Ijciet 06 09_017

http://www.iaeme.com/IJCIET/index.asp 189 editor@iaeme.com
International Journal of Civil Engineering and Technology (IJCIET)
Volume 6, Issue 9, Sep 2015, pp. 189-204, Article ID: IJCIET_06_09_017
Available online at
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=6&IType=9
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
© IAEME Publication
TRANSIENT ELASTO-PLASTIC RESPONSE
OF BRIDGE PIERS SUBJECTED TO
VEHICLE COLLISION
Dr. Avinash S. Joshi
M.B. Gharpure, Engineers and Contractors, Pune-411004, Maharashtra, INDIA
Dr. Namdeo A.Hedaoo
Associate Professor, Department of Civil Engineering,
Govt. College of Engineering, Pune, Maharashtra, INDIA
Dr. Laxmikant M. Gupta
Professor, Department of Applied Mech,
Visvesvaraya National Institute of Technology, Nagpur, Maharashtra, INDIA
ABSTRACT
Dynamic loading of structures often causes excursions of stresses well into
the inelastic range. Bridge piers subjected to collision from an errant truck is
one such loading. Owing to heavy traffic conditions coupled with lesser space,
authorities are unable to provide enough setbacks around the piers, thus
subjecting them to the hazard of a vehicle collision. The present study
investigates the dynamic nonlinear response of bridge pier subjected to a
collision. A Finite Element Analysis is carried out using a developed code in
MATLAB. Dynamic nonlinearity in the material, i.e. concrete is studied. An
elasto-plastic response of the pier is obtained by varying the pier geometry,
approach velocity of the vehicle and the grade of concrete in pier. The results
reveal several quantities. Using these results an attempt is made to quantify
the likely damage to the pier post collision. The study is intended to investigate
the effect of change in grade of concrete, effect of change in speed and mass of
the colliding vehicle considering material nonlinearity.
Keywords: collision, Drucker-Prager yield criterion, plasticity, bridge piers
Cite this Article: Dr. Avinash S. Joshi, Dr. Namdeo A.Hedaoo and Dr.
Laxmikant M. Gupta. Transient Elasto-Plastic Response of Bridge Piers
Subjected To Vehicle Collision. International Journal of Civil Engineering
and Technology, 6(9), 2015, pp. 189-204.
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=6&IType=9

Dr. Avinash S. Joshi, Dr. Namdeo A.Hedaoo and Dr. Laxmikant M. Gupta
1. INTRODUCTION
Heavy trucks have become important in local and national freight transport with the
rapid improvement of road networks and highways, especially in developing
countries. The vehicle capacities have also increased. Thus the function and the safety
of conventional transport are subjected to a risk of an errant vehicle colliding with a
bridge structure, especially bridge piers. Although heavy goods vehicle (HGV)
collision with bridge piers is a relatively rare type of loading it could have severe
consequences such as loss of life, repair costs and enormous losses due to disruption
of traffic. The forces involved are of enormous magnitude. The problem has worsened
with traffic density increasing and severe space crunch in major cities. The minimum
offset distances are very often encroached, increasing the risk of a collision. This
paper addresses the effects of a dynamic force generated due to a vehicle (truck)
collision on a bridge pier. The force-time history is one of the inputs to the program.
Several geometries of piers with different grades of concrete are analyzed using finite
element analysis capable of handling material nonlinearity that may be introduced in
the pier due to a collision. This is to identify the effect on the response of the pier due
to shape and grade of concrete. An idealized collision scene is shown in Fig.1
2. DIMENTIONAL DETAILS OF PIERS
The types of piers selected are as given in Table 1. Broadly three types of piers were
selected viz., wall type, solid circular and hollow circular piers. The sizes selected are
in accordance with the present specifications and the sizes obtained as a result of
customary design of bridges so as to represent a significant number of bridge support
systems.
Figure 1 Simplified Sketch of a Collision Scene
Table 1 Dimensional details of Pier
Sr.No. Referencing Description Dimensions in (m)*
(Fig.2)
1 W1 Wall pier - 1 1.00 x 5.00 x 7.50 (ht.)
2 W2 Wall pier - 2 1.50 x 5.00 x 7.50 (ht.)
3 SC1 Solid circular pier - 1 1.50ϕ x 7.50 (ht.)
4 SC2 Solid circular pier - 2 2.00ϕ x 7.50 (ht.)
5 HC1 Hollow circular pier - 1 2.00ϕouter (1.00 ϕinner) x 7.50 (ht.)
6 HC2
Hollow circular pier - 2 2.50ϕouter (1.50 ϕinner) x 7.50 (ht.)
Tapering to 2.00ϕouter (1.00 ϕinner)
at top
Sketches of piers are shown in Fig.2 along with the axis orientation. The collision
force is considered to act in the ‘x’ direction i.e. the traffic direction. Bridge piers
have caisson or pile foundations. These are generally buried and hence offer a great

Transient Elasto-Plastic Response of Bridge Piers Subjected To Vehicle Collision
deal of fixity to the pier. The superstructure and its inertia effect are considered in the
dynamic analysis and are suitably considered in the algorithm. The partial fixity
offered by the resistance of bearings is accommodated by applying lateral spring
elements capable of resisting displacement at the top, limited to the frictional
resistance offered by bearings. Wall piers have considerable length (5 m and 6 m).
The impact force is applied eccentrically. For the Finite element analysis a 3D-8
Noded, isoparametric brick element is employed. This is used for both, the wall piers
as well as circular pier. Hollow piers generally have thick walls, (0.5 meters in this
case), and hence the use of a thin shell element is not found to be suitable. Fig.3 and 4
show the discretization of the pier. The aspect ratio of each element is nearly equal to
one. Three grades of concrete are considered for each pier i.e. Grade 40, 50 and 60
MPa. The intention in varying the grade of concrete is to quantify the effect on the
response of piers (Details as per Table 1). An idealized stress-strain curve for
concrete is adopted and identical behavior is assumed in tension and compression.
3. FORCE-TIME HISTORIES AND VEHICLE
CHARACTERISTICS
This study considers two types of Force-time histories. They are briefly described
here along with some notable points. Commercial truck classification is determined
based on the vehicle's gross weight rating (GVWR). Force-time histories of class 6
and class 8 are considered from the above mentioned rating.
1.5m
7.50m
6.0m
6.0m
PLAN
PIER-W2
Y
Z
Z
X
1.0m
5.0m
5.0m
7.50m
SIDEELEVATION
PIER-W1
7.50m
1.5Øm
Y
X
X
Z
1.5m
PIER-SC1
2.0m
2.0Øm
7.50m
PIER-SC2 PIER-HC2PIER-HC1
Ø1.0m
Ø1.5m
Ø2.0m
Ø2.5m
1.0m
2.0m
1.5m
2.5m
Ø1.0m
1.0Øm
Ø2.0m
7.50m
2.0Øm
ELEVATION
PLAN
Figure 2 Orientation and Dimensional Details of Piers

Figure 3 Discretization of Wall Type Pier Figure 4 Discretization of Circular Pier
3.1 Type-1
Force-time history for a Medium Truck (MT) with Gross Vehicle Weight (GVW) as
11900 kgs (Cabin Load = 4590 kgs) and having wheel base 3600 x 4200mm. The
force-time history was obtained with simulation techniques using LS-DYNA. The
deceleration curve is obtained for a full frontal impact of 48 kph (kilometers per hour)
on a rigid barrier. As crash tests are carried on rigid barriers, the dynamic force
generated is maximum taking into consideration the plastic deformation of the
vehicle, while neglecting the flexibility of the barrier. Although flexibility of the
barrier matters, several studies note its significance to be less in collision analysis
[1,2].
0 20 40 60 80 100 120
-50
-40
-30
-20
-10
0
10
20
DECELERATION(G)
TIME IN MILISECONDS
DECELERATION
FULL FRONTAL CRASH TEST RESULT FOR MEDIUM TRUCK
WITH RIGID BARRIER
G = a/g, therefore a=G*g
g=9.81m/sec^2
Figure 5 Deceleration Curve (MT)
0 20 40 60 80 100 120
-4
-2
0
2
4
6
8
10
12
14
VELOCITY CURVE FROM ACCELERATION CURVE
Velocityinm/s
Time in miliseconds
VELOCITY IN m/sec
Recoil of vehicle
at 0.075 seconds
Figure.6 Velocity Curve
Fig.5 shows the deceleration curve obtained. Y axis is a dimensionless quantity
‘G’ i.e. ratio of (a/g). The actual acceleration or deceleration of the colliding vehicle is
the product of value on the Y-axis and Gravitation acceleration i.e. 9.81m/sec2. X
axis is time in millisecond (10-3
seconds). The Velocity curve is shown in Fig.6.
Recoil of the vehicle is marked at time t= 0.075 seconds from the start of collision.
The Force-Time history is shown in Fig.7 and considering the force till recoil of the
vehicle commences.
Impact force at different speeds (i.e. 40, 50 and 60kph) is derived from the force-
time history (Fig.7). To cater to the variation in force due to variation in the speed of
vehicle, the force is increased proportional to the speed. For this, the force-time
history is considered as base. This is derived from the DOT-Texas report where in a

direct correlation between the force and the speed of the vehicle which is
approximately linear is concluded.
3.2 Type-2
Force-time history for a 30 ton, Large and Single Unit Truck (SUT) is identified [3].
A complex finite element model of the vehicle, closely representing the actual vehicle
is adopted. The Force-time history due to Impact of a SUT (65000 lb = 29545kgs say
30000kgs) with a rigid cargo on a 1.0 m diameter pier has been used in the present
work. This is reproduced in Fig.8.
Based on the findings of the report [3] some of the salient points used in the present
study are enumerated.
 The results of the analyses indicate that the diameter of pier does not have significant
effect on the impact force exerted by a given truck and speed.
 Three different speeds were simulated and all the analyses showed a direct correlation
(approximately linear) between the impact force (maximum and the second peak) and
the impact speed.
Using above conclusions of the report under reference, force-time histories which
are employed in this part of the study are built.
-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0
100
200
300
400
500
600
FORCE-TIME HISTORY FOR MEDIUM SIZED TRUCK
Collisionforceint
Time in seconds
Force (t)
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
-500
0
500
1000
1500
2000
2500
FORCEint
TIME in seconds
Foce due to SUT
Mass = 30000 kgs
Velocity= 50mph
FORCE-TIME HISTORY FOR A LARGE, SUT-RIGID BALLAST
ON 1m DIA. PIER, 50 mph
Fig.7. Force-Time History for Large Truck Fig.8. Impact Force-Time Curve for
Medium Truck
4. REFERENCING OF INDENTIFICATION
In all 234 cases were analyzed. Thus the data generated after analysis required a
robust identification nomenclature. The same is illustrated below with an example.
 W1G50MTV60 : Denotes Wall pier type 1 with Grade 50, Impacted by Medium
Truck with Velocity 60 kph
 SC1G40LTV40: Denotes Solid Circular pier type 1 with Grade 40, Impacted by
Large Truck with Velocity 40 kph
 HC2G60LTV50: Denotes Hollow Circular pier type 2 with Grade 60, Impacted by
Large Truck with Velocity 50 kph
5. MESH SIZE AND CRITICAL TIME STEPPING
It is well known that, finer the meshing of the structure, more accurate is the result
obtained. This is truer for non-linear problems. A separate study is conducted on a

representative sample and the results compared in the light of coarse and fine
meshing. Employing a finer mesh increases the running time of the program to a great
extent. The meshing size is adopted without sacrificing much on accuracy of the
results and at the same time giving due importance to the computational time required
to get the desired results. Similarly, different time stepping is adopted for the dynamic
force due to Medium and Large Truck collisions so as to yield stable results. A few
small, yet significant trials were conducted adopting different time intervals.
Observing the stability of the results a time stepping of 0.0005 second is adopted for
analyzing the pier for the force time history due to Medium Truck (MT) collision. The
Force-time history for Large Trucks (LT) records steep variation. This compelled the
use of a smaller time interval i.e. 0.00025 seconds.
6. TRANSIENT MATERIAL NONLINEARITY AND OTHER
RELATED
INFORMATION
Several subroutines interconnected are developed in MATLAB including automatic
meshing. Explanation of the general assumptions and the theory used is enumerated
below. Dynamic loading of structures may create stresses well into the inelastic range.
Therefore, although under ideal conditions, the nonlinear effects are investigated. For
structural materials with limited ductility, such as concrete or rock-like materials, the
rate of straining can completely change the material response. However, in attempting
to perform an analysis of a dynamically-loaded engineering structure, the material
model is considered to be idealized. The Iterative Newton-Raphson (N-R) solution
method, an incremental-iterative solution technique, is used [4]. This technique is
carried out by applying the external load as a sequence of sufficiently small
increments so that the structure can be assumed to respond linearly within each
increment [5]. The Drucker-Prager yield criterion is adopted. The Drucker-Prager
Yield constitutive law is expressed as
(1)
where, J1 is the first stress invariant, J2 is the second invariant of the Deviatoric
stresses, α and k’ are material parameters. The yield surface has the form of a circular
cone. In order to make the Drucker-Prager circle coincide with the outer apices of the
Mohr-Coulomb hexagon at any section the equations are.
(2)
And
(3)
Here the parameters ‘c’ is cohesion in concrete and angle of internal friction. The
relation between these material parameters in terms of the compressive and the tensile
strength of concrete [6] are given as:-
(4)

(5)
where, fc is compressive strength of concrete and ft is the tensile strength. The
tensile strength is assumed to one tenth of the compressive strength. As the yield
criterion records plasticity at a gauss point the contribution to stiffness is suitably
reduced. This reduction is done through a flow rule [4]. For an elasto-plastic solution
the material stiffness is continually varying. The element stiffness is recomputed for
second iteration for each load step except the first. This reduced the computing time
considerably without any adverse effect on the accuracy of the results. For numerical
computations it is convenient to re-write the yield function in terms of alternative
stress invariant [7]. Its main advantage is that it facilitates the computer coding of the
yield function and the flow rule in a general form and necessitates only the
specifications of three constants for any criterion.
For the Drucker-Prager criterion the flow vector is expressed as,
(6)
where the vectors a1, a2 and a3 are derivatives of the stress invariants J1 and J2’
with respect to stress [7].
(7)
(8)
And
(9)
Calculating a3 using equation (9) is not required as for Drucker-Prager the
multiplying constant C3 is 0 [7]. The multiplying constants for the Drucker-Prager
yield criterion are given as
C1 = 3α, for α refer eq. (2)
C2=1.0 and C3=0.
C1, C2, C3 are constants defining the yield surface in the form suitable for
numerical analysis.
For a transient analysis, the Newmark method is adopted to iterate to a solution. The
algorithm adopts a step-by-step integration method [8]. The iterative equations in
dynamic non-linear analysis use implicit time integration. It is observed that since the
inertia of the system renders its dynamic response we get a “more smooth” response
as compared to static analysis. Convergence for dynamic non-linear analysis is rapid
as compared to a static non-linear analysis [9]. The algorithm or step-by-step
integration, i.e. the Newmark scheme for Non-linear analysis is given below [8].

Initial calculations-
1. Form the linear stiffnesss matrix K, mass matrix M and damping matrix C; initialize 0
u,
0
ů, 0
ü
2. Calculate the following constants for Newmark method
θ = 1.0 δ ≥ 0.50 α ≥ 0.25(0.5+δ)2
a0 = 1/ (αΔt2
) a1 = δ / αΔt a2 = 1 / αΔt a3 = 1/(2α) – 1
a4 = δ / α – 1 a5 = Δt (δ/α -2) / 2 a6 = a0 a7 = -a2
a8 = -a3 a9 = Δt(1 - δ) a10 = δΔt
3. Form Effective linear stiffness matrix:
K*
= K + a0 M + a1 C
4. For each time step
(A) In linear Analysis
(i)Form Effective load vector
(ii)Solve for displacement increments
(iii)Go to C.
(B) In Nonlinear Analysis
(i) If a new stiffness matrix is to be formed, update K*
for nonlinear
stiffness effects to obtain K*t
(ii) Form effective load vector
(iii) Solve for displacement increments using latest K*t
(iv) If required, iterate for dynamic equilibrium; then initialize
u(0)
= u, i=0
a) i = i+1
b) Calculate (i-1) approximation to accelerations, velocities, and
displacements
;
;
c) Calculate (i-1) effective out-of-balance loads:
d) Solve for ith
correction to displacement increments:

e) Calculate new displacement increments:
f) Check for iteration convergence , if
If convergence u = ui
and go to C
If no convergence and i < nitem : go to (a); otherwise restart using
new stiffness matrix and / or smaller time step size.
C. Calculate new Accelerations, Velocities and Displacements
7. RESULTS AND DISCUSSION
The results obtained from the several trials are presented here.
7.1 Effect of Grade of concrete on performance
Fig. 9, Fig. 10, Fig. 11 and Fig.12 show displacement of selected node within the
patch of the collision for various impact velocities and shapes of pier. Each graph
includes the response of a particular pier impacted by a particular vehicle at selected
velocity for all three grades. The effect of grade on the response of the pier in terms of
displacement is evident. The figures also show a forced elastic response along with
the elasto-plastic response of piers. Fig.13 shows the percentage reduction in the
displacement of the piers as grade is increased.
For large truck collisions a few analyses show unstable or non-converging
solutions. The non-converging solution is due to the enormous accumulation of
stresses resulting in increased plasticity and subsequent reduction in stiffness. A
reduction in stiffness means increased displacements for next iteration. This snowball
effect leads to an unstable solution. A distinct reduction in response can be seen as the
grade and size of pier increases. Thus it is inferred that an unstable solution in a
particular case indicates extreme damage to the pier.
0 100 200 300 400 500 600 700 800 900 1000
-2
0
2
4
6
8
10
12
14
x 10
-3
TIME STEP (t=0.00025secs)
Displacement(m)ofselectednode
Pier: W1 ,Impact of LTV40
Grade 40-Elasto-Plastic
Grade 50-EP
Grade 60-EP
Grade 40-Elastic
Grade 50-E
Grade 60-E
Unstable solution
0 200 400 600 800 1000 1200
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
x 10
-3
Displacement(m)ofselectednode
Pier: W1 ,Impact of MTV40
Grade 50-EP
Grade 60-EP
Grade 40-Elastic
Grade 50-E
Grade 60-E
Fig.9. Pier W-1, LT at 40 kph Fig.10. Pier W-1, MT at 40 kph

0 200 400 600 800 1000 1200
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
66
x 10
-3
Defelection(m)ofselectednode
Pier: SC1 ,Impact of MTV40
Grade 50-EP
Grade 60-EP
Grade 40-Elastic
Grade 50-E
Grade 60-E
0 200 400 600 800 1000 1200
-5
-2.5
0
2.5
5
7.5
10
12.5
15
17.5
2020
x 10
-4
Defelection(m)ofselectednode
Pier: HC1 ,Impact of MTV40
Grade 50-EP
Grade 60-EP
Grade 40-Elastic
Grade 50-E
Grade 60-E
Fig.11. Pier SC-1,MT at 40 kph Fig.12. Pier HC-1, MT at 40 kph
W-1 W-2 SC-1 SC-2 HC-1 HC-2
0
4
8
12
16
20
24
28
32
36
40
0
4
8
12
16
20
24
28
32
36
40
PercentageReductioninDisplacement
overGrade40
Type of pier
Grade 50 : Impact from Medium trucks
Grade 60 : Impact from Medium trucks
Grade 50 : Impact from Large trucks
Grade 60 : Impact from Large trucks
Fig.13. Graph Showing Reduction in Displacement Over Increasing Grade
7.2. Effect of collision on the time period of the pier
Fig. 14, Fig. 15 and Fig. 16 show the increase in the time period due to induced
plasticity in the pier. Each graph gives the velocity (speed) of the vehicle, e.g. MTV40
indicates Medium Truck with velocity of 40 kph striking the pier. The first value
indicates the time period of the pier before collision i.e. when the pier is completely in
the elastic domain. The mass of the superstructure is unchanged at 5x105 kgs (500
tones). The effect of change of grade can be judged.

Elastic MTV 40 MTV 50 MTV 60 LTV 40 LTV 50 LTV 60
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
Effect on Time period - post collision : Wall piers
Mass contribution from superstructure=500 t
Timeperiodinseconds
Type of vehicle and speed
W1-Grade 40
W1-Grade 50
W1-Grade 60
W2-Grade 40
W2-Grade 50
W2-Grade 60
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Effect on Time period - post collision : Solid Circular piers
Timeperiodinseconds
SC1-Grade 40
SC1-Grade 50
SC1-Grade 60
SC2-Grade 40
SC2-Grade 50
SC2-Grade 60
Fig.14. Effect on Time Period for Wall
Piers
Fig.15. Effect on Time Period for Solid
Circular Piers
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
Effect on Time period - post collision : Hollow Circular piers
Timeperiodinseconds
HC1-Grade 40
HC1-Grade 50
HC1-Grade 60
HC2-Grade 40
HC2-Grade 50
HC2-Grade 60
Fig.16. Effect on Time Period for Hollow Circular Piers
7.3. Effect of mass of the superstructure
Results are obtained to quantify the effect of mass of the superstructure on the natural
frequency (fn). These are tabulated in Table 2. The change in the natural frequency for
both pre and post collision is given. Also a percentage reduction indicates that for
wall piers the influence of mass is negligible. For solid circular piers this effect is less
pronounced than for hollow piers which show a maximum effect of mass of the
superstructure on the natural frequency after collision. For this study, only the
Medium Truck is considered with a speed of 50 kph. The piers considered are of type
W-1, SC-1 and HC-1.

Table 2: Effect of mass from superstructure on the natural frequency
Type of
piers
Mass of
superstructure
fn (pre-collision) fn (post-collision) Percentage
reduction in fn
cycles per second cycles per second
Wall piers
500 t 2.422 2.198 9.246
1000 t 1.731 1.567 9.446
1500 t 1.418 1.291 8.954
2000 t 1.231 1.113 9.589
Solid
circular
piers
500 t 1.739 1.256 27.783
1000 t 1.235 0.924 25.195
1500 t 1.010 0.779 22.859
2000 t 0.875 0.671 23.369
Hollow
circular
piers
500 t 2.909 2.394 17.693
1000 t 2.070 1.803 12.900
1500 t 1.693 1.509 10.866
2000 t 1.468 1.322 9.945
Fig.17, Fig.18, Fig.19 indicate the effect of mass of superstructure on
displacement. The elastic displacements are plotted along with the transient elasto-
plastic displacements. Displacement trajectory for a node within the patch of the
loading is plotted. The dynamic effects of the mass of superstructure on the time
period are also reflected in these displacement graphs. The letters ‘EP’ in the graphs
denote “Elasto-Plastic” response.
0 150 300 450 600 750 900 1050
-0.0005
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
0.0040
0.0045
Displacementofselected,impactednode
Time step , dt = 0.0005 seconds
Mass from superstructure=500t
Elasto-Plastic reponse
1000 t - Elasto-Plastic
500 t -Elastic
1000 t -Elastic 2000 t- Elastic
1500 t- Elastic
Wall pier : Effect of mass of superstructure on the response
Elasto-Plastic and Elastic, Node within the area of impact
Fig.17. Effect on Displacement w.r.t Mass of Superstructure for Wall Piers

0 200 400 600 800 1000
-0.001
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Solid Circular pier : Effect of mass of superstructure on the response
Elasto-Plastic reponse 1000 t - Elasto-Plastic
500 t -Elastic 1000 t -Elastic
1500 t- Elastic
2000 t- Elastic
Fig.18. Effect on Displacement w.r.t Mass of Superstructure for SC Piers
0 200 400 600 800 1000
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
Hollow Circular pier : Effect of mass of superstructure on the response
Elasto-Plastic reponse
500 t -Elastic
1000 t -Elastic
1500 t- Elastic
2000 t- Elastic
Fig.19. Effect on Displacement w.r.t Mass of Superstructure for HC Piers
8. PROGRESSION OF PLASTICITY
A history of induction of plasticity at every gauss point at all time intervals is stored.
This made it possible to extract the progression of plasticity as the dynamic analysis
progresses with the given forcing function. The progression of plasticity is calculated
as a percentage of the total gauss points recording plasticity. Although plasticity is not
a direct measure of damage it can be considered as an indicator for initialization of
damage. Hence a rough estimate of quantification of damage can be perceived. Fig.
20 shows wall type pier -1, with grade 60 subjected to collision from Medium truck at
velocity of 60kph. The darker elements indicate plasticity. Fig. 21 shows the same
pier with axis rotated to show the plasticity on the other face of the pier. Similarly
Fig.22 shows solid circular pier. Fig. 23 shows a hollow pier and Fig. 24 shows the
same pier with axis rotated. Only a few are presented here. Fig. 25 and Fig. 26 show
the progression of plasticity. Although the overall shape of the graphs remains same

for all the types of piers the maximum plasticity induced differs with grade and
dimensions. This is shown in Fig. 27 and Fig. 28 for Medium Truck and Large Truck
respectively. Reduction in plasticity due to change in grade and dimensions can be
judged on observing these figures. It can also be seen that collisions from large truck
proves to be severe for most of the piers selected for the study.
-2-1012
0
1
2
3
4
5
6
0
1
2
3
4
5
6
7
8
Z-axis
Collision area is
encircled nodes
Darker elements
indicate plasticity
Load step-1000
W1-G60-MTV50
X-axis-Impact dirn
Height(y-axis)
-2-1012
0
1
2
3
4
5
6
0
1
2
3
4
5
6
7
8
Load step-1000
Darker elements
indicate plasticity
Collision area is
encircled nodes
X-axis-Impact dirn
W1-G60-MTV50
Z-axis
Height(y-axis)
-2-1012 -2-1012
0
1
2
3
4
5
6
7
8
Z-axis
Collision area is
encircled nodes
Darker elements
indicate plasticity
LOAD STEP-500
SC1-G50-MTV50
X-axis(Impact dirn)
Height(y-axis)
Fig.20. W1-G60-MTV60 Fig.1.W1-G60-MTV60(axis
rotated)
Fig. 22.SC1-G50-MTV60
-2-1012 -2-1012
0
1
2
3
4
5
6
7
8
Z-axis
Collision area is
encircled nodes
Darker elements
indicate plasticity
LOAD STEP-1000
HC1-G60-MTV50
X-axis(Impact dirn)
Height(y-axis)
-2
-10
1
2
-2 -1 0 1 2
0
1
2
3
4
5
6
7
8
Collision area is
encircled nodes
Darker elements
indicate plasticity
LOAD STEP-1000
X-axis(Impact dirn)
HC1-G60-MTV50
Z-axis
Height(y-axis)
Fig.23. HC1-G60-MTV60 Fig.24. HC1-G60-MTV60 (axis rotated)

0 20 40 60 80 100
-5
0
5
10
15
20
25
30
35
40
45
W2-MTV40
W2-MTV50
W2-MTV60
W1-MTV40
W1-MTV50
PercentageofGausspointsrecordingplasticity
Time Step , dt = 0.0005 seconds
W1-MTV60
WALL PIER - GRADE 40, Impact from Medium Truck
0 100 200 300 400 500
-10
0
10
20
30
40
50
60
70
80
90
WALL PIER - GRADE 40, Impact from Large Truck
W2-LTV40
W2-LTV50
W2-LTV60
W1-LTV40
W1-LTV50
PercentageofGausspointsrecordingplasticity
Time Step , dt = 0.00025 seconds
W1-LTV60
Fig.25. Gauss Points Recording
Plasticity, Wall Piers, Grade-40,
Medium Trucks
Fig.26. Gauss Points Recording
Plasticity, Wall Piers, Grade-40, Large
Trucks
SPD
40,G
R
40
SPD
40,G
R
50
SPD
40,G
R
60
SPD
50,G
R
40
SPD
50,G
R
50
SPD
50,G
R
60
SPD
60,G
R
40
SPD
60,G
R
50
SPD
60,G
R
60
0
10
20
30
40
50
60
70
80
90
100
PercentageofGuasspointsrecordingplasticity
A
W-1
W-2
SC-1
SC-2
HC-1
HC-2
SPD:SPEED kph
GR: GRADE Mpa
Impact from Mediun Truck
SPD
40,G
R
40SPD
40,G
R
50SPD
40,G
R
60SPD
50,G
R
40SPD
50,G
R
50SPD
50,G
R
60SPD
60,G
R
40SPD
60,G
R
50SPD
60,G
R
60
10
20
30
40
50
60
70
80
90
100
Impact from Large Truck
SPD:SPEED kph
GR: GRADE Mpa
PercentageofGuasspointsrecordingplasticity
A
W-1
W-2
SC-1
SC-2
HC-1
HC-2
Fig.27. Maximum Plasticity for Fig.28. Maximum Plasticity for
Medium Truck Collision Large Truck Collision
9. CONCLUSION
The transient elasto-plastic response of concrete piers of several shapes, sizes and
grades subjected to two force-time histories are presented.
 For the selected piers it can be observed that increasing the grade of concrete has a
significant influence on the response of the pier to such high dynamic force especially
in the elasto-plastic range. The response reduces by in a range of 12% to 15% and
20% to 24% as grade is increased from M40 to M50 and from M40 to M60
respectively for medium truck collisions. Similarly, response reduces by in a range of
16% to 20% and 25% to 30% as grade is increased from M40 to M50 and from M40
to M60 respectively for large truck collisions (Ref.Fig.13).

 The time period shows significant reduction as the velocity and mass of the vehicle
increases (Refer Fig 14,15 and 16).
 The effect of mass of the superstructure too is investigated and the reduction in the
displacement and the time period is evident. Its pronounced effect in the elasto-plastic
analysis is brought forward with the results being presented alongside the results
obtained by an elastic analysis. Referring to Table 2, it can be seen that the
percentage reduction in the natural frequency of the pier remains at 9% irrespective of
the mass increase. For solid circular piers it drops from 27% to 23% with increasing
mass but for a hollow pier the effect is more pronounced as it records values from
17% to 9% (nearly half) for increasing mass.
 For a few analyses, as noted earlier, the solution was non-converging. On
observation it is due to large strains and subsequent reduction in stiffness of the
element leading to a non-converging solution. This may be interpreted as an
indication of severe damage to the pier.
 An upsurge in the trajectory of progression of plasticity can be seen for Large Truck
collisions (ref. Fig 26). This is due to that part of the force-time history recording the
impact of the cargo. Referring to Fig 27 and 28 it can be concluded that plasticity
induced is significantly less for Medium Truck collisions while collision from a
Large Truck proves to be very severe for most of the selected piers.
REFERENCES
[1] El-Tawil, S., “Vehicle collision with Bridge Piers”, Final report to the Florida
Department of Transportation for Project BC-355-6, 2004, FDOT/FHWA
publication.
[2] El-Tawil, S., Soverino, and E.S., Fonseca P., “Vehicle Collision with Bridge
Piers.”, Journal of Bridge Eng., ASCE, 2005 , pp. 345-353.
[3] Buth, C. Eugene, William, F., Brackin, Michael S., Dominique Lord, Geedipally,
Srinivas, R., and Akram Y. Abu-Odeh, “Analysis of Large Truck Collisions with
Bridge Piers: Phase 1. Report of Guidelines for Designing Bridge Piers and
Abutments for Vehicle Collisions”, 2010.
[4] Owen, D.R.J. and Hinton, E., “Finite Elements in Plasticity: Theory and
Practice”, Pineridge Press Ltd., Swansea, U.K., 1980, pp. 431-463.
[5] Arnesen, A., Sorensen, S.I. and Bergan, P.G., “Nonlinear Analysis of Reinforced
concrete”, Computers and Structures, 1978, vol.12, pp. 571-579.
[6] Cela, J.J.L., “Analysis of Reinforced concrete structures subjected to dynamic
loads with a viscoplastic Drucker-Prager Model”, Journal of Applied
Mathematical Modeling, 1997, pp 495-515.
[7] Nayak, G.C. and Zienkiewicz, O.C., “Elasto-Plastic stress analysis. A
generalization of various constitutive relations including strain softening”, Int.
Journal for Numerical Methods in Engineering, 1792, vol. 5, pp. 113-135.
[8] Bathe, K.J., Ozdemir, H. and Wilson, E.L.,“ Static and Dynamic Geometric and
Material Nonlinear Analysis”, Report no. UCSESM 74-4, University of Berkley,
California, 1974.
[9] Bathe, K.J., “Finite Element Procedures”, Prentice Hall of India Private Limited,
New Delhi, 2003, pp. 827-828.
[10] Adnan Ismael, Mustafa Gunal and Hamid Hussein. Use of Downstream-Facing
Aerofoil-Shaped Bridge Piers to Reduce Local Scour. International Journal of
Civil Engineering and Technology, 5(11), 2014, pp. 44 - 56.

Ijciet 06 09_017

Recommended

Recommended

More Related Content

What's hot

What's hot (17)

Viewers also liked

Viewers also liked (20)

Similar to Ijciet 06 09_017

Similar to Ijciet 06 09_017 (20)

More from IAEME Publication

More from IAEME Publication (20)

Recently uploaded

Recently uploaded (20)

Ijciet 06 09_017