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Preliminary bridge health evaluation using the pier vibration frequency
Construction and Building Materials 102 (2016) 552–563
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Preliminary bridge health evaluation using the pier vibration frequency Kuo-Wei Liao a,⇑, Min-Yuan Cheng a, Yung-Fang Chiu b, Jin-Han Lee a a b
Department of Civil & Construction Engineering, National Taiwan University of Science & Technology, Taipei, Taiwan Harbor and Marine Technology Center of the Ministry of Transportation and Communications, Taiwan
h i g h l i g h t s A critical frequency ratio (Rc) is proposed for preliminary bridge health evaluation. Rc is a velocity and structural strength-dependent indicator. SPT-N value does not have a significant influence on Rc. Scour depth has a significant impact on the bridge frequency. Scour depth is one of the most essential factors for bridge safety.
a r t i c l e
i n f o
Article history: Received 7 April 2015 Received in revised form 17 October 2015 Accepted 2 November 2015 Available online 13 November 2015 Keywords: Bridge safety Frequency Pushover analysis Capacity envelope Demand envelope
a b s t r a c t Frequency is often considered to be an ideal safety indicator in bridge health monitoring. Many scholars have related bridge scour depth to frequency. However, few of previous studies have focused on finding a threshold frequency that corresponds to the limit state of bridge failure. Thus, this study focuses on finding a critical frequency ratio (Rc) that can be used to distinguish between undamaged and damaged bridges. The bridge limit state was calculated by subtracting the demand from the capacity of a bridge. Both the capacity and demand envelopes of a bridge were determined using pushover analyses. Nonlinear behaviors of core/cover concrete and steel reinforcement were considered. In the end of this study, a range of frequency ratio is proposed to be an on-site indicator to evaluate bridge safety. Although the proposed safety frequency ratio is consistent with a field observation, the parameters used in this study are from Taiwan only. In addition, the proposed safety ratio only considered bridges with a single pier against flood hazard. The limits should also be considered for further applications. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Over the long-term operation of a bridge, damage to the bridge components is inevitable from natural hazards or deterioration. Serious damage may affect the performance of a bridge, which influences traffic operation and may even lead to the loss of lives. To ensure road users’ safety and prolong the service life of bridges, routine inspection is necessary. The inspection results are often used to determine the repair schedule, range and degree. Therefore, an inspection program plays an important role in a bridge maintenance plan. Among the many natural hazards, floods have drawn a great deal of attention after several Typhoon strikes in Taiwan. Morakot in 2009 caused serious damage in the southern part of Taiwan. Thus, a need exists for the government to build a ⇑ Corresponding author at: Department of Civil & Construction Engineering, National Taiwan University of Science & Technology, No. 43, Sec. 4, Keelung Rd., Taipei 106, Taiwan. E-mail address: [email protected] (K.-W. Liao). http://dx.doi.org/10.1016/j.conbuildmat.2015.11.011 0950-0618/Ó 2015 Elsevier Ltd. All rights reserved.
maintenance list of all existing bridges. Because the number of bridges is large, composing an ideal maintenance list is a difficult task. Thus, effectively evaluating the bridge health condition is an important issue in Taiwan. Engineers in Taiwan often use a preliminary inspection evaluation form (PIEF) to help with the evaluation process. The PIEF is a visual inspection process that is generally considered to be the first step in a bridge safety evaluation. If the overall assessment score from the PIEF does not meet a predefined standard, the evaluation should proceed to an advanced investigation (e.g., finite element analysis, nondestructive or destructive inspection) to ensure the safety of the bridge. Because many bridge failures accompany pier scour, 60% of the PIEF items are highly related to the scour factor. The scour factor plays a key role in a bridge safety evaluation against floods. On the other hand, visual inspections may have potential problems. For example, without help from any instrument, engineers usually cannot accurately estimate the bridge condition. To be
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exact, the scour depth is usually below the water level, so it is difficult to access. In addition, the scour hole is often filled after floods, which makes it difficult to estimate the impact of the infill materials on the bridge safety. A subjective judgment is inevitable in visual inspections, and therefore, there is a need to develop an alternative method to monitor a long list of bridges on a regular basis. Many nondestructive inspection techniques can be utilized to find the local characteristics of bridges, such as using stress waves to inspect the crack depth and concrete cover thickness. Elsaid and Seracino [1] investigated the potential application of using the change in the mode shape curvature and flexibilitybased deflections in scour detection. They concluded that horizontally-displaced mode shapes were sensitive to scour. In addition, it was found that there is a relation between the natural frequencies and the scour level. Foti and Sabia [2] have monitored foundation scour by using measurements of traffic-induced vibrations. They claimed that the modes of vibration for the superstructure can be used to identify piers that have scour-affected foundation conditions. In addition, they found that the maximum value of the variance of single signals can be used to detect piers that were affected by foundation scour. Ju [3] developed a finite element method, which accounted for the effect of the soil-fluidstructure interaction, to calculate scoured bridge natural frequencies. Prendergast et al. [4] placed accelerometers on a bridge pile above the waterline to track the depth of scour by observing changes in the natural frequency of the vibration of the foundation piles. Bayraktar et al. [5] built an equation between natural frequency and main bridge span using experimental approach. As shown in the literature, one can use the changes in the frequency of a bridge component (e.g., pier or superstructure) to determine the safety condition of the bridge. However, few of the earlier studies attempted to find the threshold value of the frequency to distinguish between a safe and unsafe bridge. Thus, this study aims to develop a methodology to determine a critical frequency ratio that can be used for the task of regular maintenance. A performance function is first defined to determine the critical bridge condition. A series of pushover analyses are implemented to determine the capacity envelope. The bridge demand is the hydrodynamic pressure that acts on the bridge. Based on the code specification in Taiwan, one can compute the pressure for different scouring situations. When the bridge capacity is equal to the demand, the limit state is called the damaged condition of the bridge. The frequency ratio of critical to undamaged bridges is the critical (safety) frequency ratio. If the measured ratio is less than the critical ratio, the monitored bridge is vulnerable to a flood hazard and a more advance analysis should be conducted to ensure
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the bridge safety. In a field test, this study uses the ambient vibration technique to measure the vibration frequency of an identical bridge before and after a flood season and determines the bridge safety condition based on the safety frequency ratio. The results show that the proposed critical frequency ratio can be used as a safety index to detect the bridge condition. There are many technologies that have been developed to measure the frequency of a bridge, for example, indirect measurement [6] and auto regressive moving average [7]. Because this study only focuses on finding the critical frequency ratio of a bridge, investigating technologies for measuring bridge frequency is beyond the scope of the current study. In addition, only simply supported bridges with a single pier, multiple spans and located in the Gaoping River Basin were studied, so application outside of this limitation should be limited. 2. Calculation of the critical frequency ratio To calculate the critical frequency ratio, one first needs to establish a bridge performance function (limit state function). The simplest performance function is capacity minus demand, which was adopted in this study. A performance equals to zero indicates that the considered bridge has reached the ultimate damaged condition. For a given bridge, its capacity (lateral resistance) and demand (external force applied to the bridge by a flood) depend on the scour depth. In other words, the capacity and demand of a bridge vary for different scour depths. In this study, the capacity envelope of a bridge is computed from pushover analyses, and the demand envelope is calculated based on the code requirement in Taiwan. The determinations of capacity and demand are described below. 2.1. Determination of capacity The seismic capacity curve of a bridge structure is often represented as the curve of the base shear and top displacement obtained from a pushover analysis. This study adopts this idea and defines the capacity of the bridge flood resistance as the yield point in the pushover analysis. Note that the yield force in a pushover curve is actually the base shear of the interested pier. Because the capacity of a bridge varies depending on the scour depth, pushover analyses are performed using different scour depths, as shown in Fig. 1, and the yield points of various pushover analyses are connected to form a curve, i.e., the capacity envelope of a bridge against floods, as shown in Fig. 2. 2.2. Determination of demand The demand of a bridge against floods is defined as the total base shear force that results from the required hydrodynamic pressure. Based on the ‘‘Highway Bridge Design Criteria” (2009), the hydrodynamic pressure is calculated by Eq. (1).
P av g ¼
Fig. 1. Pushover analyses with different scour depth.
52:5KðV av g Þ2 1000
ð1Þ
where Pavg: average hydrodynamic pressure (tf/m2); Vavg: average stream velocity (m/s); and K: a constant coefficient; 1.4 for a blunt nosed pier, 0.7 for a round pier, and 0.5 for a tip pier when the angle is less than or equal to 30°. Please note that theoretically, a critical frequency ratio (Rc) should not depend on any code. A ‘‘true” hydrodynamic pressure must be used. Practically, it is not easy to define a specific hydrodynamic pressure for Taiwan or elsewhere. Based on the study of Wu et al. [15], for bridges in Gaoping River Basin, the code-based hydrodynamic pressure has a similar trend to the average pressure from a 3D fluid-solid-coupled modeling. Thus, the
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envelope of the bridge. The flowchart of determining the demand envelope is provided in Fig. 4. Fig. 5a shows several demand points for the bridge and its corresponding demand envelope curve. Fig. 5b is an example of the capacity and demand envelops for a given bridge. 2.3. Pushover analysis
Fig. 2. An illustration of capacity envelope.
pressure equation defined in Taiwan code is used to calculate the bridge demand. Application outside of the Gaoping River Basin should be cautious. To compute the corresponding base shear of Eq. (1), the distribution of the hydrodynamic pressure on the bridge is needed. For a bridge with a single pier, the pressure distribution is shown in Fig. 3, where the maximum pressure Pmax is twice as large as Pave. The hydrodynamic pressure has a triangular distribution and changes from Pmax at the top of the stream level to zero at the streamline. The base shear force can be calculated by multiplying the pressure by the area of the stream flowing across the bridge pier. If the stream elevation is higher than the bottom of the bridge girder, the effect of the superstructure must also be considered. At this point, the hydrodynamic pressure on the superstructure is uniformly distributed with a value of Pmax. After obtaining the base shear, the next step is to locate the resulting displacement. The pushover analysis provides a force and displacement relationship for the bridge. Thus, once the base shear is calculated, this study uses a pushover curve to find the corresponding displacement. Similar to the situation of calculating the capacity, the bridge demand also depends on the scour depth. Thus, this procedure is repeated to locate several demand points, where a pair is the base shear and corresponding displacement, to form the demand
Fig. 3. The water pressure distribution.
As explained in Sections 2.1 and 2.2, the pushover analysis is used to calculate both the capacity and demand of a bridge. SAP2000 is adopted to conduct the pushover analysis and detailed implementation is provided in this section. Many studies about the pushover analysis of bridges have been published, but few of them use the foundation scouring depth in the evaluation process. The common pushover analysis assumes the foundation to be a fixed support. Because this study mainly evaluates the bridge safety after a scour occurred, the soil property of the foundation is also considered in the pushover analysis. A detailed plastic hinge property of the pier and soil property are provided below. 2.3.1. Modelling of the pier To accurately simulate the pier behavior, core/cover concrete and steel are considered separately. The concrete model proposed by Mander et al. [8] is used to simulate the confined (core) and unconfined (cover) concrete. For the confined concrete under a quasi-static monotonic loading, the longitudinal compressive concrete stress (fc) is given by:
Fig. 4. The flowchart of determining the demand.
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area of the longitudinal reinforcement to the area of the core of the section, qs is the ratio of the volume of the transverse confining steel to the volume of the confined concrete core, fyh is the yield strength of the transverse reinforcement, and x in Eq. (2) is calculated by:
x¼
ec ecc
ð5Þ
where ec is the longitudinal compressive concrete strain. ecc is calculated by:
"
!# 0 f cc 1þ5 0 1 f co
ecc ¼ eco
ð6Þ 0
where eco is corresponding unconfined concrete strain of the f co and is 0.002, as suggested by Mander et al. [8]. r in Eq. (2) is calculated by:
c¼
Ec Ec Esec
ð7Þ
qffiffiffiffiffiffi 0 where Ec ¼ 5000 f co MPa is the tangent modulus of elasticity of the concrete and 0
Fig. 5a. Illustration of demand points and demand envelope curve.
Esec ¼ 0
fc ¼
f cc xr r 1 þ xr
ð2Þ
0
where f cc is the compressive strength for the confined concrete and can be determined by: 0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 0 0 7:94f l fl 1:254 þ 2:254 1 þ 0 2 0 f co f co
0
f cc ¼ f co 0
ð3Þ 0
where f co is the unconfined concrete compressive strength and f l is the effective confining stress on the concrete and is calculated by: 0
fl ¼
2 0 1 2ds s 2ð1 qcc Þ
qs f yh
ð4Þ
where s0 is the clear vertical spacing between the spiral or hoop bars, ds is the diameter of the spiral or bars, qcc is the ratio of the
f cc
ð8Þ
ecc
A similar approach can be applied to simulate the cover concrete. For the case of ec < 2eco, one can use Eq. (9) to simulate the cover concrete behavior. 0
fc ¼
f c xco r r 1 þ xr
ð9Þ
where xco ¼ ec =eco , c ¼ Ec =ðEc Esec Þ and Esec ¼ f c =eco . For the case 0
ec P 2eco , 0
fc ¼ fc
2r r 1 þ xr
1
ec 2eco esp 2eco
ð10Þ
where esp = 0.005 is the spalling strain. For the case of
fc ¼ 0
ð11Þ
The steel stress–strain curve used in this study is described below. For the case of 0 6 es 6 ey
f s ¼ Es es
ð12Þ
For the case of
ey 6 es 6 esh
fs ¼ fy
ð13Þ
For case of
"
fs ¼ fy
Fig. 5b. Illustration of the cricital point (the cross point).
ec P esp ,
esh 6 es 6 esu ,
fu fy fu fy fy
!
eu es eu esh
2 # ð14Þ
where fs is the steel stress, Es is the elastic modulus of the steel, es is the steel strain, fy is the yield stress of the steel, fu is the ultimate steel stress, esh is the strain hardening strain, and eu is the ultimate steel strain. To account for the effect of the axial load on the characteristics of the column plastic hinge, the bridge capacity and demand are determined through two consecutive pushover analyses. In the first pushover analysis, the column plastic hinge behaviors are described by the combination of flexural and shear behaviors under the dead load condition (referred to as the DL curve, Fig. 6). The flexural behavior can be established using the aforementioned stress–strain curves of concrete and steel together with moment–curvature analysis. The column shear behavior is built using the stress–strain curves and Eq. (15). The second pushover
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2.3.2. Modelling of the soil property Because soil plays an important role in bridge safety against floods, it is necessary to account for the soil property in the numerical model. The corresponding equivalent soil springs are set around the pile according to their depths. The stiffness/coefficient of soil spring varies significantly for different empirical equations. One convenient and reasonable approach is to use the value of the standard penetration test for calculating the stiffness, which is adopted here. Simulation of the soil property is described below. For a pile foundation, to determine the equivalent soil spring, the first step is to calculate the horizontal subgrade reaction coefficient (kH). kH can be calculated by Eq. (16) [11]:
0:75 9 > kH ¼ kH0 B30H > = pffiffiffiffiffiffiffiffiffi > BH ¼ D=b > qffiffiffiffiffiffi > > 4 ; HD b ¼ k4EI
ð16Þ
where kH0 is the baseline value of the horizontal subgrade reaction coefficient (kg/cm3), BH is the equivalent pile length perpendicular to the loading direction, D is the pile diameter, EI is the flexural stiffness of the pile, and 1/b is the pile depth under consideration. kH0 is computed by: Fig. 6. Determination of the flexural behaviors used in the second pushover analysis.
kH0 ¼
1 30
ED
ED ¼ 2ð1 þ mD ÞGD ct GD ¼ 10g V 2SD
analysis is similar to the first one except that the flexural and shear behaviors are different. The flexural and shear behaviors are established again (referred to as the UL curve, Fig. 6), but the ultimate vertical load, which corresponds to the peak base shear in the first pushover analysis, replaces the dead load [9,10]. The flexural and shear behaviors used in the second pushover analysis are determined by combing the DL and UL curves. To be exact, the final yield point is the average of the yield points of the DL and UL curves, and the final ultimate point is the ultimate point on the UL curve. For instance, the flexural behavior under a dead load and ultimate axial loading is represented by curves ABDLCDL and ABULCUL, respectively (Fig. 6). The final curve is ABC. Point B is the average between points BDL and BUL, whereas point C is coincident with point CUL. That is, this study determines the flexural behavior in the second pushover analysis by combing curves from two moment-rotation analyses. The first and the second moment-rotation analysis are conducted under the dead load and ultimate vertical load, respectively. A similar rule is applied to determine the final shear behavior.
Vn ¼ Vc þ Vs qffiffiffiffi 0 V c ¼ 0:166ðk þ FÞ f c Ae ðMPaÞ
ð15Þ
9 > =
ð17Þ
> ;
where ED is the soil dynamic elastic modulus, vD is the soil dynamic Poisson’s ratio, GD is the soil dynamic shear modulus, ct is the soil weight per volume (t/m3), g is the acceleration due to gravity, and VSD is the shear wave speed and can be calculated by:
V SD ¼ C v V s
ð18Þ
where Cv is the modification factor and is calculated by:
Cv ¼
0:8ðV s < 300 m=sÞ
ð19Þ
1:0ðV s P 300 m=sÞ
where Vs is the average shear wave speed and can be obtained by:
(
Vs ¼
8N1=3 ðsandÞ
ð20Þ
100N1=3 ðclayÞ
where N is the value of the standard penetration test. The horizontal subgrade reaction coefficient (kH), geometry of the foundation (e.g., the location of each pile) and distribution of the soil layers can be used to obtain the equivalent spring coefficient for each direction using simple force equilibrium equations. Please note that an ultimate soil spring stress, described in Eq. (21), is specified in code [14] and is adopted to form a linear elastic-perfectly plastic soil spring.
pffiffiffiffiffiffi
rP ¼ r0Z K P þ 2c K P
ð21Þ
where r0Z is the effective vertical stress (tf/cm ), K P is the coefficient of passive earth pressure, c is the cohesion of soil (tf/cm2). 2
Ash f y da Vs ¼ s where k is the rotation ductility factor, which equals 1 before flexural yielding and decreases linearly to zero at the flexural ultimate condition; F is the axial load effect on the shear strength, which equals N/(13.8Ag) (MPa) for a compressive axial load and N/(3.5Ag) (MPa) for a tensile axial load, where N is the axial force and Ag is the gross cross-sectional area; Ae is the effective area, which equals 0.8 Ag; Ash is the area of the shear reinforcement; d is the effective sectional depth; s is the vertical spacing; and a = 1 for a rectangular section and p/2 for a circular section.
2.3.3. Location of the plastic hinge The plastic hinge typically occurs in the substructure under external forces, which dominates the pushover curve of a bridge. Wang et al. [12] indicated that it is very difficult to predict the plastic hinge location on the substructure because the highly nonlinear relationship between the maximum bending moment and the distribution of the soil layers. On the other hand, Foti and Sobia [2] stated that plastic hinges are typically located at pier ends. The plastic hinge is only allowed to occur at the pier end, pile top or pile end if the corresponding moments reach the yielding limit.
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K.-W. Liao et al. / Construction and Building Materials 102 (2016) 552–563 Table 1 Assessment items used in the preliminary evaluation procedure in Taiwan.
Fig. 7. Determination of the critical frequency by interpolating between the frequencies (bridges with scour depth of 6 and 9 m).
Therefore, this study does not consider the plastic hinge location effect.
Assessment Item
Item
Weight (a)
No. No. No. No. No. No. No. No. No. No. No. No. No.
Upstream river dam or reservoir facilities Foundation type Bending or narrowing of the river Eroded river bed Material on the river bed Location of the main channel Hydraulic drop effect Attack angle of flow Area ratio of bridge to cross section Foundation exposure depth Effective pier diameter Protection for river bank Protection for river bed
7 7 6 8 2 5 5 7 5 20 13 5 10
1 2 3 4 5 6 7 8 9 10 11 12 13
score and is the most dominating parameter which is consistent with the assumption of the current study. However, the preliminary evaluation table only provides an overall weighting among parameters. For each bridge, a further investigation should be individually conducted. This study adopts the Bayesian network (BN) to perform the sensitivity analysis. Reasons for adopting BN are:
2.4. Definition of the critical frequency ratio The critical frequency ratio is defined in Eq. (22).
Rc ¼
fa fb
ð22Þ
where Rc is the critical frequency ratio, fa is the frequency corresponding to a damaged bridge, and fb is the frequency corresponding to an undamaged bridge. The calculation of fb is relatively easy. fb is the first frequency of the original bridge (with no scour) obtained from a modal analysis. As mentioned in Section 2.2, the cross point in Fig. 5(b) represents a bridge that is in its limit state. As shown in Fig. 7, the capacity and demand envelopes intersect between a scour depth of 6 m and 9 m. fa is then calculated by interpolating between the two frequencies. If field frequency measurements are performed before and after a main flood event, a ratio can be calculated as shown in Eq. (23).
R¼
f ma f mb
ð23Þ
where R is the measured frequency ratio of the considered bridge and fma and fmb are the frequencies measured after and before a flood, respectively. If the measured R ratio is less than Rc, this indicates that the bridge may have serious damage and that a more advanced analysis should be performed to ensure its safety. 3. Parametric study and a case study This study considers the scouring depth as the most important parameter for a bridge against floods. As a result, a scouringinduced critical frequency ratio (Rc) is proposed as a bridge health indicator. However, whether the scouring depth is the most dominating parameter is questionable and should be verified first. In Taiwan, a two-steps evaluation is often used to examine the bridge safety condition. The first step is basically a visual examination based on a comprehensive table as shown in Table 1. It is seen that Table 1 includes 13 parameters that have potential influence on the bridge failure. Using this preliminary form as our analyzing basis, a sensitivity analysis is conducted to examine the relative importance of these 13 parameters for bridges in Gaoping River Basin. Based on the weight (a in Table 1) of 13 parameters, it is seen that foundation exposure depth weights 20% of the overall
(1) Many parameters affect bridge safety. The BN can break down these various parameters and makes them as nodes in the network, allowing one to focus on the causal relationship between two nodes at each time. Thus, the problem’s complexity is reduced. (2) The BN allows users to update the bridge safety through Bayesian learning. Because the preliminary evaluation table has not been revised for a decade, any recorded event in this period should be considered. That is, in the proposed BN, the role of preliminary evaluation table is considered as the expert’s opinions and the recorded information is the new evident. To construct a BN, the first step is to establish the network topology of BN that reflects causal relationships of various nodes. The second step is to establish the conditional probability between the nodes. Based on the existing preliminary evaluation table, this study used the AgenaRisk software to establish the proposed BN and its topology is shown in Fig. 8. A BN-based sensitivity analysis is performed for 16 bridges in Gaoping River Basin. A typical Tornado graph is displayed in Fig. 9. Table 2 displays the statistics for the top four dominating factors in 16 bridges. As expected, the dominating factor varies among different bridges. However, it is seen that the scouring depth is the most influencing factor in bridge health evaluation. Although the scouring depth has been identified as the most influencing factor for bridges in Gaoping River Basin, analyzing all bridges in this area following the proposed procedure is still a heavy task. For this situation, the bridges are classified into groups to capture the influence of each parameter and, at the same time, to increase the analysis efficiency. According to the proposed approach for the critical frequency ratio, it is easy to recognize that the most important influencing parameters are the pier diameter, pier length, pile diameter and pile length. Based on the literature and our earlier survey of bridges in Taiwan, the range of these parameters is displayed in Table 3. Note that these ranges only provide an estimation for bridges with a single pier in Taiwan. In the parametric analysis, five types of bridges were analyzed to determine the effects of each parameter on the critical frequency ratio. The principle of classification is described as follows: (1) the factors with the most significant influence on the pushover analysis are selected as control variables, including the pier
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Fig. 8. Bayesian network used in current study.
Table 2 Statistics for the top four dominating factors in 16 bridges.
*
Factor
1st
2nd
3rd
4th
Total
Foundation exposure depth Descending river bed River bank erosion Water blocking ratio
10* 6 0 0
4 6 6 0
2 4 10 0
0 0 0 16
16 16 16 16
Total
16
16
16
16
Number of appearance in this place.
Fig. 9. A typical Tornado graph for bridge health in Gaoping River Basin.
diameter, pier length, pile diameter and pile length; (2) except for the parameter of pile length, only the maximum and minimum values in the statistics are selected; and (3) the pile length is divided into four different values (i.e., 40, 20, 15 and 10) to examine the scour effect more thoroughly, and the grouping result is shown in Table 4. According to Table 4, the first type represents bridges with maximum capacity, and the second type of bridge has the second largest capacity. In other words, the larger the type number, the smaller the bridge capacity. When performing pushover analysis, each type of bridge is evaluated with five different SPT-N values. Because the SPT-N value is usually less than 50 unless the soil condition is close to a rock condition, the SPT-N values considered in this study are divided into five different values of 10, 20, 30, 40 and 50. In addition, for each representative bridge, the SPT-N value is assumed to be constant along the depth. Analysis results of the five types of bridges were consistent with many fundamental concepts of bridge engineering, which indicates that the proposed algorithm that determines the critical frequency ratio was well defined. Details are provided below. 3.1. Capacity analysis of five types of representative bridges Because the analysis processes for five types of bridges are similar, only the analysis procedure of the first type of bridge is provided below. As indicated in Table 4, the first type of bridge has a pier diameter of 4.2 m, pier length of 7.7 m, pile diameter of 1.5 m and pile length of 40 m. To examine the scour effect thoroughly,
Fig. 10. Bridge model built by SAP2000.
five scour depths at 0 (no scour), 3, 6, 9 and 12 m were considered. In the pushover analysis, if there is no scour (i.e., no foundation exposure), the potential plastic hinge is set at the bottom of the bridge pier. If the foundation is exposed, the probable positions of the plastic hinge are the end of the pier, pile top and bottom (i.e., the fixed end of the pile). Fig. 10 shows the bridge model in SAP2000. Because the in-plane stiffness is comparative large, a rigid diaphragm in-plane is adopted to simulate the bridge deck. The connections between pier and deck are modelled as hinge or roller depending on design drawing. The mass of the superstructure is included in the model to accurately estimate the pier frequency. Fig. 11 shows the capacity curves at various scour depths when the standard penetration SPT-N value is 50, where the y-axis unit is ton and the x-axis unit is cm.
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559
Fig. 11. Capacity curves of the first type bridge at various scour depths (top to bottom: 0, 3, 6, 9 and 12 m).
Table 3 General ranges of bridge sizes in Taiwan. Item
Pier dia.
Value (meter)
Pier length
Pile dia.
Pile length
Max.
Min.
Max.
Min.
Max.
Min.
Max.
Min.
4.2
1.8
13.4
7.7
1.5
0.4
40
10
Fig. 13. Capacity curves for type I (top) and II (bottom) bridges (N = 50 and no scour).
SPT-N value will have a higher capacity. Combining this information with Table 5, one can conclude that a higher bridge vibration frequency means a higher bridge capacity. Fig. 13 displayed the lateral capacity for type I and II bridges when SPT-N is 50 and there was no scour. It was observed that the difference was slight. However, when the foundation exposure was large, 12 m for example, the difference between these two types of bridge was significant, as shown in Fig. 14. It was seen that the scour depth dominated the lateral capacity of the bridge.
3.2. Demand analysis of five types of representative bridges
Fig. 12. Capacity curves of the first bridge type without scour.
When the SPT-N value is small (soil strength is low) and the structure is strong, the bridge may lose its lateral resistance before any plastic hinge is formed. When the SPT-N value is large (soil strength is high), losing lateral capacity is often accompanied with the formulation of a plastic hinge. The former case should usually be avoided because engineers expect the ultimate strength of a bridge to be controlled by the bridge itself. Table 5 displayed the vibration frequency and cycle at different scour depths for different SPT-N values. For example, when SPT-N = 50, the vibration frequencies of the bridge structure are 2.7233, 1.8412, 1.2694, 0.92068 and 0.69888 Hz for scour depths of 0, 3, 6, 9, 12 m, respectively. When scour occurred, the vibration frequency of the bridge decreased as the foundation exposure increased. Fig. 12 shows the capacity curves of the first bridge type without scour, and the SPT-N values are from 10 to 50. As expected, a bridge with a higher
As introduced in Section 2.2, the demand point consists of the pairing of the base shear and corresponding displacement that can be obtained by the code specification and pushover analysis, respectively. The demand points are therefore influenced by the stream velocity, scour depth and soil property (SPT-N value). To investigate the effects of these influencing factors on the lateral demand, three stream velocities based on the past record were selected for the demand analysis: 6.49, 3.82 and 1.15 m/s, which are the largest, average and smallest velocity of 100-year floods in the Gaoping River Basin [13]. In addition, five different scour depths of 0, 3, 6, 9 and 12 m were considered, and five different N-values were used for the soil property. The demand points of the first type of bridge at varied velocities, scour depths and SPT-N values are shown in Table 6, where only SPT-N = 10 and 50 are displayed. The demand envelope was drawn by connecting the points in these tables, as shown in Fig. 15 (case of SPT-N = 50 only). According to Fig. 15, when the velocity is high, the demand force of a bridge is greater than when the velocity is moderate, which indicates that a bridge is more likely to be damaged by a higher velocity attack. If the velocity is not large enough, more lateral displacement was needed to cause bridge failure. Based on Fig. 15, for the first type of bridge, if the water velocity is low, the bridge will function well no matter what the value of SPT-N is. These numerical results align with a general physical understanding of bridge engineering.
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K.-W. Liao et al. / Construction and Building Materials 102 (2016) 552–563 Table 5 Frequencies and periods of the first type bridge with varied SPT-N values and scour depths.
Fig. 14. Capacity curves for type I (top) and II (bottom) bridges (N = 50 and scour depth = 12 m).
Table 4 Five representative bridges analyzed in this study (unit: meter).
Pier dia Pier length Pile dia Pile length
Type I
Type II
Type IIII
Type IV
Type X
4.2 7.7 1.5 40
4.2 7.7 1.5 20
1.8 13.4 0.4 20
1.8 13.4 0.4 15
1.8 13.4 0.4 10
Fig. 15. Capacity and demand envelopes for the type I bridge with N = 50.
3.3. Discussion of the critical frequency ratio Rc Based on Section 2.4, fa and fb are needed to compute the value of Rc. fb can be obtained from the modal analysis of an undamaged bridge (i.e., no scour); fa is the frequency of the bridge that
Scour depth
SPT-N-value
Period (s)
Frequency (Hz)
0m
10 20 30 40 50
0.46 0.42 0.39 0.38 0.37
2.18 2.41 2.55 2.65 2.72
3m
10 20 30 40 50
0.64 0.60 0.57 0.56 0.54
1.55 1.68 1.75 1.80 1.84
6m
10 20 30 40 50
0.89 0.84 0.82 0.80 0.79
1.12 1.18 1.22 1.25 1.27
9m
10 20 30 40 50
1.20 1.15 1.12 1.10 1.09
0.83 0.87 0.89 0.91 0.92
12 m
10 20 30 40 50
1.56 1.50 1.47 1.45 1.43
0.64 0.67 0.68 0.69 0.70
corresponds to the intersection point of the capacity and demand envelopes. Tables 7–11 display the results of fa, fb and Rc of five representative bridges, respectively. The Rc value varies with the properties of the bridge, namely, the Rc is influenced by the structural strength. Specifically, the Rc value is smaller when a bridge has a higher stiffness. For example, the Rc values of types 1 and 2 bridges were smaller than those of bridge types 3, 4 and 5. The proposed Rc suggested that a bridge with a higher stiffness has more room to withstand a loss of structural integrity. For the first type of bridge, Rc values ranged from 0.64 to 0.71 and 0.41 to 0.45 for the case of high and moderate velocities, respectively. The higher stream velocity resulted in a larger Rc value, which indicated that the bridge was more likely to be damaged under a high velocity attack even though the structural strength of the bridge did not decrease significantly. On the other hand, if the bridge was scoured by a moderate velocity attack, the bridge may be damaged when the Rc value decreases greatly. According to Tables 7–11, the SPT-N value only has a moderate effect on the Rc value (it is only a little significant for the first type of bridge). Because the proposed Rc depends on fa and fb, which are influenced by the SPT-N value simultaneously, it is expected that the SPT-N value will not be a key factor on Rc; this agrees with the results shown here (see Tables 7–11). To derive a simple, clear, convenient and convincing rule for bridge evaluation based on results from the parametric study, the 5 representative bridges are further categorized into two groups: group A and B. The type I and II bridges are considered as group A. On the other hand, other bridge types are considered as group B. Three scenarios are used to determine the Rc value: the worst, mean value and average of the former two scenarios. Table 12 displays the analyzed results. It is seen that the suggested Rc ranging from 0.70 to 0.92 if the average scenario is adopted. Although the application of current study is limited to bridges in Gaoping River Basin, the Rc range found here is similar to the earlier research conducted by Nishimura [16]. Nishimura [16] collected 1000 first mode frequencies for bridges in Japan. The threshold values of Rc based on the on-site measurement ranges
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K.-W. Liao et al. / Construction and Building Materials 102 (2016) 552–563 Table 6 The demand pairs of type I bridge with N = 10 and 50. Scour depthd
N = 10 H F
0 3 6 9 12 a b c d
N = 50
a
b
500 940 1230 1525 1820
a
M c
b
L c
D
F
D
2 9 21 45 90
170 370 580 810 1065
1 7 12 25 49
a
Ha
b
D
15 30 50 75 100
0 0 1 2 5
F
c
F
b
500 940 1230 1525 1820
Ma D
c
2.0 6.7 17 39 77
La
b
c
F
D
170 370 580 810 1065
0 3 8 21 46
Fb
Dc
15 30 50 75 100
0 0 1 2 4
H, M and L denotes the velocity of stream are high, medium and low, respectively. F denotes the force (ton). D denotes the displacement (cm). Unit: meter.
Table 7 The Rc values of bridge type I.
Table 10 The Rc values of bridge type IV.
Stream velocity
SPT-N values
fb (Hz)
fa (Hz)
Rc
Stream velocity
SPT-N values
fb (Hz)
fa (Hz)
Rc
High
10 20 30 40 50
2.18 2.41 2.55 2.65 2.72
1.55 1.67 1.59 1.74 1.74
0.71 0.69 0.62 0.66 0.64
Medium
10 20 30 40 50
0.75 0.77 0.78 0.79 0.79
0.72 0.73 0.75 0.76 0.76
0.96 0.95 0.96 0.96 0.95
Medium
10 20 30 40 50
2.18 2.41 2.55 2.65 2.72
0.98 1.05 1.10 1.12 1.13
0.45 0.44 0.43 0.42 0.42
Low
10 20 30 40 50
0.75 0.77 0.78 0.79 0.79
0.62 0.62 0.63 0.61 0.63
0.83 0.81 0.81 0.77 0.80
Table 8 The Rc values of bridge type II.
Table 11 The Rc values of bridge type V.
Stream velocity
SPT-N values
fb (Hz)
fa (Hz)
Rc
Stream velocity
SPT-N values
fb (Hz)
fa (Hz)
Rc
High
10 20 30 40 50
1.95 2.18 2.32 2.42 2.50
1.50 1.66 1.75 1.82 1.88
0.77 0.76 0.76 0.75 0.75
Medium
10 20 30 40 50
0.71 0.74 0.75 0.76 0.77
0.67 0.71 0.72 0.74 0.74
0.95 0.96 0.96 0.97 0.97
Medium
10 20 30 40 50
1.95 2.18 2.32 2.42 2.50
1.41 1.55 1.62 1.68 1.74
0.72 0.71 0.70 0.69 0.70
Low
10 20 30 40 50
0.71 0.74 0.75 0.76 0.77
0.63 0.62 0.61 0.62 0.62
0.89 0.84 0.81 0.82 0.81
Table 9 The Rc values of bridge type III.
Table 12 Rc values in three different scenarios.
Stream velocity
SPT-N values
fb (Hz)
fa (Hz)
Rc
Scenario
Group A
Group B
All bridges
Medium
10 20 30 40 50
0.76 0.78 0.79 0.80 0.80
0.73 0.74 0.75 0.76 0.77
0.95 0.95 0.95 0.95 0.96
The worse The mean value The average of the first two
0.77 0.64 0.70
0.97 0.88 0.92
0.77–0.97 0.64–0.88 0.70–0.92
Low
10 20 30 40 50
0.76 0.78 0.79 0.80 0.80
0.61 0.60 0.62 0.60 0.60
0.80 0.77 0.78 0.75 0.75
from 0.7 to 0.85. It is seen that the proposed parametric study is able to deliver a reasonable range for Rc. 3.4. Case study The Jhih-Sin Bridge located in the Yi-Lan county was selected to evaluate its flood-resistant capacity using the proposed critical
frequency ratio. The Jhih-Sin Bridge consists of two independent bridges for the northbound and southbound lanes, and both of them are pre-stress concrete bridges with six spans. The span length is approximately 25.6 m, and a single pier was used in the substructure for both directions. The north- and southbound bridges were built in 1989 and 1997, respectively. Only the southbound bridge was analyzed in this study, as shown in Fig. 16. The frequency before and after a flood season (i.e., the summer in 2014) are measured to calculate the frequency ratio (R in Section 2.4). Because the design drawing of this bridge is incomplete, the Rc value is determined according to the earlier parametric study. The pier length and diameter of the Jhih-Sin Bridge are 8 and 3.5 m, respectively. Such size is closer to Type I or II bridges. The
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(2)
(3)
(4)
(5) (6)
(7) Fig. 16. Jhih-Sin Bridge.
Table 13 The fma, fmb and R values of the Jhih-Sin Bridge. fmb
fma
R
Rc
4.35
4.1
0.94
0.77
highest Rc from above two bridges is selected (i.e., 0.77) to verify the bridge safety. The ambient method was used to measure the bridge frequency in the field. Table 13 displayed the frequencies before and after the flood season. Based on the results, the corresponding R value is approximately 0.94 (4.1/4.35), which was larger than the Rc, indicating that this bridge does not need an advance analysis to ensure its safety. From the field observation, no serious damage was found during 2014, which meets the conclusion of the proposed preliminary safety evaluation process. 4. Conclusion Frequency is often considered to be an ideal safety indicator in bridge health monitoring. However, few of past studies have focused on finding a threshold frequency to detect the bridge limit state. This study focused on finding a critical frequency ratio (Rc) as a safety indicator that can be used to distinguish undamaged and damaged bridges. Rc was governed by the bridge capacity and demand that were determined using pushover analyses. 54 bridges in Gaoping River Basin were damaged during the attack of Typhoon Morakot (2009). The government spent 5 years to recover all the damaged bridges. To maintain the retrofitted or undamaged bridges in a healthy condition in the future, a priority maintenance list is needed. In such case, an efficient method is often adopted. The developed evaluation procedure fulfills such goal and can provide a preliminary health inspection for bridges in Gaoping River Basin. A more advance and expensive technique such as pushover analysis should be used if the measured frequency is below Rc. Based on calculations of the bridge capacity, demand and Rc, several conclusions were drawn below: (1) When the SPT-N value is small and the structure is strong, the bridge may lose its lateral resistance before any plastic
hinge is formed. When the SPT-N value is large, losing lateral capacity is often accompanied with the formulation of a plastic hinge. The vibrational frequency of a bridge decreased as the foundation exposure increased. A higher capacity of the bridge results in a higher bridge vibrational frequency. The scour depth has a significant impact on the bridge frequency. A larger demand force is observed for the case of a high velocity, which results in a larger Rc value. This means that a bridge attacked by a high velocity is more likely to be damaged. The Rc value is smaller for a bridge with a higher stiffness, which means the proposed Rc suggests that this bridge has more room to withstand the loss of structural integrity. It is seen that the SPT-N value does not have a significant influence when using frequency as a bridge safety index. The proposed critical frequency ratio (Rc) is a velocity and structural strength-dependent indicator. Therefore, the safety threshold for a bridge is not a deterministic number and varies depending on the stream velocity and the strength of a bridge. Based on the average scenario, the suggested Rc ranges from 0.70 to 0.92. This range is consistent with earlier study conducted in Japan, indicating it is a reasonable and should be useful for engineers in bridge preliminary health investigation.
A single bridge test in the field does not provide a great deal of information regarding the accuracy of the proposed methodology. However, the results of the parametric study are very encouraging and it is possible that the proposed method can provide a reasonable and efficient tool for a preliminary bridge health evaluation against floods. Please note that the proposed safety ratio only considered bridges with a single pier against a flood hazard. This limit should be considered when applying the method. Acknowledgements The authors would like to thank the Harbor and Marine Technology Center, Ministry of Transportation and Communications and National Science Council for financially supporting this study. References [1] A. Elsaid, R. Seracino, Rapid assessment of foundation scour using the dynamic features of bridge superstructure, Constr. Build. Mater. 50 (2014) 42–49. [2] S. Foti, D. Sabia, Influence of foundation scour on the dynamic response of an existing bridge, J. Bridge Eng. 16 (2) (2011) 295–304. [3] S.H. Ju, Determination of scoured bridge natural frequencies with soil– structure interaction, Soil Dyn. Earthqauke Eng. 55 (2013) 247–254. [4] L.J. Prendergast, D. Hester, K. Gavin, J.J. O’Sullivan, An investigation of the changes in the natural frequency of a pile affected by scour, J. Sound Vib. 332 (2013) 6685–6702. [5] A. Bayraktar, T. Türker, A.C. Altunisik, Experimental frequencies and damping ratios for historical masonry arch bridges, Constr. Build. Mater. 75 (2015) 234– 241. [6] C.W. Lin, Y.B. Yang, Use of a passing vehicle to scan the fundamental bridge frequencies: an experimental verification, Eng. Struct. 27 (13) (2005) 1865– 1878. [7] S. Ólafsson, R. Sigbjörnsson, Application of ARMA models to estimate earthquake ground motion and structural response, Earthquake Eng. Struct. Dyn. 24 (7) (1995) 951–966. [8] J.B. Mander, M.J.N. Priestly, R. Park, Theoretical stress–strain model for confined concrete, J. Struct. Eng. ASCE 114 (8) (1988) 1804–1826. [9] Y.C. Sung, K.Y. Liu, C.K. Su, I.C. Tsai, K.C. Chang, A study on pushover analyses of reinforced concrete columns, J. Struct. Eng. Mech. 21 (1) (2005) 35–52. [10] Y.C. Ou, H.D. Fan, N.D. Nguyen, Long-term seismic performance of reinforced concrete bridges under steel reinforcement corrosion due to chloride attack, Earthquake Eng. Struct. Dyn. 42 (2013) 2113–2127. [11] I.C. Tsai, D.W. Chang, J.S. Hwang. Studies of seismic design specifications for highway bridges. ministry of transportation and communications Taiwan area expressway engineering Bureau, 1997.
K.-W. Liao et al. / Construction and Building Materials 102 (2016) 552–563 [12] H. Wang, C.Y. Wang, C.H. Chen, C. Lin, T.R. Wu, An integrated evaluation on flood resistant capacity of bridge foundations, Sino-Geotech 127 (2011) 51–60. [13] Ministry of Economic Affairs, Regulation Master Plan of Gaoping River, Water Resources Agency, Taiwan, 2008. [14] Ministry of Economic Affairs. Seismic design specifications for highway bridges. Ministry of Transportation and Communications Taiwan Area Expressway Engineering Bureau, 2015.
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[15] T.R. Wu, H. Wang, Y.Y. Ko, J.S. Chiou, S.C. Hsieh, C.H. Chen, C. Lin, C.Y. Wang, M.H. Chuang, Forensic diagnosis on flood-induced bridge failure. II: framework of quantitative assessment, J. Perform. Constr. Facil. 28 (1) (2014) 85–95. [16] A. Nishimura, Examination of bridge substructure for integrity, Jpn Rail Eng. 114 (1990) 13–17.
Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Preliminary bridge health evaluation using the pier vibration frequency Kuo-Wei Liao a,⇑, Min-Yuan Cheng a, Yung-Fang Chiu b, Jin-Han Lee a a b
Department of Civil & Construction Engineering, National Taiwan University of Science & Technology, Taipei, Taiwan Harbor and Marine Technology Center of the Ministry of Transportation and Communications, Taiwan
h i g h l i g h t s A critical frequency ratio (Rc) is proposed for preliminary bridge health evaluation. Rc is a velocity and structural strength-dependent indicator. SPT-N value does not have a significant influence on Rc. Scour depth has a significant impact on the bridge frequency. Scour depth is one of the most essential factors for bridge safety.
a r t i c l e
i n f o
Article history: Received 7 April 2015 Received in revised form 17 October 2015 Accepted 2 November 2015 Available online 13 November 2015 Keywords: Bridge safety Frequency Pushover analysis Capacity envelope Demand envelope
a b s t r a c t Frequency is often considered to be an ideal safety indicator in bridge health monitoring. Many scholars have related bridge scour depth to frequency. However, few of previous studies have focused on finding a threshold frequency that corresponds to the limit state of bridge failure. Thus, this study focuses on finding a critical frequency ratio (Rc) that can be used to distinguish between undamaged and damaged bridges. The bridge limit state was calculated by subtracting the demand from the capacity of a bridge. Both the capacity and demand envelopes of a bridge were determined using pushover analyses. Nonlinear behaviors of core/cover concrete and steel reinforcement were considered. In the end of this study, a range of frequency ratio is proposed to be an on-site indicator to evaluate bridge safety. Although the proposed safety frequency ratio is consistent with a field observation, the parameters used in this study are from Taiwan only. In addition, the proposed safety ratio only considered bridges with a single pier against flood hazard. The limits should also be considered for further applications. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Over the long-term operation of a bridge, damage to the bridge components is inevitable from natural hazards or deterioration. Serious damage may affect the performance of a bridge, which influences traffic operation and may even lead to the loss of lives. To ensure road users’ safety and prolong the service life of bridges, routine inspection is necessary. The inspection results are often used to determine the repair schedule, range and degree. Therefore, an inspection program plays an important role in a bridge maintenance plan. Among the many natural hazards, floods have drawn a great deal of attention after several Typhoon strikes in Taiwan. Morakot in 2009 caused serious damage in the southern part of Taiwan. Thus, a need exists for the government to build a ⇑ Corresponding author at: Department of Civil & Construction Engineering, National Taiwan University of Science & Technology, No. 43, Sec. 4, Keelung Rd., Taipei 106, Taiwan. E-mail address: [email protected] (K.-W. Liao). http://dx.doi.org/10.1016/j.conbuildmat.2015.11.011 0950-0618/Ó 2015 Elsevier Ltd. All rights reserved.
maintenance list of all existing bridges. Because the number of bridges is large, composing an ideal maintenance list is a difficult task. Thus, effectively evaluating the bridge health condition is an important issue in Taiwan. Engineers in Taiwan often use a preliminary inspection evaluation form (PIEF) to help with the evaluation process. The PIEF is a visual inspection process that is generally considered to be the first step in a bridge safety evaluation. If the overall assessment score from the PIEF does not meet a predefined standard, the evaluation should proceed to an advanced investigation (e.g., finite element analysis, nondestructive or destructive inspection) to ensure the safety of the bridge. Because many bridge failures accompany pier scour, 60% of the PIEF items are highly related to the scour factor. The scour factor plays a key role in a bridge safety evaluation against floods. On the other hand, visual inspections may have potential problems. For example, without help from any instrument, engineers usually cannot accurately estimate the bridge condition. To be
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exact, the scour depth is usually below the water level, so it is difficult to access. In addition, the scour hole is often filled after floods, which makes it difficult to estimate the impact of the infill materials on the bridge safety. A subjective judgment is inevitable in visual inspections, and therefore, there is a need to develop an alternative method to monitor a long list of bridges on a regular basis. Many nondestructive inspection techniques can be utilized to find the local characteristics of bridges, such as using stress waves to inspect the crack depth and concrete cover thickness. Elsaid and Seracino [1] investigated the potential application of using the change in the mode shape curvature and flexibilitybased deflections in scour detection. They concluded that horizontally-displaced mode shapes were sensitive to scour. In addition, it was found that there is a relation between the natural frequencies and the scour level. Foti and Sabia [2] have monitored foundation scour by using measurements of traffic-induced vibrations. They claimed that the modes of vibration for the superstructure can be used to identify piers that have scour-affected foundation conditions. In addition, they found that the maximum value of the variance of single signals can be used to detect piers that were affected by foundation scour. Ju [3] developed a finite element method, which accounted for the effect of the soil-fluidstructure interaction, to calculate scoured bridge natural frequencies. Prendergast et al. [4] placed accelerometers on a bridge pile above the waterline to track the depth of scour by observing changes in the natural frequency of the vibration of the foundation piles. Bayraktar et al. [5] built an equation between natural frequency and main bridge span using experimental approach. As shown in the literature, one can use the changes in the frequency of a bridge component (e.g., pier or superstructure) to determine the safety condition of the bridge. However, few of the earlier studies attempted to find the threshold value of the frequency to distinguish between a safe and unsafe bridge. Thus, this study aims to develop a methodology to determine a critical frequency ratio that can be used for the task of regular maintenance. A performance function is first defined to determine the critical bridge condition. A series of pushover analyses are implemented to determine the capacity envelope. The bridge demand is the hydrodynamic pressure that acts on the bridge. Based on the code specification in Taiwan, one can compute the pressure for different scouring situations. When the bridge capacity is equal to the demand, the limit state is called the damaged condition of the bridge. The frequency ratio of critical to undamaged bridges is the critical (safety) frequency ratio. If the measured ratio is less than the critical ratio, the monitored bridge is vulnerable to a flood hazard and a more advance analysis should be conducted to ensure
553
the bridge safety. In a field test, this study uses the ambient vibration technique to measure the vibration frequency of an identical bridge before and after a flood season and determines the bridge safety condition based on the safety frequency ratio. The results show that the proposed critical frequency ratio can be used as a safety index to detect the bridge condition. There are many technologies that have been developed to measure the frequency of a bridge, for example, indirect measurement [6] and auto regressive moving average [7]. Because this study only focuses on finding the critical frequency ratio of a bridge, investigating technologies for measuring bridge frequency is beyond the scope of the current study. In addition, only simply supported bridges with a single pier, multiple spans and located in the Gaoping River Basin were studied, so application outside of this limitation should be limited. 2. Calculation of the critical frequency ratio To calculate the critical frequency ratio, one first needs to establish a bridge performance function (limit state function). The simplest performance function is capacity minus demand, which was adopted in this study. A performance equals to zero indicates that the considered bridge has reached the ultimate damaged condition. For a given bridge, its capacity (lateral resistance) and demand (external force applied to the bridge by a flood) depend on the scour depth. In other words, the capacity and demand of a bridge vary for different scour depths. In this study, the capacity envelope of a bridge is computed from pushover analyses, and the demand envelope is calculated based on the code requirement in Taiwan. The determinations of capacity and demand are described below. 2.1. Determination of capacity The seismic capacity curve of a bridge structure is often represented as the curve of the base shear and top displacement obtained from a pushover analysis. This study adopts this idea and defines the capacity of the bridge flood resistance as the yield point in the pushover analysis. Note that the yield force in a pushover curve is actually the base shear of the interested pier. Because the capacity of a bridge varies depending on the scour depth, pushover analyses are performed using different scour depths, as shown in Fig. 1, and the yield points of various pushover analyses are connected to form a curve, i.e., the capacity envelope of a bridge against floods, as shown in Fig. 2. 2.2. Determination of demand The demand of a bridge against floods is defined as the total base shear force that results from the required hydrodynamic pressure. Based on the ‘‘Highway Bridge Design Criteria” (2009), the hydrodynamic pressure is calculated by Eq. (1).
P av g ¼
Fig. 1. Pushover analyses with different scour depth.
52:5KðV av g Þ2 1000
ð1Þ
where Pavg: average hydrodynamic pressure (tf/m2); Vavg: average stream velocity (m/s); and K: a constant coefficient; 1.4 for a blunt nosed pier, 0.7 for a round pier, and 0.5 for a tip pier when the angle is less than or equal to 30°. Please note that theoretically, a critical frequency ratio (Rc) should not depend on any code. A ‘‘true” hydrodynamic pressure must be used. Practically, it is not easy to define a specific hydrodynamic pressure for Taiwan or elsewhere. Based on the study of Wu et al. [15], for bridges in Gaoping River Basin, the code-based hydrodynamic pressure has a similar trend to the average pressure from a 3D fluid-solid-coupled modeling. Thus, the
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envelope of the bridge. The flowchart of determining the demand envelope is provided in Fig. 4. Fig. 5a shows several demand points for the bridge and its corresponding demand envelope curve. Fig. 5b is an example of the capacity and demand envelops for a given bridge. 2.3. Pushover analysis
Fig. 2. An illustration of capacity envelope.
pressure equation defined in Taiwan code is used to calculate the bridge demand. Application outside of the Gaoping River Basin should be cautious. To compute the corresponding base shear of Eq. (1), the distribution of the hydrodynamic pressure on the bridge is needed. For a bridge with a single pier, the pressure distribution is shown in Fig. 3, where the maximum pressure Pmax is twice as large as Pave. The hydrodynamic pressure has a triangular distribution and changes from Pmax at the top of the stream level to zero at the streamline. The base shear force can be calculated by multiplying the pressure by the area of the stream flowing across the bridge pier. If the stream elevation is higher than the bottom of the bridge girder, the effect of the superstructure must also be considered. At this point, the hydrodynamic pressure on the superstructure is uniformly distributed with a value of Pmax. After obtaining the base shear, the next step is to locate the resulting displacement. The pushover analysis provides a force and displacement relationship for the bridge. Thus, once the base shear is calculated, this study uses a pushover curve to find the corresponding displacement. Similar to the situation of calculating the capacity, the bridge demand also depends on the scour depth. Thus, this procedure is repeated to locate several demand points, where a pair is the base shear and corresponding displacement, to form the demand
Fig. 3. The water pressure distribution.
As explained in Sections 2.1 and 2.2, the pushover analysis is used to calculate both the capacity and demand of a bridge. SAP2000 is adopted to conduct the pushover analysis and detailed implementation is provided in this section. Many studies about the pushover analysis of bridges have been published, but few of them use the foundation scouring depth in the evaluation process. The common pushover analysis assumes the foundation to be a fixed support. Because this study mainly evaluates the bridge safety after a scour occurred, the soil property of the foundation is also considered in the pushover analysis. A detailed plastic hinge property of the pier and soil property are provided below. 2.3.1. Modelling of the pier To accurately simulate the pier behavior, core/cover concrete and steel are considered separately. The concrete model proposed by Mander et al. [8] is used to simulate the confined (core) and unconfined (cover) concrete. For the confined concrete under a quasi-static monotonic loading, the longitudinal compressive concrete stress (fc) is given by:
Fig. 4. The flowchart of determining the demand.
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area of the longitudinal reinforcement to the area of the core of the section, qs is the ratio of the volume of the transverse confining steel to the volume of the confined concrete core, fyh is the yield strength of the transverse reinforcement, and x in Eq. (2) is calculated by:
x¼
ec ecc
ð5Þ
where ec is the longitudinal compressive concrete strain. ecc is calculated by:
"
!# 0 f cc 1þ5 0 1 f co
ecc ¼ eco
ð6Þ 0
where eco is corresponding unconfined concrete strain of the f co and is 0.002, as suggested by Mander et al. [8]. r in Eq. (2) is calculated by:
c¼
Ec Ec Esec
ð7Þ
qffiffiffiffiffiffi 0 where Ec ¼ 5000 f co MPa is the tangent modulus of elasticity of the concrete and 0
Fig. 5a. Illustration of demand points and demand envelope curve.
Esec ¼ 0
fc ¼
f cc xr r 1 þ xr
ð2Þ
0
where f cc is the compressive strength for the confined concrete and can be determined by: 0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 0 0 7:94f l fl 1:254 þ 2:254 1 þ 0 2 0 f co f co
0
f cc ¼ f co 0
ð3Þ 0
where f co is the unconfined concrete compressive strength and f l is the effective confining stress on the concrete and is calculated by: 0
fl ¼
2 0 1 2ds s 2ð1 qcc Þ
qs f yh
ð4Þ
where s0 is the clear vertical spacing between the spiral or hoop bars, ds is the diameter of the spiral or bars, qcc is the ratio of the
f cc
ð8Þ
ecc
A similar approach can be applied to simulate the cover concrete. For the case of ec < 2eco, one can use Eq. (9) to simulate the cover concrete behavior. 0
fc ¼
f c xco r r 1 þ xr
ð9Þ
where xco ¼ ec =eco , c ¼ Ec =ðEc Esec Þ and Esec ¼ f c =eco . For the case 0
ec P 2eco , 0
fc ¼ fc
2r r 1 þ xr
1
ec 2eco esp 2eco
ð10Þ
where esp = 0.005 is the spalling strain. For the case of
fc ¼ 0
ð11Þ
The steel stress–strain curve used in this study is described below. For the case of 0 6 es 6 ey
f s ¼ Es es
ð12Þ
For the case of
ey 6 es 6 esh
fs ¼ fy
ð13Þ
For case of
"
fs ¼ fy
Fig. 5b. Illustration of the cricital point (the cross point).
ec P esp ,
esh 6 es 6 esu ,
fu fy fu fy fy
!
eu es eu esh
2 # ð14Þ
where fs is the steel stress, Es is the elastic modulus of the steel, es is the steel strain, fy is the yield stress of the steel, fu is the ultimate steel stress, esh is the strain hardening strain, and eu is the ultimate steel strain. To account for the effect of the axial load on the characteristics of the column plastic hinge, the bridge capacity and demand are determined through two consecutive pushover analyses. In the first pushover analysis, the column plastic hinge behaviors are described by the combination of flexural and shear behaviors under the dead load condition (referred to as the DL curve, Fig. 6). The flexural behavior can be established using the aforementioned stress–strain curves of concrete and steel together with moment–curvature analysis. The column shear behavior is built using the stress–strain curves and Eq. (15). The second pushover
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2.3.2. Modelling of the soil property Because soil plays an important role in bridge safety against floods, it is necessary to account for the soil property in the numerical model. The corresponding equivalent soil springs are set around the pile according to their depths. The stiffness/coefficient of soil spring varies significantly for different empirical equations. One convenient and reasonable approach is to use the value of the standard penetration test for calculating the stiffness, which is adopted here. Simulation of the soil property is described below. For a pile foundation, to determine the equivalent soil spring, the first step is to calculate the horizontal subgrade reaction coefficient (kH). kH can be calculated by Eq. (16) [11]:
0:75 9 > kH ¼ kH0 B30H > = pffiffiffiffiffiffiffiffiffi > BH ¼ D=b > qffiffiffiffiffiffi > > 4 ; HD b ¼ k4EI
ð16Þ
where kH0 is the baseline value of the horizontal subgrade reaction coefficient (kg/cm3), BH is the equivalent pile length perpendicular to the loading direction, D is the pile diameter, EI is the flexural stiffness of the pile, and 1/b is the pile depth under consideration. kH0 is computed by: Fig. 6. Determination of the flexural behaviors used in the second pushover analysis.
kH0 ¼
1 30
ED
ED ¼ 2ð1 þ mD ÞGD ct GD ¼ 10g V 2SD
analysis is similar to the first one except that the flexural and shear behaviors are different. The flexural and shear behaviors are established again (referred to as the UL curve, Fig. 6), but the ultimate vertical load, which corresponds to the peak base shear in the first pushover analysis, replaces the dead load [9,10]. The flexural and shear behaviors used in the second pushover analysis are determined by combing the DL and UL curves. To be exact, the final yield point is the average of the yield points of the DL and UL curves, and the final ultimate point is the ultimate point on the UL curve. For instance, the flexural behavior under a dead load and ultimate axial loading is represented by curves ABDLCDL and ABULCUL, respectively (Fig. 6). The final curve is ABC. Point B is the average between points BDL and BUL, whereas point C is coincident with point CUL. That is, this study determines the flexural behavior in the second pushover analysis by combing curves from two moment-rotation analyses. The first and the second moment-rotation analysis are conducted under the dead load and ultimate vertical load, respectively. A similar rule is applied to determine the final shear behavior.
Vn ¼ Vc þ Vs qffiffiffiffi 0 V c ¼ 0:166ðk þ FÞ f c Ae ðMPaÞ
ð15Þ
9 > =
ð17Þ
> ;
where ED is the soil dynamic elastic modulus, vD is the soil dynamic Poisson’s ratio, GD is the soil dynamic shear modulus, ct is the soil weight per volume (t/m3), g is the acceleration due to gravity, and VSD is the shear wave speed and can be calculated by:
V SD ¼ C v V s
ð18Þ
where Cv is the modification factor and is calculated by:
Cv ¼
0:8ðV s < 300 m=sÞ
ð19Þ
1:0ðV s P 300 m=sÞ
where Vs is the average shear wave speed and can be obtained by:
(
Vs ¼
8N1=3 ðsandÞ
ð20Þ
100N1=3 ðclayÞ
where N is the value of the standard penetration test. The horizontal subgrade reaction coefficient (kH), geometry of the foundation (e.g., the location of each pile) and distribution of the soil layers can be used to obtain the equivalent spring coefficient for each direction using simple force equilibrium equations. Please note that an ultimate soil spring stress, described in Eq. (21), is specified in code [14] and is adopted to form a linear elastic-perfectly plastic soil spring.
pffiffiffiffiffiffi
rP ¼ r0Z K P þ 2c K P
ð21Þ
where r0Z is the effective vertical stress (tf/cm ), K P is the coefficient of passive earth pressure, c is the cohesion of soil (tf/cm2). 2
Ash f y da Vs ¼ s where k is the rotation ductility factor, which equals 1 before flexural yielding and decreases linearly to zero at the flexural ultimate condition; F is the axial load effect on the shear strength, which equals N/(13.8Ag) (MPa) for a compressive axial load and N/(3.5Ag) (MPa) for a tensile axial load, where N is the axial force and Ag is the gross cross-sectional area; Ae is the effective area, which equals 0.8 Ag; Ash is the area of the shear reinforcement; d is the effective sectional depth; s is the vertical spacing; and a = 1 for a rectangular section and p/2 for a circular section.
2.3.3. Location of the plastic hinge The plastic hinge typically occurs in the substructure under external forces, which dominates the pushover curve of a bridge. Wang et al. [12] indicated that it is very difficult to predict the plastic hinge location on the substructure because the highly nonlinear relationship between the maximum bending moment and the distribution of the soil layers. On the other hand, Foti and Sobia [2] stated that plastic hinges are typically located at pier ends. The plastic hinge is only allowed to occur at the pier end, pile top or pile end if the corresponding moments reach the yielding limit.
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K.-W. Liao et al. / Construction and Building Materials 102 (2016) 552–563 Table 1 Assessment items used in the preliminary evaluation procedure in Taiwan.
Fig. 7. Determination of the critical frequency by interpolating between the frequencies (bridges with scour depth of 6 and 9 m).
Therefore, this study does not consider the plastic hinge location effect.
Assessment Item
Item
Weight (a)
No. No. No. No. No. No. No. No. No. No. No. No. No.
Upstream river dam or reservoir facilities Foundation type Bending or narrowing of the river Eroded river bed Material on the river bed Location of the main channel Hydraulic drop effect Attack angle of flow Area ratio of bridge to cross section Foundation exposure depth Effective pier diameter Protection for river bank Protection for river bed
7 7 6 8 2 5 5 7 5 20 13 5 10
1 2 3 4 5 6 7 8 9 10 11 12 13
score and is the most dominating parameter which is consistent with the assumption of the current study. However, the preliminary evaluation table only provides an overall weighting among parameters. For each bridge, a further investigation should be individually conducted. This study adopts the Bayesian network (BN) to perform the sensitivity analysis. Reasons for adopting BN are:
2.4. Definition of the critical frequency ratio The critical frequency ratio is defined in Eq. (22).
Rc ¼
fa fb
ð22Þ
where Rc is the critical frequency ratio, fa is the frequency corresponding to a damaged bridge, and fb is the frequency corresponding to an undamaged bridge. The calculation of fb is relatively easy. fb is the first frequency of the original bridge (with no scour) obtained from a modal analysis. As mentioned in Section 2.2, the cross point in Fig. 5(b) represents a bridge that is in its limit state. As shown in Fig. 7, the capacity and demand envelopes intersect between a scour depth of 6 m and 9 m. fa is then calculated by interpolating between the two frequencies. If field frequency measurements are performed before and after a main flood event, a ratio can be calculated as shown in Eq. (23).
R¼
f ma f mb
ð23Þ
where R is the measured frequency ratio of the considered bridge and fma and fmb are the frequencies measured after and before a flood, respectively. If the measured R ratio is less than Rc, this indicates that the bridge may have serious damage and that a more advanced analysis should be performed to ensure its safety. 3. Parametric study and a case study This study considers the scouring depth as the most important parameter for a bridge against floods. As a result, a scouringinduced critical frequency ratio (Rc) is proposed as a bridge health indicator. However, whether the scouring depth is the most dominating parameter is questionable and should be verified first. In Taiwan, a two-steps evaluation is often used to examine the bridge safety condition. The first step is basically a visual examination based on a comprehensive table as shown in Table 1. It is seen that Table 1 includes 13 parameters that have potential influence on the bridge failure. Using this preliminary form as our analyzing basis, a sensitivity analysis is conducted to examine the relative importance of these 13 parameters for bridges in Gaoping River Basin. Based on the weight (a in Table 1) of 13 parameters, it is seen that foundation exposure depth weights 20% of the overall
(1) Many parameters affect bridge safety. The BN can break down these various parameters and makes them as nodes in the network, allowing one to focus on the causal relationship between two nodes at each time. Thus, the problem’s complexity is reduced. (2) The BN allows users to update the bridge safety through Bayesian learning. Because the preliminary evaluation table has not been revised for a decade, any recorded event in this period should be considered. That is, in the proposed BN, the role of preliminary evaluation table is considered as the expert’s opinions and the recorded information is the new evident. To construct a BN, the first step is to establish the network topology of BN that reflects causal relationships of various nodes. The second step is to establish the conditional probability between the nodes. Based on the existing preliminary evaluation table, this study used the AgenaRisk software to establish the proposed BN and its topology is shown in Fig. 8. A BN-based sensitivity analysis is performed for 16 bridges in Gaoping River Basin. A typical Tornado graph is displayed in Fig. 9. Table 2 displays the statistics for the top four dominating factors in 16 bridges. As expected, the dominating factor varies among different bridges. However, it is seen that the scouring depth is the most influencing factor in bridge health evaluation. Although the scouring depth has been identified as the most influencing factor for bridges in Gaoping River Basin, analyzing all bridges in this area following the proposed procedure is still a heavy task. For this situation, the bridges are classified into groups to capture the influence of each parameter and, at the same time, to increase the analysis efficiency. According to the proposed approach for the critical frequency ratio, it is easy to recognize that the most important influencing parameters are the pier diameter, pier length, pile diameter and pile length. Based on the literature and our earlier survey of bridges in Taiwan, the range of these parameters is displayed in Table 3. Note that these ranges only provide an estimation for bridges with a single pier in Taiwan. In the parametric analysis, five types of bridges were analyzed to determine the effects of each parameter on the critical frequency ratio. The principle of classification is described as follows: (1) the factors with the most significant influence on the pushover analysis are selected as control variables, including the pier
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Fig. 8. Bayesian network used in current study.
Table 2 Statistics for the top four dominating factors in 16 bridges.
*
Factor
1st
2nd
3rd
4th
Total
Foundation exposure depth Descending river bed River bank erosion Water blocking ratio
10* 6 0 0
4 6 6 0
2 4 10 0
0 0 0 16
16 16 16 16
Total
16
16
16
16
Number of appearance in this place.
Fig. 9. A typical Tornado graph for bridge health in Gaoping River Basin.
diameter, pier length, pile diameter and pile length; (2) except for the parameter of pile length, only the maximum and minimum values in the statistics are selected; and (3) the pile length is divided into four different values (i.e., 40, 20, 15 and 10) to examine the scour effect more thoroughly, and the grouping result is shown in Table 4. According to Table 4, the first type represents bridges with maximum capacity, and the second type of bridge has the second largest capacity. In other words, the larger the type number, the smaller the bridge capacity. When performing pushover analysis, each type of bridge is evaluated with five different SPT-N values. Because the SPT-N value is usually less than 50 unless the soil condition is close to a rock condition, the SPT-N values considered in this study are divided into five different values of 10, 20, 30, 40 and 50. In addition, for each representative bridge, the SPT-N value is assumed to be constant along the depth. Analysis results of the five types of bridges were consistent with many fundamental concepts of bridge engineering, which indicates that the proposed algorithm that determines the critical frequency ratio was well defined. Details are provided below. 3.1. Capacity analysis of five types of representative bridges Because the analysis processes for five types of bridges are similar, only the analysis procedure of the first type of bridge is provided below. As indicated in Table 4, the first type of bridge has a pier diameter of 4.2 m, pier length of 7.7 m, pile diameter of 1.5 m and pile length of 40 m. To examine the scour effect thoroughly,
Fig. 10. Bridge model built by SAP2000.
five scour depths at 0 (no scour), 3, 6, 9 and 12 m were considered. In the pushover analysis, if there is no scour (i.e., no foundation exposure), the potential plastic hinge is set at the bottom of the bridge pier. If the foundation is exposed, the probable positions of the plastic hinge are the end of the pier, pile top and bottom (i.e., the fixed end of the pile). Fig. 10 shows the bridge model in SAP2000. Because the in-plane stiffness is comparative large, a rigid diaphragm in-plane is adopted to simulate the bridge deck. The connections between pier and deck are modelled as hinge or roller depending on design drawing. The mass of the superstructure is included in the model to accurately estimate the pier frequency. Fig. 11 shows the capacity curves at various scour depths when the standard penetration SPT-N value is 50, where the y-axis unit is ton and the x-axis unit is cm.
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559
Fig. 11. Capacity curves of the first type bridge at various scour depths (top to bottom: 0, 3, 6, 9 and 12 m).
Table 3 General ranges of bridge sizes in Taiwan. Item
Pier dia.
Value (meter)
Pier length
Pile dia.
Pile length
Max.
Min.
Max.
Min.
Max.
Min.
Max.
Min.
4.2
1.8
13.4
7.7
1.5
0.4
40
10
Fig. 13. Capacity curves for type I (top) and II (bottom) bridges (N = 50 and no scour).
SPT-N value will have a higher capacity. Combining this information with Table 5, one can conclude that a higher bridge vibration frequency means a higher bridge capacity. Fig. 13 displayed the lateral capacity for type I and II bridges when SPT-N is 50 and there was no scour. It was observed that the difference was slight. However, when the foundation exposure was large, 12 m for example, the difference between these two types of bridge was significant, as shown in Fig. 14. It was seen that the scour depth dominated the lateral capacity of the bridge.
3.2. Demand analysis of five types of representative bridges
Fig. 12. Capacity curves of the first bridge type without scour.
When the SPT-N value is small (soil strength is low) and the structure is strong, the bridge may lose its lateral resistance before any plastic hinge is formed. When the SPT-N value is large (soil strength is high), losing lateral capacity is often accompanied with the formulation of a plastic hinge. The former case should usually be avoided because engineers expect the ultimate strength of a bridge to be controlled by the bridge itself. Table 5 displayed the vibration frequency and cycle at different scour depths for different SPT-N values. For example, when SPT-N = 50, the vibration frequencies of the bridge structure are 2.7233, 1.8412, 1.2694, 0.92068 and 0.69888 Hz for scour depths of 0, 3, 6, 9, 12 m, respectively. When scour occurred, the vibration frequency of the bridge decreased as the foundation exposure increased. Fig. 12 shows the capacity curves of the first bridge type without scour, and the SPT-N values are from 10 to 50. As expected, a bridge with a higher
As introduced in Section 2.2, the demand point consists of the pairing of the base shear and corresponding displacement that can be obtained by the code specification and pushover analysis, respectively. The demand points are therefore influenced by the stream velocity, scour depth and soil property (SPT-N value). To investigate the effects of these influencing factors on the lateral demand, three stream velocities based on the past record were selected for the demand analysis: 6.49, 3.82 and 1.15 m/s, which are the largest, average and smallest velocity of 100-year floods in the Gaoping River Basin [13]. In addition, five different scour depths of 0, 3, 6, 9 and 12 m were considered, and five different N-values were used for the soil property. The demand points of the first type of bridge at varied velocities, scour depths and SPT-N values are shown in Table 6, where only SPT-N = 10 and 50 are displayed. The demand envelope was drawn by connecting the points in these tables, as shown in Fig. 15 (case of SPT-N = 50 only). According to Fig. 15, when the velocity is high, the demand force of a bridge is greater than when the velocity is moderate, which indicates that a bridge is more likely to be damaged by a higher velocity attack. If the velocity is not large enough, more lateral displacement was needed to cause bridge failure. Based on Fig. 15, for the first type of bridge, if the water velocity is low, the bridge will function well no matter what the value of SPT-N is. These numerical results align with a general physical understanding of bridge engineering.
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K.-W. Liao et al. / Construction and Building Materials 102 (2016) 552–563 Table 5 Frequencies and periods of the first type bridge with varied SPT-N values and scour depths.
Fig. 14. Capacity curves for type I (top) and II (bottom) bridges (N = 50 and scour depth = 12 m).
Table 4 Five representative bridges analyzed in this study (unit: meter).
Pier dia Pier length Pile dia Pile length
Type I
Type II
Type IIII
Type IV
Type X
4.2 7.7 1.5 40
4.2 7.7 1.5 20
1.8 13.4 0.4 20
1.8 13.4 0.4 15
1.8 13.4 0.4 10
Fig. 15. Capacity and demand envelopes for the type I bridge with N = 50.
3.3. Discussion of the critical frequency ratio Rc Based on Section 2.4, fa and fb are needed to compute the value of Rc. fb can be obtained from the modal analysis of an undamaged bridge (i.e., no scour); fa is the frequency of the bridge that
Scour depth
SPT-N-value
Period (s)
Frequency (Hz)
0m
10 20 30 40 50
0.46 0.42 0.39 0.38 0.37
2.18 2.41 2.55 2.65 2.72
3m
10 20 30 40 50
0.64 0.60 0.57 0.56 0.54
1.55 1.68 1.75 1.80 1.84
6m
10 20 30 40 50
0.89 0.84 0.82 0.80 0.79
1.12 1.18 1.22 1.25 1.27
9m
10 20 30 40 50
1.20 1.15 1.12 1.10 1.09
0.83 0.87 0.89 0.91 0.92
12 m
10 20 30 40 50
1.56 1.50 1.47 1.45 1.43
0.64 0.67 0.68 0.69 0.70
corresponds to the intersection point of the capacity and demand envelopes. Tables 7–11 display the results of fa, fb and Rc of five representative bridges, respectively. The Rc value varies with the properties of the bridge, namely, the Rc is influenced by the structural strength. Specifically, the Rc value is smaller when a bridge has a higher stiffness. For example, the Rc values of types 1 and 2 bridges were smaller than those of bridge types 3, 4 and 5. The proposed Rc suggested that a bridge with a higher stiffness has more room to withstand a loss of structural integrity. For the first type of bridge, Rc values ranged from 0.64 to 0.71 and 0.41 to 0.45 for the case of high and moderate velocities, respectively. The higher stream velocity resulted in a larger Rc value, which indicated that the bridge was more likely to be damaged under a high velocity attack even though the structural strength of the bridge did not decrease significantly. On the other hand, if the bridge was scoured by a moderate velocity attack, the bridge may be damaged when the Rc value decreases greatly. According to Tables 7–11, the SPT-N value only has a moderate effect on the Rc value (it is only a little significant for the first type of bridge). Because the proposed Rc depends on fa and fb, which are influenced by the SPT-N value simultaneously, it is expected that the SPT-N value will not be a key factor on Rc; this agrees with the results shown here (see Tables 7–11). To derive a simple, clear, convenient and convincing rule for bridge evaluation based on results from the parametric study, the 5 representative bridges are further categorized into two groups: group A and B. The type I and II bridges are considered as group A. On the other hand, other bridge types are considered as group B. Three scenarios are used to determine the Rc value: the worst, mean value and average of the former two scenarios. Table 12 displays the analyzed results. It is seen that the suggested Rc ranging from 0.70 to 0.92 if the average scenario is adopted. Although the application of current study is limited to bridges in Gaoping River Basin, the Rc range found here is similar to the earlier research conducted by Nishimura [16]. Nishimura [16] collected 1000 first mode frequencies for bridges in Japan. The threshold values of Rc based on the on-site measurement ranges
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K.-W. Liao et al. / Construction and Building Materials 102 (2016) 552–563 Table 6 The demand pairs of type I bridge with N = 10 and 50. Scour depthd
N = 10 H F
0 3 6 9 12 a b c d
N = 50
a
b
500 940 1230 1525 1820
a
M c
b
L c
D
F
D
2 9 21 45 90
170 370 580 810 1065
1 7 12 25 49
a
Ha
b
D
15 30 50 75 100
0 0 1 2 5
F
c
F
b
500 940 1230 1525 1820
Ma D
c
2.0 6.7 17 39 77
La
b
c
F
D
170 370 580 810 1065
0 3 8 21 46
Fb
Dc
15 30 50 75 100
0 0 1 2 4
H, M and L denotes the velocity of stream are high, medium and low, respectively. F denotes the force (ton). D denotes the displacement (cm). Unit: meter.
Table 7 The Rc values of bridge type I.
Table 10 The Rc values of bridge type IV.
Stream velocity
SPT-N values
fb (Hz)
fa (Hz)
Rc
Stream velocity
SPT-N values
fb (Hz)
fa (Hz)
Rc
High
10 20 30 40 50
2.18 2.41 2.55 2.65 2.72
1.55 1.67 1.59 1.74 1.74
0.71 0.69 0.62 0.66 0.64
Medium
10 20 30 40 50
0.75 0.77 0.78 0.79 0.79
0.72 0.73 0.75 0.76 0.76
0.96 0.95 0.96 0.96 0.95
Medium
10 20 30 40 50
2.18 2.41 2.55 2.65 2.72
0.98 1.05 1.10 1.12 1.13
0.45 0.44 0.43 0.42 0.42
Low
10 20 30 40 50
0.75 0.77 0.78 0.79 0.79
0.62 0.62 0.63 0.61 0.63
0.83 0.81 0.81 0.77 0.80
Table 8 The Rc values of bridge type II.
Table 11 The Rc values of bridge type V.
Stream velocity
SPT-N values
fb (Hz)
fa (Hz)
Rc
Stream velocity
SPT-N values
fb (Hz)
fa (Hz)
Rc
High
10 20 30 40 50
1.95 2.18 2.32 2.42 2.50
1.50 1.66 1.75 1.82 1.88
0.77 0.76 0.76 0.75 0.75
Medium
10 20 30 40 50
0.71 0.74 0.75 0.76 0.77
0.67 0.71 0.72 0.74 0.74
0.95 0.96 0.96 0.97 0.97
Medium
10 20 30 40 50
1.95 2.18 2.32 2.42 2.50
1.41 1.55 1.62 1.68 1.74
0.72 0.71 0.70 0.69 0.70
Low
10 20 30 40 50
0.71 0.74 0.75 0.76 0.77
0.63 0.62 0.61 0.62 0.62
0.89 0.84 0.81 0.82 0.81
Table 9 The Rc values of bridge type III.
Table 12 Rc values in three different scenarios.
Stream velocity
SPT-N values
fb (Hz)
fa (Hz)
Rc
Scenario
Group A
Group B
All bridges
Medium
10 20 30 40 50
0.76 0.78 0.79 0.80 0.80
0.73 0.74 0.75 0.76 0.77
0.95 0.95 0.95 0.95 0.96
The worse The mean value The average of the first two
0.77 0.64 0.70
0.97 0.88 0.92
0.77–0.97 0.64–0.88 0.70–0.92
Low
10 20 30 40 50
0.76 0.78 0.79 0.80 0.80
0.61 0.60 0.62 0.60 0.60
0.80 0.77 0.78 0.75 0.75
from 0.7 to 0.85. It is seen that the proposed parametric study is able to deliver a reasonable range for Rc. 3.4. Case study The Jhih-Sin Bridge located in the Yi-Lan county was selected to evaluate its flood-resistant capacity using the proposed critical
frequency ratio. The Jhih-Sin Bridge consists of two independent bridges for the northbound and southbound lanes, and both of them are pre-stress concrete bridges with six spans. The span length is approximately 25.6 m, and a single pier was used in the substructure for both directions. The north- and southbound bridges were built in 1989 and 1997, respectively. Only the southbound bridge was analyzed in this study, as shown in Fig. 16. The frequency before and after a flood season (i.e., the summer in 2014) are measured to calculate the frequency ratio (R in Section 2.4). Because the design drawing of this bridge is incomplete, the Rc value is determined according to the earlier parametric study. The pier length and diameter of the Jhih-Sin Bridge are 8 and 3.5 m, respectively. Such size is closer to Type I or II bridges. The
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(2)
(3)
(4)
(5) (6)
(7) Fig. 16. Jhih-Sin Bridge.
Table 13 The fma, fmb and R values of the Jhih-Sin Bridge. fmb
fma
R
Rc
4.35
4.1
0.94
0.77
highest Rc from above two bridges is selected (i.e., 0.77) to verify the bridge safety. The ambient method was used to measure the bridge frequency in the field. Table 13 displayed the frequencies before and after the flood season. Based on the results, the corresponding R value is approximately 0.94 (4.1/4.35), which was larger than the Rc, indicating that this bridge does not need an advance analysis to ensure its safety. From the field observation, no serious damage was found during 2014, which meets the conclusion of the proposed preliminary safety evaluation process. 4. Conclusion Frequency is often considered to be an ideal safety indicator in bridge health monitoring. However, few of past studies have focused on finding a threshold frequency to detect the bridge limit state. This study focused on finding a critical frequency ratio (Rc) as a safety indicator that can be used to distinguish undamaged and damaged bridges. Rc was governed by the bridge capacity and demand that were determined using pushover analyses. 54 bridges in Gaoping River Basin were damaged during the attack of Typhoon Morakot (2009). The government spent 5 years to recover all the damaged bridges. To maintain the retrofitted or undamaged bridges in a healthy condition in the future, a priority maintenance list is needed. In such case, an efficient method is often adopted. The developed evaluation procedure fulfills such goal and can provide a preliminary health inspection for bridges in Gaoping River Basin. A more advance and expensive technique such as pushover analysis should be used if the measured frequency is below Rc. Based on calculations of the bridge capacity, demand and Rc, several conclusions were drawn below: (1) When the SPT-N value is small and the structure is strong, the bridge may lose its lateral resistance before any plastic
hinge is formed. When the SPT-N value is large, losing lateral capacity is often accompanied with the formulation of a plastic hinge. The vibrational frequency of a bridge decreased as the foundation exposure increased. A higher capacity of the bridge results in a higher bridge vibrational frequency. The scour depth has a significant impact on the bridge frequency. A larger demand force is observed for the case of a high velocity, which results in a larger Rc value. This means that a bridge attacked by a high velocity is more likely to be damaged. The Rc value is smaller for a bridge with a higher stiffness, which means the proposed Rc suggests that this bridge has more room to withstand the loss of structural integrity. It is seen that the SPT-N value does not have a significant influence when using frequency as a bridge safety index. The proposed critical frequency ratio (Rc) is a velocity and structural strength-dependent indicator. Therefore, the safety threshold for a bridge is not a deterministic number and varies depending on the stream velocity and the strength of a bridge. Based on the average scenario, the suggested Rc ranges from 0.70 to 0.92. This range is consistent with earlier study conducted in Japan, indicating it is a reasonable and should be useful for engineers in bridge preliminary health investigation.
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