Engineering, 2009, 2, 75-90
doi:10.4236/eng.2009.12009 Published Online August 2009 (http://www.SciRP.org/journal/eng/).
A Non-Linear 3D FEM to Simulate Un-Bonded Steel
Reinforcement Bars under Axial and Bending Loads
Rami HAWILEH1, Adeeb RAHMAN2, Habib TABATABAI3
1Department of Civil Engineering, American University of Sharjah, Sharjah,United Arab Emirates.
2Department of Civil Engineering & Mechanics, University of Wisconsin-Milwaukee, Milwaukee, WI, USA.
3Department of Civil Engineering and Mechanics, University of Wisconsin-Milwaukee, Milwaukee, WI, USA.
Received May 21, 2009; revised July 1, 2009; accepted July 10, 2009
Abstract
This paper presents development of 3D non-linear finite element model to simulate the response and predict
the behavior of un-bonded mild steel bars under axial and bending loading. The models were successfully
analyzed with the finite element software ANSYS, taking into account the nonlinear material properties of
the reinforced mild steel bars. A bending strain relationship is derived based on a parametric study involving
multiple nonlinear finite element models. A mild steel fracture criterion based on low-cycle fatigue models is
proposed to control the total (elastic and plastic) strains in the mild steel bar below a maximum permissible
limit. In addition, FE predictions of bar elongation due to strain penetration reasonably agreed with a pro-
posed empirical equation by Raynor and Lehman. It was concluded that the equation proposed by Raynor
and Lehman is considered valid for estimating the additional unbounded length and can be used in both
analysis and design.
Keywords: Finite Element, Combined Axial and Bending Loading, Steel Rebar, Precast Hybrid Frame.
1. Introduction
In the Precast Seismic Structural Systems (PRESSS)
research [1], a total of five different seismic structural
systems made from precast concrete elements were pro-
posed. These systems formed various parts of the struc-
tural framing in the PRESSS Phase III experimental
building that was tested at the University of California at
San Diego.
The structural system identified as the unbonded
post-tensioned frame with damping reinforced mild steel
bars (hybrid frame) performed very well in the PRESSS
evaluation. Hawileh et al. [2] developed a nondimen-
sional design procedure for this type of hybrid connec-
tions. The hybrid frames contain precast elements
(beams and columns) that are connected by unbonded
post-tensioning steel and partially debonded reinforce-
ment bars, both of which contribute to the overall mo-
ment resistance. An important feature of the connection
between beam and column is the hybrid combination of
mild steel and post-tensioning steel where the mild steel
is used to dissipate energy by yielding in tension and
compression and the post-tensioning steel is used to
clamp the beam against the column. The post-tensioning
force would act as a clamping/restoring force to bring the
frame back to its original configuration after an earth-
quake, and would provide for shear resistance through
friction developed at the beam-column interface.
Hawileh et al. [3] developed a 3-D nonlinear finite
element (FE) model of a hybrid frame system that pre-
dicted the experimental results of Cheok and Stone [4].
From the results of the analyses, it was determined that
the mild steel bars in a hybrid frame exhibit significant
inelastic axial and bending strains. Once the gap at the

R. HAWILEH ET AL
76
d
b
/ 2
dy
dx
dS
O
d

R
B'
B
A
'
Figure 1. Deformed segment of the mild steel bar.
beam-column interface opens, relatively high levels of
repetitive plastic strains develop in the mild steel bar.
This prompted a need for investigating the behavior and
performance of reinforcing steel bars when subjected to
high amplitude levels of plastic strains, considering both
bending and axial strains, respectively. The relatively
high inelastic strains in the finite element model show
the potential vulnerability of reinforced mild steel bars to
low-cycle fatigue failure. It should be noted that both the
PRESSS [1] and National Institute of Standards and
Technology (NIST) [4] test reports indicate bar fractures
during cyclic testing.
Mander and Panthaki [5] studied the behavior of rein-
forcing steel bars under low-cycle fatigue subjected to
axial-strain reversals with strain amplitudes ranging from
yield to 6%. The low-cycle fatigue behavior of steel bars
subjected to bending strain reversals with variable am-
plitudes were studied by Liu [6]. In addition, a mild steel
fracture criterion under combined axial and bending
strains is proposed in this paper.
2. Bending Strain Calculations for Bars
Liu [6] presented the following procedure for calculating
strain-displacement relationships for bars subjected to
bending when the material is stressed beyond the elastic
range. Consider the deformed bar segment shown in
Figure 1.
Assuming that the cross section of bar is symmetric
(i.e. elastic and plastic neutral axes are at the center of
the bar), the bending strain and the radius of curvature
can be calculated as follows:

















R
d
dR
Rdd
d
R
dsBA b
b
b
1
2
2
ds 

 (1)
where, R = radius of curvature, d = angle of rotation, b
= maximum bending strain, and db = bar diameter.
However,

2
22222 2
11
dy
dsdx dydxdxy
dx



 




 (2)
21/2
(1 )ds dxy  (3)
but,
tan
dy
ydx


(4)
Differentiate both sides of the above equation
2
2
2sec
dy d
ydx
dx


  (5)



2
1tan d  ydx


 (6)
2
1tan 1
ydx y
d
y

2
d
x


 

(7)
knowing that,s = R

dd, and therefore
ds
d
R


1 (8)
By inserting d and ds values from Equations 4 to 7,
Equation 9 yields:

2
3
2
2
1
2
2
1
)1(
1
1
y
y
ydx
dx
y
y
R









 (9)
3. Axial Strain Calculations for Bars
In Figure 2, the solid line CD is the initial unbonded
length of the reinforcing bar in a hybrid frame. The
dashed line DE shows the path that the end D of the un-
bonded segment of the mild steel bar would take as it
moves from D to E.
Assuming that the center of rotation “O” for the open-
ing of the joint at the beam-column interface is at the
neutral axis of the beam when the hybrid frame is sub-
jected to a design interface rotation of des, the following
can be written using the same assumptions as in the
PRESSS program (See Figure 2 and Figure 3).
where,
R = (1--des)hg (10)

R. HAWILEH ET AL
77
Lsu
DUCT GROUT
y
xCONCRETE
C
E
D
''
'''
CD
E
O

des
Lsu X
Y
R = OA
''+ 
des
/2
Figure 2. Path of point D of the unbonded segment of the mild steel bar.
Figure 3. Location of the center of rotation at des [1]. Figure 4. Isoparametric view of the entire model.

R. HAWILEH ET AL
78
des = distance from the compression face of the beam to
the neutral axis at  des divided by the height of the beam
distance from the center of the mild steel bar to the near-
est face divided by the height of the beam hg = beam-
height  des = maximum (design) interface rotation Lsu =
unbonded length of the mild steel bar at each interface
Let:
a,a
s
u
s
L


 (1 1)
where s is the horizontal gap opening at the location of
the tension bar at the beam-column interface (Figure 3)
and a,a is the axial bar strain assuming that there is no
vertical movement at the end of the bar as a result of
rotation.
des des
(1)
gdes
sR h

   (12)
des
a,a a,a
(1 )
s
g
des
su
h
L




  (13)
and,
,
tan
s
udes
aa
L
R


  (14)


22
1
22
2
,
,
(1 )
1
des g su
des g
desa a
aa
ODh L
h

 

  



(15)
However,
cos2
des
XDE




 




(16)
des DE OD

 (17)


,
1
22
2
,
1
cos 2
desgdes
aa
des
desa a
h
X



 










(18)
and,
sin2
des
YDE



  




(19)


,
1
22
2
,
1
sin 2
desgdes
aa
des
desa a
h
Y
















(20)
tan
su
Y
Ls


  (21)


1
22
2
,
1
22
2
,
sin 2
tan
1cos
2
des
desa a
des
desa a

 
 








 
 


 


(22)
The axial strain in the reinforcing bar (axial) can be
written as:
22
()
s
u
axial
su
LX YL
L


su (23)

1
22
2
,
12cos 2
des
axialdesaa


 










1
2
22
,1 desaa




(24)
4. The FE Model
As discussed in the previous Section of this paper, the
deformed geometry of the bar is needed to allow calcula-
tion of bending strains. Therefore, it is important to de-
termine the inelastically deformed shape of the reinforc-
ing bar at des. A FE model using ANSYS [7] is devel-
oped to predict the inelastically deformed shape of the
bar. Once an equation for the deflected shape of the bar
is derived, the bending and axial strains of the bar can be
calculated based on the derived equations of the previous
Sections.
In addition, a parametric study involving multiple
nonlinear FE models using the was performed to inves-
tigate and simulate the combined axial and bending be-
havior of the mild steel bar for six different cases listed
in Table 1.
The undeformed geometry of the mild steel bar at the
beam-column interface is modeled as shown in Figure 4.
The dimensions of the model are listed in Table 2.
Figure 5. BEAM188 3-D linear finite strain beam [7].

R. HAWILEH ET AL
79
Table 1. Study cases.
No. (1-des-)h  des Lsu (inches)٭'' (rad.) X (inches)٭ Y (inches)٭ X/Lsu
1 20 0.02 10 0.4636 0.398 0.204 0.0398
2 20 0.04 20 0.7854 0.784 0.816 0.0392
3 30 0.02 15 0.4636 0.597 0.306 0.0398
4 30 0.04 30 0.7854 1.176 1.224 0.0392
5 20 0.01 5 0.245 0.200 0.051 0.04
6 20 0.02 20 0.7854 0.396 0.404 0.0198
٭Note: 1 inch = 25.4 mm.
Table 2. Model dimensions.
Material Width (in)٭ Height (in)٭ Diameter (in)٭ Area (in2)٭
Mild Steel Bar _ _ 1 0.785
Grout _ _ 3 6.281
Concrete Block 7 7 _ 41.932
٭ Note: 1 inch = 25.4 mm; 1 sq in. = 0.000645 m2.
The partially unbonded bar and two concrete blocks rep-
resenting parts of the beam and column are modeled. The
grout around the bar in the bonded portion is also mod
eled. The length of the concrete block is assumed to be
20 inches (508 cm) on each side of the unbounded length
of the bar to provide sufficient bar bonded length for all
study cases.
BEAM188 [7] element is used to model the entire
model. The geometry, node locations, and the coordinate
system for the ANSYS element BEAM 188 are shown in
Figure 5. The cross section details of the beam elements
are provided separately. The cross sectional dimensions
and properties of the mild steel bar, duct grout, and con-
crete are shown in Figures 6 through 8
4.1. Material Properties
Components of this structure consist of the following-
materials: Concrete, Grout, and Grade 60 Reinforcing
bar Grade 60. In order to simulate and analyze this struc-
ture, exact material properties and relevant coefficients
should be defined properly. The concrete compressive
strength f′c is 7400 psi (51 MPa) and the modulus of
elasticity for all the materials used in the model is listed
in Table 3. The stress-strain curves for the mild steel bar
and reinforced concrete are displayed in Figure 9 and
Figure 10. Other defined coefficients include:
Density (  ): The densities of reinforced mild steel and
concrete were assumed to be 490 lb/ft3 (7850 kg/m3) and
150 lb/ft3 (2400 kg/m3), respectively.
Poisson’s Ratio (  ): The Poisson’s Ratio () was as-
sumed to be 0.3 for steel and 0.2 for concrete
4.2. Boundary Conditions
The entire concrete block nodes on the left side of the
bar’s unbonded length were restrained (fixed) in all 6
directions (Figure 4). All of the concrete block nodes at
the right end of the unbonded mild steel bar were simul-
taneously deformed to achieve the following concurrent
displacements and rotation: A horizontal displacement
X; a vertical displacement Y; and a rotation des.

R. HAWILEH ET AL
80
Figure 6. Mild steel bar cross section.
Figure 7. Duct grout cross section.

R. HAWILEH ET AL
81
Figure 8. Concrete cross section.
#8, Grade 60 Reinforcing Mild Steel Bar
0
20
40
60
80
100
120
00.02 0.04 0.06 0.080.1
Strain (in/in)
Stress (ksi)
Figure 9. Stress-strain curve of grade 60 mild steel bar [3].

R. HAWILEH ET AL
82
Stress-Strain Model of unconfined concrete for f `
c
= 7400 psi
0
1000
2000
3000
4000
5000
6000
7000
8000
00.00050.001 0.0015 0.0020.0025 0.003
Strain (in/in)
Stress (psi)
Series 1
Figure 10. Stress-strain model of concrete [3].
Figure 11. Deflected shape of the model.

R. HAWILEH ET AL
83
Table 3. Material properties.
Material # Material
Name
Modulus of Elasticity
E
s
(
ksi
)
٭
1 Concrete 4900
3 Mild Steel 29,000
5 Duct Grout 3500
٭Note: 1 ksi = 6.895 MPa.
The grout and reinforcing bar elements were not re-
strained in either block. The values of these applied dis-
placements for the six different cases are listed in Table
1. The relative magnitudes of these displacements are
consistent with the deformations in precast/prestressed
hybrid frames.
5. Results
The FE model provides full fields of stress and strain
throughout the model. Figure 11 shows the deflected
shape of the entire model. The total (elastic + plastic)
axial strains and for the reinforced mild steel bar, grout,
and concrete for the first study case are shown in Figures
12 to 14. The results for the bar axial and bending strains
for the six study cases are tabulated in Table 4.
The above figures show the development of plastic
strains in the mild steel bar with its highest value at the
fixed end of the bar as shown in Figure 12. The vertical
deformation across the unbonded segment of the bar is
plotted in Figure 15 for the first study case and in Figure
16 for all the six model study cases.
In all six model cases, the vertical deformation data
for the bars were fitted into a 3rd order polynomial. The r2
value of this regression analysis is 1.0 in all cases, which
means that the regression results provide perfect fit for
the finite element results. The equations for the vertical
deflection, slope, second derivative, curvature, and
bending strain take the following form:
y = ax3 + bx2 + cx + d (25)
y = 3ax2 + 2bx + c (26)
y = 6ax + 2b (27)

3
22
1
1
y
Ry






3
22
2
16 +2
13 2
ax b
R
ax bx









(28)
A general equation for the bending strain in the bar
can be determined using constants a, b, c, and d based on
the prescribed boundary conditions as follows:
Boundary Conditions:
At x = 0
y = 0  d = 0
y = 0  c = 0
At x = Lsu
y = y
y = des

3
2
des su
su
L
aL
 y


 (29)
2
3dessu
su
yL
bL
 
 (30)
Using y and Lsu values from Equations 14 to 21,

1
22
2
,
2
,
2()sin()
2
1
des
desdesaa
desgdes
aa
a
h

 





 









(31)

1
22
2
,
,
3() sin()
2
1
des
desa ades
desg des
aa
bh

 













(32)
According to the FE results the values of both the
curvature and bending strain are maximum at the fixed
end of the bar (at x =0).
Therefore, Substituting Equations 27 and 28 in Equa-
tion 29 and considering that x = c = d = 0, yields:
max
12b
R
 

 (33)
,max
max
1
2
b
b
dbd
R


 

 b
(34)
where, b,max is the maximum bending strain along the
length of the bar. Substituting for b from Equation 33,

R. HAWILEH ET AL
84
Figure 12. Axial total strain distribution of unbonded mild steel bar.
Figure 13. Axial total strain distribution for concrete block.

R. HAWILEH ET AL
85
Figure 14. Axial total strain distribution for grout.
Vertical Displacement vs x
y = -0.0002x
3
+ 0.0035x
2
+ 0.0001x - 5E-05
R
2
= 1
-0.050
0.000
0.050
0.100
0.150
0.200
024681012
x (in)
y (in)
Case 1
多项式 (Case 1)
Figure 15. Vertical deformation of the unbonded segment of the bar.

1
22
2
,
,max
,
3() sin()
2
1
des
desa ades
b
b
g
des des
aa
d
h

 











 



(35)
1
22
2
,max ,
3() sin()
2
des b
bdesaa des
su
d
L

 

 


 

 

 



(36)
Table 5 compares the predicted a, b, and b,max values
with the finite element analysis results. It can be noticed
that from Table 5 that the predicted values for b,max
(Equation 37) is slightly greater than those of the FE
or,

R. HAWILEH ET AL
86
results for all model cases. This is because the grout is
modeled in the finite element model which will results in
reducing the bending strain in the bar. The grout effect is
not included in the predicted equations (Equations 32, 33,
and 37).A more simplified equation for calculating the
bar’s bending strain (b,max) at des can be estimated from
Equation 36 as follows.
,max ,
2.4 b
baa
g
d
h





(37)
6. Low-Cycle Fatigue Life Relationships for
Mild Steel Bars
High-cycle fatigue and low-cycle fatigue fractures are
two different modes of damage in members subjected to
cyclic loading. Generally, low-cycle fatigue lives range
from one up to 105 cycles and high-cycle fatigue cycle
lives are greater than 105 cycles [8].
Manson and Coffin [9] proposed the following empirical
equation to estimate the general fatigue life of a material:
(2 )(2 )
2
fb
fff
N
E





c
N
(38)
where,
ε
Δ = total strain range (max

-min

)
f
σ = fatigue strength coefficient
f
N
= number of cycles to failure
E = modulus of elasticity
b = fatigue strength exponent (ranges from -0.05 to
-0.15)
f
ε = fatigue ductility coefficient
c = fatigue ductility exponent (ranges from -0.5 to -0.8)
The first term in the above equation represents the
elastic strain component (high-cycle fatigue) and the
second term represents the plastic strain component
(low-cycle fatigue).
The mild steel bar in hybrid frames is subjected to
large inelastic deformations. Koh and stephens [10]
found that for most low-cycle fatigue analyses the elastic
part can be neglected, for which Equation 39 can be sim-
plified.
(2 )
2
c
ff
N
 
 (39)
Mander [11] experimentally evaluated the low-cycle
fatigue behavior of reinforcing steel bars subjected to
cyclic axial strain amplitudes ranging from yield to 6%.
He evaluated the experimental results with existing
low-cycle fatigue models found in the literature. The
experimental data were fit to existing fatigue equations.
As a result, low-cycle fatigue life relationships were de-
veloped for reinforcing steel bars. The following fatigue
life relationships are all based on the work of Mander
[11].
The relationship between plastic-strain amplitude (ap)
and low cycle fatigue life for axial deformations of steel
bars is as follows:
0.5
= 0.08(2)
2
p
ap f
N





(40)
where,
p,max p,min
-
plastic strain amplitude =
2
ap


 (41)
pp
= range of plastic strain = -
,maxp,min



(42)
,max
,min
= maximum plastic strain in a cycle
= minimum plastic strain in a cycle
= number of cycles to failure
p
p
f
N


The total axial strain amplitude of the mild steel de-
formed reinforcement subjected to strain cycles ranging
from zero to s,max can also be calculated using the above
low cycle fatigue equations as follows [11]:
0.448
0.0795(2 )
2
a
N




 f
(43)
where,
s,max s,min
-
2
a


 (44)
,max = maximum total strain in a cycle s

,min= 0 s

s,max 2a

  (45)
Although many of the tests performed by Mander [11]
were on ASTM A615 reinforcing bars, the author rec-
ommends the above equation for all steel types. Addi-
tional tests on ASTM A706 bars are being performed by
the writers of this paper [12].
In a separate study by Chin Liu [6], mild steel bar
specimens were subjected to bending reversals and
low-cycle fatigue life relationships were developed. The
results of constant displacement amplitude tests were
plotted and a best fit relationship of the following form
was derived:
0.32
0.23(2 )
2
b
bp f
N





(46)
where,
bp
b,max b,min
= plastic bending strain amplitude
-
= 2


(47)

R. HAWILEH ET AL
87
Vertical Displacement vs x
0.000
0.200
0.400
0.600
0.800
1.000
1.200
1.400
0510 15 20 25 30 35
x (in)
y (in)
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Figure 16. Vertical deformation of the unbonded segment of the bar.
Plastic Strain vs 2N
f
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0510 15 20 25
2N
f
Plastic strain Amplitude
(
ap
or 
ap
)
axial
bending

ap

bp
Figure 17. Plastic strain-life relationship.
b
b,maxb,min
= range of bending plastic strain = -

 (48)
,max
,min
= maximum bending plastic strain in a cycle
= minimum bending plastic strain in a cycle
= number of cycles to failure
b
b
f
N


Figure 17 displays the plastic strain-life relationships
under both axial and bending loadings.
It can be noticed from Figure 17 that for a given num-
ber of cycles to failure, low cycle bending fatigue can
achieve higher maximum strain amplitude than low cycle
axial fatigue. In other words, for the same maximum
strain amplitude in the bar, specimens subjected to cyclic
bending strains would have longer life than pure axial
strains. This difference is expected to occur because un-
der bending, only the extreme fibers of the cross section
develop maximum strain.
The low-cycle fatigue equations under bending action
can be converted to equivalent fatigue equations under

R. HAWILEH ET AL
88
Table 4. FEA results.
Case
Study
No.
Average
Axial
Strain
avg
Maximum
Bending
Strain
b
total
total =
avg + b
(b / avg)
(%)
1 0.0398 0.0035 0.0433 8.8
2 0.0392 0.0038 0.0430 9.7
3 0.0398 0.0025 0.0423 6.2
4 0.0392 0.0026 0.0418 6.7
5 0.0400 0.0029 0.0429 7.2
6 0.0198 0.0019 0.0217 9.5
axial loading [6]. If the number of cycles to failure is
assumed to be the same for both the axial and bending
equations, the effective strain amplitude under axial
loading can be calculated as follows:

1
2
2
c
b
bfb f
N
 
 (49)

2
2
2
c
a
afa f
N



 (50)

11
12cc
ba
fb fa









(51)

2
1
c
c
b
afa
fb

 

 



(52)
where, b = strain amplitude under cyclic bending
a = effective strain amplitude under cyclic axial load-
ing
c1 = fatigue ductility exponent under bending
c2 = fatigue ductility exponent under axial loading
Based on the above, the effective strain amplitude un-
der bending for the mild steel bar is:
1.4
0.63( )
2a

b


 (53)
and,
1.4
,max21.26() aa b
 
  (54)
The effective axial strain amplitude and the correspond-
ing maximum effective axial strain for the six FE model
cases are listed in Table 6.
It can be noticed from Table 6 that the value of the ef-
fective axial strain amplitude (a) is much smaller than
the axial strain in the bar. However, the bars in the study
by Liu [6] were subjected to bending reversals only
without subjecting the bar to plastic axial strain
throughout the section. Equation 53 is not directly appli-
cable to bars in hybrid frames because the additional
axial strain due to bending in hybrid frames is superim-
posed on a fully plastic section from the axial loading
effects.
This issue requires an experimental evaluation pro-
gram to address the combined axial-bending strain ef-
fects on low-cycle fatigue of bars. In lieu of this experi-
mental evaluation, the proposed conservative approach is
to find the maximum total strain variation in the mild
steel bar by adding the maximum bending strain with the
axial strain as follows [8]:
totalaxial b



 (55)
The total can be conservatively used to estimate
low-cycle fatigue life using Mander’s recommended
equation (Equation 44) for the axial strain plus the pro-
posed equation by Lieu for the bending strain (Equation
50).
Table 5. Predicted results compared with FEA results.
a b b,max Model
Case
No.
FE
10-5
Calculated
10-5
FE
10-3
Calculated
10-3
FE
10-3
Calculated
10-3
Calculted/FE
(%)
1 -20 -20.8 3.5 4.12 3.5 4.12 18
2 -9 -10.4 3.8 4.12 3.8 4.12 8
3 -8 -9.24 2.5 2.75 2.5 2.75 10
4 -4 -4.62 2.6 2.75 2.6 2.75 6
5 -30 -41.6 4.3 4.12 4.3 4.12 4
6 -5 -5.1 1.9 2 1.9 2 5

R. HAWILEH ET AL
89
Table 6. Effective axial strain calculations.
Case Study
No. x/Lsu b a
1 0.0398 0.0035 0.00046
2 0.0392 0.0038 0.00052
3 0.0398 0.0025 0.00028
4 0.0392 0.0026 0.00030
5 0.0399 0.0029 0.00035
6 0.0198 0.0019 0.00019
7. Effect of Grout in Duct
Raynor and Lehman [13] studied the bond characteristics
of bars grouted in light-gauge metal ducts. Growth in the
unbonded length of the bar was noticed under high cyclic
strain. This leads to a non-uniform strain distribution at
each end of the unbonded length of the bar as shown in
Figure 12. Raynor and Lehman [13] represented the bar
elongation due to strain penetration by an equivalent
additional unbonded length Lua as follows:
1.5
0.81( )
()
s
usy
ua
bg
f
f
L
df

 (56)
where,
fsu = steel bar ultimate strength
fsy = steel bar yield strength (ksi)
fg = grout strength (ksi)
Thus the average axial strain would be equal to:
2
avg
s
uu
L
LL



where,

22
s
us
LLX YL
u


Table 7 compares the above equation that calculates the
average strain in the bar due to strain penetration with
the FE results for the six study cases. The assumed grout
strength and the yield strength of the bar are as follows:
f’g = 8 ksi (55 MPa)
fsy = 60.9 ksi (420 MPa)
fsu = 85 ksi (586 MPa)
db = 1 in. (25.4 mm)
mm) (21.84 in. 86.0
)8(
)1()9.6085( 81.0
5.1




ua
L
It can be noticed from Table 7 that the FE results and
the empirical equation that calculates the additional
equivalent unbonded length in the bar are very close
(within 15%). Therefore the equation proposed by
Raynor and Lehman [13] is considered valid and can be
used in both analysis and design.
8. Conclusions
This paper presented finite element results and analytical
derivation to document the behavior of reinforced mild
steel subjected to a combined axial and bending loading.
A bending strain relationship is derived based on a pa-
rametric study involving multiple nonlinear finite ele-
ment models. The proposed axial and bending strain
equations are based on the deflected shape of the bar.
A mild steel fracture criterion is also proposed based
on the results of the parametric study to control the total
(elastic and plastic) strains in the mild steel bar below a
maximum permissible limit. The applied total strain was
conservatively used to estimate low-cycle fatigue life for
the axial strain plus the bending strain.
a
(57)
Table 7. Average strain in the bar due to grout effect.
Model
Case No.
Lsu X (in.)٭ Y (in.)٭ L (in.)٭
(in.)٭ avg = L/(Lsu+2Lua) avg (FE)
(FE/Eq.)
%
1 10 0.398 0.204 0.4 0.034 0.036 5.8
2 20 0.784 0.816 0.8 0.037 0.037 0
3 15 0.597 0.306 0.6 0.036 0.037 1
4 30 1.176 1.224 1.2 0.0378 0.0375 1
5 5 0.2 0.051 0.2 0.03 0.035 16
6 20 0.396 0.404 0.4 0.018 0.019 5
٭Note: 1 inch = 25.4 mm.

R. HAWILEH ET AL
90
Finally, the bar elongation due to strain penetration by
an equivalent additional unbonded length Lua was inves-
tigated. It was found that the finite element model pre-
dictions and the proposed empirical equation proposed
by Raynor and Lehman in calculating the additional
equivalent unbonded length in the bar are in very good
agreement (within 15%). It can be concluded the equa-
tion proposed by Raynor and Lehman is considered valid
for estimating the additional unbounded length (devel-
opment length), and can be used in both analysis and
design.
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