: S(1) = S0 -^tweh2 = -3.68*10-6m4
2
MT S
>12s0
>12s1
J12s2
tf Cro
MT SM
tw Cro
MT SM
= - 1.01N/m
= 0.07N/m2
2
tf C
= - 1.15N/m
2
The distributions of the stress On, Static moments (and secondary shear stresses On over the section due to the non-uniform torsion are shown in Fig. 10.
Figure 10. Stress (ail) , Static moments (Sra), secondary shear stress (ai2s) (N\mm2).
A combined graph for angle of twist for closed and open sections is shown in Fig. 11 for both uniform
and restrained torsion. The rotation at vertex yj = a due to non-uniform torsion are 50 percent of the
rotation for uniform torsion of I section and 40 percent of the rotation for uniform torsion of Chunnel section. For closed channels like rectangular hollow cross-section, the rotations in uniform and non-uniform torsions are considered negligible as their magnitudes are very small compared to the open cross sections as shown in Fig. 11.
Figure 11. The combined normalised graphs of Pi for all restrained section types.
In this study, the characteristic number for torsion is the main criteria to study the behaviour of a bar
with restrained torsion and the variation of the uniform, non-uniform angle of twist, Mtp, Mts and Mro
on the axis of the bar are expressed as a function of the characteristic number for torsion and all results are compared with different studies [14, 15, 27]. From Fig. 11, the normalization graph clearly presents the combined graph for angle of twist for closed and open sections for both uniform and restrained torsion. For closed channels like rectangular hollow cross-section, the rotations are very small and are considered negligible but comparing to different studies [29, 43] it shows that the effect of warping must be considered in the case of non-uniform torsion of closed-section beams.
4. Conclusion
In this study, the characteristic number for torsion was adopted to study the behaviour of a bar with restrained torsion. The resulting responses from all cases using different value for the characteristic number for torsion were analyzed and compared. According to the previous findings, it is concluded that:
1. When considering fixed support conditions Vlasov torsion must be accounted as the behaviour for St. Venant and Vlasov torsion existing for all section types.
2. In this study, we used the hyperbolic than polynomial shape functions as it gave the exact results as the analytical solution.
3. With the increasing of 9 value, MTp reduces fast and the torsional moment is dominated by St. Venant in a major part of the beam but for 9 = 1 both torsion mechanisms contribute to Mt. The
distribution of Mro varies with the different values of 9 and 0*a consequently the normal warping stress differs along the beam based on the position of restrained torsion and section type.
4. The angle of twist of non-uniform torsion differs from uniform torsion by 50, 1.8, and 41 percents for the I-section, rectangular tube, and channel section, respectively.
5. Normal stresses dominate the stress within the bar as the cross-section is restrained and Vlasov shear stresses are very small and vanished when the normal stress reached maximum. Contradictory to this, at the free end the shear stress is dominated by St. Venant shear stresses and its maximum value is located at the mid of the flange.
6. The primary shear stress is considerably lower for the closed than open sections. The secondary shear stress is depending on the static moments of the warping function.
7. For closed channels like rectangular hollow cross-section, the rotation and warping are very small and are considered negligible but comparing to different studies it shows that the effect of warping must be considered in the case of non-uniform torsion of closed-section beams.
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Contacts:
Vera Galishnikova, GalishnikovaVV@mgsu.ru Tesfaldet Hadgembes Gebre, tesfaldethg@gmail.com
Received 27.09.2020. Approved after reviewing 08.04.2021. Accepted 04.06.2021.
