transient elasto-plastic response of …...transient elasto-plastic response of bridge ... ......

http://www.iaeme.com/IJCIET/index.asp 189 [email protected]

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.

1.5 m

7.5

0 m

6.0 m

6.0 mPLAN

PIER - W2

Y

Z

Z

X

1.0m

5.0 m

5.0 m

7.5

0 m

SIDE ELEVATION

PIER - W1

7.5

0 m

1.5Ø m

Y

X

X

Z

1.5m

PIER - SC 1

2.0m

2.0 Øm

7.5

0 m

PIER - SC 2 PIER - HC 2PIER - HC 1

Ø 1.0m

Ø 1.5mØ 2.0m

Ø 2.5m

1.0 m

2.0 m

1.5 m

2.5 m

Ø 1.0m

1.0 Øm

Ø 2.0m

7.5

0 m

2.0 Øm

ELEVATION

PLAN

Figure 2 Orientation and Dimensional Details of Piers

http://en.wikipedia.org/wiki/Gross_vehicle_weight_rating



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

DEC

ELER

ATI

ON

(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

Ve

loci

ty in

m/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

Co

llis

ion

fo

rce

in

t

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

FOR

CE

in t

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 0u,

0ů,

0ü

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

14x 10

-3

TIME STEP (t=0.00025secs)

Dis

pla

cem

en

t (m

) o

f sele

cte

d n

od

e

Pier: W1 ,Impact of LTV40

Grade 40-Elasto-PlasticGrade 50-EPGrade 60-EPGrade 40-ElasticGrade 50-EGrade 60-E

Unstable solution

0 200 400 600 800 1000 1200-0.5

0

0.5

1

1.5

2

2.5

3

3.5x 10

-3


Dis

pla

cem

en

t (m

) o

f sele

cte

d n

od

e

Pier: W1 ,Impact of MTV40

Grade 40-Elasto-Plastic

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

66x 10

-3


Defe

lecti

on

(m

) o

f sele

cte

d n

od

e

Pier: SC1 ,Impact of MTV40


0 200 400 600 800 1000 1200

-5

-2.5

0

2.5

5

7.5

10

12.5

15

17.5

2020x 10

-4


Defe

lecti

on

(m

) o

f sele

cte

d n

od

e

Pier: HC1 ,Impact of MTV40


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

Percen

tag

e R

ed

ucti

on

in

Dis

pla

cem

en

t

ov

er G

ra

de 4

0

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

0.75Effect on Time period - post collision : Wall piers

Mass contribution from superstructure=500 tT

ime p

erio

d i

n s

eco

nd

s

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

Mass contribution from superstructure=500 t

Tim

e p

erio

d i

n s

eco

nd

s


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

0.75Effect on Time period - post collision : Hollow Circular piers

Mass contribution from superstructure=500 t

Tim

e p

erio

d i

n s

eco

nd

s


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

Dis

pla

cem

en

t o

f sele

cte

d,

imp

acte

d n

od

e

Time step , dt = 0.0005 seconds

Mass from superstructure=500t

Elasto-Plastic reponse

1000 t - Elasto-Plastic

1500 t - Elasto-Plastic2000 t - Elasto-Plastic

500 t -Elastic1000 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


Dis

pla

cem

en

t o

f se

lecte

d,

imp

acte

d n

od

e



Elasto-Plastic reponse 1000 t - Elasto-Plastic



500 t -Elastic 1000 t -Elastic1500 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


Dis

pla

cem

en

t o

f se

lecte

d,

imp

acte

d n

od

e



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

01

23

45

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

He

igh

t (y

-ax

is)

-2-1012

01

23

45

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

He

igh

t (y

-ax

is)

-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)

Heig

ht

(y-a

xis

)

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)

Hei

gh

t (y

-axi

s)

-2-10

12

-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

Hei

gh

t (y

-axi

s)

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

Percen

tag

e o

f G

au

ss p

oin

ts r

eco

rd

ing

pla

stic

ity

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

90WALL PIER - GRADE 40, Impact from Large Truck

W2-LTV40

W2-LTV50

W2-LTV60

W1-LTV40

W1-LTV50

Percen

tag

e o

f G

au

ss p

oin

ts r

eco

rd

ing

pla

stic

ity

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 4

0, G

R 4

0

SPD 4

0, G

R 5

0

SPD 4

0, G

R 6

0

SPD 5

0, G

R 4

0

SPD 5

0, G

R 5

0

SPD 5

0, G

R 6

0

SPD 6

0, G

R 4

0

SPD 6

0, G

R 5

0

SPD 6

0, G

R 6

0

0

10

20

30

40

50

60

70

80

90

100

Per

cen

tag

e o

f G

ua

ss p

oin

ts r

eco

rdin

g p

last

icit

y

A

W-1

W-2

SC-1

SC-2

HC-1

HC-2

SPD:SPEED kph

GR: GRADE Mpa

Impact from Mediun Truck

SPD 4

0, G

R 4

0

SPD 4

0, G

R 5

0

SPD 4

0, G

R 6

0

SPD 5

0, G

R 4

0

SPD 5

0, G

R 5

0

SPD 5

0, G

R 6

0

SPD 6

0, G

R 4

0

SPD 6

0, G

R 5

0

SPD 6

0, G

R 6

0

10

20

30

40

50

60

70

80

90

100

Impact from Large Truck

SPD:SPEED kph

GR: GRADE Mpa

Percen

tag

e o

f G

ua

ss p

oin

ts r

eco

rd

ing

pla

stic

ity

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.

transient elasto-plastic response of …...transient elasto-plastic response of bridge ... ......

Documents