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DOI 10.14311/CEJ.2021.02.0030 409
TORSIONAL SHEAR STRESS WITH ARBITRARY CROSS-
SECTIONS IN HOMOGENEOUS ISOTROPIC ELASTIC
MATERIAL USING FINITE ELEMENT METHOD
Dang-Bao T an1,2
1. VSB–Technical Uni e si y o Os a a, Facul y o Ci il Enginee ing, Depa men
o S uc u es, Lud íka Podéš ě 1875/17, 708 00 Os a a, Czech Republic;
dang.bao. an@ sb.cz
2. Thu Dau Mo Uni e si y, Facul y o A chi ec u e, Depa men o Ci il
Enginee ing, T an Van On 06, 75000 Binh Duong P o ince,
Vie nam;bao d@ dmu.edu. n
ABSTRACT
De e mining he shea s ess o a s uc u al elemen caused by o sion is a i al p oblem. The
analy ical solu ion o he Sain -Venan o sion is only sui able o simple c oss-sec ions. The
nume ical me hods o e alua e he shea s ess due o o sion o complica ed c oss-sec ions is
indispensable. Many scien is s ha e s udied he o sion p oblem wi h a ious nume ical me hods.
This pape aims o p esen an e icien ini e elemen me hod o assessing he shea s ess wi h
a bi a y c oss-sec ions in homogeneous iso opic elas ic ma e ial due o o sion. MATLAB is he
language o p og amming he nume ical me hod. The alida ion examples we e pe o med o show
he eliabili y and e iciency o he au ho ’s nume ical me hod.
KEYWORDS
To sional shea s ess, Isopa ame ic eigh -noded quad ila e al elemen s, Sain -Venan
o sion, Wa ping unc ion
INTRODUCTION
To sion in he s uc u e occu s due o asymme ical loads, ei he by geome ic dimensions o
by in e connec ions be ween membe s. In many cases, o sion can be go e ning design ac o s. I
is, he e o e, essen ial o accu a ely de e mine he shea s ess caused by o sion. Sain -Venan
analysed he o sion p oblem using he semi-in e se me hod, assuming an unknown displacemen
o sa is y he equilib ium equa ions and bounda y condi ions. P and l hen in oduced he s ess
unc ion o he Sain -Venan o sion and he me hod o memb ane analogy [1]. Howe e , de e mining
he shea s ess due o o sion by he analy ic solu ion is complica ed o a complex single connec ed
domain o mul iply connec ed domain. A single connec ed domain is a domain whe e he c oss-
sec ion is bounded by a closed. The mul iply connec ed domain is he domain whe e he c oss-
sec ion is bo de ed by se e al closed.
Nowadays, s uc u al elemen s such as beams and columns a e mo e and mo e complica ed
in hei shapes. Hence he use o nume ical me hods o de e mine shea s ess due o o sion is
indispensable. Va ious nume ical me hods o assess o sional shea s ess ha e been pe o med by
many esea che s [2-22]. Ely, J. F., and Zienkiewicz, O. C. [2] i s sol ed Poisson’s equa ion o
P and l’s s ess unc ion using he ini e di e ence me hod and in es iga ed he ec angula sec ion
wi h and wi hou holes. He mann, L. R. [3] u ilized he ini e elemen me hod o calcula e he wa ping
unc ion o he o sion o i egula sec ional shapes. Based on he Hellinge –Reissne p inciple, Xiao,
Q. Z., e al. [4] de eloped a 4-node elemen wi h ou s ess pa ame e s o de e mine he shea
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s ess o he polygonal sec ion. G u mann, F. e al. [5] used he ini e elemen me hod o e alua e
shea s ess using he wa ping unc ion, which is mo e con enien han using P and l’s s ess
unc ion when conside ing a mul iply connec ed domain. G u mann’s nume ical me hod has been
implemen ed in o an enhanced e sion o he p og am FEAP [6], which used isopa ame ic ou -
noded quad ila e al elemen s. Fialko, S.Y. e al. [7] de eloped a nume ical me hod using cons an
iangula elemen s o sol e he Sain Venan p oblem o o sion, and o sionless bending o p isma ic
ba s is ealized in he SCAD so wa e [8]. Jog, S. e al. [9] p esen ed a ini e-elemen o mula ion o
Sain Venan o sion wi h a mul iply-connec ed domain in an aniso opic ma e ial. Recen ly, Behesh i,
A. [10] de i ed a ini e elemen om s ain-g adien elas ici y o o sion o p isma ic ba s wi h minimal
dimensions.
Besides he ini e di e ence and ini e elemen me hod, many au ho s ha e applied o he
nume ical me hods such as he bounda y elemen me hod (BEM) [11-16], line elemen -less me hod
(LEM) [17-18], a null- ield in eg al echnique [19], and ini e- olume me hod [20-21] o analyse he
o sion p oblem. Ka sikadelis, J. T. e al. [11] examined he o sion o gene al composi e ba s by
BEM while Gaspa i, D. e al. [12] ackled o ho opic beams wi h a polygonal c oss-sec ion. Ba one,
G. e al. [13] implemen ed he Complex Va iable Bounda y Elemen Me hod o examine he o sion
p oblem in he single connec ed domain. Lee, J.W. e al. [14] ob ained a new BEM om he gene al
Cauchy in eg al o mula de i ed om he Bo el–Pompeiu o mula o analyse he o sion p oblem.
Pa adiso, M. e al. [15] achie ed an e icien me hod o de e mine he wa ping unc ion pa ame e
wi h he gene al c oss-sec ion. Chen, K. H. e al. [16] in oduced a new e o es ima ion echnique in
BEM o op imise he o sion p oblem wi h a mul iply-connec ed domain. Di Paola, M. e al. [17]
p oposed LEM o deal wi h shea s ess in o sion p oblem wi h iso opic ma e ial and a bi a y c oss-
sec ion. San o o, R. [18] handled he Sain Venan o sion p oblem o o ho opic beams wi h a
gene al c oss-sec ion by LEM. Chen, J-T. e al. [19] in oduced he null- ield in eg al echnique o
analyse he o sion p oblem o ci cula c oss-sec ions wi h ound holes. Chen, H. e al. [20-21]
de eloped a new ini e- olume based on Bansal and Pinde a’s wo k o in es iga e Sain Venan ’s
o sion p oblems o homogeneous and composi e p isma ic ba s wi h mul iply connec ed domain.
In summa y, all o he wo ks show he easibili y and e ec i eness in academia. Cu en ly, o
he au ho ’s knowledge, G u mann’s me hod [5] has been de eloped in o he FEAP p og am [6 ] o
he Uni e si y o Cali o nia, Be keley, and Allplan B idge [22]. Fialko’s me hod [7] is simila o
G u mann, de eloped as a module in he SCAD comme cial so wa e [8]. I means he me hods o
G u mann and Fialko a e p ac ical. Howe e , due o he use o he ou -noded quad ila e al elemen
in FEAP and he cons an s ain iangle elemen in SCAD, o achie e high accu acy, i is necessa y
o mesh e y smoo hly, which a ec s he calcula ion speed. So, his esea ch aims o de elop a new
nume ical me hod (NMB) based on he wo k o G u mann by using he isopa ame ic eigh -noded
quad ila e al elemen . The alida ion examples we e pe o med o show he eliabili y and e iciency
o NMB.
FINITE ELEMENT METHOD PROCEDURE
Figu e 1 shows he a bi a y c oss-sec ion o p isma ic beam in homogeneous iso opic elas ic
ma e ial, he longi udinal axis is he x-axis, he c oss-sec ion deno ed

is in yz plane. The mul iply
connec ed domain

is bounded by
1 2 1
, ,..., ,
nn
   
. S is he cen e o g a i y. On
1 2 1
, ,..., ,
nn
   
he igh -handed o hogonal basis sys em is de ined wi h angen ec o and
ou wa d no mal ec o
[ , ]T
yz
nnn
. Wi h he o ien a ion o he associa ed coo dina e s is uniquely
de ined.
The displacemen ield
x y z
[u ,u ,u ]T
u
is exp essed as [1]
T
x
u


,
yx
uz


,
,
zx
uy


(1)
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whe e
x

: o sion angle,
x
d
dx



,
( , )
Tyz

: wa ping unc ion o o sion. He e, he cons ain o he
wa ping o o sion is
0.
TdA




(2)
Fig. 1 - C oss-sec ion o a p isma ic ba .
The shea s esses a e gi en as
T
xy Gz
y








,
.
T
xz Gy
z








(3)
The pola second momen o a ea can be w i en as
.
TT
T
I y y z z dA
zy



   

   

   

   


(4)
The go e ning di e en ial equa ion o he Sain -Venan o sion is exp essed as
22
22
0.
TT
in
yz


  

(5)
Meanwhile, he bounda y condi ion is gi en as
( 1,2,..., ),
TT
y z y z i
n n n z n y on i n
yz


    

(6)
whe e
y
dz
nds

,
.
z
dy
nds

(7)
The go e ning di e en ial equa ion (5) is ans o med in o he weak o m by using he
Gale kin’s me hod as below
( , ) ( ) 0,
TT
Tyz
G dA n z n y ds
y y z z
   
  
 

   
    

   


(8)
wi h es unc ion
1()H


.
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The equa ion (8) is app oxima ed by using he ini e elemen me hod. G u mann’s me hod
used isopa ame ic ou -noded quad ila e al elemen s. To imp o e he compu a ion speed, NMB
used isopa ame ic eigh -noded quad ila e al elemen s. The
[ , ]T
yzx
, he unknown unc ion
T

and he es unc ion

a e in e pola ed in he local coo dina e sys em as ollows
 
8
1
,
hii
i
N



xx
,
 
8
1
( ) ,
T h T
ii
i
N
   


,
 
8
1
( ) , ,
hii
i
N
   


(9)
whe e h deno es he app oxima e solu ion o he ini e elemen me hod,
 
,
i
N

deno es he shape
unc ion o he elemen . Figu e 2 shows he isopa ame ic eigh -noded quad ila e al elemen used in
NMB. The shape unc ions o his elemen can be desc ibed as ollows [23, 24]
12
34
22
56
2
78
11
( , ) (1 )(1 )( 1 ), ( , ) (1 )(1 )( 1 ),
44
11
( , ) (1 )(1 )( 1 ), ( , ) (1 )(1 )( 1 ),
44
11
( , ) (1 )(1 ), ( , ) (1 )(1 ),
22
11
( , ) (1 )(1 ), ( , ) (1 )(1
22
NN
NN
NN
NN
x h x h x h x h x h x h
x h x h x h x h x h x h
x h x h x h x h
x h x h x h x
= - - - - - = + - - + -
= + - - + + = - + - + +
= - - = + -
= - + = - - 2).h
(10)
Fig 2 -. Isopa ame ic eigh -noded quad ila e al elemen .
Inse ing he de i a i es o
()
Th

and
h

in o he equa ion (8) eads as
88
11
1
( ) 0.
numel e T e
ij j i
ij
e
KF





(11)
He e, deno es he assembly ope a o wi h numel he o al numbe ini e elemen s. The
s i ness pa
e
ij
K
o he nodes i and j and he igh hand
e
i
F
ead
11
11
11
,
( , ) ( , ) ( , ) ( , ) ( , ) .
e
j j j j
ei i i i
ij e
Q
Pjj
eii
ij p q p q p q p q p q p q
pq
N N N N
N N N N
K dA d d
y y z z y y z z
NN
NN
K w w y y z z

         
  

   
   
   
   
   
       
   





   

  

J
J
(12)
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11
11
11
,
( , ) ( , ) ( , ) .
e
ei i i i
ie
Q
P
eii
i p q p q p q p q
pq
N N N N
F z y dA z y d d
y z y z
NN
F w w z y
yz

     
  

   
   
   
   
   
   






  

J
J
(13)
whe e
J
deno ed as Jacobian ma ix de ined as
,
yz
yz
Jxx
hh
éů
¶¶
ęú
ęú
¶¶
ęú
=ęú
¶¶
ęú
ęú
¶¶
ëű
(14)
p
w
and
q
w
a e he weigh s and
p

and
q

a e he in eg a ion poin s o he Gaussian in eg a ion
echnique. NMB uses 3 x 3 Gauss quad a u e de i ed om he 1D case, whe e he quad a u e poin s
a e loca ed a
3/5
, 0 and
3/ 5
, and he co esponding weigh s a e equal o 5/9, 8/9, and 5/9,
espec i ely (see [23, 24]). The alue
T
i

o one a bi a y nodal poin i has o be alue 0.
VALIDATION EXAMPLES
The objec i e o his sec ion is o demons a e he eliabili y and e ec i eness o NMB. Fo
his pu pose, ou alida ion examples de i ed om [1, 5, 9] we e examined, and hei esul s a e
compa ed wi h hose analyzed by NMB implemen ed in MATLAB R2015a.
A ba o squa e c oss-sec ion subjec ed o he o sion momen
1
T
M
kN.m wi h he leng h
o he edge 2 m is analysed in he i s example. Jog, C.S. e al. [9] in es iga ed his p oblem using
16 isopa ame ic nine-noded quad ila e al elemen s. To achie e con e gence alues o he shea
s ess and he pola second momen o a ea, he squa e c oss-sec ion is di ided in o 16, 64, 256
elemen s wi h uni o m mesh by NMB, espec i ely. Figu e 3 shows he disc e iza ion o he squa e
c oss-sec ion wi h 16 eigh -noded quad ila e al elemen s. The poin s, A, B, C, D, co espond wi h
he coo dina es (0, 1), (2, 1), (0, 0), (0, 2), espec i ely.
The analy ical esul s o maximum shea s ess and pola second momen o a ea ob ained
om [1, 9] a e
max 0.592


kPa,
2.24923
T
I
m4, espec i ely. Table 1 shows he compa ison o he
esul s o he maximum shea s ess, he pola second momen o a ea be ween he me hods. The
esul s o NMB-16 elemen s and Jog, C.S. e al. [9] a e he same. When he squa e c oss-sec ion is
e ined in o 256 elemen s, NMB is in good ag eemen wi h analy ical solu ions.
Tab. 1 - The maximum shea s ess
max

and he pola second momen o a ea
T
I
o squa e c oss-
sec ion
Fac o s
Analy ical solu ions
Jog, C.S. e al. [9]
NMB- 16 elemen s
NMB- 256 elemen s
()
max
a

[kPa]
0.592
0.6173
0.619174
0.601461
()b
T
I
[m4]
2.24923
2.2519
2.25187
2.24925
E o (a), (%)
-
4.273
4.590
1.598
E o (b), (%)
-
0.1187
0.1173
0.000
Figu e 4 depic s he dis ibu ion o shea s ess
xz

o squa e c oss-sec ion, which dec eases
he magni ude g adually om he midpoin o he edge o he cen e o g a i y along he y-axis [1].

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Figu e 5 and 6 p esen he a ia ion o he shea s ess
xz

along he AB and CD segmen ,
espec i ely. Wi h 16 elemen s, NMB and Jog, C.S. e al. [9] canno cap u e p ecisely he shea
s ess dis ibu ion along he CD segmen . When he mesh is ine enough, in his case, 256 elemen s,
he esul s o NMB and heo y ha monize e y well.
Fig. 3 - Disc e iza ion o he squa e c oss-
sec ion wi h 16 elemen s.
Fig. 4 - Dis ibu ed shea s esses
xz

o
squa e c oss-sec ion.
Fig. 5 - The a ia ion o shea s ess
xz

along
he AB segmen .
Fig. 6 - The a ia ion o shea s ess
xz

along
he CD segmen .
The second example is an equila e al iangle subjec ed o he o sion momen
1
T
M
kN.m
wi h he heigh o he iangle 0.2 m. The analy ical esul s o maximum shea s ess
max

and pola
second momen o a ea
T
I
a e 1.62380 MPa and 6158.40 cm4, espec i ely [1]. The iangula c oss-
sec ion was disc e ized 4, 8, 14, 37, 57, 109, 658 eigh -noded quad ila e al elemen s wi h non-
uni o m mesh o ob ain he con e gence esul s.
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Tab. 2 - The maximum shea s ess
max

and he pola second momen o a ea
T
I
o iangula
c oss- sec ion
Fac o s
Analy ical solu ion
NMB- 658 elemen s
E o , (%)
max

[MPa]
1.62380
1.6247
0.055
T
I
[cm4]
6158.40
6158.40
0.000
I is clea om Table 2 ha he esul s o he maximum shea s ess, he pola second momen
o a ea ob ained om NMB a e in good ag eemen wi h he heo e ical solu ion [1]. Figu e 7 shows
he iangula c oss-sec ion di ided by 14 eigh -noded quad ila e al elemen s. The poin s, E, F,
co espond wi h he coo dina es (0,
0.2
3
), (0.2,
0.2
3
), espec i ely.
Figu e 8 depic s he s ess dis ibu ion
xy

o iangula c oss-sec ion, whe e magni ude
alues do no exis along wi h segmen EF [1]. Figu e 9 p esen s he s ess dis ibu ion
xz

o
iangula c oss-sec ion. I can obse e om Figu e 10 ha he esul o he a ia ion o shea s ess
xz

along he EF segmen s o NMB and analy ical me hod is e y well ma ched.
Fig. 7 - Disc e iza ion o he iangula c oss-
sec ion wi h 14 elemen s.
Fig. 8 - Dis ibu ed shea s esses
xy

o
iangula c oss-sec ion.
Fig. 9 - Dis ibu ed shea s esses
xz

o
iangula c oss-sec ion.
Fig. 10 - The a ia ion o shea s ess
xz

along
he EF segmen wi h NMB-109 elemen .
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The hi d example conce ns a ba o squa e c oss-sec ion wi h hole applied o he o sion
momen
1
T
M
kN.m. Figu e 11 shows he geome ical dimensions and he meshing o 96
isopa ame ic eigh -noded quad ila e al elemen s by NMB o he squa e c oss-sec ion wi h hole. The
poin s, G, H, I, J, co espond wi h he coo dina es (0, 1), (1.6, 1), (0, 0), (0, 2), espec i ely.
Jog, C.S. e al. [9] analysed his p oblem using 96, 1536, 6144 isopa ame ic nine-noded
quad ila e al elemen s, espec i ely. Figu e 12 depic s he a ia ion o pola second momen o a ea
co esponding o 96, 1536, 6144 disc e iza ion elemen s in es iga ed by NMB. In Table 3, he
con e gence o he pola second momen o a ea alue by NMB is p esen ed as a compa ison wi h
ha ob ained om Jog, C.S. e al. [9]. Figu e 13 and 14 p esen he s ess dis ibu ion
xy

,
xz

o
squa e c oss-sec ion wi h hole, espec i ely. Figu e 15 and 16 plo he a ia ion o shea s ess
xz

along he segmen s, GH, IJ, espec i ely, e alua ed wi h he wo me hods, Jog, C.S. e al. [9], and
NMB. F om he igu es and able men ioned abo e, he eliabili y o he NMB is once mo e e i ied.
Fig. 11 - The dimension and he disc e iza ion
o 96 elemen s o he squa e c oss-sec ion
wi h hole.
Fig. 12 - The a ia ion o pola second momen
o a ea co esponding o disc e iza ion elemen s.
Tab. 3 - The pola second momen o a ea o squa e c oss-sec ion wi h hole
Fac o s
Jog, C.S. e al.- 6144 elemen s [9]
NMB- 6144 elemen s
E o , (%)
T
I
[m4]
1.707
1.70718
0.01054
A icle no. 30
THE CIVIL ENGINEERING JOURNAL 2-2021
DOI 10.14311/CEJ.2021.02.0030 417
Fig. 13 - Dis ibu ed shea s esses
xy

o he
squa e c oss-sec ion wi h hole.
Fig. 14 - Dis ibu ed shea s esses
xz

o he
squa e c oss-sec ion wi h hole.
Fig. 15 - The a ia ion o shea s ess
xz

along
he GH segmen .
Fig. 16 - The a ia ion o shea s ess
xz

along
he IJ segmen .
Acco ding o Figu e 17, he b idge c oss-sec ion is an example o mul iply connec ed domains
was conside ed he ou h example. Wi h his example, he pola second momen o a ea o he
b idge c oss-sec ions neglec ed he can ile e pa ob ained by he analy ical solu ion is 40.0 m4 [5].
A compa ison wi h a heo e ical explana ion is no possible. The compa ison be ween NMB and
FEAP [5,6], implemen ed G u mann’s nume ical me hod by using isopa ame ic ou -noded
quad ila e al elemen , was pe o med on he con e gence speed. The b idge c oss-sec ion was
di ided in o 50000 nodes wi h a uni o m mesh by FEAP o ob ain he con e gence o he esul s.
In NMB, he b idge c oss-sec ion is meshed wi h 80, 406, 1892, 4634, 11414 nodes o
in es iga e he con e gence esul . Figu e 18 shows he b idge c oss-sec ion di ided in o 80 nodes
by NMB. Figu e 19 depic s he con e gence o he pola second momen o a ea ob ained by NMB
wi h 4634 nodes.