A new method of characterizing the damage of high strength concrete structures is presented, which is based on the deformation energy double parameters damage model and incorporates both of the main forms of damage by earthquakes: first time damage beyond destruction and energy consumption. Firstly, test data of high strength reinforced concrete (RC) columns were evaluated. Then, the relationship between stiffness degradation, strength degradation, and ductility performance was obtained. And an expression for damage in terms of model parameters was determined, as well as the critical input data for the restoring force model to be used in analytical damage evaluation. Experimentally, the unloading stiffness was found to be related to the cycle number. Then, a correction for this changing was applied to better describe the unloading phenomenon and compensate for the shortcomings of structure elasticplastic time history analysis. The above algorithm was embedded into an IDARC program. Finally, a case study of high strength RC multistory frames was presented. Under various seismic wave inputs, the structural damages were predicted. The damage model and correction algorithm of stiffness unloading were proved to be suitable and applicable in engineering design and damage evaluation of a high strength concrete structure.
Structural concrete is a material with quite outstanding cumulative damage characteristics; however, damage development and damage accumulation under different conditions are fundamental for understanding structural failure. Seismic damage greatly influences the bearing capacity of a building or structure during the followup service period and remaining life. Structural damage and its accumulation occur under dynamic loading, and a reasonable damage model must be constructed so that the damage can be described.
The structure damage caused by earthquakes is closely related to the maximum structural deformation and the low cycle fatigue effect caused by the accumulation of damage. In this paper, we have adopted a model based on two major forms of seismic damage [
In this paper, we use the stiffness degradation trilinear model (DT3 model, Figure
Degrading trilinear model (DT3).
For these structures, the elasticplastic time history analyses of the structure are carried out in order to determine the effect of accumulated damage on performance. Since the relationship between the unloading stiffness and ductility coefficient is also unknown, IDARC [
Programming flowchart.
We chose the Park and Ang’s double parameters damage model for this paper [
Park and Ang represented the effect of cyclic loading on structural damage by the parameter
When calculating the damage index, the loaddeformation curve for each test is traced up to the failure point; at the point of failure (
Damage evaluation rules from Park and Ang model based on value of
Damage model  Intact  Slight damage  Medium damage  Serious damage  Collapsed 

Park and Ang model 
0–0.4 
0.4–1.0 
>1.0 
Park and Ang proposed their model based on a large number of test data, covering the following range of parameters:
More recently, high strength concrete (
Relevant tests of high strength concrete columns have been performed by authors, including component design, seismic tests, design of loading devices, and evaluation of the test data to determine the loading mechanism, failure mode,
Relationship between stiffness degradation and ductility coefficient.
Relationship between strength degradation and ductility coefficient.
Defining
Calculation of
Specimen 






Double parameters damage model  Modified model 

mm  kN  mm  kN⋅mm  kN⋅mm  
TS1  6.51  223  2.74  8292.0  1451.73  0.104  6.994  0.941 
TS2  6.25  236  2.41  6053.0  1475  0.144  10.028  0.929 
TS3  6.25  269  2.43  9619.0  1681.25  0.101  9.857  0.950 
TS4  5.62  241  2.18  5120.0  1354.42  0.153  6.625  0.912 
Comparison of values of
Formula (
Through research on the restoring force model of high strength reinforced concrete columns, we have established a restoring force model which can adequately explain the nonlinear characteristics, using the principle of simplified calculation, in addition to considering the hysteresis characteristics of component, using the degrading trilinear model (Figure
The data at key points on the skeleton curve were determined experimentally and then used to initialize an analysis program that is implemented in IDARC.
The IDARC program for structural damage analysis had been used to simulate the earthquake response of a frame structure collocated with a high strength concrete column. To better simulate the dynamic performance of a high strength concrete structure, the dependency of the unloading stiffness and ductility coefficient is analyzed after being introduced into the IDARC program. Loading and unloading standards can be expressed as follows: loading
Unloading during structural vibration is complicated, as structure unloading stiffness matrix changes with time, leading to a time varying parameter system. At present, structure elasticplastic time history analysis generally does not consider the influences that the unloading stiffness and motion state have on each other. These influences mean that the dynamic equations represented by tangent stiffness and secant stiffness are not equivalent, as the matrix is constantly changing during this process.
From the deformation properties and the restoring force model tests of high strength concrete columns, Figure
Measured
The unloading stiffness degradation used to determine the damage index conforms to observed evolution of structure damage. As the cyclic stress presenting in structural member, the strain
In formula (
Formula (
Figure
Measured
In this paper, we model a structure comprising a sixfloor frame of castinsitu high strength reinforced concrete. The strength grades of the concrete beams are C60–C80 and the main reinforcement and lateral reinforcement are HRB400
The axial forces applied to each column in the structure are computed by elastic analysis. This paper adopts two indices to characterize the wave, the design characteristic period of site structure
El Centro, P0151 and P0994 seismic waves.
IDARC is from the Earthquake Engineering Research Centre of the State University of New York at Buffalo and is widely used for various types of nonlinear dynamic response time history analysis and damage analysis of structures. Details of our IDARC analysis will be given below.
This corrective force is then applied for the next time step of the analysis. The unbalanced forces are computed when moments, shears, and stiffness are being updated in the hysteretic model. We selected a step length of 0.02 seconds, to produce smaller changes at each step and thus avoid larger residual forces, which could lead to unstable calculations.
This paper selects high strength RC structures as an example to investigate dynamic analysis, study the seismic performance, and evaluate the degree of damage. Under seismic action (3 waves of structure PEER ground motion, 2 waves of EPDA ground motion, 1 wave of artificial seismic wave and the SATWE program), the maximum displacement of each floor and the maximum angular displacement between the floors are shown in Figure
Maximum floor displacement and story drift angle of building in seismicity basic intensity 8 area.
The implemented program provides the structural damage index and plastic hinge distribution for each seismic wave. The structural plastic hinge distribution and sequence of their appearances under earthquake ground motion PEER are shown in Figure
Plastic hinge distribution of building.
Damage index statistics of frame.
The sheardisplacement hysteresis curves for the bottom of the structure are shown in Figure
Base sheardrift hysteretic curve of building first floor.
The modified parameter damage model was used to calculate damage parameters for all columns in each floor of the frame structure under the earthquake waves. The results are shown in Table
Damage index of middle columns in each floor under seismic waves.
Earthquake wave 




IDARC  Modified damage model  

mm  kN  mm  kN⋅mm  
P  1  173  362  102 

0.205  0.65 
0  2  135  419  60.8 

0.181  0.77 
1  3  124  111  44.7 

0.275  0.76 
1  4  58.6  245  14.6 

0.198  0.79 
5  5  27.8  191  19.6 

0.152  0.64 
6  140.1  404  65 

0.134  0.56  


P  1  70.1  346  41.9 

0.081  0.5 
0  2  73.1  373  43.3 

0.069  0.65 
9  3  60  341  43.4 

0.131  0.65 
9  4  45.1  245  29.4 

0.078  0.68 
5  5  32.5  113  15.2 

0.06  0.48 
6  72.7  352  40.6 

0.05  0.47  


E  1  148  276  69 

0.135  0.53 
L  2  96.5  294  68.3 

0.11  0.73 
3  98.1  221  66.1 

0.162  0.7  
4  28.8  330  19.8 

0.1  0.71  
5  26.7  184  13.5 

0.085  0.53  
6  30.8  212  15.8 

0.081  0.53 
In this paper, based on the deformationenergy damage model and using experimental test data, we determined the expressions for the coefficient of the cyclic load effect and values required for the double parameter damage model of members of a high strength reinforced concrete column member. Ductility is the key index of highperformance concrete, as found by the analysis of the relationships between stiffness degradation, strength degradation, and ductility. A model of the restoring force was used; this model allowed extrapolation of the data to the point of failure to obtain the restoring force. A trilinear model was employed to describe damage.
Generally, the elasticplastic time history analysis does not consider the relationship between unloading stiffness and motion state which influence each other. Thus, we evaluate the structure elasticplastic time history analysis by introducing the unloading correction coefficient, which is a parameter that characterizes the change with time of the unloading stiffness matrix structure, and solve the equivalent problem between the structural dynamic equation indicated by tangential stiffness and secant stiffness. This gives a good description of the elasticplastic structure vibration dynamic process, which can be used to describe the concrete structure unloading in the dynamic elasticplastic state. We then implemented these modifications into the elasticplastic time history analysis in IDARC program.
Presenting the case study of a high strength RC frame, the member dynamic response characteristics of the restoring force was calculated from the material average strengths, where the longitudinal bar and stirrup bar were high strength bar and the concrete strength of the beamcolumn was high. The structural damages were observed under various seismic waves, including two strong artificial seismic wave records from the PEER library, the El Centro wave, two natural seismic waves, and one artificial seismic wave from EPDA. Dynamic analysis was used to analyze the seismic performance and evaluate the degree of damage. Under a strong earthquake, the maximum displacement and angular displacement between floors are consistent. The maximum angular displacement is more than limits by code, but there is no obvious weak floor; the damage to the base of the structure accumulates up until the maximum impact of reaction. The distribution of the damage index reflects the seismic design criterion of “strong column, weak beam,” nearly forming a symmetrical beam hinge system. The lateralshear hysteresis curve is showing the yielding and the stiffness degradation degree. The displacements only appear to be exceeding initially and to be later small generally.
To improve the calculation efficiency, the implementation of the modified damage model in IDARC program was developed and performed well for evaluating highly constrained high strength reinforced concrete structures. Comparison and further validation analysis and evaluation of damage with more software and programs will be done in future work, and more structural members data is necessary in evaluating the more precise correlation between damage model parameters and exact damage to structures.
The authors of the paper declare that there is no conflict of interests regarding the publication of this paper. The authors do not have a direct financial relation with the commercial identity that might lead to a conflict of interests for any of the authors.
This work is supported by Natural Science Foundation of China (Grant no. 50578066), Natural Science Foundation of Fujian Province of China (Grant no. 2011J01320), and National Special Fund for the Territorial Resources in the Public Interest (no. 2011110202). The authors sincerely thank them for their support and funding.