NONLINEAR FINITE ELEMENT ANALYSIS OF REINFORCED CONCRETE STRUCTURES SUBJECTED TO IMPACT LOADS A Thesis Submitted to the Graduate School of Engineering and Sciences of İzmir Institute of Technology in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in Civil Engineering by Neriman Çare CAĞALOĞLU October 2010 İZMİR
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NONLINEAR FINITE ELEMENT ANALYSIS OF REINFORCED CONCRETE STRUCTURES
SUBJECTED TO IMPACT LOADS
A Thesis Submitted to
the Graduate School of Engineering and Sciences of
İzmir Institute of Technology
in Partial Fulfillment of the Requirements for the Degree of
MASTER OF SCIENCE
in Civil Engineering
by
Neriman Çare CAĞALOĞLU
October 2010
İZM İR
ii
We approve the thesis of Neriman Çare CAĞALOĞLU _________________________ Assist. Prof. Selçuk SAATCI Supervisor _________________________ Assist. Prof. Gürsoy TURAN Committee Member _________________________ Assoc. Prof. Alper TAŞDEMİRCİ Committee Member 04 October 2010 _________________________ Prof. Gökmen TAYFUR Head of the Department of Civil Engineering
_________________________ Assoc. Prof. Talat YALÇIN
Dean of the Graduate School of Engineering and Sciences
iii
ACKNOWLEDGEMENTS
First of all I would like to thank to my supervisor Assistant Professor Selçuk
Saatci who gave me the opportunity to do this research, guided with his valuable
experiences and knowledge. He never let me down, when I went him with my questions
and my problems about my research. Without his support and helps, it would be
impossible for me to complete this research.
Moreover, I am so thankful for two people (who are not with us right now) in
my life; one is my father who encouraged me to begin this project and second is my
grandmother that supported me by her prayers and love all the time. And of course to
my mother, brother and my best friends Gülbinay Can and İnci Bağdaş Tok for their
sympathy and concern.
Finally, my thanks are to Mayor of Urla Municipality Mr.Selçuk
Karaosmanoğlu, all staff of Urla Municipality and the whole people I cannot remember
now for their support at my hard times during researches.
iv
ABSTRACT
NONLINEAR FINITE ELEMENT ANALYSIS OF REINFORCED CONCRETE STRUCTURES SUBJECTED TO IMPACT LOADS
Design of reinforced concrete structures against extreme loads, such as impact
and blast loads, is increasingly gaining importance. However, due to the problem’s
complicated nature, there exists no commonly accepted methodology or a design code
for the analysis and design of such structures under impact loads. Therefore, engineers
and researchers commonly resort to the numerical methods, such as the finite element
method, and utilize different methods and techniques for the analysis and design.
Although each method has its advantages and disadvantages, usually engineers and
researchers persist on their method of choice, without evaluating the performance of
other methods available. In addition, there is no significant study in the literature
comparing the methods available that can guide the engineers and researchers working
in the area. This study compares the performance of some numerical methods for the
impact analysis and design with the help from actual impact test results in the literature.
Computer programs VecTor2 and VecTor3 were selected for nonlinear finite element
methodology, which were based on the Modified Compression Field Theory. Impact
tests conducted on reinforced concrete beams were modeled and analyzed using these
programs. Moreover, same beams were modeled also using a single degree of freedom
spring system method. The results obtained from both approaches were compared with
each other and the test results, considering their accuracy, computation time, and ease of
use.
v
ÖZET
DARBE YÜKLERİNE MARUZ KALAN BETONARME YAPILARIN DOĞRUSAL OLMAYAN SONLU ELEMANLAR YÖNTEMİ İLE
ANAL İZİ
Betonarme yapıların darbe ve patlama yükleri gibi ani ve yüksek şiddetli yüklere
karşı tasarımı günümüzde gittikçe daha önem kazanan bir konu haline gelmiştir. Ancak
problemin karmaşıklığından ötürü betonarme yapıların bu tür yüklere karşı tasarımı için
genel kabul görmüş bir yöntem veya detaylı bir teknik yönetmelik bulunmamaktadır.
Dolayısıyla, darbe yüklerine maruz kalan betonarme yapıların tasarım ve analizi için
yaygın olarak sonlu elemanlar yöntemi gibi sayısal yöntemlere başvurulmakta ve
birbirinden çok farklı sonlu eleman metotları ve betonarme modelleme teknikleri
kullanılmaktadır. Kullanılan her değişik yöntemin avantajları ve dezavantajları
bulunmakla birlikte, bu konu üzerinde çalışan araştırmacı ve mühendisler genellikle
analizlerini kendi seçtikleri belli bir yöntem ile yapmakta, farklı yöntemleri denemeye
gerek duymamaktadır. Literatürde yaygın kullanılan farklı yöntemlerin karşılaştırılması
ve değerlendirilmesi ile ilgili bir çalışma yoktur. Bu tezde, darbe yüklerine karşı
betonarme yapıların analizi için yaygın olarak kullanılan bazı sayısal yöntemler
literatürde yer alan darbe testleri ışığında karşılaştırılarak birbirlerine karşı avantaj ve
dezavantajları ortaya konulmuştur. Doğrusal olmayan sonlu elemanlar yöntemi olarak
Değiştirilmi ş Basınç Alanı Teorisi tabanlı VecTor2 ve VecTor3 programları seçilmiş ve
betonarme kirişler üzerinde yapılan darbe testleri bu programlarla modellenmiştir. Aynı
kirişler başka bir yöntem olarak tek serbestlik dereceli yaylar olarak modellenmiş ve
analizleri yapılmıştır. Elde edilen sonuçlar birbirleriyle ve test sonuçlarıyla
karşılaştırılmış, kullanılan yöntemler sonuçların doğruluğu, çözüm süresi ve kullanım
kolaylıkları bakımından değerlendirilmişlerdir.
vi
TABLE OF CONTENTS
LIST OF FIGURES ....................................................................................................... viii
LIST OF TABLES .......................................................................................................... xii
LIST OF SYMBOLS ..................................................................................................... xiii
Figure 3.4. Finite element model for VT2 ...................................................................... 39
Figure 3.5. Comparison of observed and computed responses, SS0a ............................ 40
Figure 3.6. Comparison of observed and computed responses, SS1a ............................ 41
Figure 3.7. Comparison of observed and computed responses, SS1b ............................ 41
Figure 3.8. Comparison of observed and computed responses, SS2a ............................ 41
Figure 3.9. Comparison of observed and computed responses, SS2b ............................ 42
Figure 3.10. Comparison of observed and computed responses, SS3a .......................... 42
Figure 3.11. Comparison of observed and computed responses, SS3b .......................... 42
Figure 3.12. Finite element model for VT2 (A24&B24) ................................................ 45
Figure 3.13. Comparison of observed and computed responses, A24 - V=1 m/s .......... 46
Figure 3.14. Calculated crack profile at peak point, A24 - V=1 m/s .............................. 46
Figure 3.15. Comparison of observed and computed responses, A24 - V=3 m/s .......... 46
Figure 3.16. Calculated crack profile at peak point, A24 - V=3 m/s .............................. 47
Figure 3.17. Comparison of observed and computed responses, A24 - V=4 m/s .......... 47
Figure 3.18. Calculated crack profile at peak point, A24 - V=4 m/s .............................. 47
Figure 3.19. Comparison of observed and computed responses, B24 - V=1 m/s ........... 48
Figure 3.20. Calculated crack profile at peak point, B24 - V=1 m/s .............................. 48
Figure 3.21. Comparison of observed and computed responses, B24 - V=3 m/s ........... 49
Figure 3.22. Calculated crack profile at peak point, B24 - V=3 m/s .............................. 49
Figure 3.23. Comparison of observed and computed responses, B24 - V=4 m/s ........... 49
Figure 3.24. Calculated crack profile at peak point, B24 - V=4 m/s .............................. 50
Figure 3.25. Finite element model for VT2 (A48 and B48) ........................................... 51
Figure 3.26. Comparison of observed and computed responses, A48 - V=1 m/s .......... 52
Figure 3.27. Calculated crack profile at peak point, A48 - V=1 m/s .............................. 52
Figure 3.28. Comparison of observed and computed responses, A48 - V=3 m/s .......... 52
Figure 3.29. Calculated crack profile at peak point, A48 - V=3 m/s .............................. 53
Figure 3.30. Comparison of observed and computed responses, A48 - V=4 m/s .......... 53
Figure 3.31. Calculated crack profile at peak point, A48 - V=4 m/s .............................. 53
Figure 3.32. Comparison of observed and computed responses, B48 - V=1 m/s ........... 54
Figure 3.33. Calculated crack profile at peak point, B48 - V=1 m/s .............................. 54
Figure 3.34. Comparison of observed and computed responses, B48 - V=3 m/s ........... 54
x
Figure 3.35. Calculated crack profile at peak point, B48 - V=3 m/s .............................. 55
Figure 3.36. Comparison of observed and computed responses, B48 - V=4 m/s ........... 55
Figure 3.37. Calculated crack profile at peak point, B48 - V=4 m/s .............................. 55
Figure 3.38. Finite element model for VT3 .................................................................... 57
Figure 3.39. Comparison of observed and computed responses, SS0a .......................... 58
Figure 3.40. Comparison of observed and computed responses, SS1a .......................... 58
Figure 3.41. Comparison of observed and computed responses, SS1b .......................... 59
Figure 3.42. Comparison of observed and computed responses, SS2a .......................... 59
Figure 3.43. Comparison of observed and computed responses, SS2b .......................... 59
Figure 3.44. Comparison of observed and computed responses, SS3a .......................... 60
Figure 3.45. Comparison of observed and computed responses, SS3b .......................... 60
Figure 3.46. Comparison of observed and computed responses, SS0a .......................... 61
Figure 3.47. Comparison of observed and computed responses, SS1a .......................... 61
Figure 3.48. Comparison of observed and computed responses, SS1b .......................... 62
Figure 3.49. Comparison of observed and computed responses, SS2a .......................... 62
Figure 3.50. Comparison of observed and computed responses, SS2b .......................... 62
Figure 3.51. Comparison of observed and computed responses, SS3a .......................... 63
Figure 3.52. Comparison of observed and computed responses, SS3b .......................... 63
Figure 3.53. Effect of damping on computed response of A24, V=3 m/s (VT2) ........... 65
Figure 3.54. Effect of damping on computed response of SS3b (VT3) ......................... 65
Figure 3.55. Finite element model for VT2 (B24 - half) ................................................ 67
Figure 3.56. Finite element model for VT2 (B24 - full) ................................................. 67
Figure 3.57. Comparison of observed and computed responses, B24 - V=4 m/s ........... 68
Figure 3.58. Three-dimensional view of finite element meshes ..................................... 68
Figure 3.59. (a) Section for M1 (b) Section for M2 (c) Section for M3 ............... 69
Figure 3.60. Comparison of model M1, M2, and M3 (MS0) ......................................... 70
Figure 4.1. Simple mechanical model of two-mass system ............................................ 72
Figure 4.2. Divided response of a two-mass system ....................................................... 72
Figure 4.3. Hard and soft impact .................................................................................... 73
Figure 4.4. Hard impact .................................................................................................. 73
Figure 4.5. Load-deformation pattern of contact zone R2 (∆u) ...................................... 74
Figure 4.6. Characteristic load-deformation behavior of the struck body R1 (u1) .......... 74
Figure 4.7. Examples of assumed static load-deflection relationship ............................. 76
Figure 4.8. Two mass model for hard impact ................................................................. 77
xi
Figure 4.9. Comparison of observed and computed responses, SS0a ............................ 78
Figure 4.10. Comparison of observed and computed responses, SS1a .......................... 78
Figure 4.11. Comparison of observed and computed responses, SS1b .......................... 79
Figure 4.12. Comparison of observed and computed responses, SS2a .......................... 79
Figure 4.13. Comparison of observed and computed responses, SS2b .......................... 79
Figure 4.14. Comparison of observed and computed responses, SS3a .......................... 80
Figure 4.15. Comparison of observed and computed responses, SS3b .......................... 80
xii
LIST OF TABLES
Table Page
Table 2.1. Concrete material table .................................................................................... 5
Table 2.2. Transverse Reinforcement ratios and stirrup spacing for beams. .................. 22
Table 2.3. Transverse reinforcement ratios and stirrup spacing for beams .................... 23
Table 2.4. Material and behavioral models used for concrete ........................................ 29
Table 2.5. Material and behavioral models used for steel reinforcement ....................... 29
Table 2.6. Peak values as obtained from the tests and analyses ..................................... 33
Table 3.1. Transverse reinforcement rations and stirrup spacing for beams .................. 35
Table 3.2. Cylinder test results ....................................................................................... 35
Table 3.3. Steel coupon test results ................................................................................. 36
Table 3.4. Material densities ........................................................................................... 36
Table 3.5. List of static design values for specimen ....................................................... 38
Table 3.6. Peak values as obtained from the tests, VT2 (first impact) ........................... 43
Table 3.7. Peak values as obtained from the tests and VT2 analyses for A24 ............... 48
Table 3.8. Peak values as obtained from the tests and VT2 for B24 .............................. 50
Table 3.9. Peak values as obtained from the tests, VT2 (first impact) ........................... 53
Table 3.10. Peak values as obtained from the tests, VT2 (first impact) ......................... 55
Table 3.11. Peak values as obtained from the tests and VT3 ......................................... 60
Table 3.12. Peak values as obtained from VT2 and VT3 ............................................... 63
Table 3.13. Damping properties used in analyses (VT2) ............................................... 65
Table 3.14. Damping properties used in analyses (VT3) ............................................... 66
Table 4.1. The input file parameters ............................................................................... 78
Table 4.2. Peak values as obtained from the tests and SM analyses .............................. 80
xiii
LIST OF SYMBOLS
a Maximum aggregate size
cd Compression softening, strain softening factor
ct Compression softening coefficient ��� Material stiffness matrix ���� Concrete material matrix ����� Concrete material stiffness matrix relative to principal axes ���� Reinforcement component material stiffness matrices ����� Reinforcement material stiffness matrix relative to principal axes
Ec Elastic modulus of concrete
Es Elastic modulus of reinforcement ��� Secant modulus of concrete in the principal tensile direction ��� Secant modulus of concrete in the principal compressive direction ��� Secant modulus of reinforcement
F A function (to be discussed) of the principal stress state (σxp, σyp, σzp)
fc Ultimate uniaxial compressive strength
fc’ Peak compressive stress obtained from a standard cylinder test
fcb Ultimate biaxial compressive strength
fcx Stress in concrete in x -directions
fcy Stress in concrete in y-directions
fc1 Stress in concrete in principal 1-directions
fc2 Stress in concrete in principal 2-directions
fp Peak concrete compressive stress ���� Local stress at a crack of reinforcement parallel to the x-direction ����� Local stress at a crack of reinforcement parallel to the y-direction
fsx Average stress of axial stress in the x-direction
fsy Average stress of axial stress in the y-direction
fyx Yield strength of reinforcement in the x-direction
fyy Yield strength of reinforcement in the y-direction
ft Ultimate uniaxial tensile strength
f’ t Concrete cracking strength
fy Yield stress
xiv
f1 Ultimate compressive strength for a state of biaxial compression superimposed
on hydrostatic stress state ���
f2 Ultimate compressive strength for a state of uniaxial compression
superimposed on hydrostatic stress state ��� �̅� Secant shear modulus of concrete
m1, m2 Collision of two masses
R1, R2 A contact spring
S Failure surface (to be discussed) expressed in terms of principal stresses and
five input parameters ft, fc, fcb, f1 and f2
sθ Average crack spacing perpendicular to the crack
smx Average crack spacing in the x-direction
smx Average crack spacing in the y-direction ��� Coordinate transformation matrix
u1, u2 Displacement of two masses (m1, m2) �� , �� Acceleration vector
where; ���� : local stress at a crack of reinforcement parallel to the x-direction ����� : local stress at a crack of reinforcement parallel to the y-direction
�� : angles between the normal to the crack and the reinforcement in the x-direction ��� : angles between the normal to the crack and the reinforcement in the y-direction
y
x
f sy
f c1
f sx
σ x
τ xy
σ y
τ xy
12
θ ni
Figure 2.9. Average stress between crack
Since it is a principal plane, shear stresses are absent from the section in Figure
2.10. However, as the reinforcement generally crosses the crack at a skewed angle, local
shear υci, are present on the crack surface as shown in Figure 2.10. Static equivalency of
average and local stresses in the direction tangential to the crack determines the local
This situation is also called Soft Impact (Figure 4.3a) where the kinetic energy
of the striking body is completely transferred into deformation energy of the striking
body, while the rigidly assumed resisting structure remains undeformed.
73
Figure 4.3. Hard and soft impact (Source : CEB 1988)
The opposite response is called Hard Impact (Figure 4.3b) and occurs when the
striking body is relatively rigid. In this case the kinetic energy of the striker is to a large
extent absorbed by deformation of the struck body, which normally is the structure.
Accordingly, the beams subjected to impact loads in this study exposed hard impact
conditions. Hard impact conditions require the local behavior of the target body to be
considered, as well as its general deformations (Figure 4.4).
The spring R2 represents the contact characteristics of between the structure and
the striking body. The general load-deformation pattern in the contact zone of a solid is
shown in Figure 4.4. This pattern may be influenced by strain rate effects. Elastic
compression take place in the range of 0 < ∆� < ∆� followed by an elasto-plastic
situation for ∆� < ∆� < ∆� where permanent internal damage occurs. For ∆� >∆� a further compaction or even liquefaction may follow with very high values of u�/u�∆�� .
m2
m1
u2
u1
R2 (∆u)
R1 (∆u)
∆u=u2-u1
Figure 4.4. Hard impact
74
∆u
Fc
∆u ∆u1 2
Figure 4.5. Load-deformation pattern of contact zone R2 (∆u)
u
R
Figure 4.6. Characteristic load-deformation behavior of the struck body R1 (u1)
The stiffness and strength of the structure is represented by the spring R1.
Although R1 can be linearly elastic, for a typical reinforced concrete structure, a
nonlinear force-deformation relationship is more realistic as shown in Figure 4.6.
Further details can also be incorporated into the behavior of the structure, such as the
loading-unloading relationships and hysteresis rules. In this study, hysteresis rules
defined by Takeda et al. (1970) are used to define the behavior of spring R1. The static
response was idealized by definining a primary curve for initial loading and a set of
rules for reversals as described in Figure 4.7. Using such a set of loading-unloading
rules, the structural response under dynamic loads can be defined in considerable detail.
As mentioned earlier, the mass m2 represents the mass of the striking body. Mass
m1, representing the mass of the structure, requires calculation of a “participating mass”,
based on the estimated shape of deformation of the structure under the impact load, as
given in Equation 4.5 and 4.6.
75
3 = v 3w ∙ Φ(x) dx{M
(4.5)
|(t) = v }(t, 5) ∙ Φ (x) dx{M
(4.6)
3w : distributed mass
Once all properties are determined, the response under the impact loading can be
determined by the simultaneous solution of Equations 4.1 and 4.2 through a finite
difference scheme defined in time domain, as shown below.
3�� + r = 0 (4.7) 3�� + r − r = 0 (4.8)
where
r = r(�) ; r = r(Δ�) ; Δ� = � − � (4.9)
Therefore, using the finite difference method for nonlinear springs,
~X] = ~X − Δt3 ∙ r(Δ�X ) (4.10)
�X] = �X + ~X] ∙ Δt (4.11)
~X] = ~X + Δt3 ∙ �(rΔ�X ) − (r�X)� (4.12)
�X] = �X + ~X] ∙ Δt (4.13)
and with the initial conditions at t = 0 ;
~ = 0 ; � = 0 ; ~ = ~M ; � = 0 (4.14)
76
PY
PCr
DCr DY
(a)
2
31 6
7
4
8
5
9
14
1511
16 1213
20
1718
1910b
(b)10a
1
2
3
4
5
6
7
8
9
1011
12
13
(c)
Figure 4.7. Examples of assumed static load-deflection relationship
For this study, a FORTRAN program was developed to calculate the structural
response based on the hysteresis rules defined by Takeda et al. and carry out the time-
stepping algorithm described in Equations 4.7 to 4.13. The listing for this code is given
in the Appendix A.
77
4.3. Impact Analysis of Reinforced Saatci Beams using CEB
Formulations
In this section, impact tests carried out by Saatci are simulated with the
procedure described in the preceding section. The approximation made to predict the
behavior of a struck beam is shown in Figure 4.8.
The input file for the developed FORTRAN program includes Analysis
Parameters, Structural Parameters, and Impacting Mass Parameters as shown in Table
4.1. Total duration of response was taken as 0.25 s and number of time steps was taken
as 20000. Structural Parameters, such as cracking and yielding points, are taken from
static test results presented previously in Section 2.4. It should be noted that these
parameters can be acquired from the VecTor or ANSYS analysis. To consider the strain
rate effects caused by the rapid loading under impact, cracking loads were multiplied
with 1.5, yield loads were multiplied with 1.2, and ultimate loads were multiplied with
1.2, based on the estimated strain rates and recommendations by CEB. Impacting
masses were 211 kg for a lighter drop-weight (beams identified as a-series), and 600 kg
for heavier drop-weight (beams identified as b-series). The contact velocity 8 m/s was
assigned as an initial velocity for the drop-weights. The contact stiffness was
determined by calculating the stiffness of the drop-weight according to its structural
properties as 50 kN/mm. The local crushing of concrete was ignored in determining the
contact stiffness.
m1
R2
R1
m2
v2,0
m2
v2,0
Figure 4.8. Two mass model for hard impact
78
Table 4.1. The input file parameters
Analysis Parameters Structural Parameters Impacting Mass Parameters
Total duration (s)
No. of time steps
Record every Nth time step
Cracking load (kN)
Cracking displacement (mm)
Yield Load (kN)
Yield Displacement (mm)
Ultimate load (kN)
Ultimate displacement (mm)
Effective mass (kg)
Impacting mass (kg)
Contact velocity (m/s)
Contact stiffness (kN/mm)
The following figures present the comparisons of mid-span displacements, as
observed in tests and computed with Spring Model (SM). Peak displacements are
summarized in Table 4.2.
Figure 4.9. Comparison of observed and computed responses, SS0a
Figure 4.10. Comparison of observed and computed responses, SS1a
-5
0
5
10
15
20
25
30
0 50 100 150 200 250
Dis
plac
emen
t (m
m)
Time (msec)
TESTSM
-15
-10
-5
0
5
10
15
20
25
0 50 100 150 200 250
Dis
plac
emen
t (m
m)
Time (msec)
TESTSM
79
Figure 4.11. Comparison of observed and computed responses, SS1b
Figure 4.12. Comparison of observed and computed responses, SS2a
Figure 4.13. Comparison of observed and computed responses, SS2b
-10
0
10
20
30
40
50
60
70
0 50 100 150 200 250
Dis
plac
emen
t (m
m)
Time (msec)
TESTSM
-10
-5
0
5
10
15
20
0 50 100 150 200 250Dis
plac
emen
t (m
m)
Time (msec)
TESTSM
-10
0
10
20
30
40
50
60
0 50 100 150 200 250
Dis
plac
emen
t (m
m)
Time (msec)
TESTSM
80
Figure 4.14. Comparison of observed and computed responses, SS3a
Figure 4.15. Comparison of observed and computed responses, SS3b
Table 4.2. Peak values as obtained from the tests and SM analyses
Test
TEST RESULTS SM RESULTS
Max. Displacement
(mm)
Time at Max. Displacement
(ms)
Max. Displacement
(mm)
Time at Max. Displacement
(ms)
Error in Max. Displ. (%)
SS0a 9.32 10.83 23.90 13.14 -156.49
SS1a 12.08 8.75 19.93 10.76 -65.00
SS1b 39.55 16.25 59.30 17.39 -49.95
SS2a 10.54 10.42 17.88 9.89 -69.60
SS2b 37.86 16.25 51.59 15.39 -36.26
SS3a 10.70 6.25 17.83 10.03 -66.60
SS3b 35.29 15.83 51.59 15.39 -46.17
-10
-5
0
5
10
15
20
0 50 100 150 200 250Dis
plac
emen
t (m
m)
Time (msec)
TESTSM
-10
0
10
20
30
40
50
60
0 50 100 150 200 250
Dis
plac
emen
t (m
m)
Time (msec)
TESTSM
81
As shown in figures, the results of SM are shape wise very similar to the actual
response observed in the test for moment critical beams (SS1a, SS1b, SS2a, SS2b,
SS3a, and SS3b). However, in general, SM model highly overestimated the response.
This can be attributed to two facts: First, the rules defined by Takeda et al. are based on
ductile members which have a definite yield plateau as shown in Figure 4.7a. However,
these beams, even if they were expected to be ductile under static loads, mostly
exhibited shear dominant behavior under impact loading before undergoing large
deformations. This is observed more clearly in SS0a. Secondly, the factors applied to
the static properties to consider the strain rate effects were chosen through rough
assumptions based on the expected strain rates. However, the actual strain rates may
vary during the response and well exceed the assumed values, causing a stiffer response
compared to the expected one.
Thus, it can be said that, although it is much simpler and quick to apply, SM
models failed to predict the impact response of reinforced concrete beams accurately.
This method has been further developed for better predictions (Fujikake et al. 2009).
However, use of complicated methodologies to better define the structural response and
the effect of strain rates costs the attractiveness of the method as a simple tool. Hence,
nonlinear finite element methods seem to be more suitable for accurate predictions.
82
CHAPTER 5
CONCLUSION
The study presented in this thesis had several focus points. The main objective
was to evaluate the numerical methods available for predicting the behavior of
reinforced concrete (RC) structures subjected to impact loads. For this purpose, a two-
dimensional nonlinear finite element reinforced concrete analysis program called
VecTor2, a three-dimensional nonlinear finite element reinforced concrete analysis
program called VecTor3 and a three-dimensional finite element program called ANSYS
were used for analyses. A separate program employing simple mass-spring models were
also developed in FORTRAN.
Shear-critical beams were selected for testing the methods, since modeling the
shear behavior of RC structures presents a challenge. Moreover, shear mechanisms are
known to dominate the impact behavior of reinforced concrete structures. Hence, if a
method was found to be successful in modeling the shear dominant impact behavior, it
is safe to claim that such a method would be successful in modeling the impact behavior
of ductile members as well. Results of the experimental impact studies found in the
literature were modeled with the mentioned methods and results were compared to
evaluate the methodologies. In general, programs VecTor2 and VecTor3 performed
well in estimating the impact response. The modeling methodology for RC employed in
these programs, the Modified Compression Field Theory (MCFT), was known as one of
the more successful methods in predicting the static shear behavior of RC structures,
and it performed well in modeling the shear dominant impact behavior as well. Damage
profiles, peak displacements and displacement characteristics were captured with good
accuracy. Both two- and three-dimensional finite element applications performed alike,
with limited dissimilarities due to their different way of handling the impact load. Thus,
it can be said that the methodology would perform successfully in the structures where a
two-dimensional analysis is not possible. Static analyses were also carried out with
another well-known program ANSYS. Such analyses were also intended to be used to
determine the static properties of the structures to be used in mass-spring models.
However, ANSYS did not perform well in ductile members and severely
83
underestimated the deformation capacity of the members. Hence, the results of these
analyses could not be further used.
Simple mass-spring models were also tried for modeling the shear-critical
beams. However, they performed rather poorly with the material models chosen.
Although possible enhancements to such models were reported in the literature, they
were not applied in this study due to their complicated nature and still questionable
accuracy.
84
REFERENCES
ANSYS Academic Research. Help System,Coupled Field Analysis Guide. 2009.
Belytschko, T., Liu, W. K., and Moran B., Nonlinear Finite Elements for Continua and Structures. John Wiley&Sons, 2000.
Bentz, E. C. Augustus: Finite Element Post-Processor fot VecTor2 and TRIX. Toronto: University of Toronto, 2003.
CEB Comite Euro-International Du Beton. Concrete Structures Under Impact and Impulsive Loading. Dubrovnik: Comite Euro-International Du Beton, 1988.
Chen, W. Plasticity in Reinforced Concrete. New York: McGraw-Hill, Inc., 1982.
Fanning, P. Nonlinear Models of Reinforced and Post-tensioned Concrete Beams. Electronic Journal of Structural Engineering, 2001: 111-119.
Fujikake, K., Bing L., and Soeun S. Impact Response of Reinforced Concrete Beam and Its Analytical Evaluation. Journal of Structural Engineering Vol. 135 No. 8, Aug. 2009: 938-950.
JSCE. Japan Concrete Standard . (in Japanese), 1996.
Kishi, N., Mikami, H., Matsuoka K. G., and Ando T. Impact Behavior of Shear-Failure-Type RC Beams without Shear Rebar. International Journal of Impact Engineering (Elsevier Science Ltd.), 2002: 955-968.
Palermo, D., and Vecchio F. J. Behavior of Three-Dimensional Reinforced Concrete Shear Walls. ACI Structural Journal 99, No. 1, Jan-Feb. 2002: 81-89.
Saatci, S. Behaviour and Modelling of Reinforced Concrete Structures Subjected to Impact Loads. Toronto: Univercity of Toronto PhD Thesis, 2007.
Saatci, S., and Vecchio F. J. Nonlinear Finite Element Modeling of Reinforced Concrete Structures under Impact Loads. ACI Structural Journal (American Concrete Institute) 106, No. 5, Sep.-Oct. 2009: 717-725.
Selby, R. G. Three Dimensional Constitutive Relations for Reinforced Concrete. Toronto: University of Toronto PhD Thesis, 1993.
85
Takeda, T., Sozen M. A., and Nielsen N. N. Reinforced Concrete Responce to Simulated Earthquakes. Journal of the Structural Division, Proceeding of the American Society of Civil Engineers Vol.96, Dec. 1970: 2757-2773.
Vecchio, F. J. Disturbed Stress Field Model for Reinforced Concrete:Formulations. Journal of Structural Engineering 126, No. 9, 2000: 1070-1077.
Vecchio, F. J., and Collins M. P. The Modified Compression-Field Theory of Reinforced Concrete Elements Subjected to Shear. ACI Journal 83, No. 2, Mar.-Apr. 1986: 219-231.
Vecchio, F. J., and Wong P. S. VecTor2 & FormWorks User's Manual. 2002.
Willam, K. J., and Warnke E. D.. Constitutive Model for the Triaxial Behavior of Concrete. Italy: International Association for Bridge and Structural Engineering. Vol. 19, 1975.
86
APPENDIX A
THE PROGRAM CODE IN FORTRAN
The program code in FORTRAN written by the supervisor Assist. Prof. Selçuk
Saatci used in Chapter 4.
C PROGRAM SPRING
C
C This program calculates the response of a two-mas s spring system
C simulating a hard impact
C The hysteresis rules for RC member are taken from
C Takeda, Sozen and Nielsen (1970), "Reinforced Con crete Response to
Simulated Earthquakes", ASCE Journal of the Structu ral Division
C
C v2.0
C June 17, 2010
C
IMPLICIT NONE
C
C COMMON /SPRING1/ R1,CRKLD, CRKDISP, YLDLD, Y LDISP,