CONCRETE LIBRARY OF JSCE NO. 34, DECEMBER 1999
A STUDY ON THE FUNDAMENTAL CHARACTERISTICS OF
LATTICE MODEL FOR REINFORCED CONCRETE BEAM ANALYSIS
(Reprinted from Journal of Materials, Concrete Structures and Pavements, No.599/V-40, August 1998)
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Tilda-Ellfl TANABE
A new modification of the lattice model is described by the authors. This new method depends
ftmdamintally on the calculation of minimum total potential energy for the structure at each calculation
increment, starting from the elastic stage up to the failure stage. The angle of inclination of the diagonals and the
appropriate discretization of the truss member are very important parameters affecting the results of the lattice
model, and these issues are studied in the paper. The applicability of the Modified Lattice Model is examined by
comparison with proposed shear strength equations and existing experimental data.
Key Words: shear-resisting mechanism, modified lattice model, arch element, total potential energy, and
subdiagonal element.
Fawzy M. BEHAIRY, is an Assistant Professor in the Civil Engineering Department, Assiut University, Assiut,
Egypt. He earned his doctoral degree from Nagoya University in March 1999. His research interests include nonlinear and post-peak analysis of RC structures, as well as the modification of the constitutive equations using
FEM. Another interest is the seismic analysis of 3D RC structures. After graduation, he served as a Research
Associate at the Technical Research Institute, Maeda Corporation, Tokyo, Japan. He is a Member of the JSCE and
JCI.
Junichiro NIWA is a professor of civil engineering at Tokyo Institute of Technology, Tokyo, Japan. He received
his doctoral degree from the University of Tokyo in 1983. He has served as secretary of the JSCE Concrete
Committee since April 1999. He specializes in the mechanics of concrete structures, including shear problems,
fractures, and non-linear FEM analysis.
Tada-aki TANABE is a professor in the department of Civil Engineering at Nagoya University, Nagoya Japan.
He obtained his Dr. Eng. from Tokyo University in 1971. His research interests include thermal stress behavior,
the characteristics of water migration, and constitutive equations for young and hardened concrete. He is a
member of the JSCE, JCI, RELEM, and IYB SE.
"—"]O9"'
1
. INTRODUCTION
It is generally agreed that the truss analogy is applicable to the analysis of shear resistance in reinforced
concrete structures. There are several different truss models that can be used to analyze the shear resisting
mechanism in reinforced concrete beams, but each model still has some problems that need to be investigated. For
example, the Lattice Model, which was first proposed by Niwa et al. [14] and extended later by the authors into
three dimensions [6], [7], has several fundamental points that require clarification
while other points need
modification. In this model, the arch member is a very important concept, because after yielding of the shear
reinforcement the model can explain the increase in shear capacity, while the simple trass model cannot,
especially in the case of deep beams. An arch element has certain important effects on shear carrying capacity
[14]. The thickness of the arch element is determined by minimizing the total potential energy for the whole
structure, but there is no physical explanation for the minimisation of total potential energy, and once the value of
arch element thickness is determined in the elastic stage, it remains unchanged throughout the loading history. The
thickness of the arch element maybe changed during the loading stages, but any change is simply neglected.
In this paper, we first clarify this
minimization at every loading stage.
issue and then demonstrate improved accuracy by performing the
Also studied is the change in thickness of the arch element with the corresponding load carrying capacity at
different loading stages. Further, a rational explanation for the strain incompatibility
through the width of a beam
caused by separating the arch memberand the truss memberwithin one beam will be explained. Experimentally,
it is found that a two-dimensional stress analyses is not adequate for reinforced concrete members [1 1]. With this
clarified, the fundamental characteristics
of the arch element mechanism for the shear resistance of reinforced
concrete membersare discussed; in particular, the strain values between the arch and diagonal elements in the
samecross section are shown to be unequal. The strains may not be uniform in the direction of member width.
The third issue to be clarified is the most appropriate direction for discretization
of subdiagonal members.The
approuch is to determine a suitable procedure for applying the Modified Lattice Model such that a similar
response to the experimental results is obtained while changing the spacing of shear reinforcement and the
subdiagonal angle. Finally, the "Modified Lattice Model" is used to simulate the shear failure of reinforced
concrete beams. The changing stress states of each memberinside the beam are investigated.
The authors attempt to give a rational mechanism or rational explanation for all previous problems.
Furthermore, based on the modified and rational model, we give some numerical calculation results that may be
useful in actual applications.
2.
OUTLINE ON THE MQDIFIE LATTICE MODEL
The chosen element discretization
and structural geometry of the Modified Lattice Model is illustrated
in
Fig.l. The reasoning behind this truss discretization
will be verified in the following sections. The reinforced
concrete beam is simulated as simple truss components under bending and shear. The compressive stress in the
upper part of the beam is resisted by concrete in the form of a horizontal strut with a cross-sectional area equal to
the area of the upper rectangle in Fig.l. The tensile stress in the lower part is born by the lower steel in the form of
horizontal reinforcing members as well as by the horizontal concrete fibers in the lower section with a crosssectional area equal to the lower rectangular area in Fig.l. To resist the shear forces inside the beam, the truss
model has diagonal concrete tension and compression memberswith the areas shownin Fig.l; these can be fixed
after the value of "t" is determined as will be shown in section 3. For the vertical members, the effect of the
concrete is not considered because the resistance of concrete to tension is already incorporated in the diagonal
tension member. This is in addition to the vertical steel members, which represent the shear reinforcement in the
web.Figure 2 shows the cross section ofa concrete beammodeled as a Modified Lattice Model.
-no-
Flexural
compression
memb
er.
Width of arch member
Width of truss
memb
er
b(l-t)/
bt b(i-ty
Flexural tension
memb
er
L/2
b
Fig. l Modified Lattice
Fig.2 Cross section of concrete beam
in the Modified Lattice Model
Model
In Fig.l, the thick solid line represents the arch element, which is assumed to be a flat, slender element
connecting the nodes at each end of the beam with the area shown in Fig.2. In this analysis, the arch element and
the diagonal elements are separated and each has its own stress and strain distribution.
The reason for this element
separation is that structural action is normally a combination of series and parallel couplings between the cracking
zones and the uncracked (elastic) zones. In the Modified Lattice Model, we simulate these zones with continuous
pairs oftension and compression members.The arch memberis considered a very important element in this study,
since it represents the core of the beam [13]. The AIJ design code [1] and many other codes [4] assume two
dimensional stress fields; but if the member section is wide enough, the stress may not be uniform in the direction
of memberwidth. It was also found experimentally by Ichinose [1 1] that the values of beam strains or stresses are
not uniform across the width in the same cross section. This means that the stress or strain diagram is not constant
in the direction of beam width [1 1]. Thus, in this model, we separate the arch element and the diagonal element,
and each is given its own stress and strain distribution.
The arch element has the ability to resist a large portion of
the applied load [13], so it is very important to look for changes in the area of the arch element at different loading
stages, as will be shown in section 3. In the Modified Lattice Model, the diagonal tension member of concrete
resists the principal tensile stress resulting from shear force. The stress-strain relation of the tension member is
assumedto be expressed as in Eq. (1) and Eq. (2) [8], [9] and as shown in Fig.3.
Forthe descendingbranch
Where
sraad
ayare
(fr
the
(1)
csr
of
£crj
strain
E
O ",
For the ascending branch (sr < ecr}
o> =(l-a)/,
and
the
stress
exp|
of
the
/?
--1
scr
tension
ecr is the strain at the time of concrete cracking, and Ec is the modulus of elasticity
+ <tft
element
(2)
respectively,
of the concrete. The stress-
strain behavior of concrete in tension is taken to be elastic, as shown in Eq. 1, and gradual softening is assumed
thereafter as shown in Eq. (2). In Eq. (2), "m"value can be varied to simulate an appropriate softening slope and
the value of a can be appropriately
assumed to simulate the appropriate residual stress [3], [10]. Here in this
-Ill-
calculation m =0.5 and a = 0.0 are taken based on the fracture
energy concept, taking the length of each member as the
characteristic length.
Or
The diagonal compression member and the arch member
must resist the diagonal compression caused by shear. To
account for the compression-softening behavior of crushed
concrete, the model proposed by Collins et al. [18] is adopted.
In this model, the softening coefficient was made a.function of
the transverse tensile strain. Thus, the tension and-compression
membersare considered as a pair together. Equation (3) shows
the compressive stress-strain relationship
of concrete in this
study. The stress-strain relationship
for reinforcing bars is
assumed to be elasto-plastic
for the case of tension and
compression members.
c-
=-nfc
Where the peak-softening coefficient
is
T;=
2
1
Fig. 3 Tensile stress-strain
curve of concrete
I
c
\
(3a)
.EnJ
1 .0
\SnJ
< 1.0
(3b)
à"34f ^
0 .8 - 0.3
U0
And the strain at peak stress s0 - -0.002.
3.
ADOPTION
OF MINIMUM TOTAL POTENTIAL
ENERGY
The effect of total potential energy during calculations with the Modified Lattice Model has a significant effect
on the final results. It has been found that there is a relation between the area of the arch element and the
corresponding total potential energy of the structure. Niwa et al. [14] showed that if the ratio of the width of the
arch element is assumed to be "t", the value of "t" is determined by minimizing the total potential energy for the
whole structure. In this work however,it is found that this thickness increases gradually during loading from the
elastic stage up to complete failure of the beam. This means that the area of the diagonal members falls gradually
as the various different loading stages are reached.
A physical explanation for the adoption of minimum total
potential energy maybe given firstly using a very simple spring
model as shown in Fig. 4. In this model, the cross-sectional area
of the spring consists of two different materials with different
moduli of elasticity,
Ej and E2, under a concentrated load P.
Assumethe stiffest portion lies in the middle of the cross section
with a stiffness value Ep The total potential energy for this
model is obtained by Eq. (4). Substituting
for the values
ofO"and S , we obtain the total potential energy as a parameter,
d
p
E2
E i E 2
(1 - t)b -2
tb
p
ependent on the area of each material part within the cross
sectionas
showninEq.
(5).
n -v'
_,. A ^
..
Fig. 4 Cross- section
1
f
ofa spring
1
2
0
/
80
' .'" " " "
" " "* ｻ .
L oad
Arch element
60
' - 10 0
Arch element
Truss element
si
-Trusselement
40 :
-200
P .B K ig y
20
©
X
p
-300
" **" *
.
;__
.- -
" .* * *
0
-400
0. 1
0.3
(s) Commonelement
0.5
0.7
0.9
"t"Values
Fig. 5 Triangular model
Fig. 6 Behaviour ofT.P. energy and
applied load for various "t" values
;r= l/2J0i£0^
n
+1/2/02^2
(4)
-^"
=
-Au2(l12\E\tll+E2(l-t)li\
(5)
From Eq. (5) the total potential energy is known to decrease monotonically with increasing area of the stiffer
portion. Therefore, the stiffer portion should occupy the whole area in order to make the potential minimum.
However,our beam element is not exactly in the same category.
So, we illustrate the real situation using the model shown in Fig.5. This model is a triangular shape under a
concentrated load P. The cross-sectional area of the sides 1 and 2 has been divided into two different materials
with two different moduli of elasticity,
EI and E2, representing the truss element and the arch element,
respectively. The ratio of the width of the arch element is assumed to be "t" to the total width of the member.
Member3 is a commonmaterial with a definite modulus of elasticity. The total potential energy of the structure is
calculated from Eq. (6).
n
(6)
=\l2\asdv-Pu
Where, u is the vertical displacement at the loaded point of the structure under the applied load "P". Take
dnl a =Qto obtain a "t" value corresponding to the minimum total potential energy and substitute it in the energy
Table 1 Outline of experimental data.
N o
C ro ss
b
S ec.
cm
5.& S -
h
cm
4 2 .5
d
a /d
fc
M Pa
A ,
fy
M Pa
A w
Iw Y
M P a
s
D
R
cm
2 0 .3
...2 .1 5
3 1 .0
cm
2 3 .1
5 30
cm
1 .4 2
5 30
cm
1 3 .3
2
R
2 0 .3
4 5 .7
3 8 .9
2 .0 0
2 4 .6
2 4 .5
3 20
1 .4 2
3 20
1 8 .3
3
R
3 0 .0
3 5 .0
3 0 .0
3 .5 0
2 3 .7
1 2 .2
4 19
0 .5 6
3 14
l l .0
4
T
3 0 .0
1 5 .0
3 5 .0
3 0 .0
3 .5 0
2 3 .7
1 2 .2
4 19
0 .5 6
3 14
l l .0
5
R
4 5 .0
6 0 .0
5 2 .5
2 .8 6
4 3 .9
9 5 .7
3 83
1 .4 3
355
2 5 .0
6
R
4 5 .0
6 0 .0
5 2 .5
2 .8 6
6 6 .2
9 5 .7
3 83
1 .4 3
355
1 5 .0
*In this table,
width=30.0cm,
R means rectangular section and T means T-shaped
flange depth=7.5cm and web width =15cm.
-773-
section.
No.4: flange
equation. Figure 6 shows the relation beween the total potential energy and applied load "P" for different values
of "t". From this figure, we find that the point corresponding to minimum total potential energy corresponds to the
maximum
applied load at a particular value of "t". This means that, if this value of "t" is used, we obtain the
stiffest beam with a minimum potential energy. So, in the Modified Lattice Model, the total potential energy by
applying Eq. (7) is calculated for different values of"t" starting from 0. 1 ~ 0.9 with a very small increment.
r
.P.£.=EM7
In Equation (7),
T.P.E. is the total potential
(7)
-M
=1
energy of the structure, whereft
is the internal
force inside each
memberof the structure, S is the displacement of each individual member, n is the number of truss members,p
is the value of the external concentrated load, and A is the displacement at the loaded point. By minimising these
values of total potential energy we can obtain the corresponding "t" value. From this value, wecan calculate the
area of the arch element and the subdiagonal elements at each step of the calculation. So, when weconsider the
total potential energy and the corresponding area of the different elements, weobtain stiffer calculated results with
the real response of the beam, which converges with the experimental results along the different loading stages, as
wewill see in the next section.
4
. APPROPRIATE
DISCRETIZATION
METHOD FOR TRUSS MEMBER
V
In this section, a clarification of the appropriate
discretization
of lattice members, which may
initially
predetermined to have a 45-degree angle,
is given. To investigate
the extent of the
discretization,
three different truss models with
different numbers of diagonal pairs along the depth
of the beam are investigated.
The three different
forms are shown in Fig.7. To determine the most
appropriate discretization
model among the three
previous forms of truss model in Fig.7, six
different
reinforced
concrete
beams are
investigated
as examples and the results of the
Modified Lattice Model are compared with the
experiment results. An outline of the experimental
data utilized in examining the Modified Lattice
Model using the three different forms of truss in
Fig.7 is presented in Table 1.
V
(b) Model (2)
The calculated
results using the Modified
Lattice Model and the normal lattice model can be
compared clearly in Fig.8 (Leonhardt' s experiment
[12];
No.4 in Table 1) and Fig.9 (Ohuchi's
experiment [16]; No.5 in Table 1). The number of
subsection
diagonal
members increases from
model (1) to model (2) and to model (3). After
cracking, the neutral axis of the beam starts to
moveupward during the development of cracks.
The height of the
cracks depends on the cross
( c)
Model (3)
Fig.7 Forms of model of simulation
-114~
0
.64
0 .48
0.3
M
odi
0 .32
S>
C
0.15
racking Load |
Mod.
Norm.
L.M.
Exp.Leonhardt
0
0.002
0.004
0 .16
L.M.
à" à"
0.006
0.004
0.008
0.008
Displacement (m)
0.012
Fig.9 Comparison with experiment (No.5)
0 .2 8
0
.65
/
0 .2 1
e a
2
ｫ
& > 0 .14
e a
T3
-ca
2 0 .0 7
/ " "x
/
^ m
0.52
'"x
¥'I:,I -.
t
¥
/*
<" '
0.39
M o d .l M oodd.l2--
0.13
M o d .2 M o d . 3-
M o d .3
.45
s
0.26
u
0
¥
/
¥ /
I
/
H
0
0.02
Displacement (m)
Fig.8 Comparison with experiment (No. 4)
0 .3
0.016
0.6
0.75
0 .3
0 .9
"t" Values
0.45
0.6
0.75
0 .9
"t" Values
Fig. 10 Comparison of "t" values for the
three models (No.2)
Fig. 1 1 Comparison oft" values for
the three models (No. 5)
section of the beam and the steel reinforcement ratio. So, in the case of model (1), if a crack occurs, it means that
the depth of the crack equals the overall depth of the beam. In this case, complete failure takes place suddenly.
However,it is not realistic because experimentally failure does not occur suddenly. In the case of model (2), if
cracks occur, it means the depth of the crack equals half of the depth of the beam and failure is not experienced
suddenly as in the previous case. This situation looks logical and is close to the experimental behavior of
reinforced concrete beams. In the case of model (3), the depth of the first crack equals 1/3 of the overall depth of
the beam. In this case, the development of cracks is not like the experimental behavior of the beam. Thus we
conclude that the numerical results are closest to the experimental results in the case of model (2), as shown in
Fig.8 and Fig.9. The distances of the vertical shear reinforcements are changed amongthe three different models;
that which has a nother effect on the outpot results. So in conclusion, model (2) is the preferable means of
implementing the Modified Lattice Model. Comparing the results for the three models, wefind the cracking load
decreases in order frommodel (1) to model (3). In the case of model (1), the elastic energy of the failed elements
is much higher than that in model (2). Also in the case of model (2) it is much higher than in the case of model
(3). The reason for this is the increasing of number of subsection diagonal members. The strain energy falls with
the decrease in the original length of the failed elements. However,the ultimate loads using these three models are
almost the same because of the similarity in fracture energy for the three different models. From these
experimental results, we can conclude that the Modified Lattice Model can capture displacement behavior
-775-
adequately and it exhibits almost the same response as the
original beam. In particular, the displacement at the peak is
closer to the experimental results than any other truss
model.
The changing thickness of the arch element is illustrated
in Fig.10 (Clark's experiment [5]) and Fig. ll (Ohuchi's
experiment [16]) for beams No.2 and No. 5 in Table 1,
respectively.
According to the Modified Lattice Model,
analysis, the thickness
of the arch element increases
gradually from the elastic stage, in which it remains
constant, until complete failure of the beam. After the peak
point, the depth of the arch element decreases due to crack
extension. Thus, the arch element thickness increases
gradually in order to maintain the effectivness of the arch
element up to the failure point. As has been discussed
above, the most appropriate truss discretization
is model
(2). This suggests that the probable arch width is around
0.4b early in loading, increasing with the load up to 0.7b.
5. APROPRIATE
SUBDIAGONAL
ANGLE
FOR THE MODIFIED LATTICE MODEL
To study the direction of initial cracking in each of the
solved beams according to the three different models in
Fig.7, three different values of subdiagonal inclination
angle in the Modified Lattice Model are suggested. These
are 51, 45, and 26 degrees, as shown in Fig.12 (a), (b), and
(c) respectively.
Figurte 13 and Fig.14 show the loaddisplacement diagram for beam No.2 [5] in Table 1 for
these subdiagonal angles and also for the different models
( c)
Subdiagonal
Fig. 12 Three different
0 angle
=26°
subdiagonal
angles
0.3
0 .25
0.24
0 .2
Mod. I/
Mod.
0.18
a
0 .15
0.12
0 .1
0.06
0 .05
0.
0
0.001
6
0.0032
Displacement
Fig. 13 Load-displacement
0=51°
and
0.0048
0.003
0.0064
(m)
diagram in case of
45° (No.2)
0.006
Displacement
Fig. 14 Load-displacement
9=26°
and
(m)
diagram in case of
45° (No.2)
116
mentioned in Fig.7. The numerical results using the Modified Lattice Model are compared with experimental
results. It is found that the results using model (2) with a diagonal angle 45 degrees are very close to the
experimental results. With any other inclination angle, the load-displacement
relationship
diverges up or down
from the experimental results. In the case of a 5 1-degree angle, the diagonal members are shorter than in the 45degree case, so the load-displacement
relatipnship
follows a lower path than the relation using a 45-degree
diagonal angle, as shown in Fig.13. But in the case of the same model using a 26-degree angle, the results follow
a higher path than in the case of a 45-degree angle, as in Fig.14. Actually, this happens because the increasing
length of the diagonal member means more elastic energy and greater structure stiffness. This behavior is
common
to all three different models, and also for each angle of inclination.
6
. EXAMINATION
OF THE APPLICABILITY
OF THE MODIFIED
LATTICE
MODEL
To examine the applicability
of the Modified Lattice Model, many different beams are calculated numerically
using the second model in Fig.7 (b). The shear strength is calculated for each beam and compared with the basic
shear strength equation.
6.1 For beams with web reinforcement
Various reinforced concrete beams with different parameters are analyzed using the Modified Lattice Model.
The shear strength of each beam is compared with Eq. (8), which is considered the basic method of calculating
shear strength in the truss analogy [17].
1 .3
p
1 .3
l af -
1 .2
w
mi n
1 .1
ｧ
2
1
1 .2
.L .M .
1 .1
M o d L .M
D
tf o a i "
0 .9
2 0 .9
t i
N o rm .L .
0 .8
0 .8
n 7
°^5
25
20
30
35
1.5
40
2.5
Pw(%)
jC (MPa)
( a)
3.5
(b) Shear capacity with pw (%)
for beams with web R.
Shear capacity with fc (MPa)
for beams with web R.
1 .3
1 .2
1 .1
a
D
tfn
H
0 .9
o d .L .
s
m
H
. No
.L .M .
0 .8
25
30
35
40
45
50
n 7
55
1
2
3
4
5
a/d
d (cm)
(c) Shear capacity with depth
for beams with web R.
(d) Shear capacity with
a/d Ratio
Fig. 15 Change of shear carrying capacity
for beams with web reinforcement
-777-
6
v
(8)
-V +V
vy~vc^¥s
Where, Vcis the shear capacity
of linear members without shear reinforcement,
is the shear capacity of the shear reinforcing
77
obtained by Eq. (9) [15],
steel and obtained by Eq. (10) [7].
Aonrl/3,.1/3^-0.25
Kc=0.20/c
1.4
^
mg,
pw rf
L
,
(9)
0.75+-^-pwt/
(10)
*s~AwJwy
Where, fc is the compressive
effective
strength of concrete (MPa),
pw is the reinforcement
depth of the concrete beam (m), a/d is the shear span-effective
reinforcement over the intervals,
and Vs
ratio (=100 4s l (bwd}, d is the
depth ratio,
fwy is the yield strength of the shear reinforcement,
Aw is the area of shear
and z =d 11.15. To examine
the applicability
of the Modified Lattice Model to beams with web reinforcement, comparisons are carried out
using Eq. (8) and also with the normal Lattice Model 14).
Figure 15 shows how the results of predicted shear capacity using the normal Lattice Model, the Modified
Lattice Model, and Eq. (8) change with some parameters, taking beam No.l [2] in Table 1 as a definite example
1.3
B S
1.2
wor"
1.1
u
*Ia
l
ST 1-2
|1.1
o d .L .M
ll
|
la
J
H H
N o
i
.L .M .
U
H
0.9
-^5
20
25
30
35
oa . l g ^ &
fl BB B
N orm r !icr
0.9
0.8
0.8
°
1.3
0.7
40
1.5
2 .5
p w (% )
/c fMPaj
( a)
(b) Shear capacity with pw (%)
Shear capacity with fc (MPa)
1 .3
1 .3
e a
1 .2
1 .2
w
ｧ
1 .1
T3
I
d .L .
l
* ｻ^
m
￨
K
B
N o tm
0 .9
.L .
0 .8
0 .7
2 5
30
35
40
4 5
o -
1 .1
o
T3
I
l
L
<D
a
0 .9
CO
J
o .8
0 .7
50
M o d .L .M .
0*
1
2
*0
TSto r
&
3
4
*
.L .M .
5
6
a /d
d (cm)
(c) Shear capacity with the depth
(d) Shear capacity with the ratio (a/dj
Fig. 16 The change of shear carrying
capacity for beams without
webreinforcement
118
-
for cross-sectional area. These parameters are the concrete strength, reinforcement ratio, effective depth, and also
the shear span-depth ratio of the beam. As seen in Fig.15, the shear capacity as calculated by the normal Lattice
Model is generally smaller than that by Eq. (8), but the predicted results using the Modified Lattice Model are
much closer to Eq. (8) and are admissible. Also from Fig.15, the tendency of the prediction results using the
Modified Lattice Model is still not exactily similar to Eq (8); hence the maximumdifference varies from0.96 to
1.08, but in the case of the normal Lattice Model the range is from 0.88 to 1.17.
6
.2 For beams without web reinforcement
To examine the applicability
of the suggested Modified Lattice Model to concrete beams without web
reinforcement, numerical calculations are performed and compared with Eq. (9) [1 5] which has been accepted as a
basis of the design equation for the JSCE. The results are compared with Eq. (9) and also with the normal Lattice
Model [14]. Figure 16 shows how the results for predicted shear capacity using the normal Lattice Model, the
Modified Lattice Model, and Eq. (9) change, with some different parameters. Taking the dimension of the crosssectional area of beam No. 6 [16] in Table 1 as an example. These parameters are concrete strength,
reinforcement ratio, effective depth, and the shear span-depth ratio. As seen from Fig.16, the shear capacity
predicted by the normal Lattice Model is smaller than that
by Eq. (9), but the variation for the Modified Lattice
Model is much smaller and is admissible.
Also from
Fig.16, the tendency of the predicted results using the
Modified Lattice Model is also still not similar to Eq. (9);
since the maximumrange varies from0.96 to 1.03, but in
the case of normal Lattice Model the range is from 0.88 to
1.1. The predicted shear failure mode by the Modified
Lattice Model is failure of the diagonal tension member,
which corresponds,
to the experimental
results.
Fig. 17 Illustrative
beam example (No.2)
Consequently, it can be concluded that the prediction of
shear capacity by the Modified Lattice Model is adequate.
7
. APPLICATION
OF THE MODIFIED
LATTICE MODEL TO SHEAR FAILURE
SIMULATI ON
To investigate
the changing stress states in each
member within a reinforced concrete beam using the
Modified Lattice Model, Clark's experiment (No.2 in
Table 1) was chosen as the subject for a solved example.
Figure 17 shows the Modified Lattice Model for
reinforced concrete beam No. 2 in Table 1. The stresses
in diagonal concrete members and stirrups and the stress
in the arch member are examined. From the results of a
simulation
of this beam using the Modified Lattice
Model, it is found that at the primary cracking stage, the
concrete elements in the bottom chord start to crack first
as shown in Fig.18 (a). Then diagonal cracking initiates
as shown in Fig.18 (b). The stirrups then begin to yield.
Although the stirrups begin yielding and the diagonal
( a)
x
Cracks in bottom members
V
-
(d) Cracks in diagonal
members
Fig. 18 The propagation
of cracks
119
tension elements have cracked, the beam still continues to carry a load up to complete failure. This is because of
the existence of the arch element, which continues to bear the load up to the end of loading with some stirrups. At
the final stage, all the stirrups yield. At that time, the arch element is immediately crushed. Fromthis simulation
of beam failure, we can categorize it as a shear failure.
According to this simulation, and considering the objectivity
of the post processing of the calculated results
and the simple representation
of the shear resisting mechanism, the Modified Lattice Model can be said to
simulate the shear failure mode very smartly. Although the Modified Lattice Model is a simulation in which the
compatibility
condition, the equilibrium condition, and the constitutive
equations used are much simplified
as
compared with normal FEM, it. ia able to capture -the shear behaviour of concrete beams reasonably well
throughout changes to the shear resisting mechanism.
8.
CONCLUSIONS
In the proposed Modified Lattice Model, a reinforced concrete beam subjected to shear force is converted into
simple truss and arch members by considering the minimum total potential energy for the structure at each step of
loading. A nonlinear incremental analysis is performed. The conclusions obtained from this research are as
follows:
1.
2.
3.
4.
5.
6.
By minimizing the total potential energy of the reinforced concrete beam, we obtain only a single value for
the thickness of the arch element, and this corresponds to the stiffest case for the structure. This is quite
similar to the original response obtained in experimental analysis.
The Modified Lattice Model tends to estimate beam stiffness closer to the experimental results. Furthermore,
the predicted displacement at the peak is almost identical to the experimental results.
The thickness of the arch member, which plays a very important role in the Modified Lattice Model,
increases gradually as the displacement of the loading point increases once diagonal cracks are initiated and
until complete failure of the beam.
The applicability
of the Modified Lattice Model has been examined for beams with and without web
reinforcement under different parametric conditions. Predictions of shear strength by the Modified Lattice
Model are very close to the basic shear strength equations as accepted by the Standard Specification
of JSCE.
Also, comparing with the experimental results, it gives satisfactory accuracy.
In the case (2) with 45-degree subdiagonal members, the position of the neutral axis is in reasonable
agreement with experimental work. So, model (2) with two pairs of subdiagonal members and with a 45degree angle of inclination is an appropriate discretization
in implementing Modified Lattice Model analysis.
Using different forms of the Modified Lattice Model, the ultimate load remains almost constant but the
cracking load decreases depending on the strain energy of the cracked element.
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2.
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4.
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