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© 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 1
NEGATIVE SHEAR LAG EFFECT IN CONTINUOUS DOUBLE T
BEAM BY 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-2021-0005
Abs ac . This pape p esen s he use o a ini e elemen
me hod (FEM) o analyze he nega i e shea lag e ec on
p es essed con inuous double T beam. Howe e , he
nume ical me hod can be applied o i) any c oss-sec ion,
ii). he mos common ypes o suppo s, such as ixed,
pinned, olle , iii) and any applied load, concen a ed, o
dis ibu ed passed h ough he shea cen e o he c oss-
sec ion. The cha ac e is ics 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 a linea
isopa ame ic elemen , is de eloped o compu e he
de lec ion, o a ion angle, bending wa ping pa ame e ,
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.
Keywo ds
Shea lag, es ained wa ping, p es essing, load, ini e
elemen me hod
1. In oduc ion
Acco ding o he elemen a y beam heo y, when he beam
elemen is unde load, he longi udinal no mal s esses a e
assumed o be p opo ional o he dis ance om he neu al
axis. Howe e , in p ac ice, hese s esses a e nonuni o mly
dis ibu ed o e he wid h o he c oss-sec ion. This
phenomenon is called shea lag.
Many au ho s ha e esea ched he shea lag
phenomenon. Reissne , E. [1] es ablished a displacemen
ield along he axis o he beam, aking in o accoun he
e ec o shea lag, which is exp essed by a pa abolic
unc ion, and used he p inciple o minimum po en ial
ene gy o ob ain he lange s ess a he c oss-sec ion.
Du ing he pas se e al yea s, many au ho s [2-15] based
on hinned wall beam heo y [16] imp o ed a spanwise
displacemen [1], which conside s he wa ping o langes,
o analyze shea lag due o lexu e. Mos s udies ocus on
concen a ed and uni o mly dis ibu ed loads. The e ec o
p es essed load on shea lag has only been pe o med in a
ew s udies [7, 11]. Chang, S. T. [7] de i ed an analy ical
me hod om [2] and [1] o calcula e he shea lag o simply
suppo ed p es essed conc e e. Zhou, S. J. [11] p oposed
a new FEM o analyze he shea lag in p es essed conc e e
box gi de s.
The p ac ical design o he beam s uc u e equi es
as e and easie pa ame e adjus men han he 3D
modelling using shell o solid. Some au ho s [17-21] ha e
ans o med he 3D analysis o he shea lag phenomenon
in o sepa a ed 2D c oss-sec ional and 1D modelling. El
Fa mi, R. [17, 18] de i ed he nonuni o m shea and
o sional wa ping o a bi a y homogeneous c oss-sec ion
om he Sain Venan beam p oblem ex ension. Le
Co ec, V. and Filippou, F. C. [19] de ined he axial
displacemen due o wa ping by in e pola ion a wa ping
deg ees o eedom numbe on he c oss-sec ion o
es ablish FEM o mula ion o shea o sional wa ping in
he elas ic and elas oplas ic analysis o beams. Fe adi,
M.K. e al. [20] ob ained FEM ha accu a ely cap u es
no mal s ess due o es ained wa ping by ep oducing
c oss-sec ional wa ping as a linea combina ion o wa ping
modes. Dika os, I. C. and Sapoun zakis, E. J. [21]
p oposed a heo y o de e mine nonuni o m wa ping due
o lexu e o composi e beam wi h a bi a y c oss-sec ion
using he Bounda y Elemen Me hod (BEM).
The pu pose o his pape is o in es iga e he nega i e
shea lag e ec due o p es essing load in con inuous
double T beam by es ablishing a nume ical me hod using
FEM, based on displacemen and s ain ields de i ed om
Dika os, I. C. and Sapoun zakis, E. J. [21]. A 2D FEM
based on he Gale kin app oach is ob ained om
compu ing he wa ping unc ion o he co esponding
ellip ic di e en ial equa ions. The o he kinema ical
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a iables o he beam a e calcula ed om he p inciple o
i ual wo k by de eloping a 1D FEM.
2. Fo mula ion he p oblem
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 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. The
coo dina e sys em is Sxyz h ough he shea cen e o he
c oss-sec ion S. CXYZ is he pa allel sys em wi h Sxyz
h ough he cen e o g a i y C.
Fig. 1: C oss-sec ion o a p isma ic beam.
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 exposed o he a bi a y
dis ibu ed o concen a ed loads, ans e se loading pz(x)
along he z-di ec ion, bending momen mY(x), and wa ping
momen ()
P
CY
mx
ϕ
along wi h he Y di ec ion. The c oss-
sec ion is assumed wi h no dis o ion.
The geome ic cons an s o he beam a e de ined as
2
2
,
,
,
,
PP
CY CY
YY
ZZ
PP
CY CY
Ad
IZd
IYd
I
d
ϕϕ
ϕϕ
Ω
Ω
Ω
Ω
=Ω
=Ω
=Ω
=Ω




(1)
whe e A is he a ea o c oss-sec ion, ,
YY
I
Z
Z
I
a e he
second momen s o a ea wi h espec o Z, Y-axis,
espec i ely,
P
P
CY CY
I
ϕϕ
is he wa ping cons an .
P
CY
ϕ
de ines he shea wa ping unc ion wi h espec o
cen e o g a i y C, which is ob ained om
,
PP
CY CY
Z
ϕφ
=−
(2)
whe e (,)
P
CY yz is de e mined om he ellip ical
di e en ial equa ions
2in
0on ,
P
PZ
CY
YY
PP
CY CY
yz n
AZ
I
nn
yz
φ
φφ

∇=− Ω



∂∂
+=Γ
∂∂
(3)
whe e P
ZZ
A
kA= is de ined as he p ima y shea a ea, kZ
is he shea co ec ion ac o ob ained om he de ini ion
gi en in [22]
In addi ion, he e alua ed wa ping unc ion
P
CY
φ
om
(3) con ains an in eg a ion cs, which can be ob ained om
G u mann, F. [23]
1.
sP
CY
cdA
A
φ
Ω
=−  (4)
The displacemen ield is exp essed as [21]
(, ,) (, ,) (, ,)
() () (,),
(, ,) 0,
(, ,) (),
PS
P
YYCY
uxyz u xyz u xyz
x
Zxyz
xyz
wxyz wx
θηϕ
=+
=+
=
=
(5)
whe e ,,u w
a e he o al displacemen s co esponding
wi h he axis x, y, z
,(,,)
PS
uuxyz
a e he p ima y and seconda y axial
displacemen , espec i ely,
()
Y
x
θ
is he angle o o a ion o he sec ion abou he Y
axis,
Y
η
is he bending wa ping pa ame e .
The s ess ield ob ained om he heo y o elas ici y
as [21]
()
()()
()()
,,
,, ,
,, ,
,
,
,
PS
xx xx
PS
xy xy
PS
xz xz
P
xx xx Y x Y x CY
PP SP
xy xy z y CY y z CY y
PP SP
xz xz z z CY z z CY z
EEZE
GGZ G
GGZ G
σσ
ττ
ττ
σεθ ηϕ
τγγ ϕ γϕ
τγγ ϕ γϕ
== +
== + +
== + +
1424314243
1444244431442 443
144424443 14243
(6)
whe e ,
P
Z
xY
w
γ
θ
=+
, ,
S
Z
YxY
w
γ
ηθ
=− −
a e de ined as
he p ima y and seconda y shea s ains, espec i ely. I is
emphasized ha he s ess componen
x
x
σ
is composed o
i) he classic no mal s ess, P
x
x
σ
, de e mined om he
enginee ing beam heo y and ii) he wa ping no mal s ess,
S
x
x
σ
, caused by wa ping o he c oss-sec ion is he p ima y
eason o he shea lag phenomenon.
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The bending momen , he wa ping momen , he
p ima y shea o ce, he seconda y shea o ce a e deno ed
,
Y
M
,
P
CY
M
ϕ
,
P
z
Q S
z
Q, espec i ely ob ained as [21]
()()
()
()
()()
()()
,
,
,,
22
,,
,,
22
,,
,
,
,
.
PPP
CY CY CY
Yxx YYYx
P
xx CY Y x
PPP
ZxyCYyxzCYz
P
ZCYyCYz
SPP
ZxyCYyxzCYz
SP P
ZCYy CYz
MZdEI
MdEI
Qd
G
Qd
Gd
ϕϕϕ
σθ
σϕ η
τφ τφ
γφ φ
τϕ τϕ
γϕ ϕ
Ω
Ω
Ω
Ω
Ω
Ω
=Ω=
=Ω=

=+Ω


=+



=+Ω


=− + Ω








(7)
3. FEM p ocedu es
3.1. De lec ion wx, o a ion angle 𝜽𝒀 and
bending wa ping pa ame e 𝜼𝒀
The p inciple o i ual wo k igno ing olume ic o ces is
used o es ablish he s i ness ma ix o he 1D beam
elemen . Assume he beam elemen includes wo end
nodes, 1, 2, The symbol (.)
δ
deno es he i ual
quan i ies. The in e nal i ual wo k is
()
,
ixxxxxyxyxzxz
V
WdV
σδε τδγ τδγ
=++
 (8)
whe e V is he olume o a p isma ic beam.
Subs i u ion he Eq. (6) and Eq. (7) o Eq. (8), he
in e nal i ual wo k can be ew i en as
()() ()
,, ,,
00
PP
CY CY
LL
iYY YxYx YxYx
WEI dxEI dx
ϕϕ
δθ θ δη η
=+ +

()()
0
L
PPP
ZZZ
GA dx
δγ γ
+

()()
0
()
L
PSS
ZZZ
GA A dx
δγ γ
−
(9)
The ex e nal i ual wo k is
()
0() P
CY
L
ezxYY Y
Span
Wpxwmmdx
ϕ
δδθδη
=+++

14444444244444443
,1
()
xyz
end node
u wd
δδδ
Ω++ Ω

14444244443 +
,2
(),
xyz
end node
u wd
δδδ
Ω++ Ω

14444244443
(10)
whe e x, y, z a e he componen s o ac ion ec o applied
on he la e al su ace o he beam which is ela ed o he
end nodes- ex e nal loads as [21]
()
ˆˆ
() , () ,
ˆ,
P
CY
zzYx
P
xCY
px d mx Zd
m d
ϕ
ϕ
ΩΩ
Ω
=Ω = Ω
=Ω

 (11)
Using he exp ession Eq. (11), he Eq. (10) can be
ep esen ed as
e
W=000
() P
Cy
LLL
zx YY Y
pxwdx m dx m dx
ϕ
δδθδη
++ +

222
111
ˆˆˆ
.
P
CY
zi xi Yi Yi Yi
i
iii
pw m m
ϕ
δδθδη
===
++
 (12)
To de i e he elemen s i ness ma ix, he a iables wx,
𝜃, and 𝜂 need o be in e pola ed wi hin each elemen . wx,
𝜃 and 𝜂 a e independen a iables. As a esul , any kind
o 𝐻 shape unc ion can be used o he p esen beam. We
use 1D linea isopa ame ic shape unc ion o bo h
a iables [24, 25]. Tha is,
[]
[]
[]
1
12
2
1
12
2
1
12
2
() () ,
() () ,
() () ,
x
x
x
Y
Y
Y
Y
Y
Y
w
wN N w
NN
NN
ξξ
θ
θξξ
θ
η
ηξξ
η

=


=


=

(13)
whe e 1,
x
w 2,
x
w 1,
Y
θ
2,
Y
θ
Y1,
η
Y2
η
a e he nodal
displacemen s, o a ion angle, bending wa ping pa ame e
a he beam end nodes (1) and (2), espec i ely
()
1
11,
2
N
ξ
=−
()
2
11.
2
N
ξ
=+ (14)
The co esponding elemen nodal deg ees o eedom
is 11Y1 2 2Y2
{,,,,,}.
T
xY x Y
ww
θη θη
=d (15)
Subs i u ing Eq. (13-14) o Eq. (9) and Eq. (12) leads
o he elemen s i ness ma ix and elemen o ce ec o as
1234
,
eeeee
=+++KKKKK
(16)
2
123 ()()()
,
zY P
CY
e eee e e
ip im im
ϕ
=+++ + +F FFFF F F (17)
whe e
1
e=K1
11
1
T
YY
E
IJd
ξ
−
BB
() ()
11
,
T
YY i i i
i
E
Iw J
ξξ
=BB (18)
() ()
1
222
1
22
,
PP
CY CY
PP
CY CY
eT
T
ii i
i
EI Jd
EI w J
ϕϕ
ϕϕ
ξ
ξξ
−
=
=


KBB
BB (19)
() ()
1
333
1
33
,
ePT
Z
PT
Zi i i
i
GA Jd
GA w J
ξ
ξξ
−
=
=


KBB
BB
(20)
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() ()
1
444
1
44
()
() ,
ePT
Z
PT
Zii i
i
GA A Jd
GA A w J
ξ
ξξ
−
=−
=−


KBB
BB
(21)
1
11
1
eT
Z
pJd
ξ
−
=
FN
()
1,
T
iiZ
i
wpJ
ξ
=N (22)
1
22
1
eT
Y
mJd
ξ
−
=
FN
()
2,
T
iiY
i
wmJ
ξ
=N (23)
1
33
1P
CY
eT
mJd
ϕ
ξ
−
=
FN
()
3,
P
CY
T
ii
i
wmJ
ϕ
ξ
=N (24)
()
,
z
e
ip
F()
,
Y
e
im
F()
P
CY
e
im
ϕ
F a e he ans e se o ces, bending
momen s, wa ping momen s a he end nodes i,
espec i ely,
,
2
L
J= (25)
[]
[]
[]
11, 2,
21,2,
31,1 2,2
41,112,22
11 2
21 2
312
0000,
00 00 ,
00,
,
00 00,
0000,
00 00 ,
xx
xx
xx
xx
NN
NN
NN NN
NNNNNN
NN
NN
NN

=

=

=

=− − −

=
=
=
B
B
B
B
N
N
N
(26)
,
i
wi
ξ
a e he weigh s and he coo dina e o in eg a ion
poin s o he Gaussian in eg a ion echnique. In he p esen
s udy, o a oid he shea locking, we use one-poin Gauss
quad a u e (2,
i
w=0)
i
ξ
=.
Assembling he elemen s i ness ma ix and load
ec o s in he sys em ma ix equa ion gi en below
..=Kd F (27)
3.2. Wa ping unc ion,
P
CY
φ
,
P
CY
ϕ
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 ion (3) is ans o med o weak o m as
0.
PP P
CY CY Z
YY
A
dd
yy zz I
φφ
ηη η
ΩΩ


∂∂
∂∂
+Ω−Ω=




∂∂ ∂∂ 


(28)
The wa ping unc ion alue,
P
CY
φ
, in Eq. (28) is
app oxima ed by he FEM, which is p esen ed in [22].
Finally, he alue
P
CY
ϕ
is calcula ed om Eq. (2).
4. Valida ion example
In his sec ion, a compu e code is de eloped in he
MATLAB R2015 based on he o mula ions desc ibed in
he p e ious sec ions. A p es essed con inuous beam wi h
a double T c oss-sec ion was analyzed using his code. The
beam's geome ical cha ac e is ics, he bounda y and
loading condi ions a e shown in Fig. 2.
Poin s B1, B2, and B3, a e on he uppe pla e and ha e
he y coo dina es -1.55 m, -0.825 m, and 0, espec i ely.
The s aigh endons a e posi ioned uni o mly a he cen e
o he slab. On he plan, endons a e a anged a he in e nal
suppo wi h a o al leng h o 2.8 m. Assume he e a e six
endons, each o which is s essed wi h a o ce o 140 kN.
Modulus o elas ici y E= 30 x 103 MPa, and he Poisson
a io 𝜈= 0.2.
Fig. 2: Con inuous beam, he load, and he geome y o he c oss-sec ion, uni s [mm].
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Fig. 3: The dis ibu ion o he classic no mal s ess in c oss-sec ion a x=14 m, uni s [kPa].
Fig. 4: The dis ibu ion o he wa ping no mal s ess in c oss-sec ion a x=14 m, uni s [kPa].
Fig. 5: The dis ibu ion o he o al no mal s ess 𝜎 in c oss-sec ion a x=14 m, uni s [kPa].
This example was analyzed by employing 840 axial
elemen s and 126 elemen s in he c oss-sec ion. The
geome ic cons an o he c oss-sec ion de e mined om
he p esen s udy a e as A= 0.287 m2,
Z
k= 0.2606, YY
I=
0.00977 m4,
P
P
CY CY
I
ϕϕ
= 0.00248 m4.
The p es essed load applied o he beam is
ans o med in o wo equi alen loads, Px= 840 kN, and
My= 115.9 kNm, shown in Fig 2. ETABS 2018 [26] is used
o simula e 3D model included beam, shell, and endon
elemen s. The shell and beam elemen s a e di ided in o
9856, 700 elemen s, espec i ely.
Fig. 3, Fig. 4, Fig.5 show he con ou plo o he classic
s ess, he wa ping no mal s ess, and he o al no mal
s ess 𝜎 on he c oss-sec ion p edic ed by he p esen
s udy a he posi ions x= 14m, espec i ely. In Fig. 6, he
a ia ion o he o al no mal s ess 𝜎 along he wid h o
he uppe lange o he c oss-sec ion a he posi ion x= 14
m is shown and compa ed be ween he esul om i)
Enginee ing beam heo y, ii) 3D simula ion ETABS 2018.
In Table 1, he no mal s ess
x
x
σ
in he poin s B1, B2,
and B3 p edic ed by he p esen s udy a e gi en and
compa ed wi h he esul s om 3D simula ion. F om he
abo e Figu es and Tables, he in luence o he shea lag
phenomenon is appa en , and he p esen s udy's accu acy
can be e i ied.
Fig. 6: The a ia ion o he o al no mal s ess 𝜎 along he wid h in
he uppe lange a in e nal suppo .
Tab.1: Compa ison o he no mal s ess 𝜎 [MPa] a he poin s B1, B2,
B3
Me hods Poin B1 Poin B2 Poin B3
P esen s udy 3.9924 3.9487 4.0066
ETABS 2018 [26] 3.96 3.93 4.025
E o (%) 0.818 0.475 0.457

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© 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 6
5. Conclusions
In his pape , FEM is de eloped o analyze he nega i e
shea lag e ec due o p es essing in con inuous double T
beam. Howe e , he nume ical me hod can be used o
a bi a y c oss-sec ions wi h mos bounda y condi ions
and load ypes. A h ee-span con inuous beam subjec ed o
he p es essing load a he in e nal suppo was analyzed
and compa ed wi h he esul s om 3D simula ion
so wa e. I was obse ed ha he p esen s udy could
p edic he nega i e shea lag phenomenon accu a ely due
o p es essing load.
Acknowledgemen 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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