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Measurement of existing prestressing force in concrete structures through an embedded vibrating beam strain gauge
Measurement 83 (2016) 10–19
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Measurement of existing prestressing force in concrete structures through an embedded vibrating beam strain gauge S. Biswal, A. Ramaswamy ⇑ Civil Engineering Department, Indian Institute of Science, Bangalore 560012, India
a r t i c l e
i n f o
Article history: Received 24 August 2015 Received in revised form 12 January 2016 Accepted 18 January 2016 Available online 27 January 2016 Keywords: Existing prestressing force Vibrating wire strain gauge Crack reopening load Digital image correlation Prestressed concrete structures
a b s t r a c t A steel frame is designed to measure the existing prestressing force in the concrete beams and slabs when embedded inside the concrete members. The steel frame is designed to work on the principles of a vibrating wire strain gauge and in the present study is referred to as a vibrating beam strain gauge (VBSG). The existing strain in the VBSG is evaluated using both frequency data on the stretched member and static strain corresponding to a fixed static load, measured using electrical strain gauges. The evaluated strain in the VBSG corresponds to the existing stain in the concrete surrounding the prestressing strands. The crack reopening load method is used to compute the existing prestressing force in the concrete members and is then compared with the existing prestressing force obtained from the VBSG at that section. Digital image correlation based surface deformation and change in neutral axis monitored by putting electrical strain gauges across the cross section, are used to compute the crack reopening load accurately. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Prestressed concrete structures are widely used in the civil engineering field because of the advantages offered such as reduced self weight and cross section of structural elements and reduced deformations. However losses occur in prestressing strands due to time dependent (creep, shrinkage) deformation of concrete and relaxation of steel strands along with the short term losses in concrete (elastic shortening, wedge slip, wobble, friction) and in steel strands. Significant loss level in the prestressing strands can result in compromising safety and serviceability of prestressed concrete structures. Hence it is necessary to predict the loss of prestress in concrete structures over their design life, so as to take decisions on the serviceability and safety of these structures. However as mentioned in
⇑ Corresponding author. E-mail addresses: [email protected] (S. Biswal), ananth@ civil.iisc.ernet.in (A. Ramaswamy). http://dx.doi.org/10.1016/j.measurement.2016.01.031 0263-2241/Ó 2016 Elsevier Ltd. All rights reserved.
[1] since future prestress loss estimates are based on robust analytical models it necessary to conduct experiments for short term measurements based on which the parameters of the analytical model are updated for more realistic prediction of future prestress losses. Hence a cost effective and easy to use system to measure the existing prestress forces at various time instances is necessary. A stress wave based determination of existing tensile force in prestressing strands is given in [2–4] to measure the existing stresses in steel strands. Fiber optic sensors are used in [5,6] to monitor the existing stresses in prestressing strands. However as mentioned in [7] the fiber optic sensors are prone to mechanical damage during their installation on the prestressing strands as a consequence of mishandling, misalignment, pinching or bending at the tendon anchorages. A change in acoustoelastic wave velocity due to change in stress in strands is used as a technique [8] to measure the existing prestress force in steel strands. However the velocity of the applied wave is not much affected by the change in stress in strands. Variation of
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longitudinal stress wave velocity with respect to applied stress level on the strands is studied in [9] for measuring existing prestress in strands. The longitudinal stress wave can be used to measure the existing stress if the applied stress level is lower than about 40% of the ultimate strength of the tendon. However in reality the applied stress level is about 70–80% of the ultimate tensile strength. A relationship between the strand force and the strain in the core wire is developed in [10], so that by measuring the strain in the core wire by stress wave velocity the strand force can be estimated. However the stress condition in the core wire and that on the circumferential wires is different. Measurement of existing stresses using induced magnetic field is given in [11]. A more elaborate study on the effect of grouting on the techniques mentioned in the above studies [4–6,8–11] is required. A sensitivity based finite element model updating technique is developed in [12] for the inverse analysis of the existing prestress force in each element of prestressed girders. In the above [12] method it is assumed that the flexural rigidity of each element is known and this method fails to give unique estimates when both the flexural rigidity and the prestress force are not known. Two commonly used methods of measuring the existing prestressing force in concrete structures are through vibrating wire strain gauges (VWSG) [13,14] and crack re-opening method [15]. While existing stress in concrete through VWSG is a non-destructive technique, existing stress from crack re-opening method is a semidestructive method. The principle of working of VWSG are given in [13,14]. By knowing the strain in the wire and assuming the strain in concrete to be equal to the strain in the wire of the VWSG embedded at the same level of concrete the existing strain in concrete is calculated. The existing stress in the concrete at the level of VWSG is obtained from the calculated strain times the modulus of elasticity of concrete. However VWSG’s are consumables, since they are embedded inside the concrete and they cannot be reused, and the VWSG’s are very expensive for small scale experiments on laboratory as the data acquisition system associated with the VWSG’s is very costly. In case of crack reopening method [15] the structure is loaded till crack appears and is unloaded thereafter. Displacement measuring gauges are then fixed across the crack and the structure is again loaded till the crack re-opens. Knowing the load corresponding to the crack re-opening and equating the total bending stress across the cracked section, the existing stress in the structure is evaluated. However the plot of load versus crack opening displacement is smooth and it is difficult to find the exact load at which the crack re-opens. In the present study a steel frame is designed to be used in the experiments to measure the existing prestressing force in the concrete structures. The steel frame is called as vibrating beam strain gauge (VBSG) since it uses the same principle as the VWSG to measure the existing strain from the measured natural frequencies [13,14]. However in this study the VBSG is used to measure the existing prestressing force from the frequency data as well as from static responses of the stretched element of the VBSG. The major advantage of the proposed VBSG over VWSG is the
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cost which is much less than the VWSG. Data acquisition systems used for the static response measurements and frequency measurements are also much cheaper than the data acquisition system used along with VWSG. The VBSG can be used for measuring existing stresses in concrete structures in any laboratory or industries, where facilities are available for measuring static responses or frequencies only. Unlike the crack reopening method, and similar to the VWSG, the VBSG is a non-destructive tool for measuring the existing prestressing force in the concrete structures. The VBSG is validated against a test prism, discussed in the later section. The VBSG is further embedded in post-tensioned beam and slab and the existing prestressing force obtained is then compared with the existing prestressing force obtained from crack re-opening load. In the present study two methods are used to determine the crack reopening load. The first method is based on deformation results from digital image correlation (DIC) [17] and the second method is based on change in neutral axis of structural member with respect to crack evolution. 2. Experimental determination of existing prestressing force in concrete structures using VBSG The VBSG is embedded in concrete at the same level as the prestressing strands and very close to the strands so that the strain in the strands is same as the strain in concrete at the same level which result in same strain in the main steel bar of the VBSG.The geometric details of the VBSG are given in the section below. 2.1. Test frame set up and instrumentation The idea behind the proposed test frame set up is to embed a steel bar inside the prestressed concrete before casting at the same level as the prestressing strands and after the concrete hardens, be able to load and measure responses, so as to back calculate the existing strain in the steel bar. The test frame set up is made up of stainless steel sections. The steel bar is fixed at both ends by putting threads at the ends and tightened against circular plates. To ensure that the strain in steel bar of the VBSG to be same as that of the concrete at the same level the steel bar is extended at both the ends and tightened against circular plates as shown in Fig. 1a. A vertical steel bar is welded at the center of the main steel bar to facilitate access to the main steel bar for loading and taking responses. For ease in loading a rectangular hollow pipe is cut and welded on the top of the secondary steel bar as shown in Fig. 1a. Strain gauge is fixed at the center of the main steel bar (Fig. 1b) and then covered with beeswax (Fig. 1c) to prevent moisture coming in contact with the strain gauge. Use of water-proof sensors, such as fiber optic sensors can be useful in eliminating the need for the water proofing. However the cost to be paid for the waterproofing sensors is much higher over the benefit obtained, also minimizing the major advantage over the existing VWSG. PVC pipes are then put around the steel bars so as to provide free space for the deformation of the bars under loading. Water sealant is then applied at the joints of the
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Fig. 1. Vibrating beam strain gauge.
pipes and wrapped with PVC tape to prevent water flow from the cement paste into the pipes as shown in Fig. 1d. Since, the PVC pipe around the horizontal bar and the PVC T-connector are wrapped with PVC tape, there is no direct contact (no bond) between concrete and the PVC pipe and the PVC T-connector. It can be assumed that the PVC tape wrapped around the PVC pipe and the PVC Tconnector has negligible strength for small relative displacement between the two, allowing smooth transfer of strain from the surrounding concrete to the VBSG. The length of the horizontal bar of the T-frame in the VBSG is similar to the length of the commercially available VWSG, apart from the diameter of the VBSG which is slightly larger than the VWSG. The length of the vertical bar of the T-frame can be reduced if aligned advantageously along the shortest dimension. Though the crosssectional area of the VBSG is more than the VWSG, it is substantially small compared to the surface area of the concrete structure. However the cross-sectional area of the VBSG is not negligible and does put a constraint on the spacing of VBSGs over the span of the prestressed concrete structure. 2.2. Finite element modeling using Abaqus The VBSG frame is modeled in FE software Abaqus [18]. The horizontal and the vertical steel beams are modeled
with beam element B33 and the top rectangular plates are modeled using shell elements S4 as shown in Fig. 2b. A restart analysis is done for evaluating the responses of the frame with initial strain on the bottom horizontal beam. In the first step of restart analysis an initial strain is applied on the bottom beam followed by a nonlinear static analysis and in the second step a linear perturbation analysis is done to get the static responses and modal properties. 2.3. Experimental determination of existing prestressing force in concrete structures In this study two methods are used to determine the existing prestressing force in the concrete structures with the proposed VBSG test frame set up. In the first method the first natural frequency of the stretched bar of the VBSG is used whereas in the second method static displacement and strains at the center of the stretched bar under known loads is used for the determination existing prestressing force as explained below. 2.3.1. Based on principle of vibration of a beam with initial strain As shown in Fig. 3, if 0 is the initial strain in the beams, m is the mass per unit length, L is length, A is the crosssection, I is the moment of inertia and E is the elastic
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Fig. 2. Vibrating beam strain gauge geometric details and FE model.
2.3.2. Based on principle of static deformation of transverse loaded beam with initial strain As shown in Fig. 4, if 0 is the initial strain in the beams, L is length, A is the cross-section, I is the moment of inertia, E is the elastic modulus of the beam and Q is the static load applied at the center of the beam, deflection at the center of the fixed–fixed beam [20] is given by
Fig. 3. Vibration of fixed–fixed beam.
modulus of the beam, the first natural frequency of the fixed–fixed beam [19] is given by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 AL2 f ji ¼0 ¼ f ji ¼0 1 þ 4p2 I
where f ji ¼0 ¼ k
sffiffiffiffiffiffiffiffiffi EI 2 mL4
ð1Þ
0
f ji ¼0 ¼4 f ji ¼0
3
!2
15
where yji ¼0
QL3 ¼ ; 192EI
kL u¼ 2
ð3Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 A and k ¼ I
Similarly the strain at the center [20] is given by
j ¼
and k ¼ 4:73
i
By knowing the first natural frequency corresponding to the first bending mode of the bottom beam along with the geometric and material properties of the frame structure the existing strain in the bottom member of the frame is given by
2
12 u 3 yji ¼0 ¼ yji ¼0 3 sin u cos u u u 2 2
4p 2 I
ð2Þ
AL2
0
¼ ji ¼0
where
j ¼0 ¼ i
12 u 3 sin u cos u u 3 u 2 2
QLy 8EI
and y ¼
ð4Þ
d 2
Since the static displacement and strains at the center of the beam are functions of the existing strain in the beam element by measuring the displacement and strains we can estimate the existing strain in the beam element as given in the above Eqs. (3) and (4) respectively. By knowing the existing strain at a section in the concrete member, the existing prestressing force is calculated by multiplying the existing strain with the modulus of elasticity and cross sectional area of the concrete member.
Q 3. Experimental test set up
L/2
0,
L, A, I, E
Fig. 4. Static deformation of fixed–fixed beam.
In this study a test prism is cast with the VBSG being embedded in it for validation of VBSG using the principle of vibration of axially loaded beam. The VBSG’s are then used in beam and slab for checking the effectiveness of the proposed VBSG.
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3.1. Validation using test prism For validating the efficiency of the VBSG a test prism is cast. The dimensional details of the prism is given in Fig. 5a. A strain gauge is fixed on one of the faces of the prism to compare the surface strain to that on the main bar of the VBSG. The test prism is loaded sequentially along the length with different load levels as shown in Fig. 5b and at each load level the natural frequency of the VBSG is measured. The measured frequency is then compared with the frequency estimated from the FE modeling of the VBSG (shown in Fig. 2b) in finite element software Abaqus [18] with known parameters of the model. 3.2. VBSG in post-tensioned beam The VBSG is used in the post-tensioned beam cast in our laboratory. The beam is of 2400 mm long with width of 250 mm and depth of 250 mm. The beam is posttensioned concentrically with four strands of 12.7 mm diameter each. Each strand is loaded to 0.7 times of its ultimate tensile strength which is 187 KN. The VBSG is placed at the same level as the center of steel duct containing the strands and very close to the duct pipe containing the strands and at a distance of 500 mm from left end of the beam as shown in Fig. 6. 3.3. VBSG in post-tensioned slab The post-tensioned slab used in the validation of VBSG has length of 1400 mm, width of 1000 mm and depth of 500 mm. Three sets of 12.7 mm diameter with seven core strands and twelve numbers in each set are used with an eccentricity of 175 mm from the top surface. Each strand is loaded up to 0.7 times of its ultimate tensile strength of 187 KN. The first, second and third sets are placed at 200 mm, at 500 mm and at 800 mm respectively from
the left end of the cross-section and along the width. Three VBSGs are used one near each prestressing duct at a distance of 400 mm from the left end along the length of the slab and at the same level as the center of steel duct containing the strands as shown in Fig. 7. 3.4. Prestress force measurement from crack reopening method Crack reopening method [15] is commonly used for determining existing prestress force in prestressed concrete structures. In the crack reopening method the structure is first loaded till the cracks appear and then unloaded. Displacement measuring instruments are then fixed across the cracks and loaded again till the cracks reopen. The load is plotted against the crack opening displacement. The knee in the load versus crack reopening plot correspond to the decompression load at the section where the crack exists. By knowing the decompression load the existing prestress force at that section is calculated as
T ¼A
MDL Z
ð5Þ
where T is the existing prestress force, A is the crosssectional area, M DL is the bending moment corresponding to the decompression load and Z is the section modulus at the section crack occurs. However it is always difficult to extract the knee from the load versus crack reopening plot and a small error in the estimated decompression load will result in a large error in the prestress force evaluation. 3.4.1. Crack reopening load from DIC Digital image correlation (DIC) is used to get a more accurate evaluation of the load at which crack reopens [16]. The basics of DIC is explained in [17]. In DIC a grid of pixels is constructed and at each nodes of the grid the relative displacement among the nodes is determined.
Fig. 5. Test prism for validating VBSG.
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Fig. 6. VBSG in beam.
Fig. 7. VBSG in slab.
The relative displacement between two nodes near the two opposite faces of the crack is termed as the crack mouth opening displacement (cmod). The experimental test set up and instrumentation for estimating the decompression load in the prestressed concrete beam is shown in Fig. 8. The decompression load is estimated by evaluating the image number corresponding to the knee in the cmod versus image number plot as shown in Fig. 8. Similar test set up is also used for the prestressed concrete slab for estimating the corresponding decompression load.
3.5. Crack reopening load from change in neutral axis method The process of crack reopening results in shift of the position of centroid of stiffness which in turn changes the position of neutral axis. In this study the change in neutral axis method is used for determining the decompression load at which the crack reopens. The experimental test set up and instrumentation for evaluating the decompression load in the prestressed concrete slab is shown in Fig. 9. The decompression load corresponds to the knee
Fig. 8. Cracking load from DIC.
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Fig. 9. Cracking load from change in N.A.
in the load versus neutral axis plot as shown in Fig. 9. Similar test set up is also used for the prestressed concrete beam for estimating the decompression load. 4. Results Frequency of the VBSG embedded in the test prism is measured before applying the axial load, to update the parameters of the finite model of the VBSG (shown in Fig. 2b) in ABAQUS [18]. The estimated strain mjf m , from the frequency measurement f m , of the VBSG in the test prism at different axial load levels are then compared with the actual strain m , measured from the strain gauge mounted on one side of the test prism. The error in the estimated strain Erjmjf m , is the relative absolute error (in percentage) in the estimated strain mjf m , from the frequency measurement f m , in the VBSG relative to the measured strain m , on the surface of the concrete prism and can be written as
Erjmjf m ¼
jm mjf m j
m
100
ð6Þ
From the results given in Table 1, it is clear that the VBSG is able to estimate existing strain in the test prism with an average error of 1.35% of the measured strains. The post-tensioned beam and slab are kept inside the humidity chamber as shown in Fig. 10 with controlled temperature at 40 °C and relative humidity at 70%. The specimens are taken out at certain time intervals so as to measure the subsequent loss of prestress force with time and after the measurements are taken the specimens are again put back inside the humidity chamber. In this study three measurements are taken on VBSG embedded in the
Table 1 Validation of VBSG using test prism. P appl in KN
m in microstrain
f m in Hz
mjf
in microm strain
% Erjmjf m
50 100 150 200 250
57.6 115.5 173.1 230.5 288.5
40.3 39.7 39.1 38.6 38.0
56.2 113.5 170.7 227.4 284.9
2.43 1.73 1.33 1.34 1.2
beam, one right after stressing and the other two at 78 days and 233 days from the day of stressing respectively. Similarly three measurements are taken on the VBSG embedded in the slab, one after the stressing, accounting the sequencing process in slab and the other two at 104 days and at 222 days from the day of stressing respectively. Three VBSG’s are embedded in the slab one along each prestressing strand duct pipes. In this study only results of the VBSG along the central duct of the slab is reported. Strain and frequency measurements are taken on VBSG before stressing of the beam and slab so as to update the material parameters of the finite element model of VBSG modeled using Abaqus. The total prestress force in the beam and slab versus the strain corresponding to a static load of 10 Newton (small so as not to introduce permanent strain in the VBSG), and frequency of the VBSG embedded in the beam and slab modeled in Abaqus are shown in Fig. 11 for the VBSG in beam and in Fig. 12 for the VBSG in slab. The plots in Figs. 11 and 12 are obtained from simulations run in ABAQUS [18] for response of the VBSG under external excitation with the same initial strain in the VBSG as across the concrete cross-section, wherein the VBSG is embedded. The maximum load up to which simulations are run are based on the number of prestressing strands and the total number of strands used in the concrete structure. In the prestressing beam four prestressing strands are placed with a maximum load of 4 187 ¼ 748 KN. Since the simulations are run in Abaqus a load range from 0 to 800 KN is taken. Similarly in the prestressing slab 3 sets of 12 strands are placed with a maximum load of 3 12 187 ¼ 6732 KN. In the simulation in Abaqus a load range of 0–8000 KN is taken, though the simulation up to the exact maximum load was sufficient. In the finite element model of VBSG the initial strain is taken as the strain induced in the concrete structure at the level of the VBSG due to the prestress force applied at the two ends of the beam and slab respectively. Since the prestressing strands and the VBSG are placed eccentrically in the post-tensioned slab, the curvature of the slab estimated at the level of VBSG is used as initial rotation in the finite element model in ABAQUS (shown in Fig. 2b) at the ends of the VBSG along with the initial axial strain. The total prestress in the beam and slab are estimated by interpolating the corresponding strain and frequency data
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Fig. 10. Beam and slab inside humidity chamber.
Fig. 11. Total tension in beam w.r.t. strain and frequency in VBSG.
Fig. 12. Total tension in slab w.r.t. strain and frequency in VBSG.
of the VBSG in Fig. 11 for the beam and in Fig. 12 for the slab. The comparison of estimated total prestressing force T f m , from frequency measurement f m , and total prestressing force T m , from strain measurement m , in the beam and in the slab against the average of the measured forces T mjav g , from the DIC technique T mjDIC , and from the change in neutral axis technique T mjNA , are given in Table 2 for
the beam and in Table 3 for the slab. The error in the estimated existing prestressing force from the frequency measurement in the VBSG ErjT f m , in Tables 2 and 3 is the relative absolute error (in percentage) with respect to the average of the measured prestressing force T mjav g , from the DIC T mjDIC , and change in N.A technique T mjNA , respectively, and is given by
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Table 2 VBSG implemented in post-tensioned beam. ts in days
T mjDIC in KN
T mjNA in KN
T mjav g: in KN
f m in Hz
T jf m in KN
% ErjT jf m
m
1 78 233
– 413 367
– 422 375
525 417.5 371
38.78 39.17 39.35
531.9 426.4 381.9
1.31 2.13 2.94
24.55 24.03 23.82
in micro strain
T jm in KN
% ErjT jm
530.3 422.8 377.4
1.01 1.27 1.73
Table 3 VBSG implemented in post-tensioned slab. ts in days
T mjDIC in KN
T mjNA in KN
T mjav g: in KN
f m in Hz
T jf m in KN
% ErjT jf m
m
1 104 222
– 3660 3330
– 3680 3360
4600 3670 3345
38.31 38.79 38.97
4666 3742 3423
1.43 1.96 2.33
25.79 25.02 24.74
ErjT f m ¼
jT mjav g T f m j 100 T jav g
ð7Þ
The error in the estimated existing prestressing force from the strain measurement in the VBSG ErjT m , in Tables 2 and 3 is the relative absolute error (in percentage) with respect to the average of the measured prestressing force T mjav g , from the DIC and change in N.A technique respectively, and is given by
ErjT m ¼
jT mjav g T m j 100 T mjav g
ð8Þ
From the results given in Tables 2 and 3, it is clear that the proposed VBSG is able to estimate the existing prestress force in the concrete structures. The error in the estimated tension force from strain data ErjT m , is less compared to the estimated tension force from the measured frequency data ErjT f m , on the VBSG. Also the error increases with reduction in existing prestressing force in concrete members. The increase in error with reduction in existing force may be attributed to the sensitivities of the strain gauges and the accelerometers used for measuring strains and frequencies as well as the least count of the measurements recorded in the data acquisition along with the electronic noise associated with the measured response signal, which is the same irrespective of the absolute measurement value. The measurement performance of the VBSG is more dependent on the sensitivity of the measuring system used, than the actual size of the VBSG. 5. Conclusion The steel frame designed on the working principle of a vibrating wire strain gauge is able to estimate the existing prestressing force in the concrete structures. The average error in estimating the existing prestressing force is around 1.9% from frequency measurements and 1.4% from static strain measurements. Since, both the strain and frequency measurements on the VBSG are able to estimate the existing prestressing force with similar accuracy, in the real application any one of the measurement is sufficient to
in micro strain
T jm in KN
% ErjT jm
4649 3721 3401
1.07 1.39 1.67
estimate the existing prestressing force and both the measurements are not required simultaneously. For applying VBSG in real structures, its size may be chosen so as to have negligible effect on the overall strength of the corresponding structure. The size of the VBSG also depends on the sensitivity and least count of the strain and frequency measuring instruments available. The proposed VBSG can be used directly in the newly constructed prestressed concrete structures for estimating the subsequent prestress losses with time. For existing structures the VBSG can be fixed on the outer surface of the prestressed beam and slab, and by knowing the current condition with some well known NDT techniques, the subsequent prestress loss with time can estimated. Since, in the present study the beam and slab are kept inside humidity chamber in a controlled condition with the same temperature and humidity, the temperature effect on the VBSG is not considered (change in temperature from initial time to subsequent time instances is zero). For implementation in real life structures, the temperature correction on the VBSG has to be considered. It is difficult to estimate the current health of prestressed concrete structures from measured global responses using finite element model updating technique, when both the concrete material properties and the existing prestressing force are not known. Though the error involved using VBSG (1.4–1.9%) in the prestress force estimation is little higher than that of the vibrating wire strain gauge (error of 0.5%), the VBSG with the error in prestress force as measurement uncertainty along with the measured global responses, can accurately estimate the current health of the prestressed concrete structure. A future study is focused on using the estimated prestress force from the VBSG with uncertainty, along with the global responses on the prestressed concrete structure to find out the current health of prestressed concrete structures. Acknowledgment The authors would like to acknowledge the Department of Atomic Energy (DAE), India, for the financial support to conduct experiments, used in this study (under the project Sanction No. 2012/34/45/BRNS-2428).
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Measurement of existing prestressing force in concrete structures through an embedded vibrating beam strain gauge S. Biswal, A. Ramaswamy ⇑ Civil Engineering Department, Indian Institute of Science, Bangalore 560012, India
a r t i c l e
i n f o
Article history: Received 24 August 2015 Received in revised form 12 January 2016 Accepted 18 January 2016 Available online 27 January 2016 Keywords: Existing prestressing force Vibrating wire strain gauge Crack reopening load Digital image correlation Prestressed concrete structures
a b s t r a c t A steel frame is designed to measure the existing prestressing force in the concrete beams and slabs when embedded inside the concrete members. The steel frame is designed to work on the principles of a vibrating wire strain gauge and in the present study is referred to as a vibrating beam strain gauge (VBSG). The existing strain in the VBSG is evaluated using both frequency data on the stretched member and static strain corresponding to a fixed static load, measured using electrical strain gauges. The evaluated strain in the VBSG corresponds to the existing stain in the concrete surrounding the prestressing strands. The crack reopening load method is used to compute the existing prestressing force in the concrete members and is then compared with the existing prestressing force obtained from the VBSG at that section. Digital image correlation based surface deformation and change in neutral axis monitored by putting electrical strain gauges across the cross section, are used to compute the crack reopening load accurately. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Prestressed concrete structures are widely used in the civil engineering field because of the advantages offered such as reduced self weight and cross section of structural elements and reduced deformations. However losses occur in prestressing strands due to time dependent (creep, shrinkage) deformation of concrete and relaxation of steel strands along with the short term losses in concrete (elastic shortening, wedge slip, wobble, friction) and in steel strands. Significant loss level in the prestressing strands can result in compromising safety and serviceability of prestressed concrete structures. Hence it is necessary to predict the loss of prestress in concrete structures over their design life, so as to take decisions on the serviceability and safety of these structures. However as mentioned in
⇑ Corresponding author. E-mail addresses: [email protected] (S. Biswal), ananth@ civil.iisc.ernet.in (A. Ramaswamy). http://dx.doi.org/10.1016/j.measurement.2016.01.031 0263-2241/Ó 2016 Elsevier Ltd. All rights reserved.
[1] since future prestress loss estimates are based on robust analytical models it necessary to conduct experiments for short term measurements based on which the parameters of the analytical model are updated for more realistic prediction of future prestress losses. Hence a cost effective and easy to use system to measure the existing prestress forces at various time instances is necessary. A stress wave based determination of existing tensile force in prestressing strands is given in [2–4] to measure the existing stresses in steel strands. Fiber optic sensors are used in [5,6] to monitor the existing stresses in prestressing strands. However as mentioned in [7] the fiber optic sensors are prone to mechanical damage during their installation on the prestressing strands as a consequence of mishandling, misalignment, pinching or bending at the tendon anchorages. A change in acoustoelastic wave velocity due to change in stress in strands is used as a technique [8] to measure the existing prestress force in steel strands. However the velocity of the applied wave is not much affected by the change in stress in strands. Variation of
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longitudinal stress wave velocity with respect to applied stress level on the strands is studied in [9] for measuring existing prestress in strands. The longitudinal stress wave can be used to measure the existing stress if the applied stress level is lower than about 40% of the ultimate strength of the tendon. However in reality the applied stress level is about 70–80% of the ultimate tensile strength. A relationship between the strand force and the strain in the core wire is developed in [10], so that by measuring the strain in the core wire by stress wave velocity the strand force can be estimated. However the stress condition in the core wire and that on the circumferential wires is different. Measurement of existing stresses using induced magnetic field is given in [11]. A more elaborate study on the effect of grouting on the techniques mentioned in the above studies [4–6,8–11] is required. A sensitivity based finite element model updating technique is developed in [12] for the inverse analysis of the existing prestress force in each element of prestressed girders. In the above [12] method it is assumed that the flexural rigidity of each element is known and this method fails to give unique estimates when both the flexural rigidity and the prestress force are not known. Two commonly used methods of measuring the existing prestressing force in concrete structures are through vibrating wire strain gauges (VWSG) [13,14] and crack re-opening method [15]. While existing stress in concrete through VWSG is a non-destructive technique, existing stress from crack re-opening method is a semidestructive method. The principle of working of VWSG are given in [13,14]. By knowing the strain in the wire and assuming the strain in concrete to be equal to the strain in the wire of the VWSG embedded at the same level of concrete the existing strain in concrete is calculated. The existing stress in the concrete at the level of VWSG is obtained from the calculated strain times the modulus of elasticity of concrete. However VWSG’s are consumables, since they are embedded inside the concrete and they cannot be reused, and the VWSG’s are very expensive for small scale experiments on laboratory as the data acquisition system associated with the VWSG’s is very costly. In case of crack reopening method [15] the structure is loaded till crack appears and is unloaded thereafter. Displacement measuring gauges are then fixed across the crack and the structure is again loaded till the crack re-opens. Knowing the load corresponding to the crack re-opening and equating the total bending stress across the cracked section, the existing stress in the structure is evaluated. However the plot of load versus crack opening displacement is smooth and it is difficult to find the exact load at which the crack re-opens. In the present study a steel frame is designed to be used in the experiments to measure the existing prestressing force in the concrete structures. The steel frame is called as vibrating beam strain gauge (VBSG) since it uses the same principle as the VWSG to measure the existing strain from the measured natural frequencies [13,14]. However in this study the VBSG is used to measure the existing prestressing force from the frequency data as well as from static responses of the stretched element of the VBSG. The major advantage of the proposed VBSG over VWSG is the
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cost which is much less than the VWSG. Data acquisition systems used for the static response measurements and frequency measurements are also much cheaper than the data acquisition system used along with VWSG. The VBSG can be used for measuring existing stresses in concrete structures in any laboratory or industries, where facilities are available for measuring static responses or frequencies only. Unlike the crack reopening method, and similar to the VWSG, the VBSG is a non-destructive tool for measuring the existing prestressing force in the concrete structures. The VBSG is validated against a test prism, discussed in the later section. The VBSG is further embedded in post-tensioned beam and slab and the existing prestressing force obtained is then compared with the existing prestressing force obtained from crack re-opening load. In the present study two methods are used to determine the crack reopening load. The first method is based on deformation results from digital image correlation (DIC) [17] and the second method is based on change in neutral axis of structural member with respect to crack evolution. 2. Experimental determination of existing prestressing force in concrete structures using VBSG The VBSG is embedded in concrete at the same level as the prestressing strands and very close to the strands so that the strain in the strands is same as the strain in concrete at the same level which result in same strain in the main steel bar of the VBSG.The geometric details of the VBSG are given in the section below. 2.1. Test frame set up and instrumentation The idea behind the proposed test frame set up is to embed a steel bar inside the prestressed concrete before casting at the same level as the prestressing strands and after the concrete hardens, be able to load and measure responses, so as to back calculate the existing strain in the steel bar. The test frame set up is made up of stainless steel sections. The steel bar is fixed at both ends by putting threads at the ends and tightened against circular plates. To ensure that the strain in steel bar of the VBSG to be same as that of the concrete at the same level the steel bar is extended at both the ends and tightened against circular plates as shown in Fig. 1a. A vertical steel bar is welded at the center of the main steel bar to facilitate access to the main steel bar for loading and taking responses. For ease in loading a rectangular hollow pipe is cut and welded on the top of the secondary steel bar as shown in Fig. 1a. Strain gauge is fixed at the center of the main steel bar (Fig. 1b) and then covered with beeswax (Fig. 1c) to prevent moisture coming in contact with the strain gauge. Use of water-proof sensors, such as fiber optic sensors can be useful in eliminating the need for the water proofing. However the cost to be paid for the waterproofing sensors is much higher over the benefit obtained, also minimizing the major advantage over the existing VWSG. PVC pipes are then put around the steel bars so as to provide free space for the deformation of the bars under loading. Water sealant is then applied at the joints of the
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Fig. 1. Vibrating beam strain gauge.
pipes and wrapped with PVC tape to prevent water flow from the cement paste into the pipes as shown in Fig. 1d. Since, the PVC pipe around the horizontal bar and the PVC T-connector are wrapped with PVC tape, there is no direct contact (no bond) between concrete and the PVC pipe and the PVC T-connector. It can be assumed that the PVC tape wrapped around the PVC pipe and the PVC Tconnector has negligible strength for small relative displacement between the two, allowing smooth transfer of strain from the surrounding concrete to the VBSG. The length of the horizontal bar of the T-frame in the VBSG is similar to the length of the commercially available VWSG, apart from the diameter of the VBSG which is slightly larger than the VWSG. The length of the vertical bar of the T-frame can be reduced if aligned advantageously along the shortest dimension. Though the crosssectional area of the VBSG is more than the VWSG, it is substantially small compared to the surface area of the concrete structure. However the cross-sectional area of the VBSG is not negligible and does put a constraint on the spacing of VBSGs over the span of the prestressed concrete structure. 2.2. Finite element modeling using Abaqus The VBSG frame is modeled in FE software Abaqus [18]. The horizontal and the vertical steel beams are modeled
with beam element B33 and the top rectangular plates are modeled using shell elements S4 as shown in Fig. 2b. A restart analysis is done for evaluating the responses of the frame with initial strain on the bottom horizontal beam. In the first step of restart analysis an initial strain is applied on the bottom beam followed by a nonlinear static analysis and in the second step a linear perturbation analysis is done to get the static responses and modal properties. 2.3. Experimental determination of existing prestressing force in concrete structures In this study two methods are used to determine the existing prestressing force in the concrete structures with the proposed VBSG test frame set up. In the first method the first natural frequency of the stretched bar of the VBSG is used whereas in the second method static displacement and strains at the center of the stretched bar under known loads is used for the determination existing prestressing force as explained below. 2.3.1. Based on principle of vibration of a beam with initial strain As shown in Fig. 3, if 0 is the initial strain in the beams, m is the mass per unit length, L is length, A is the crosssection, I is the moment of inertia and E is the elastic
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Fig. 2. Vibrating beam strain gauge geometric details and FE model.
2.3.2. Based on principle of static deformation of transverse loaded beam with initial strain As shown in Fig. 4, if 0 is the initial strain in the beams, L is length, A is the cross-section, I is the moment of inertia, E is the elastic modulus of the beam and Q is the static load applied at the center of the beam, deflection at the center of the fixed–fixed beam [20] is given by
Fig. 3. Vibration of fixed–fixed beam.
modulus of the beam, the first natural frequency of the fixed–fixed beam [19] is given by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 AL2 f ji ¼0 ¼ f ji ¼0 1 þ 4p2 I
where f ji ¼0 ¼ k
sffiffiffiffiffiffiffiffiffi EI 2 mL4
ð1Þ
0
f ji ¼0 ¼4 f ji ¼0
3
!2
15
where yji ¼0
QL3 ¼ ; 192EI
kL u¼ 2
ð3Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 A and k ¼ I
Similarly the strain at the center [20] is given by
j ¼
and k ¼ 4:73
i
By knowing the first natural frequency corresponding to the first bending mode of the bottom beam along with the geometric and material properties of the frame structure the existing strain in the bottom member of the frame is given by
2
12 u 3 yji ¼0 ¼ yji ¼0 3 sin u cos u u u 2 2
4p 2 I
ð2Þ
AL2
0
¼ ji ¼0
where
j ¼0 ¼ i
12 u 3 sin u cos u u 3 u 2 2
QLy 8EI
and y ¼
ð4Þ
d 2
Since the static displacement and strains at the center of the beam are functions of the existing strain in the beam element by measuring the displacement and strains we can estimate the existing strain in the beam element as given in the above Eqs. (3) and (4) respectively. By knowing the existing strain at a section in the concrete member, the existing prestressing force is calculated by multiplying the existing strain with the modulus of elasticity and cross sectional area of the concrete member.
Q 3. Experimental test set up
L/2
0,
L, A, I, E
Fig. 4. Static deformation of fixed–fixed beam.
In this study a test prism is cast with the VBSG being embedded in it for validation of VBSG using the principle of vibration of axially loaded beam. The VBSG’s are then used in beam and slab for checking the effectiveness of the proposed VBSG.
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3.1. Validation using test prism For validating the efficiency of the VBSG a test prism is cast. The dimensional details of the prism is given in Fig. 5a. A strain gauge is fixed on one of the faces of the prism to compare the surface strain to that on the main bar of the VBSG. The test prism is loaded sequentially along the length with different load levels as shown in Fig. 5b and at each load level the natural frequency of the VBSG is measured. The measured frequency is then compared with the frequency estimated from the FE modeling of the VBSG (shown in Fig. 2b) in finite element software Abaqus [18] with known parameters of the model. 3.2. VBSG in post-tensioned beam The VBSG is used in the post-tensioned beam cast in our laboratory. The beam is of 2400 mm long with width of 250 mm and depth of 250 mm. The beam is posttensioned concentrically with four strands of 12.7 mm diameter each. Each strand is loaded to 0.7 times of its ultimate tensile strength which is 187 KN. The VBSG is placed at the same level as the center of steel duct containing the strands and very close to the duct pipe containing the strands and at a distance of 500 mm from left end of the beam as shown in Fig. 6. 3.3. VBSG in post-tensioned slab The post-tensioned slab used in the validation of VBSG has length of 1400 mm, width of 1000 mm and depth of 500 mm. Three sets of 12.7 mm diameter with seven core strands and twelve numbers in each set are used with an eccentricity of 175 mm from the top surface. Each strand is loaded up to 0.7 times of its ultimate tensile strength of 187 KN. The first, second and third sets are placed at 200 mm, at 500 mm and at 800 mm respectively from
the left end of the cross-section and along the width. Three VBSGs are used one near each prestressing duct at a distance of 400 mm from the left end along the length of the slab and at the same level as the center of steel duct containing the strands as shown in Fig. 7. 3.4. Prestress force measurement from crack reopening method Crack reopening method [15] is commonly used for determining existing prestress force in prestressed concrete structures. In the crack reopening method the structure is first loaded till the cracks appear and then unloaded. Displacement measuring instruments are then fixed across the cracks and loaded again till the cracks reopen. The load is plotted against the crack opening displacement. The knee in the load versus crack reopening plot correspond to the decompression load at the section where the crack exists. By knowing the decompression load the existing prestress force at that section is calculated as
T ¼A
MDL Z
ð5Þ
where T is the existing prestress force, A is the crosssectional area, M DL is the bending moment corresponding to the decompression load and Z is the section modulus at the section crack occurs. However it is always difficult to extract the knee from the load versus crack reopening plot and a small error in the estimated decompression load will result in a large error in the prestress force evaluation. 3.4.1. Crack reopening load from DIC Digital image correlation (DIC) is used to get a more accurate evaluation of the load at which crack reopens [16]. The basics of DIC is explained in [17]. In DIC a grid of pixels is constructed and at each nodes of the grid the relative displacement among the nodes is determined.
Fig. 5. Test prism for validating VBSG.
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Fig. 6. VBSG in beam.
Fig. 7. VBSG in slab.
The relative displacement between two nodes near the two opposite faces of the crack is termed as the crack mouth opening displacement (cmod). The experimental test set up and instrumentation for estimating the decompression load in the prestressed concrete beam is shown in Fig. 8. The decompression load is estimated by evaluating the image number corresponding to the knee in the cmod versus image number plot as shown in Fig. 8. Similar test set up is also used for the prestressed concrete slab for estimating the corresponding decompression load.
3.5. Crack reopening load from change in neutral axis method The process of crack reopening results in shift of the position of centroid of stiffness which in turn changes the position of neutral axis. In this study the change in neutral axis method is used for determining the decompression load at which the crack reopens. The experimental test set up and instrumentation for evaluating the decompression load in the prestressed concrete slab is shown in Fig. 9. The decompression load corresponds to the knee
Fig. 8. Cracking load from DIC.
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Fig. 9. Cracking load from change in N.A.
in the load versus neutral axis plot as shown in Fig. 9. Similar test set up is also used for the prestressed concrete beam for estimating the decompression load. 4. Results Frequency of the VBSG embedded in the test prism is measured before applying the axial load, to update the parameters of the finite model of the VBSG (shown in Fig. 2b) in ABAQUS [18]. The estimated strain mjf m , from the frequency measurement f m , of the VBSG in the test prism at different axial load levels are then compared with the actual strain m , measured from the strain gauge mounted on one side of the test prism. The error in the estimated strain Erjmjf m , is the relative absolute error (in percentage) in the estimated strain mjf m , from the frequency measurement f m , in the VBSG relative to the measured strain m , on the surface of the concrete prism and can be written as
Erjmjf m ¼
jm mjf m j
m
100
ð6Þ
From the results given in Table 1, it is clear that the VBSG is able to estimate existing strain in the test prism with an average error of 1.35% of the measured strains. The post-tensioned beam and slab are kept inside the humidity chamber as shown in Fig. 10 with controlled temperature at 40 °C and relative humidity at 70%. The specimens are taken out at certain time intervals so as to measure the subsequent loss of prestress force with time and after the measurements are taken the specimens are again put back inside the humidity chamber. In this study three measurements are taken on VBSG embedded in the
Table 1 Validation of VBSG using test prism. P appl in KN
m in microstrain
f m in Hz
mjf
in microm strain
% Erjmjf m
50 100 150 200 250
57.6 115.5 173.1 230.5 288.5
40.3 39.7 39.1 38.6 38.0
56.2 113.5 170.7 227.4 284.9
2.43 1.73 1.33 1.34 1.2
beam, one right after stressing and the other two at 78 days and 233 days from the day of stressing respectively. Similarly three measurements are taken on the VBSG embedded in the slab, one after the stressing, accounting the sequencing process in slab and the other two at 104 days and at 222 days from the day of stressing respectively. Three VBSG’s are embedded in the slab one along each prestressing strand duct pipes. In this study only results of the VBSG along the central duct of the slab is reported. Strain and frequency measurements are taken on VBSG before stressing of the beam and slab so as to update the material parameters of the finite element model of VBSG modeled using Abaqus. The total prestress force in the beam and slab versus the strain corresponding to a static load of 10 Newton (small so as not to introduce permanent strain in the VBSG), and frequency of the VBSG embedded in the beam and slab modeled in Abaqus are shown in Fig. 11 for the VBSG in beam and in Fig. 12 for the VBSG in slab. The plots in Figs. 11 and 12 are obtained from simulations run in ABAQUS [18] for response of the VBSG under external excitation with the same initial strain in the VBSG as across the concrete cross-section, wherein the VBSG is embedded. The maximum load up to which simulations are run are based on the number of prestressing strands and the total number of strands used in the concrete structure. In the prestressing beam four prestressing strands are placed with a maximum load of 4 187 ¼ 748 KN. Since the simulations are run in Abaqus a load range from 0 to 800 KN is taken. Similarly in the prestressing slab 3 sets of 12 strands are placed with a maximum load of 3 12 187 ¼ 6732 KN. In the simulation in Abaqus a load range of 0–8000 KN is taken, though the simulation up to the exact maximum load was sufficient. In the finite element model of VBSG the initial strain is taken as the strain induced in the concrete structure at the level of the VBSG due to the prestress force applied at the two ends of the beam and slab respectively. Since the prestressing strands and the VBSG are placed eccentrically in the post-tensioned slab, the curvature of the slab estimated at the level of VBSG is used as initial rotation in the finite element model in ABAQUS (shown in Fig. 2b) at the ends of the VBSG along with the initial axial strain. The total prestress in the beam and slab are estimated by interpolating the corresponding strain and frequency data
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Fig. 10. Beam and slab inside humidity chamber.
Fig. 11. Total tension in beam w.r.t. strain and frequency in VBSG.
Fig. 12. Total tension in slab w.r.t. strain and frequency in VBSG.
of the VBSG in Fig. 11 for the beam and in Fig. 12 for the slab. The comparison of estimated total prestressing force T f m , from frequency measurement f m , and total prestressing force T m , from strain measurement m , in the beam and in the slab against the average of the measured forces T mjav g , from the DIC technique T mjDIC , and from the change in neutral axis technique T mjNA , are given in Table 2 for
the beam and in Table 3 for the slab. The error in the estimated existing prestressing force from the frequency measurement in the VBSG ErjT f m , in Tables 2 and 3 is the relative absolute error (in percentage) with respect to the average of the measured prestressing force T mjav g , from the DIC T mjDIC , and change in N.A technique T mjNA , respectively, and is given by
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Table 2 VBSG implemented in post-tensioned beam. ts in days
T mjDIC in KN
T mjNA in KN
T mjav g: in KN
f m in Hz
T jf m in KN
% ErjT jf m
m
1 78 233
– 413 367
– 422 375
525 417.5 371
38.78 39.17 39.35
531.9 426.4 381.9
1.31 2.13 2.94
24.55 24.03 23.82
in micro strain
T jm in KN
% ErjT jm
530.3 422.8 377.4
1.01 1.27 1.73
Table 3 VBSG implemented in post-tensioned slab. ts in days
T mjDIC in KN
T mjNA in KN
T mjav g: in KN
f m in Hz
T jf m in KN
% ErjT jf m
m
1 104 222
– 3660 3330
– 3680 3360
4600 3670 3345
38.31 38.79 38.97
4666 3742 3423
1.43 1.96 2.33
25.79 25.02 24.74
ErjT f m ¼
jT mjav g T f m j 100 T jav g
ð7Þ
The error in the estimated existing prestressing force from the strain measurement in the VBSG ErjT m , in Tables 2 and 3 is the relative absolute error (in percentage) with respect to the average of the measured prestressing force T mjav g , from the DIC and change in N.A technique respectively, and is given by
ErjT m ¼
jT mjav g T m j 100 T mjav g
ð8Þ
From the results given in Tables 2 and 3, it is clear that the proposed VBSG is able to estimate the existing prestress force in the concrete structures. The error in the estimated tension force from strain data ErjT m , is less compared to the estimated tension force from the measured frequency data ErjT f m , on the VBSG. Also the error increases with reduction in existing prestressing force in concrete members. The increase in error with reduction in existing force may be attributed to the sensitivities of the strain gauges and the accelerometers used for measuring strains and frequencies as well as the least count of the measurements recorded in the data acquisition along with the electronic noise associated with the measured response signal, which is the same irrespective of the absolute measurement value. The measurement performance of the VBSG is more dependent on the sensitivity of the measuring system used, than the actual size of the VBSG. 5. Conclusion The steel frame designed on the working principle of a vibrating wire strain gauge is able to estimate the existing prestressing force in the concrete structures. The average error in estimating the existing prestressing force is around 1.9% from frequency measurements and 1.4% from static strain measurements. Since, both the strain and frequency measurements on the VBSG are able to estimate the existing prestressing force with similar accuracy, in the real application any one of the measurement is sufficient to
in micro strain
T jm in KN
% ErjT jm
4649 3721 3401
1.07 1.39 1.67
estimate the existing prestressing force and both the measurements are not required simultaneously. For applying VBSG in real structures, its size may be chosen so as to have negligible effect on the overall strength of the corresponding structure. The size of the VBSG also depends on the sensitivity and least count of the strain and frequency measuring instruments available. The proposed VBSG can be used directly in the newly constructed prestressed concrete structures for estimating the subsequent prestress losses with time. For existing structures the VBSG can be fixed on the outer surface of the prestressed beam and slab, and by knowing the current condition with some well known NDT techniques, the subsequent prestress loss with time can estimated. Since, in the present study the beam and slab are kept inside humidity chamber in a controlled condition with the same temperature and humidity, the temperature effect on the VBSG is not considered (change in temperature from initial time to subsequent time instances is zero). For implementation in real life structures, the temperature correction on the VBSG has to be considered. It is difficult to estimate the current health of prestressed concrete structures from measured global responses using finite element model updating technique, when both the concrete material properties and the existing prestressing force are not known. Though the error involved using VBSG (1.4–1.9%) in the prestress force estimation is little higher than that of the vibrating wire strain gauge (error of 0.5%), the VBSG with the error in prestress force as measurement uncertainty along with the measured global responses, can accurately estimate the current health of the prestressed concrete structure. A future study is focused on using the estimated prestress force from the VBSG with uncertainty, along with the global responses on the prestressed concrete structure to find out the current health of prestressed concrete structures. Acknowledgment The authors would like to acknowledge the Department of Atomic Energy (DAE), India, for the financial support to conduct experiments, used in this study (under the project Sanction No. 2012/34/45/BRNS-2428).
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