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NONUNIFORM TORSION WITHOUT SHEAR DEFORMATION
EFFECT USING FEM
Dang-Bao TRAN1,2, Ja osla NAVRÁTIL1
1 Depa men o S uc u es, Facul y o Ci il Enginee ing, VSB–Technical Uni e si y o Os a a,
Lud íka Podéš ě 1875/17, Os a a, Czech Republic.
2 Depa men o Ci il Enginee ing, Facul y o A chi ec u e, Thu Dau Mo Uni e si y, T an Van On 06,
Binh Duong P o ince, Vie nam.
dang.b[email p o ec ed], ja osla [email p o ec ed].
DOI: 10.35181/ ces-2021-0006
Abs ac . This pape p esen s he use o a ini e elemen
me hod (FEM) o analyze nonuni o m o sion wi h an
a bi a y c oss-sec ion wi h homogeneous elas ic ma e ial
wi hou shea de o ma ion e ec . Beams a e cons ained
by he mos common ypes o suppo s, such as ixed,
pinned, and olle , and a e subjec ed o any applied
o sional load, concen a ed, and dis ibu ed. The
p esen ed FEM ans o ms he 3D analysis o nonuni o m
o sion beams in o sepa a ed 2D c oss-sec ional and 1D
modeling. The geome ic cons an s o he c oss-sec ion a e
i s ly de i ed om 2D FEM, which uses a 9-node
isopa ame ic elemen . Then, a 1D FEM, which uses he
He mi ian shape unc ion, is de eloped o compu ing he
wis angle, he de i a i e o wis angle, and s ess
esul an s. Finally, he s ess ield is ob ained om he
local analysis on he 2D-c oss sec ion. A MATLAB
p og am is execu ed o alida e he nume ical me hod
h ough examples. The alida ion examples ha e p o en
he eliabili y o he au ho 's nume ical me hod o
analyzing p oblems de ined abo e.
Keywo ds
Nonuni o m o sion, hin-walled s uc u e, shea
de o ma ion e ec , wa ping es ain , ini e elemen
me hod.
1. In oduc ion
In enginee ing p ac ice, we o en encoun e beam
s uc u es loaded in o sion. When he wa ping o a
membe 's c oss-sec ion is no es ained, he s ess ield o
a p isma ic beam wi h homogeneous iso opic ma e ial can
be de i ed om he Sain -Venan heo y s ic ly [1-4]. In
p ac ice, because i) bounda y condi ions a e imposed, ii)
geome ical cha ac e is ics o he sec ion, he wa ping o
beam's c oss-sec ion is es ained, which leads o
addi ional no mal and shea s esses, which he Sain
Venan heo y does no ake in o accoun . Vlaso [5] was
he i s o o mula e he p oblem o nonuni o m o sion.
Bensco e [6] imp o ed Vlaso 's heo y, which neglec s
he shea de o ma ion e ec , leading o e o s wi h closed
c oss-sec ions [7].
Many au ho s es ablished FEM o conside nonuni o m
o sion aking in o accoun he shea de o ma ion e ec [7-
14]. Howe e , he abo e esea ch [7-14] used he
app oxima ions o Thin Tube Theo y [5] o de e mine he
ba 's geome ic cons an s, es ic ing he accu acy and
applicabili y o he o mula ions. El Fa mi [15,16]
p oposed a beam heo y wi h a nonuni o m wa ping,
including he e ec s o o sion and shea o ces o
a bi a y c oss-sec ions made o homogeneous iso opic
elas ic ma e ial. Mokos and Sapoun zakis [17] p esen ed a
nonuni o m o sion heo y conside ing he shea
de o ma ion e ec o gene al c oss-sec ions implemen ed
by Bounda y Elemen Me hod. The pu pose o his pape
is o es ablish a nume ical me hod using FEM o sol e he
nonuni o m o sion p oblem wi hou he shea de o ma ion
e ec o he p isma ic beam wi h a bi a y c oss-sec ion
unchanged h oughou he leng h wi h homogeneous
iso opic elas ic ma e ial, based on displacemen and s ain
ields de i ed om Sapoun zakis, EJ e al. [17]. This pape
is a p elimina y esea ch s ep o u u e esea ch, which
conside s he shea de o ma ion e ec in nonuni o m
o sion.
2. A b ie in oduc ion o he
heo e ical o mulas o
nonuni o m o sion
Le us conside a p isma ic beam wi h a bi a y c oss-
sec ion, cons an along he leng h L wi h he modulus o
elas ici y E, and shea modulus G. The longi udinal axis is
he x-axis, and he c oss-sec ions lie in he y–z plane. S is
he shea cen e o he c oss-sec ion. The pa allel sys em
,
S
zzz=+ S
yyy=+ in e sec s a O, he a bi a y poin .
CXYZ is he pa allel sys em wi h Sxyz h ough he cen e
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o g a i y C.
The mul iply connec ed domain Ω is bounded by n
cu es, 1,Γ 2,Γ..., 1,
n−
Γ ,
n
Γ as Fig. 1. Tangen ec o
wi h associa e coo dina e s and no mal ec o n se up he
igh -handed sys em. The beam is en o ced o he a bi a y
loads, dis ibu ed o que m (x), concen a ed o que M i(x),
concen a ed wa ping momen Mwi(x). The c oss-sec ion is
assumed wi h no dis o ion.
Fig. 1: C oss-sec ion o he p isma ic beam
The displacemen ield is exp essed as [17]
(,,)uxyz ='(,) (,),
PS
xS S
yz yz
θφ φ
+ (1)
(,) (),
x
xz z x
θ
=− (2)
(, ) (),
x
wxy y x
θ
= (3)
whe e u, , w a e he axial and ans e se displacemen s o
he beam wi h espec o Sxyz
x
θ
is he angle o wis
(,)
P
Syz
φ
is he p ima y wa ping unc ion wi h espec o
shea cen e S de i ed om
20in
on .
P
S
PP
SS
yzyzn
nnznyn
yz
φ
φφ
∇= Ω
∂∂
+=− Γ
∂∂
(4)
Mo eo e , he es ained wa ping unc ion can be
compu ed as ollows by using he ans o ma ion o
coo dina es [18]
1
(,) (,) (,) .
PP P
SOSS O
yz yz yZ zY yzdA
A
φφ φ
Ω
=+−−
 (5)
whe e (,)
P
Oyz
φ
is he wa ping unc ion wi h espec o
𝑂𝑦𝑧 sys em coo dina es, A is he a ea o he c oss-sec ion
S
S
φ
is he seconda y wa ping unc ion wi h espec o he
shea cen e S, gi en as
'''
2() in
0on ,
SP
x
SS
SS
SS
yz n
Ex
G
nn
yz
θ
φφ
φφ
∇=− Ω
∂∂
+=Γ
∂∂
(6)
The s ess ields ob ained om he heo y o elas ici y
as [17]
"() (,),
P
xx x S
Ex yz
σθφ
= (7)
xy
τ
='()
P
xy
P
S
x
Gx z
y
τ
φ
θ

∂−+


∂

144424443
,
S
xy
S
S
Gy
τ
φ
∂
∂
123
(8)
xz
τ
='()
P
xz
P
S
x
Gx z
z
τ
φ
θ

∂++


∂

144424443
,
S
xz
S
S
Gz
τ
φ
∂
∂
123 (9)
In addi ion, he shea s ess in (8), (9) can be seen as
composed o wo pa s: p ima y shea s ess and seconda y
shea s ess.
The wa ping momen , he o al wis ing momen , he
p ima y wis ing momen , he seconda y wis ing momen
a e deno ed by ,
w
M ,
M
,
P
MS
M
ob ained as [17]
w
M=P
xx S d
σφ
Ω
−Ω=
'' ,
Sx
EC
θ
− (10)
',
PP
PP P
SS
xy xz
x
M
zyd
yz
GI
φφ
ττ
θ
Ω


∂∂
=−++Ω



∂∂



=
 (11)
S
M=
PP
SS
SS
xy xz
yz
φφ
ττ
Ω

∂∂
−− =


∂∂

''' (),
S
EC x
θ
− (12)
,
P
S
MM M=+ (13)
whe e
I
is he o sional cons an , de e mined as
I
=22 ,
PP
SS
yzy z d
zy
φφ
Ω

∂∂
++ − Ω


∂∂

 (14)
S
C a e he wa ping cons an de e mined by
()
2.
P
SS
Cd
φ
Ω
=Ω
 (15)
The ela ion be ween he s ess esul an , o al wis
momen M , and he dis ibu ed o que m (x) is exp essed
as [17]
().
Mmx
x
∂=−
∂ (16)
A e some algeb a, he equilib ium equa ion o he
nonuni o m o sion p oblem o a homogeneous iso opic
ba wi hou shea de o ma ion e ec is exp essed as
42
42
.
xx
S
dd
EC GI m
dx dx
θθ
−= (17)
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3. FEM p ocedu es
3.1. Angle wis ,
x
θ
Applying Gale kin’s me hod, one o he me hods o
weigh ed esidual, he go e ning equa ion Eq. (17) is
ans o med o weak o m as
42
42
0
0,
Lxx
S
dd
IEC GI mdx
dx dx
θθ
η

=−−=



 (18)
whe e
η
is he es unc ion.
The beam is disc e ized in o a numbe o ini e
elemen s. A e some manipula ion, he weak o mula ion
o Eq. (18) can be exp essed as
22
22
1ee
nxx
S
LL
i
dd
dd
IEC dxGI
dx dx
dx dx
θθ
η
η
=

=+−




0
0,
e
L
x
w
L
ddd
mdx GI M M
dx dx dx
θηη
ηη

+− + − =



 (19)
whe e
Le is an beam elemen domain
n is he numbe o elemen s o he beam.
The He mi ian polynomial in e pola ion unc ion,
which achie es he co ec solu ion o he p oblem wi h
e ining mesh, p o es mo e lexible han he Hype bolic
in e pola ion unc ion, which gi es he mos accu a e
esul s wi h he mos analy ical solu ion [19]. We choose
cubic unc ions o he spa ial in e pola ion o he wis
angle,
x
θ
, in e ms o nodal a iables. To his end, we
conside an elemen ha has wo nodes, one a each end.
The wis angle can be exp essed as
12
112 3 24
() () () () ,
xx
xx x
dd
Hx Hx Hx Hx
dx dx
θθ
θθ θ
=+ + +
(20)
whe e
23 23
12
23 3
23 23
34
23 23
32 2
() 1 , () ,
32
() , () ,
x
xxx
Hx Hx x l
ll l
xx xx
Hx Hx
ll ll
=− + = − +
=− =−+
(21)
12
12
,,,
x
x
xx
dd
dx dx
θθ
θθ
a e he nodal deg ees o eedom.
Inse ing he Eq. (20), Eq. (21) o Eq. (19) esul s in he
s i ness ma ix o he beam elemen
12
,
ee
=+KK K (22)
whe e
111
0,
e
L
eT
S
EC dx=
KBB
(23)
222
0,
e
L
eT
GI dx=
KBB
(24)
whe e
2
22 2
3
12 4
12222
,
dH
dH dH dH
dx dx dx dx

=


B (25)
3
12 4
2.
dH
dH dH dH
dx dx dx dx

=

B (26)
The co esponding elemen nodal deg ees o eedom
is 12
12
,,, .
T
xx
xx
dd
dx dx
θθ
θθ

=

e
d (27)
The hi d e m in Eq.(19) esul s in he elemen o ce
ec o . Fo a gene ally dis ibu ed o que, we need o
compu e
1
2
03
4
() ,
e
L
H
H
mx dx
H
H



=




e
F (28)
whe e e
F is he elemen o ce ec o .
The las e m in Eq. (19) is he bounda y condi ions o
he o al wis ing momen and he wa ping momen a he
wo bounda y poin s, x = 0 and x = L, o he beam. I hese
bounda y condi ions a e known, he known o al wis ing
momen and wa ping momen a e included in he sys em
o ce ec o a he wo bounda y nodes. O he wise, hey
emain unknown. Howe e , he wis angle and he
de i a i e o he wis angle a e known as geome ic o
his case. Assembling he elemen s i ness ma ix and
ec o s leads o he sys em ma ix equa ion gi en below
.=Kd F (29)
3.2. Wa ping unc ion, ,
P
S
SS
φφ
Using Gale kin’s me hod, wi h es unc ion 1()H
η
∈Ω
and applying he Gauss-G een heo em, he go e ning
equa ions Eq. (4), Eq. (6) a e ans o med o weak o m as
PP
SS
d
yy zz
φφ
ηη
Ω

∂∂
∂∂
+Ω−


∂∂ ∂∂

(𝑛𝑧−𝑛
𝑦)𝜂𝑑𝑠
=0.
(30)
''' () 0.
SS P
SS x
S
Ex
dd
yy zz G
φφ θ
ηη φη
ΩΩ

∂∂
∂∂
+Ω− Ω=


∂∂ ∂∂


(31)
The wa ping unc ion alues, P
S
φ
, ,
S
S
φ
in Eq. (30) and
Eq. (31) a e app oxima ed by he FEM, which is p esen ed
in [3, 4].
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4. Valida ion examples
In his sec ion, he accu acy o he p oposed FEM has been
examined. A compu e code is de eloped in he MATLAB
R2015a so wa e based on he o mula ions desc ibed in
he p e ious sec ions. The ob ained esul s a e compa ed
wi h he a ailable wo ks o he li e a u e.
4.1. Example 1
As a i s example, a hin-walled beam wi h an I-shaped
c oss-sec ion shown in Fig. 2 is analyzed and compa ed
wi h he esul s ob ained by T alli, A.M. [8] and Kim, N. I.
e al. [11]. The I-sec ion beam is clamped– ee and he
concen a ed o sional momen Mx = 25 kNm is applied a
i s ee end. The leng h o he beam is 5 m. The Young’s
modulus and he Poisson’s a io a e E = 2 x 106 MPa and
𝜈= 0.3 , espec i ely. The i s es was analyzed using
16 axial elemen s and 19 elemen s in he c oss-sec ion. The
geome ic cons an s o he c oss-sec ion de e mined om
he p esen s udy a e as ollows 76
2.1559 10
S
Cm
−
=× ,
64
2.8643 10
Im
−
=× .
Fig. 2: A can ile e beam, he applied load, and he geome y o he I
c oss-sec ion, uni s [mm].
Fig. 3: The a ia ion o wis angle along he x-axis.
In Table 1, he angle o wis a he ee end p edic ed by
he p esen s udy is gi en and compa ed wi h he esul s o
o he esea che s. The a ia ions o he angle o wis along
he leng h o he hin-walled beam a e shown in Fig. 3. I
can be seen om Table 1 and Fig. 3 ha he wis angle
esul s ob ained om he p esen s udy a e in excellen
ag eemen wi h o he wo ks.
Tab.1: Compa ison o he wis angle 𝜃 a he ee end calcula ed by
di e en me hods
Me hods 𝜃 [𝑟𝑎𝑑] a x = L
P esen s udy 0.5171
T alli [8] 0.5183
Kim e al [11] 0.5173
Vlaso [11] 0.5167
Fig. 4: Dis ibu ion o he axial s ess 𝜎 in 2D c oss-sec ion a ix-end,
uni MPa,
Fig. 5: Maximum o no mal s ess 𝜎 along he axis x o he beam.
Fig. 4 shows he con ou plo o he axial s ess 𝜎 a
he clamped end o he I-sec ion beam. Fig. 5 depic s he
a ia ion along he axis x o he no mal s ess 𝜎. Fig. 6
and Fig. 7 p esen he a ia ion along he axis x o he
wa ping momen and he o sional momen s, espec i ely.
I is obse ed om Fig. 4 ha he a ia ions o he axial
s ess a e symme ic o he a ea cen e o he c oss-sec ion.
In con as o he langes, no signi ican axial s ess
appea s a he web o he I-sec ion beam. The maximum
alue o 𝜎 = 515.911 MPa occu s a he co ne s o he
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langes. In Fig. 5, he a ia ions o he maximum axial
s ess 𝜎 a he ixed-end o he c oss-sec ion along he
leng h o he beam a e shown and compa ed o he ones
ob ained om [8, 11]. I can be seen ha he axial s ess
p edic ed by he p esen s udy is in good ag eemen wi h
he esul s o T alli, A. M. [8].
Fig. 6: The a ia ion o wa ping momen Mw along he axis x o he
beam.
Fig. 7: The a ia ion o o sional momen s along he axis x o he beam.
4.2. Example 2
A can ile e ed hin-walled beam wi h a channel c oss-
sec ion depic ed in Fig.8 is analyzed in his sec ion. The
dep h, wid h, and hickness o he c oss-sec ion a e h =
0.833 m, b = 0.917 m, = 1/6 m, espec i ely. The leng h
o he beam is L = 4 m and, i is made o a ma e ial wi h
he modulus o elas ici y E = 2 x 105 MPa and he shea
modulus G = 0.77 x 105 MPa. The conside ed hin-walled
beam is subjec ed o uni o mly dis ibu ed wis ing
momen equal o mx = 4.07 x 103 kNm/m. The second
example was analyzed using in he p esen s udy by
employing 80 axial elemen s and 64 elemen s in he c oss-
sec ion. The geome ic cons an s o he c oss-sec ion
de e mined om he p esen s udy a e as ollows
6
0.0059
S
Cm=, 4
0.00407889
Im=.
Fig. 8: Can ile e beam, he applied load, and he geome y o he
channel c oss-sec ion, uni s [mm].
Shakou zadeh, H. e al. [7] used FEM based on
Bensco e 's model o sol e his p oblem and compa ed i
wi h he esul s ob ained om Vlaso 's heo y, which
neglec s he shea de o ma ion e ec . Meanwhile,
Sapoun zakis, E. J. e al. [20] used Analog Equa ion
Me hod (AEM) and AEM wi h isogeome ic analysis wi h
3 B-spline ypes: i) Quad a ic B-spline, ii) Cubic B-spline),
iii) Quad a ic B-spline o in es iga e his p oblem.
Fig. 9: The a ia ion o wis angle along he x-axis.
Fig. 10: The a ia ion o de i a i e o he wis angle along he x-axis.

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Fig. 9 and Fig.10 show he a ia ion o he angle o
wis and he de i a i e o wis angle along he x-axis,
espec i ely. The alue o he angle o wis and he
de i a i e o wis angle a he ee end, and he wa ping
momen a he ixed end a e p esen ed in Table 2 and
compa ed wi h he ones ob ained om Shakou zadeh, H.
e al [7], Sapoun zakis, E. J. e al. [20].
Fig. 11: The a ia ion o wa ping momen along he x-axis.
Fig. 12: The a ia ion o he o sional momen along he x-axis.
Tab.2: Compa ison o he wis angle, he de i a i e o wis angle, and
wa ping momen be ween he me hods
Me hods 𝜃 [ ad]
a x = L 𝜃
󰆒
[ ad/m]
a
x
= L
Wa ping momen
Mw [N.m2] a x = 0
P esen s udy -0.0429 -0.0114 -19.083 x 106
Sain - Venan model
[7]
-0.103 - -
Vlaso model [7] -0.045 -0.012 -18.33 x 106
FEM-Bensco e model
[7]
-0.045 -0.011 -18.17 x 106
AEM (50NP) [20] -0.050 -0.009 -16.75 x 106
AEM(Quad a ic B-
spline) [20]
-0.039 -0.006 -13.48 x 106
AEM(Cubic B-spline)
[20]
-0.061 -0.013 -18.29 x 106
AEM(Qua a ic B-
spline) [20]
-0.046 -0.008 -15.5 x 106
Fig. 11 and Fig.12 depic he a ia ion o he wa ping
momen and he o sional momen s along he axis,
espec i ely. I can be seen om Table 2 ha he wis
angle analyzed by he p esen s udy is in good ag eemen
wi h he esul s o Vlaso model [7], FEM-Bensco e
model [7], and AEM(Cubic B-spline) [20].
The dis ibu ion o he axial s ess 𝜎 and shea s ess
𝜏 inside he channel c oss-sec ion a he posi ion x = 0.5
m is shown in Fig. 13. The maximum no mal s ess 𝜎 =
657.865 MPa occu s on he ip o he lange. As expec ed,
he dis ibu ion o he shea s ess 𝜏, wi h he maximum
alue 205.981 MPa, dec eases om he ou e lange edge
o he inne edge [21].
(a)
(b)
(c)
Fig. 13: The no mal s ess 𝜎 and b) he shea s ess 𝜏 b) he shea
s ess 𝜏 a x = 0.5 m om he ixed-end, uni s [MPa].
5. Conclusion
In his pape , a ini e elemen me hod is de eloped o sol e
nonuni o m o sion wi hou he shea de o ma ion e ec o
he p isma ic beam wi h an a bi a y c oss-sec ion wi h
homogeneous iso opic ma e ial. Two examples we e
pe o med o alida ing he accu acy o he p esen s udy.
SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER
© 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 14
The compa ison o he esul s om he p esen s udy wi h
he co esponding esul s in he li e a u e shows ha he
p esen s udy can p edic he esponses o he non-uni o m
o sion wi h a bi a y c oss-sec ion wi hou shea
de o ma ion e ec accu a ely. Linea analysis o
nonuni o m o sion conside ing shea de o ma ion would
be one o he u u e esea ch opics.
Acknowledgmen s
The wo ks we e suppo ed by he S uden G an
Compe i ion VSB-TUO. The egis a ion numbe o he
p ojec is SP2021/77 “Nonuni o m o sion in p isma ic
beams wi h a bi a y c oss-sec ions using FEM”.
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