Abstract
The interface between anchor plate and anchorage nut can be considered as a coupling spring whose stiffness is essentially the contact stiffness related to normal force at the contact surface, thus removing the usual assumption of fixed connection. The nut threads and steel bar are also connected with a coupling spring with the stiffness determined based on the threads treated as cantilever components. An analytical model was developed to assess the axial vibration of an anchorage system consisting of prestressing fine-rolled screw-threaded steel bars (P/S FSSBs). Both laboratory model and field tests show that the contact stiffness between anchor plate and anchorage nut has good a linear relationship with the effective tension of prestressed tendons. The vibration model was used to obtain the natural frequency and tensile force of a P/S FSSB. It is shown to be feasible and practical. According to the analytical model and test method presented in this paper, the proposed tension tester for P/S FSSBs possesses the features of repeated use, rapid testing, and minimized impact on the construction and is thus suggested for wider practical applications.
1. Introduction
Prestressing fine-rolled screw-threaded steel bars (P/S FSSBs) have been extensively used as vertical prestressing tendons in concrete bridges. The FSSB anchorage system consists of a FSSB, anchor plate with nut at the top and bottom, and exposed segment at the top, as shown in Figure 1 where the anchor plate and anchorage nut are in contact with each other and the nut anchorage section is connected with the FSSB through the threads. The effective tension of the FSSB affects bridge’s safety and durability significantly. The exposed section of vertical P/S FSSB including the nut section has been analyzed as a fully fixed cantilever beam by Zhong et al. [1], in which a developed test method was used to measure the effective tension of the bar system by observing the natural frequency change in the exposed section. However, their proposed method would result in a gross error when the tension is low and thus the tensile testing cannot be continued once the exposed section is cut. Therefore, it is necessary to establish a better model to more accurately measure the tensile force in P/S FSSB.
Figure 2 shows the proposed mechanical model representing the FSSB anchorage system (Figure 1). The anchor plate-anchorage nut interface can be represented by a spring with the normal contact stiffness k. The interaction between the threads in the anchorage nut and the FSSB can also be represented by springs with the stiffness kst (see the enlarged view A in Figure 2). Each FSSB end is fixed with the anchor plate and attached to the concrete web of the prestressed concrete (PC) box-girder bridge. The acceleration sensor was installed vertically on the top nut surface and was connected to the signal analyzer, as shown in Figure 2. At the top of the FSSB exposed section, the transient excitation was applied to simulate vibration signals which can then be used to estimate the axial vibration natural frequencies of the bar by a signal analysis. It is possible to estimate the contact stiffness k value based on its relation with the axial vibration natural frequency. Based on proportional relation between contact stiffness k and normal contact force (i.e., the effective tension in P/S FSSB) [2–4], the effective tension in P/S FSSB may be determined indirectly. The model presented in Figure 2 is similar to the bolt fastening component used in mechanical engineering and pipeline engineering. The relationship between the axial vibration frequency of the mud recovery pipe of deep-water drilling rig and the water depth was studied by Wang et al. [5]. Yang et al. [6] studied the axial vibration control of ship propulsion shafting based on the mass and spring model. An approximate analytical model for the axial vibration of an induction motor was established by Medoued et al. [7], where the acceleration sensor was used to measure axial vibration signals, to analyze the change of axial vibration frequencies, and to realize the damage identification of the motor. Based on the deformation between screw rod and nut threads, a discrete model was developed to evaluate the axial vibration characteristics of a bolt fastener in a variety of mechanical connections [8–11]. 144 experiments were conducted by Tian et al. [3] to estimate the tangential and normal contact stiffnesses at the interface where transient excitations were used to estimate the natural frequencies. Using the springs to represent the contact surface (similar to Figure 2), an analytical model was developed to estimate the tangential and normal stiffnesses at the interface for different normal stress conditions [2–4]. Based on the literature review, it is deemed feasible to obtain the natural frequency to determine the normal contact stiffness by means of exercising transient excitations on the structure with similar contact surface.
2. Discrete Mechanical Model for Axial Vibration
The discrete mechanical model relies on the contacting relationships between the anchor plate and anchorage nut and between the nut threads and FSSB.
2.1. Equations of Motion
Based on Figure 2, a discrete P/S FSSB anchorage system model was developed (Figure 3) in which the exposed section of vertical P/S FSSB (Figure 2) consists of two mass blocks with the mass for each, which are connected by a coupling spring with the stiffness (Figure 3). Note that , where A is the cross-sectional area and ρ is the mass density of FSSB. The tensioning section of vertical P/S FSSB (Figure 2) consists of a mass block with the mass and a coupling spring with the stiffness (Figure 3), where . The object of this study is the anchorage system in which the anchorage nut has four laps of threads [12]. The anchorage nut section (Figure 2) and each lap of threads in the anchorage nut are discretized into two mass blocks with the masses and and two coupling springs with the stiffnesses and (Figure 3). Note that and equals to 1/4 of the mass of the anchorage nut. Each lap of threads and the anchorage nut are connected by a spring with the stiffness (see the enlarged view A in Figure 2). It is assumed that the bottom of the anchorage nut section is fixed. is the deformation of the exposed segment; , , , , and are the screw-thread displacements of the FSSB in the anchor section; and , , , , and are the counterparts of the nut (Figure 3). Regarding the studied problem, there is virtually no damping effect, and therefore, it is ignored. The equations of motion for the discrete model shown in Figure 3 can be derived using D’Alembert’s principle as follows:
In equations (1)–(11), the second-order derivatives of the displacement with respect to time are , , , , , , , , , , and . The matrix form iswhere (from equation (14)) and (from equation (15)) are the mass and stiffness matrices, respectively; is the column displacement vector (from equation (13)); is the column vector of acceleration; and 0 is the zero column vector:
2.2. Model Parameters
2.2.1. Stiffnesses of Anchorage Nut and FSSB
It is assumed that the cross-sectional areas of FSSB and anchorage nut before and after the tension are virtually unchanged. In equations (1)–(11), , , , and , where , , , and represent the nominal diameter of FSSB, the length of the tensioning section, the length of the exposed section, and the height of the anchorage nut section, respectively. is the area of the cross section of the anchorage nut, and is the modulus of elasticity of FSSB and anchorage nut.
2.2.2. Calculation of the Interaction Stiffness of Threads
Figure 4 shows the anchorage nut and steel bar sections, where dt, ds, and dv are the anchorage nut diameter, screw-thread outer diameter, and nominal diameter of FSSB, respectively. The screw-thread structure is shown in Figure 5 where , , , , , and are, respectively, the thread pitch, thread lap length, bottom thread width, thread height, tilting angle on the top thread surface, and tilting angle on the bottom thread surface. Treating the threads as cantilever components similarly to the study of Zhong et al. [13], the stiffness of thread interaction between the FSSB and the anchorage nut can be determined bywhere
(a)
(b)
(a)
(b)
Once relevant calculation parameters are obtained, the following equation can be concluded from equation (12):where ω is the system’s natural frequency.
The nontrivial solution of equation (18) is obtained under the following condition:
The unknown parameters in equation (19) include the natural vibration frequency ω and the contact stiffness . If ω is found, can be obtained from equation (19) and then the natural vibration frequencies of each mode can be determined for the model shown in Figure 2.
3. Experimental Investigation on the Natural Frequency-Tension Relationship of FSSB under Axial Vibration
3.1. Modeling and Test Scheme
The test model was a reinforced concrete (RC) cuboid with the strength grade of C50 and the dimensions of 300 cm × 360 cm × 120 cm. In the model, reserve double row ducts with five 5 cm diameter ducts spaced at ≈45 cm in each row were used. The compression bars and anchor plate of Q235 were set at both ends of the duct before pouring the concrete. The FSSB with the diameter of 32 mm and its anchoring system were used in this test. This experiment used five additional steel bars with the respective tension values of 100 kN, 200 kN, 300 kN, 400 kN, and 530 kN. Figure 6 shows the experimental model of this study.
(a)
(b)
In order to ensure that the frequency test and analysis are not affected by boundary conditions, the P/S FSSB was not installed with the force sensor. To measure the effective tension similarly to the study of Zhong et al. [14], the test device shown in Figure 7 was used. The test scheme was set up as follows: (1) first, the anchorage nut was installed and the vertical P/S FSSB was connected by a connecting sleeve, followed by the installation of the lifting jack, reaction frame, self-locking nuts, and other components; (2) a force sensor was installed between the self-locking nuts and the lifting jack, and then the magnetic stand and the dial indicator were installed; (3) the oil pump was started, the lifting jack was loaded, the vertical P/S FSSB tension was allowed to reach the predetermined threshold value, the tension was measured with a force sensor, and afterwards, the oil pump’s return valve was closed; (4) after the anchorage nut was tightened, the dial indicator value was read and the tension was measured again; lastly (5) the oil pump’s return valve was opened, the lifting jack was unloaded, the dial indicator data were recorded again, and then the reaction frame and the self-locking nuts were demolished. The effective tension was calculated according to the study of Zhong et al. [14].
3.2. Signal Testing of the Axial Vibration of FSSB
3.2.1. Test Instrument
The acceleration sensor (Brüel & Kjær 4514-B-001) with a sensitivity of 10 mV/m/s2 was used, and the analysis and acquisition system of the dynamic signal was the Hi-Techniques Synergy Data Acquisition System.
3.2.2. Test Procedure
(a) The height and mass of the anchorage nut and the mass per unit length of FSSB were measured, and the effective tension was obtained according to the above test scheme (Figure 7) and the study of Zhong et al. [14]; (b) the acceleration sensor was installed on the top anchorage nut surface (Figure 2), the signal analyzer was started, and the sampling frequency was set to 200 kHz; (c) an artificial strike hammer with its head fabricated from chloroprene rubber (shore hardness D60 and weight = 300 g) was used to strike the top part of FSSB (Figure 8); and (d) the acceleration-time histories of axial vibration of the FSSB were obtained first and then the fast Fourier transform (FFT) technique was used to convert them into the acceleration-response spectra. The spectra corresponding to the five different effective tensions are shown in Figures 9–13 with the corresponding frequencies listed in Table 1.
3.3. Relationship between the Contact Stiffness and the Effective Tension
The actual vertical P/S FSSB in a bridge is shown in Figure 1, where the bottom anchorage setup is fixed and connected with the concrete girder web. The experimental model shown in Figure 6 differs from that for the actual bridge, where the top and bottom anchorage setups are symmetrical and the same stress state is assumed between them. in equations (1)–(11) is equal to 1/2 of the length of the tension section in the model, as indicated in Table 1.
According to the anchorage system of FSSB and the relevant parameters indicated in Table 1, the stiffness and mass values of the FSSB and anchorage nut of the test model were computed as summarized in Table 2. Substituting the results from Tables 1 and 2 into equation (19) yields the corresponding effective tensions of 110.8 kN, 210.7 kN, 306.6 kN, 397.9 kN, and 522.6 kN (Table 1). The anchor plate-anchorage nut contact stiffnesses of the experimental model corresponding to the effective tensions are summarized in Table 3. Figure 14 shows the relation between the effective tension and contact stiffness for the experimental model, which is expressed by the regression equation: with R2 = 0.9990 indicating a good linear correlation.
4. Field Test for Assessment of the Contact Stiffness-Effective Tension Relationship
4.1. Engineering Background and Field Test Plan
A field experiment was carried out at Dagutan Bridge in Fujian Province, China, which involves a three-span (66 m + 120 m + 66 m) PC rigid box-girder bridge. The main girder is a single-cell box with variable heights. The vertical P/S FSSB has the nominal diameter of 25 mm, and the control tension for a single steel bar is 327.5 kN. The height of the anchorage nut is 0.065 m. The test instrument and scheme are as described in the above.
4.2. Results of Frequency Testing and Contact Stiffness Calculation
A field test of 16 P/S tendons was conducted, in which the geometric parameters are listed in Table 4 along with the effective tensions and measured frequencies. The computed stiffnesses and masses for the FSSB and anchorage nut based on the field test results are summarized in Table 5. Substituting the results from Tables 4 and 5 into equation (19), the contact stiffnesses are obtained, as shown in Table 6. The relation between the contact stiffness k and effective tension is shown in Figure 15, which is described by the regression equation: with R2 = 0.9872 indicating a good linear correlation. Note that the imperfectly vertical FSSB and uneven contact surface between the anchor plate and anchorage nut in the field contribute to some measurement errors.
5. Application of the Developed Analytical Model for the Axial Vibration of P/S FSSB Anchorage System
The application of the test principle is illustrated in Figure 16. According to the scheme shown in Figure 7, the graded tensioning of P/S FSSB generally started from 200 kN and increased by about 100–200 kN per load level until the design tension was reached. Using this testing procedure (Figure 16), the effective tension in the tensioning section l3 was calculated based on the study of Zhong et al. [14], the length of the corresponding exposed section l2 was obtained, the frequency of the exposed section ω was estimated, and the relevant parameters were calculated. The contact stiffness can be calculated by equation (19). The above procedure was repeated, and the test was completed using three additional P/S FSSBs. The relationship between the contact stiffness and effective tension can be described by with and being the fitting parameters. This process is shown in Figures 16(a) and 16(b).
After tensioning the vertical P/S FSSB, the natural frequency of axial vibration was tested according to the model shown in Figure 2. By substituting the related parameters into equation (19), the contact stiffness was obtained. The effective tension T of P/S FSSB was obtained by substituting into the relation equation between the contact stiffness and effective tension. This process is shown in Figures 16(c) and 16(d).
As seen in Figure 16, the relevant calculation procedure is integrated into the signal analysis device. The fabrication of a specialized testing device (i.e., the tension tester) for estimating the effective tension of P/S FSSBs is presented in Figure 17; this device is comprised of different systems for acquiring and analyzing signals, importing frequencies and calculating tensions. The tension tester controls the main menu of the liquid crystal panel which includes four submenus: the input of the parameter, acquisition of signals, window of the signal display, and module of tension calculation.
6. Conclusions
In this study, the contact between the anchor plate and anchorage nut is represented by a coupling spring whose stiffness is essentially the contact stiffness related to the normal force at the contacting surface, which lifts the usual assumption of fixed connection. The FSSB and nut threads are also connected by a coupling spring whose stiffness is calculated by treating the threads as cantilever components. In this study, an analytical model is proposed for assessing the axial vibration of the P/S FSSB anchorage system. Both model and field tests show that the contact stiffness between the anchor plate and anchorage nut has a good linear relationship with the effective tension of the P/S FSSB. For practical purposes, a fitted linear equation was derived based on the field test results and is suggested to describe the contact stiffness -effective tension relationship of P/S FSSB. Based on the fitted equation and the natural frequency of the axial vibration of P/S FSSB, the contact stiffness can be determined using the proposed model and then the P/S FSSB’s effective tension can be obtained indirectly. The study results show that the effective tension test method is practical and feasible. The developed tension tester has the features of repeated use, fast testing, and minimized impact on the construction; therefore, it is suggested that the proposed method be considered as a nondestructive test for the effective tension of P/S FSSBs.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 51178183), to which the authors are very grateful.