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The 8th International Conference on

Structural Health Monitoring of Intelligent Infrastructure

Brisbane, Australia | 5-8 December 2017

Assessment of Flexural Stiffness and Load Carrying Capacity Using

Substructural System S. Jamali1, KY. Koo2, THT. Chan3, A. Nguyen4, DP.Thambiratnam5

1 PhD Candidate, Queensland University of Technology – Australia, 2Lecturer, University of Exeter- United Kingdom,

3,5Professor, Queensland University of Technology – Australia, Email: tommy.chan@qut.edu.au

4 Research Fellow, Queensland University of Technology – Australia,

Abstract

The evaluation philosophy of an existing structure must differ from the design philosophy of a new

structure. Many parameters are related to the structural properties and among them, stiffness and

boundary conditions are governing factors affecting the resistance of the structure to external loading.

However, due to the aging and deterioration of structures, effects of boundary conditions change over

time and it becomes intricate to understand their current condition for structural assessment. This

study was set out to assess the feasibility of estimating flexural stiffness in beam-like structures using

substructural bending rigidity identification (SBI) method when no information about 'as-is’ fixity of

boundary conditions is available. The results of numerical and experimental analyses showed that

the SBI method accurately estimates the flexural stiffness of various beam profiles having unknown

boundary conditions. It was found that using estimated stiffness, analytical model can be tuned with

updated support fixities and structural and modal characteristics, which are then used for detection

of the critical member, worst load effect and evaluation of serviceable load carrying capacity.

Findings from this study provided evidence that SBI can be efficiently implemented as a short-term

monitoring tool with easy testing setup for in-operation structural integrity assessment of beam-type

elements.

Keywords: stiffness, beam, boundary condition, substructural system, load rating, load carrying

capacity

1. Background

Studies over the past two decades have provided important information on using structural health

monitoring (SHM) techniques for condition assessment of structural elements. By measuring the

dynamic response of a structure such as a bridge under known/unknown loading, one can obtain

vibrational parameters after analysing the data and correlating it to structural parameters, since any

change in vibrational behaviour of a structure leads to a corresponding change in structural

parameters (Chan & Thambiratnam 2011; Jamali et al. 2017a). Load carrying capacity (LCC) of

structures such as bridge girders decreases over time due to various factors such as environmental

condition, over-loading and material deterioration. On the other hand, increase in vehicular loading

requires substantially higher LCC to accommodate change in loading pattern; which directly affects

safety and maintenance costs. In Australia and many other countries, bridge standards provide

specifications for bridge load rating when safe bearing capacity of bridge is uncertain (Jamali et al.

2017b, c). For structural integrity assessment, conventional non-destructive testing such as infrared

thermography, acoustic emission, and visual inspection are conducted to determine integrity index

of a structure. Nevertheless, these tests are merely extensions of design standards and can provide

limited information about structural safety, which leads to conservative results. Therefore, current

structural conditions must be considered for evaluation of existing structures. Reviewed studies

clearly indicate the importance of stiffness as a measure of structural integrity index, akin to LCC.

To evaluate stiffness of beam-like structures as a whole system, measurement of global parameters

such as deflections, rotations and natural frequencies are involved (Hoffmann et al. 2007,

Lakshmanan et al. 2008, Gao et al. 2015). Previous investigations have found the stiffness of bridges

by simplifying the bridge as a single degree-of-freedom system using Euler beam theory (Chowdhury

1999; Higgs & Tongue 1999; Chen et al. 2002; Islam et al. 2014). Similarly, Subramaniam (1994),

Samali et al. (2007), Moore et al. (2013) and Brashaw et al. (2009) attempted to evaluate the current

stiffness of timber bridges by assuming ‘as-designed’ boundary conditions. One major issue in

previous studies that has not been dealt with is lack of information on support conditions. It is well-

established that the behaviour of a structure varies with its aging; for instance, roller supports might

change to provide partial fixity over time, which affects the load path of the structure by restraining

rotational deflection and hence giving different load distribution and capacity. Therefore fixity of

boundary conditions at structural joints such as abutments are prone to change which must be

correctly reflected in structural condition assessment. This paper presents a new approach for

flexural rigidity assessment of beam-type elements using ‘conquer and divide’ strategy. Substructural

bending rigidity identification (SBI) approach divides the beam into an internal substructural system

that correlates acceleration input at interfacial location of substructure to that centrally output through

an optimization scheme. Significant advantages of using SBI is that no prior knowledge of boundary

conditions is needed and it uses an efficient sensory system (5 sensors), which makes it simple to

setup and deliver. Due to page constraints, further details on the mathematical model development

of SBI can be found in (Jamali et al. 2017c). In this paper, implications of SBI for various beam-like

structures are assessed, and then relationship between estimated stiffness and LCC is explored. In the

next section, the rationale behind SBI’s theory is evaluated which is followed by concluding remarks.

2. Verification study

The complete process of SBI is shown below and it is detailed in (Jamali et al. 2017c). SBI layout is

prepared based on the length of the beam to capture the bending deformation under flexural mode.

Using the first bending mode, the vertical and rotational accelerations are recorded and using

nonlinear curve square fitting, response of interfacial acceleration is minimized with respect to the

central response. Minimized objective function ( ) of optimization is used for estimation of bending

stiffness using Classical beam theory. Following sections illustrate the application of SBI for

stiffness estimation and calibration of analytical model.

Figure 1. SBI process

2.1 Continuous Beam

A continuous beam with four equal spans of 8m each is divided into a series of substructural systems

meshed at 0.1m as shown in Figure 2. Band limited white noise is applied at span two to excite the

fundamental mode of vibration of the rectangular beams simulated in CSiBridge®. In traditional

approach, Leff is used for effective length of span as support-to-support spacing. In SBI, Lsub

approximates the fixity of boundary conditions using internal substructural length chosen arbitrarily

and independent of boundaries. As an example of SBI input for span one, accelerations at joints 2&3

are simulated to obtain vertical acceleration ⟨[𝑗𝑜𝑖𝑛𝑡(2)𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛+𝑗𝑜𝑖𝑛𝑡 (3)𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛

2]⟩ and rotational

acceleration at left end interfacial⟨[𝑗𝑜𝑖𝑛𝑡(2)𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛−𝑗𝑜𝑖𝑛𝑡 (3)𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛

𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑗𝑜𝑖𝑛𝑡𝑠/𝑠𝑒𝑛𝑠𝑜𝑟𝑠]⟩ , while joints 19&20 are

used for right end interfacial. The response of interfacial ends is then minimized with regard to mid-

span vertical acceleration response at joint 11. In the same way, response of each span is recorded

separately for SBI optimization.

Figure 2. Elevation view of continuous beams

After taking Fast Fourier Transform (FFT) of signals, it was noted that due to the existence of nodal

points; some higher modes are not captured as seen in power spectral density of span three (see Figure

3-a). However, only first bending mode is needed for SBI, which was accurately detected by all

substructural spans. In addition, spans located further away from excitation source had more noise

component in comparison to span two, implying that excitation source needs to be sufficiently high

to excite the early modes of the structure. This view is supported by (Memory et al. 1995) who

concluded that response of a structure to a dynamic load mobilizes a higher modulus of elasticity in

comparison with that to a static load.

Span one Span two Span three Span four

L substructure L effective

(a)

(b)

Figure 3. (a) Spectral densities of substructural system at span three (b) Optimization of bending frequency

Having the first flexural mode at 9.13 Hz, a range of 7 to 11.5 Hz was selected as bending region for

optimization using nonlinear curve-square fitting method. Figure 3(b) compares the optimization of

substructural system input-output response at span three. Clear trend of minimization in bending peak

can be observed after implementing SBI; this confirms that optimization has globally converged with

very low residuals. From the correlational analysis and using SBI formula (Jamali et al. 2017c):

23 42 47 21

4 4

2.4028 10 0.08 9.1 2 7.72.638 10 .

16 16 1.5124

subSBI

A LEI N m

(1)

Where A is the cross sectional area, is mass density and is angular frequency that is

2 bendingf . Estimated SBI for span three differs 0.49% to intact EI (2.652×107 N.m2), which gives

the fixity of support to be in pinned condition when substituted into the Classical beam theory as

shown in equation (2). Stiffness of the other spans approximated to be: span 1 = 2.644 ×107 (0.28%),

span 2 = 2.6651 ×107 (0.02%), span 4 =2.653 ×107 (0.6%); which denotes that the adequate intensity

of excitation needs to be distributed across each span for better stimulation of modal modes and

estimation of stiffness. Thus, random moving excitation would be a good alternative which is

investigated in the next case study.

2 2 7

2 2

2.638 109.13 3.159

2 2 8 192.2

SBI

eff

EIf

L m

(2)

Using updated stiffness and boundary conditions and applying load-displacement relationship for

linear system with allowable deflection, bending capacity would be:

max3

482438437.5( / ) (0.032 ) 78.03kN

eff

EIK N m m

L (3)

This means that the span remains within permissible deflection as long as the applied load does not

exceed this value. Allowable deflection can be code-defined value or specified threshold measured

from previous investigations. Taking into account the rigidity of span in load resistance and nonlinear

behaviour, a variation of ±10% from obtained value is allowed as safety margin since structure reacts

to load differently as opposed to theoretical assumptions (Nowak & Collin 2012).

2.2 Solid Concrete Girder Bridge

To assess the performance of SBI for bridge model updating and load rating, a 20m reinforced

concrete girder bridge is investigated. Layout of bridge is as illustrated in Figure 4, which consists of

two traffic lanes with parapet on each side.

Figure 4. Cross sectional view of the bridge

In order to mimic the testing condition, various truck loadings as per AS 5100.2 (Australian Standards

2004) were applied to represent the ambient traffic. To prevent interruption of traffic flow and easier

accessibility, measurement was recorded at the base of left exterior girder (G1). For the purpose of

measurements, interior girders may be considered as well. Nonetheless, due to the passage of vehicles

that induce additional mass on deck, estimation of stiffness would be less accurate; so measurement

after passage of vehicles is preferred for interior girders. Another important aspect of SBI is Lsub,

which is the length of internal substructure. Determination of Lsub is mainly dependent on the total

support-to-support length of structure (Leff), and preferably it is suggested to keep Lsub long enough

so that bending behaviour of beam is closely represented. With regard to the concrete girder, Lsub was

taken as 18.8m as internal substructure length. Figure 5 provides the spectral densities of bending

peak at interfacial (S1y/S2y) and central (Syy) locations. This isolated bending peak is used as bending

region for optimization.

(a) (b)

Figure 5. Power spectral density of G1, (a) Hertz scale, (b) Log scale

Output of optimization resulted in EI= 2.912 ×1010 N.m2, which is very close to intact EI

(2.9304×1010 N.m2). Having estimated the stiffness, analytical model could be updated to match the

modal and structural parameters obtained by SBI. After that, the deck is load rated as per

specification of TMR guidelines (Queensland Department of Transport and Main Roads 2013) for

three travel cases as illustrated in Figure 6 to find out the most adverse load effect. For the load

assessment of bridge components, two assessment vehicles are included for ease of calculation, viz.

45.5t semi-trailer as reference vehicle and 44t as coexisting vehicle with traveling speed of 45 km/hr

as follows:

Case 1: Reference vehicle is travelling along bridge centerline without any other vehicles

Case 2: Assessment vehicle is positioned adjacent to the kerb and second assessment

vehicle is positioned adjacent to the carriageway centerline

Case 3: Reference vehicle is travelling on the lane with coexisting vehicle in the other lane

Results of multi-step static analysis revealed that largest load effects for shear force and bending

moments occur at G4 and G2 respectively in case two, when reference vehicle is positioned along

the kerb and other vehicle is adjacent to the centerline of deck. What follows is an illustrative

description of shear assessment of G4 for live load case (SFLL) (Jamali et al. 2017c):

1

(1 ) A F

ULS DL DL SDL SDLLL n

LL LL

i

SF SF SFSF

SF DLA V

(4)

where

Ultimate shear capacity of girder (ϕSFLL) = 1480 kN

Dead load shear force (SFDL) = 469 kN

Superimposed dead load shear force (SFSDL) = 88 kN

Live load shear force from reference vehicle (SFLL1) = 267 kN

Live load shear force from accompanying vehicle (SFLL2) = 42 kN ×accompanying vehicle

factor (AVF)

Dynamic load allowance (DLA) = 0.4

Figure 6. Lateral position of assessment vehicles for different cases

Using appropriate load factors, the shear force ratio is found to be 0.881. This highlights that G4 has

no reserved capacity for this travel loading case and suitable management plan such as load limitation

or retrofitting is required for functional performance of the girder. It is also worth noting that similar

approach can be carried out for assessment of other load effects, for example when bending is critical

at mid-span.

2.3 Aluminum Beams

As pointed out in the introduction to this paper, SBI can efficiently estimate the stiffness of beam-

like structures with unknown boundary conditions. In an experimental study, two square hollow

section (SHS) beams made of aluminum were tested to better understand the effect of Lsub and

damage on SBI output. For the sake of stable testing, beams were seated on plank at support locations

and held by clamping without creating any specific boundary condition as shown in Figure 7.

(a)

(b)

Figure 7. Testing setup for SHS beams, (a) Small SHS, (b) Large SHS

For both beams, excitation was provided by random hammer tapping along the length of the beam

for logging acceleration data at sensor locations (S1 to S5). Figure 8 shows the optimization inputs

for small SHS as well as the spectral densities found by Welch method for each beam.

(a)

(b)

Figure 8. (a) Input-output response of sub-structural length for small SHS, (b) Spectral density of first bending mode

For both beams, Young’s modulus is 70×109 N/m2 while those estimated for small SHS was 64.5

×109 N/m2 and for large SHS was 69.8 ×109 N/m2. This discrepancy could be attributed to the spacing

of interfacial sensors for obtaining rotational acceleration. Lack of accuracy on approximation of

rotational acceleration was due to the distance between S1-S2 and S4-S5, while for large SHS; this

distance was only 6cm. Moreover, considering the relationship of Lsub in SBI equation (with power

of four), short sensor spacing for end conditions leads to longer Lsub, thus having more length of

beam covered to represent its bending behavior. Using calculated stiffness and equation (2), fixity of

supports are found to be pinned-fixed condition which assisted in updating the analytical model. By

adopting load-deflection relationship, bending capacity within linear range for large SHS is 1.6 kN.

Large SHS was then further tested as a damaged beam by placing solid mass midway between S2-

S3. It can be seen from the data in Table 1 that for a range of damages with different severities, SBI

effectively reflected the degradation of beam by estimating the reduction trend for bending frequency

and flexural stiffness. It appears that the change in global frequency affects the value in Eq. (1)

and in turn EI. Since frequency is not a very good indicator of damage (level 1 and level 2 damages),

trivial changes in EI can be seen due to small damage severity. Yet, severe damage will have

significant impact on stiffness which can be confirmed by quick visual inspection in case of real

structures. In addition, the merit of SBI is that it is dependent on global behaviour of the structure,

and as long as the global characteristics change, the optimized EI will reflect these changes as well.

Table 1. Effect of damage on big SHS

Damage state Bending

mode (Hz) EI (N.m2) ξ (average)

No damage 44 14196 1.4028

0.5 kg mass 41.2 13144 1.3833

1 kg mass 37 11600 1.3525

1.5 kg mass 35.3 10919 1.3412

3. Conclusion

Variation in loading conditions and surrounding environment change the properties of a structure, in

which such changes must be considered for structural assessment. Stiffness and boundary conditions

are important indices of structural integrity that affect the LCC. Underestimating the LCC of

structures such as bridges will cause load-posting and traffic limitation that undoubtedly affect the

road network functionality. Towards this end, this paper developed and applied a novel method for

estimating the flexural stiffness and the boundary conditions of beam-like structures. The SBI method

was assessed for different beam configurations made up of diverse materials and support conditions.

The results of this study indicate that SBI can be successfully carried out for continuous beams and

bridge girders under impact, random and moving load excitations. Furthermore, it was identified that

using the estimated stiffness, modal and structural properties of a structure along with updated

boundary conditions can be used for calibration of finite element model for further assessment.

Experimental investigations also confirmed that SBI accurately reflects the effect of damage in the

beam-like elements which could be applied to the load-displacement relationship to approximate the

linear LCC within permissible deflection range. Owing to the latest developments in sensory

equipment, for future works it is recommended that effect of using gyroscope be assessed to

determine the rotational acceleration. Above findings suggest that SBI can be considered as a cost-

effective method for in-service assessment of beam-type structures on periodic basis.

Acknowledgments

The first author is thankful for the research funds provided by the Queensland University of

Technology. The authors have no conflict of interests regarding the publication of this paper.

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