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Hussin, Manal, Chan, Tommy H.T., Fawzia, Sabrina, & Ghasemi, Negareh(2015)Finite element modelling of lamb wave propagation in prestress concreteand effect of the prestress force on the wave’s characteristic. In10th RMS Annual Bridge Conference: Bridges – Safe and Effective RoadNetwork, 2 - 3 December 2015, Ultimo, N.S.W.

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ABC2015 1

Finite Element Modelling of Lamb Wave Propagation in Prestress Concrete and Effect of the Prestress Force on the Wave’s Characteristic

Manal Hussin 1; Tommy H.T. Chan 1; Sabrina Fawzia 1; Negareh Ghasemi 1

1 School of Civil Engineering and Built Environment, Queensland University of Technology, Brisba

ne, Queensland, Australia

Abstract In order to assess the structural reliability of bridges, an accurate and cost

effective Non-Destructive Evaluation (NDE) technology is required to ensure their safe and

reliable operation. Over 60% of the Australian National Highway System is prestressed

concrete (PSC) bridges. However, most of the in-service bridges are more than 30 years old

and may experience a heavier traffic load than their original intended level.

Use of Ultrasonic waves is continuously increasing for (NDE) and Structural Health

Monitoring (SHM) in civil, aerospace, electrical and mechanical applications. Ultrasonic

Lamb waves are becoming more popular for NDE because it can propagate long distance and

reach hidden regions with less energy loses.

This paper aims to show how prestress force (PSF) of (PSC) beam could be numerically

quantified using the fundamental theory of acoustic-elasticity. A three-dimension finite

element modelling approach is set up to perform parametric studies in order to better

understand how the lamb wave propagation in PSC beam is affected by changing in the PSF

level. Results from acoustic-elastic measurement on prestressed beam are presented, showing

the feasibility of the lamb wave for PSF evaluation in PSC bridges.

Introduction

Prestressing tendons are the main load carrying components used in the prestressed concrete

(PSC) bridge and are critical to their performance [1, 2]. Therefore loss of prestress force

(PSF) as well as the presence of structure defects (e.g. corrosion and broken wires) affecting

these elements are critical for the performance of the entire structure and may lead to

catastrophic failure[3].

Manal Hussin; Tommy H.T. Chan; Sabrina Fawzia; Negareh Ghasemi

Currently, the main Non-Destructive Evaluation (NDE) methods can reveal substantial

subsurface flaws in materials are X-ray and Ultrasonic. Other techniques such as neutron

diffraction techniques, magnetic particle and eddy current can detect some subsurface

features, but only near the surface of the materials. However, these techniques are limited to

laboratory applications, time consuming and hazardous [4, 5]. Further, the instruments are not

reliable for long term or continuous monitoring [4].

Ultrasonic waves are high permeability sound waves propagation in solid, liquid and air and

vibrating at a frequency more than 20 kHz (too fast to be audible to human) [3]. Based on

that, Ultrasonic waves can be generated with Ultrasonic transducers, and can be used as an

effective tool to identify properties of solid materials, such as thickness, flaws etc. For this

reason, it can be used to evaluate the PSF in the interior of the structure as well as at the

surface due to the high permeability of Ultrasonic wave.

The PSF can be investigated directly with the help of numerical models if laboratory tests are

too costly to conduct. The simulation results can be then used to design, prepare and develop

experimental techniques to identify PSF in real structures.

In this study, the effect of the PSF on the Lamb wave propagation characteristic such as the

wave velocity and the energy of the amplitude will be investigated through a simulation

model.

Acoustic-Elastic Law

Generally, the Time-of-Flight (TOF) of an Ultrasonic wave over a fixed distance is obtained

by dividing the propagation distance by the Ultrasonic wave velocity. However, when the

compressive strength or tensile stress is applied to the material, the Ultrasonic velocity

slightly changes in proportional to the load stress, the propagation time will also change, this

is generally called the acoustic-elastic law [6-8].

The load stress can be evaluated from the transit time changes as shown in equation 1:

∆𝑉/𝑉0 = 1 + 𝐶𝐴𝑉0 𝜎 (1)

Where:

Finite Element Modelling of Lamb Wave Propagation in Prestress concrete and Effect of the Prestress force on the wave’s characteristic

3

∆𝑉 denotes velocity change due to stress, CA acoustic-elastic constant, 𝑉0 velocity without

any stress in specimen, 𝜎 prestress force.

Wave Equation in an Infinite Material

The wave equation in an infinite, isotropic material is derived from Newton’s second law [9]

as shown in equation 2:

𝐹 = 𝑚𝑚 = 𝑚�̈� (2)

Where:

F denotes force, m mass, a acceleration and �̈� the second derivative of the deformation with

respect to time. This can be rewrite as equation 3:

𝐹1,𝑡𝑡𝑡 = 𝑚𝑢1̈ = 𝜌𝑉𝑢1̈ (3)

𝑢1 ̈ denotes deformation in the x1 direction, ρ density and V volume of the object.

The force in x1 direction can be written as the sum of the normal contribution, F1,1 and the

two shear contributions F1,2 and F1,3. Looking at the force acting on an infinitesimal volume

element, the equation above becomes:

𝜌𝑢1 =̈ 𝜕𝐹1,1+𝜕𝐹1,2+𝜕𝐹1,3𝜕𝑥1𝜕𝑥2𝜕𝑥3

= 𝜕𝐹1,1𝜕𝐴1𝜕𝑥1

+ 𝜕𝐹1,2𝜕𝐴2𝜕𝑥2

+ 𝜕𝐹1,3𝜕𝐴3𝜕𝑥3

(4)

𝜕𝐴1, 𝜕𝐴2 and 𝜕𝐴3 are the area element with normal vector in the x1, x2 and x3 directions

respectively. Equation 4 can be written in term of normal stress 𝜎1,1 and shear stresses

𝜎1,2 and𝜎1,3. The equations for those on an infinitesimal element are:

Manal Hussin; Tommy H.T. Chan; Sabrina Fawzia; Negareh Ghasemi

𝜕𝜎1,1 = 𝜕𝐹1,1𝜕𝐴1

(5)

𝜕𝜎1,2 = 𝜕𝐹1,2𝜕𝐴2

(6)

𝜕𝜎1,3 = 𝜕𝐹1,3𝜕𝐴3

(7)

Using the same method to derive expressions for the deformation in x2 and x3 direction we

obtain the following set of equations for the deformations:

𝜌𝑢1̈ = 𝜕𝜌1,1𝜕𝑥1

+𝜕𝜌1,2𝜕𝑥2

+ 𝜕𝜌1,3𝜕𝑥3

(8)

𝜌𝑢2̈ = 𝜕𝜌2,1𝜕𝑥1

+𝜕𝜌2,2𝜕𝑥2

+ 𝜕𝜌2,3𝜕𝑥3

(9)

𝜌𝑢3̈ = 𝜕𝜌3,1𝜕𝑥1

+𝜕𝜌3,2𝜕𝑥2

+ 𝜕𝜌3,3𝜕𝑥3

(10)

In an isotropic material, Hook’s law can be written as in equation 11:

ρi,j = λεkkδij + 2µεij (11)

i,j,k =1,2,3

Where:

δij denotes the Kroenker delta and λ and µ are the first and second Lame parameters

respectively; these parameters are related to Young Modules E and Poisson’s ration ν as

shown in the following equations 12 and 13 respectively:

Finite Element Modelling of Lamb Wave Propagation in Prestress concrete and Effect of the Prestress force on the wave’s characteristic

5

𝜆 = 𝐸𝐸(1+𝜐)(1−2𝐸) (12)

𝜇 = 𝐺 = 𝐸2(1+𝐸) (13)

For small deformations u<<1 higher order terms in the strain tensors can be neglected and

linearity can be assumed, so strain of the materials could be approximated with infinitesimal

strain tensor as shown in equation 14:

𝜀𝑖𝑖 = 12

(𝜕𝑢𝑖𝜕𝑥𝑗

+ 𝜕𝑢𝑗𝜕𝑥𝑖

) (14)

Both the stress and the strain tensors are symmetrical around the main diagonal, as shown in

equations 15 and 16 respectively [10]:

𝜎𝑖𝑖 = 𝜎𝑖𝑖 = �𝜎11 𝜎12 𝜎13𝜎21 𝜎22 𝜎23𝜎31 𝜎32 𝜎33

� (15)

𝜀𝑖𝑖 = �𝜀11 𝜀12 𝜀13𝜀21 𝜀22 𝜀23𝜀31 𝜀32 𝜀33

� (16)

Rewriting equations 8-10 using equations 11 and 14 yields equation 17, an expression for the

displacement equation of motion [11]:

𝜌𝑢𝚤̈ = 𝜇 𝜕2𝑢𝑖𝜕𝑥𝑗𝜕𝑥𝑗

+ (𝜇 + 𝜆) 𝜕2𝑢𝑗𝜕𝑥𝑖𝜕𝑥𝑗

(17)

Considering plane wave traveling along the x-axis, two body wave solution of this form exist

for this equation. One describes plane, longitudinal waves, which have the form:

𝑢𝑥 = 𝑓(𝑥 ∓ 𝑐𝐿𝑡), 𝑢𝑦 = 𝑢𝑧 = 0 (18)

Manal Hussin; Tommy H.T. Chan; Sabrina Fawzia; Negareh Ghasemi

By substitution in the governing equation above, the wave speed in longitudinal direction can

be determined using equation 19:

𝐶𝐿 = �2𝜇+𝜆𝜌

(19)

The other solution of this form is the transvers wave solution as shown in equation 20:

𝑢𝑦 = 𝑓(𝑥 ∓ 𝑐𝑇𝑡), 𝑢𝑥 = 𝑢𝑧 = 0

𝑢𝑧 = 𝑓(𝑥 ∓ 𝑐𝑇𝑡), 𝑢𝑥 = 𝑢𝑦 = 0 (20)

By substitution in the governing equation, the wave speed in transvers direction can be

determined using equation 21:

𝐶𝑇 = �𝜇𝜌 (21)

At this point, the longitudinal and transvers waves are non-dispersive since their wave speeds

are only a function of material properties and not frequency [12]. The dispersive equations

(Rayleigh-Lamb frequency equations) are then introduced subsequently.

Theoretical Lamb Wave Dispersion Curves

Pure lamb waves are guide dispersive waves propagating in an elastic isotropic plate with

traction free boundaries [13]. Lamb wave with practical displacement in both x and y-

direction actually represent a group of a wave type including the bending wave, the Rayleigh

wave and the quasi-longitudinal wave [14].

Lamb 1971 derived the dispersion relation (or the Rayleigh-lamb frequency relations) that

can handle the transition between these types of waves and traveling across the plane of a free

plate. The author mentioned that, only some frequency-velocity pairs can propagate through

it. The frequency-velocity can be obtained from the dispersion relation, as illustrated in

equations 22 and 23:

Finite Element Modelling of Lamb Wave Propagation in Prestress concrete and Effect of the Prestress force on the wave’s characteristic

7

tan (𝑞ℎ)tan (𝑝ℎ)

= − 4𝑘2 𝑝𝑞(𝑞2−𝑘2)2 𝑆𝑆𝑚𝑆𝑡𝑆𝑐 𝑚𝑚𝑚𝑆𝑚 (22)

tan (𝑞ℎ)tan (𝑝ℎ)

= − �𝑞2−𝑘2�2

4𝑘2 𝑝𝑞 Antisymmetric motion (23)

For a given 𝜔 and derived 𝜅, the displacement can be determined from equations 22 and 23

respectively and for both symmetric and asymmetric modes. Multi symmetric and

asymmetric lamb wave modes can exist at a given single frequency. Figure 1 shows these

waveforms as displacement through a thickness cross-section.

Fig. 1 Symmetric and Asymmetric wave form [12]

Since lamb wave are dispersive, the wave speed, or phase velocity is a function of frequency.

These individual waves interact and the resulting waves propagate at a group velocity which

may be different from the induvial phase velocities. Numerical method for calculating phase

and group velocity dispersion curves are outlined by Rose [10] to solve the equations for

more details. However, LAMSS (LABORATORY FOR ACTIVE MATERIALS AND

SMART STRUCTURES) from the University of South Carolina provided free and useful

software packages to calculate these parameters such as: phase velocity, group velocity, wave

number etc. directly in conjunction with Mathlab.

Modelling Development

In order to study the effect of various PSF on the wave characteristic such as velocity and

energy of the amplitude, a 3D numerical model is build using ABAQUS software. A two-step

Manal Hussin; Tommy H.T. Chan; Sabrina Fawzia; Negareh Ghasemi

loading has been considered: a stationary loading to generate a stressed body and explicit

solvers based on the central different scheme used in simulating high–speed events (such as

Lamb wave).

The simulations have been performed with the solid cubic particles and homogenized model

were meshed with 8-node brick element. Figure 2 shows schematic of PSC beam used for the

simulation and Table I show its dimension and materials properties.

Fig. 2 The PSC beam modelled in ABAQUS

Table I. Dimensions and materials properties for the beam model

L

(m)

H

(m)

W

(m)

𝜌

(kg/m3)

E

(GPa)

µ

(GPa) v

λ

(GPa)

3 0.5 0.6 2400 34.8 14.5 0.2 9.6

In order to decrease the model size, symmetry conditions are applied so, only quarter-volume

of the beam is taken into consideration and a symmetric boundary condition is applied on all

the nodes of the symmetric plane.

For the convergence of dynamic simulation, the mesh of the structure (at least in the wave

propagation area as shown in Figure 2) should be fine enough so at least ten elements exist

per wavelength along the direction of wave propagation. Further, the integration time step

should also be sufficient small to resolve the frequency response of the structure.

Receiver 1(183.5mm) Transducer

L H W

Finite Element Modelling of Lamb Wave Propagation in Prestress concrete and Effect of the Prestress force on the wave’s characteristic

9

Hanning-window excitation signals with four and half cycles and frequency of 300 kHz is fed

directly into FEM as a forcing function to generate Lamb wave. Figure 3 shows the Hanning-

signal used to generate Lamb wave.

Fig.3 Four and half Hanning-Window excitation signal at 300 kHz

`

Simulation results

Different parameters have been studied in this research, such as acceleration and velocity

both in longitudinal and transverse directions. The study is repeated for a range of prestress

forces: 0, 10, 30, and 50 MPa. The overall purpose of the numerical model is to evaluate the

Lamb wave characteristic such as: the wave velocity and the energy of the amplitude in

various magnitudes of PSF.

Figures 4 and 5 illustrate the acceleration responses of the node in the receiver point for

different PSF in the longitudinal and the transvers directions. Figures 6, 7, 8 and 9 illustrated

the zoomed peak for symmetric (S0) and asymmetric (A0) modes in longitudinal and

transvers directions respectively.

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.0000015 0.000003

Nor

mal

ized

Am

pltu

de

Time Sec

Manal Hussin; Tommy H.T. Chan; Sabrina Fawzia; Negareh Ghasemi

The simulation results of the acceleration responses show that, the peak of the S0 and A0

mode are affected by changing the magnitudes of PSF even though the change was quite

small. However, A0 responses show a clear change in TOF rather than the S0 in both

directions. Following the acoustic–elastic law mentioned earlier in this paper, the acceleration

parameter can be quite useful to be used in the PSF identification using equation 1.

Figures 10 and 11 illustrate the velocity responses of the node in the receiver point for

different PSF in the longitudinal and the transvers directions. Figures 12, 13, 14 and 15

illustrate the zoomed peak for symmetric and asymmetric mode in longitudinal and transvers

directions.

-10000

-5000

0

5000

10000

1.00003 1.00004 1.00005 1.00006 1.00007LA1

m/s

ec2

Time sec

Longitudinal Acceletaion (LA) 0 Mpa 10 Mpa30 Mpa 50 Mpa

S0 A0

-3000

-1500

0

1500

3000

1.00003 1.00004 1.00005 1.00006 1.00007

TA2

m

/sec

2

Time in Sec

Transverse Acceleration (TA) 0 mpa 10 mpa30 mpa 50 mpa

S0 A0

7800

8000

8200

1.0000558 1.000056

S0 LA 0 Mpa 10 Mpa30 Mpa 50 Mpa

1700

1800

1900

2000

1.0000555 1.0000559

0 mpa 10 mpa30 mpa 50 mpa

S0 TA

4300

4600

4900

5200

1.0000585 1.0000589

A0 LA 0 Mpa 10 Mpa

650

750

850

950

1.0000583 1.0000585 1.0000587 1.0000589

A0 TA 0 mpa 10 mpa

Fig. 4 longitudinal Acceleration Fig. 5 Transverse Acceleration

Fig.6 Zoomed of the S0 peak LA Fig.7 Zoomed of the S0 peak TA

Fig.8 Zoomed of the A0 peak LA Fig.9 Zoomed of the A0 peak TA

Finite Element Modelling of Lamb Wave Propagation in Prestress concrete and Effect of the Prestress force on the wave’s characteristic

11

Although, the velocity responses do not show any significant change in the TOF, the energy

of the amplitudes are certainly affected by changing the PSF in the S0 and A0 modes and for

both directions.

The amplitude energy for the longitudinal directions increases with increasing the PSF due to

the fact that they are at the same direction of the PSF as shown in Figure 16.

Conversely, the transvers responses show attenuation behaviour with increasing the PSF due

to scattering caused in the wave.

-0.01

-0.005

0

0.005

0.01

1.00003 1.00005 1.00007LV m

/sec

Time Sec

Longitudinal Velocity (LV) 0 Mpa 10 mpa30 Mpa 50 Mpa

S0 A0

-0.0015

-0.0005

0.0005

0.0015

1.00003 1.00005 1.00007

TV m

/sec

Time sec

0 Mpa 10 Mpa30 Mpa 50 Mpa

Transverse Velocity ( TV)

S0 A0

0.006

0.0064

0.0068

0.0072

1.0000528 1.0000533 1.0000538

S0 LV 0 Mpa 10 mpa30 Mpa 50 Mpa

0.00145

0.0015

0.00155

0.0016

1.0000529 1.0000531 1.0000533

0 Mpa 10 Mpa30 Mpa 50 Mpa

S0 TV

0.001

0.0015

0.002

0.0025

0.003

1.0000588 1.0000593 1.0000598

A0 LV 0 Mpa 10 mpa30 Mpa 50 Mpa

-0.0008

-0.0004

0

0.0004

1.000061 1.000062 1.000063

0 Mpa 10 Mpa30 Mpa 50 Mpa

A0 TV

Fig.12 Zoomed of the S0 peak LV Fig.13 Zoomed of the S0 peak TV

Fig.14 Zoomed of the A0 peak LV Fig.15 Zoomed of the A0 peak TV

Fig. 10 Longitudinal Velocity Fig. 11 Transverse Velocity

Manal Hussin; Tommy H.T. Chan; Sabrina Fawzia; Negareh Ghasemi

Fig.16 illustrated the PSF and the wave propagation direction

Conclusions

The objective of this investigation is to evaluate Lamb wave propagation through a

prestressed concrete beam considering various prestress forces incorporated into the finite–

element code ABAQUS CAE, as a preliminary study to identify PSF in prestressed concrete

bridges using NDE techniques.

Different parameters have been studied such as the acceleration and the velocity in

longitudinal and transvers directions. It is found that the wave mechanism (wave velocity) is

varied with increasing the PSF. This phenomenon can therefore be further employed to

identify the PSF following the acoustic-elastic law.

It is well known that Lamb waves in concrete are severely attenuated due to inhomogeneous

nature of the material. Whereas the simulation results illustrated that Energy of the amplitude

is far more influenced by the changing of the PSF rather than the wave velocity. Therefor

more investigation in this area needs to be conducted to develop a general relationship

between the Energy of the amplitude and the PSF.

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PSF PSF Longitudinal direction

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