The natural frequency change of reinforced concrete (RC) members during damage when subjected to low cycle loading was studied through horizontal cyclic loading experiments. Three groups of RC flexural members were subjected to horizontal, harmonic, low cycle loading to simulate earthquake conditions. The relation of instantaneous load, instantaneous displacement, and instantaneous natural frequency during loading was deduced. Using the resulting equation, the test members’ natural frequencies at any moment during loading could be calculated accurately. Then the natural frequency change curves and their fitting equations were also obtained. The impact of loading period
It is important for engineers to assess the integrity or damage status of a structure after severe cyclic loading, such as from an earthquake. One method for diagnosing the status of structural damage relies on the structural natural frequency [
Most recent research focuses on identifying structural damage through the natural frequency change. However, there are few such investigations. Abraham et al. [
An RC flexural member was used as the research object in a low cycle loading experiment to determine the change characteristics of an RC structural member during seismic damage. By changing the section size and reinforcement, three groups of nine test members were designed with different sizes, strengths, and stiffnesses. The structures of the test members are shown in Figures
Size and reinforcement of the first group of test members.
Size and reinforcement of the second group of test members.
Size and reinforcement of the third group of test members.
Basic parameters of test members.
Group (specimen number)  Section size (mm)  Reinforcement ratio (%)  Section size of ground beam (mm)  Area 1 of stirrups  Area 2 of stirrups  Area 3 of stirrups  Amount of specimens 

1st group (Z1–9)  250 ∗ 350  1.4  400 ∗ 500  6@50  6@100  8@100  9 
2nd group (J1–9)  320 ∗ 450  1.3  400 ∗ 500  8@50  8@100  10@100  9 
3rd group (M1–9)  286 ∗ 400  1.6  400 ∗ 500  8@50  8@100  10@100  9 
The following materials were used. Concrete used in the test members was C30 (GB500102010, Ministry of Housing and UrbanRural Development of the People's Republic of China (MOHURD)) [
The loading system consisted of a hydraulic servo actuator, reaction wall, connecting pieces, and jack and auxiliary equipment, as shown in Figure
Loading system.
The common method of acquiring a test member’s frequency during structural damage is to stop loading and use special instruments to measure the test member’s frequency at regular time intervals [
A reinforced concrete seismic member’s yield moment can be approximated as [
The yield load is
Using the graph multiplication method, the yield displacement can be obtained as
The yield load and yield displacement results of each group’s specimens are shown in Table
Yield loads and yield displacements.
Specimen  Item  

Yield moment (N·mm)  Yield load (N)  Yield displacement (mm)  
First group  115.3 × 10^{6}  82.2 × 10^{3}/60  8.7 
Second group  235.9 × 10^{6}  148.7 × 10^{3}/130  15.5 
Third group  187.3 × 10^{6}  122.1 × 10^{3}/110  11.8 
Ground motion during an earthquake is a complex irregular vibration. It is difficult to study the structural damage process under an earthquake load because of the load’s irregularity, which encompasses various vibration amplitudes and frequencies [
Harmonic vibration loading scheme.
Period  Amplitude  



 

Z1/J1/M1  Z4/J4/M4  Z7/J7/M7 

Z2/J2/M2  Z5/J5/M5  Z8/J8/M8 

Z3/J3/M3  Z6/J6/M6  Z9/J9/M9 
Each group of test members had a different strength, stiffness, and size, so their damage processes differed. However, their damage processes could generally be divided into four stages.
In the first stage, the bottom of a test member cracked at the first loading cycle; cracks then developed rapidly and soon passed throughout the front and back surface of the test member. Test data showed that a test member’s carrying capacity decreased rapidly. This stage corresponds to the first 20 cycles of loading for the first group of test members, the first 10 cycles for the second group, and the first 15 cycles for the third group.
In the second stage, cracks further developed and the number of cracks increased obviously, and the concrete was partially spalled. Concrete and reinforcement still worked together to bear the load. Reinforcements yielded, and a plastic hinge was formed at the bottom of the test members. The rate of damage increased, and the carrying capacity decreased quickly. This stage corresponds to 20–60 cycles of loading for the first group of test members, 10–40 cycles for the second group, and 15–50 cycles for the third group.
In the third stage, the concrete between the main cracks crushed severely, and many concrete blocks spalled. Concrete no longer bore a load, so the load was borne solely by the steel. The plastic hinge zone further expanded. This stage corresponds to 60–100 cycles of loading for the first group of specimens, 40–70 cycles for the second group, and 50–80 cycles for the third group.
In the fourth stage, the test member was destroyed completely, a large number of concrete blocks spalled, and the steel bars were exposed. The load applied to a test member maintained a certain low value and varied little. Test members’ destruction zones concentrated at the bottom. In this stage, the height of the destruction zone (plastic hinge region), which could be clearly observed, was about 35 cm for the first group of specimens, 20 cm for the second group, and 27 cm for the third group.
Figure
Damage of Z2 at (a) 10th loading cycle; (b) 30th loading cycle; (c) 70th loading cycle; (d) 100th loading cycle.
Damage of J3 at (a) 10th loading cycle; (b) 30th loading cycle; (c) 70th loading cycle; (d) 100th loading cycle.
Damage of M5 at (a) 10th loading cycle; (b) 30th loading cycle; (c) 70th loading cycle; (d) 100th loading cycle.
A specimen’s failure mode is important for the study of its frequency change during the damage process, because different failure modes indicate different damage processes. Research shows that RC columns generally have three types of failure mode: bending, shear, and bending shear [
For the convenience of study, the failure modes of all test members should be the same. To induce bending failure in all the test members, an axial force was not applied to test members, i.e., the axial compression ratios were zero. Large shear span ratios were adopted, as shown in Table
Specimens’ axial compression ratios, shear span ratios, and stirrup ratios.
Specimen  Item  





First group  0  3.76  0.7 
Second group  0  2.92  0.85 
Third group  0  3.29  0.96 
Hysteretic curves reflect a test member’s strength degradation, stiffness degradation, and energy dissipation [
Typical hysteretic curves of some test members. (a) Z1. (b) J1. (c) M2. (d) Z4. (e) J4. (f) M4. (g) Z7. (h) J7. (i) M8.
Figure
We analyzed the impact of load and displacement on a test member’s natural frequency and determined a function of the three variables. This function enabled us to accurately obtain the realtime value of a test member’s natural frequency.
During reciprocating movement, a test member’s stiffness changed continuously with time because of damage. Stiffness is a function of the time
Equation (
The structural natural frequency (circular frequency) is [
A test member’s quality,
By substituting Equation (
Equation (
According to the loading plan (Table
The measured data of test members Z1, Z2, Z3, J1, J2, J3, M1, M2, and M3 were analyzed. Their loading amplitudes were all 30 mm, but their loading periods varied. The loading displacement
Natural frequencychange curves. (a) Z1, Z2, Z3. (b) J1, J2, J3. (c) M1, M2, M3. (d) Z4, Z5, Z6. (e) J4, J5, J6. (f) M4, M5, M6. (g) Z7, Z8, Z9. (h) J7, J8, J9. (i) M7, M8, M9.
Similarly, curves of the inherent frequency versus the number of loading cycles were obtained for test members Z4, Z5, Z6, J4, J5, J6, M4, M5, and M6, whose loading amplitudes were all 24 mm, as shown in Figures
It can be seen that the impact of the loading period (that is, the speed of a test member’s reciprocating motion) on the damage process is mainly reflected in the speed of damage. Thus, a relationship between the loading period and the damage speed was thought to exist. For further analysis, we obtained the speed of the damage process of each test member. We determined a test member’s damage rate function by taking the first derivative of the fitting equation of the frequencychange curve, as shown in Table
Frequency change equations and damage rate equations of test members.
Specimen category  Equation  

Frequency change equation  Damage rate equation  

Z1 ( 


Z2 ( 

 
Z3 ( 

 
J1 ( 

 
J2 ( 

 
J3 ( 

 
M1 ( 

 
M2 ( 

 
M3( 

 



Z4 ( 


Z5 ( 

 
Z6 ( 

 
J4 ( 

 
J5 ( 

 
J6 ( 

 
M4 ( 

 
M5 ( 

 
M6 ( 

 



Z7 ( 


Z8 ( 

 
Z9 ( 

 
J7 ( 

 
J8 ( 

 
J9 ( 

 
M7 ( 

 
M8 ( 

 
M9 ( 


Damagerate fitted surfaces of (
Equation (
Coefficients of function of
Test members  Coefficients of the function  






 
Z1, Z2, Z3  −6  0.024  4.5  −6.2 
−0.01  −0.88 
J1, J2, J3  −3.6  0.0088  2.59  −3.887 
0.0001  −0.5 
M1, M2, M3  −1.476  0.0113  0.84  −1.2 
−0.003  −0.149 
Z4, Z5, Z6  −1.6  0.012  1.1  −2.43 
−0.0059  −0.18 
J4, J5, J6  −4.735  0.019  2.97  −1.3 
−0.006  −0.48 
M4, M5, M6  −0.208  0.002  −0.04  −1.2 
−1.9 
0.014 
Z7, Z8, Z9  0.32  0.0011  −0.37  −2.7 
−0.00034  0.08 
J7, J8, J9  −0.16  −0.0015  0.18  −1.711 
0.001  −0.078 
M7, M8, M9  −0.185  0.0019  −0.02  3.6 
1.2 
0.003 
We drew the following conclusions from Equation (
The test members were regrouped according to their loading periods; that is, Z2, Z5, Z8, J2, J5, J8, M2, M5, and M8 were sorted into a group whose loading period was 2.4 s; Z1, Z4, Z7, J1, J4, J7, M1, M4, and M7 were sorted into a group whose loading period was 1.875 s; and Z3, Z67, Z9, J3, J6, J9, M3, M6, and M9 were sorted into a group whose loading period was 3.0 s. The natural frequency change curves of these three groups are shown in Figures
Natural frequencychange curves. (a) Z2, Z5, Z8. (b) J2, J5, J8. (c) M2, M5, M8. (d) Z1, Z4, Z7. (e) J1, J4, J7. (f) M1, M4, M7. (g) Z3, Z6, Z9. (h) J3, J6, J9. (i) M3, M6, M9.
The fitting equations of the curves and damage rate equations can be seen in Table
Damagerate fitted surfaces of
Coefficients of function
Test members  Coefficients of the function  






 
Z2, Z5, Z8  −0.075  −0.0077  0.0033  6.3 
−5.2 
−9.9 
J2, J5, J8  −0.37  0.009  −0.01  9.539 
−0.0002  0.0004 
M2, M5, M8  0.044  0.0019  −0.03  −1.2 
1.77 
0.0005 


Z1, Z4, Z7  −0.36  0.032  −0.0012  −0.0012  0.0002  −3.01 
J1, J4, J7  4.9  −0.0048  −0.44  −4.8 
0.0006  0.008 
M1, M4, M7  −3.257  0.0206  0.185  −3.57 
−0.0007  −0.0026 


Z3, Z6, Z9  −1.56  0.12  0.013  −0.0028  −0.00054  1.4 
J3, J6, J9  −2.578  −0.0049  0.2  5.356 
0.0005  −0.0048 
M3, M6, M9  −1.207  0.0066  0.066  −2.4 
−0.000166  −0.0012 
Figure
The previous section only analyzed the individual impact of the loading period or loading amplitude on a test member’s frequency change. We now examine how the loading period and loading amplitude work together to affect a test member’s frequency change, so as to find the natural frequencychange characteristics of an RC member under an arbitrary harmonic vibration.
The change characteristics of a test member’s frequency can be expressed as a function of one variable,
Take the nonlinear items in the above nonlinear models as new elements, as shown in Table
Variable substitution.
Equation ( 
Equation (  

Original item  New item  Original item  New item 





































 

 

 

 

 

 

 

 

 


Equations (
Present all the independent valuables which may have influence on the dependent valuable according to relevant theories and experiences.
Calculate the correlation coefficients of the independent variables to the dependent variable. Sort the independent variables according to their correlation coefficients absolute value from large to small.
Set up one element linear fitting model using the independent variable with the largest correlation coefficient absolute value. Then test the significance of this fitting equation. If the test shows that the fitting effect is significant, then go to (4). And if it is not significant, stop modeling.
Carry out the addition and elimination of the independent variables and the updating of fitting equations. The specific process is as below. According to the order of the absolute value of the correlation coefficient from large to small, the corresponding independent variables are introduced into the regression equation one by one. Each new variable is introduced, and every independent variable in the new regression equation should be tested for significance. And, also the significance of the new regression equation should be tested. If the test shows that the regression effect is not significant, the independent variable with less influence on the dependent variable is excluded and the regression equation is updated. After that, every independent variable in the updated regression equation will still need to be tested and eliminated, and then regression equation is updated again, and this is repeated this until each independent variable in the regression equation is significant. And then, the independent variable is reintroduced that has not been introduced before.
And so on, until the introduced independent variables cannot be eliminated and new independent variables cannot be introduced.
The above analysis steps were run through MATLAB and the following results were gained. The fitting coefficients in Equation (
Coefficients of quadratic polynomial model Equation (
Coefficients  Specimen  

Z1∼Z9  J1∼J9  M1∼M9  

111.072  104.704  7.25751 

0.374128375718846  0.123087222141136  −0.182407588719503 

12.3712456256323  118.886313511019  86.0463796394658 

−1.61142709539306  −10.8999491603412  1.81975050928462 

0  0.540879978562249  1.58776166995393 

−0.0205457982698535  −0.0113599390393166  −0.0093710737740441 

−0.0246603388200758  −0.0468368467841787  0.0596143780058488 

6.6373455251616 
0.00055949107587472  0.00146298647471792 

−2.72656082597941  −25.9877593894723  −24.8240056022407 

0  0.142689463548048  −0.162714814814822 
Judgment coefficient ( 
0.888816  0.718614  0.924151 
Root mean square error  4.65158  10.5467  4.27426 
And the fitting coefficients in Equation (
Coefficients of cubic polynomial model Equation (
Coefficients  Specimen  

Z1∼Z9  J1∼J9  M1∼M9  

464.342  1709.89  497.628 

−0.273120219091463  −0.801798388025358  −1.00165849326798 

−210.811670727894  −925.096418576694  −237.681795491431 

−23.6752063223750  −110.527720686002  −26.2093732898868 

18.1227919288250  85.2333666882871  21.9612377145612 

0  0  0 

0.323953069722666  0.395546851710759  0.557882540071001 

0.00059579407036310  0.00258127311544728  0.00560634872500473 

0  0  0 

0  0  0 

−0.0138495848812992  0.00032744647692856  0.0131160269775290 

−4.44132996866784  −5.86075713099332  −4.85689087155177 

0.00026004731450507  −0.0003227936704906  −0.0011537934068080 

0.0923976583244754  −1.18632392696069  0.0562704448782981 

−7.822800905166 
−0.0002285901078045  −0.0008395572236522 

5.1757805967185 
3.7955614628876 
0.00014072890415772 

−0.0137328563625555  −0.0760408388477975  −0.149006278920346 

−1.320325398082 
−6.139924628880 
−3.620413282287 

13.3407306630236  15.3022606022822  13.6034019915526 

−0.0031497457424234  0.0424940950506309  −0.0033490129536076 
Judgment coefficient ( 
0.950952  0.962327  0.986819 
Root mean square error  3.09919  3.86832  1.788886 
We can see that the coefficients
Structural natural frequencychange characteristics under low cycle loading are important to identify structural damage caused by earthquakes. Harmonic vibration was used to simulate earthquake conditions. The impact of the loading period and loading amplitude on the structural natural frequency change during damage was analyzed. It was deduced that at any moment of loading, the instantaneous load, instantaneous displacement, and instantaneous frequency of a structure followed Equation (
The higher an RC member’s stiffness, the greater the change of its natural frequency. Loading displacement has an obvious impact on the hysteretic curve, whereas the impact of the loading period is weak.
The impact of the loading period on the structural damage process is mainly reflected in the damage speed. The function
The impact of the loading amplitude on the structural damage process is also mainly reflected in the damage speed. The function
The function
In future work, frequencychange experiments on RC members will be carried out under more loading periods and loading amplitudes to further verify our conclusions. More groups of RC members will be tested with different strengths, stiffnesses, and sizes to investigate the applicability of the above conclusions. The coefficients in Equation (
This paper contains a large number of data which were obtained from the horizontal cyclic loading test carried out in this study. The original experimental data mainly include the test members’ displacements and the forces on the test members in the whole process from structural undamaged state to structural destruction state. The hysteretic curves in Figure
The authors declare that they have no conflicts of interest.
This work was financially supported by the Key Project of Chinese National Programs for Fundamental Research and Development (973 Program, Grant No. 2015CB057706), the National Natural Science Foundation of China (Grant No. 51108044), the Natural Science Foundation of Hunan Province (Grant No. 2018JJ2443), and the Research Project of the Educational Commission of Hunan Province of China (Grant No. 15C0053).