Researches on blastresistant measures for underground structures such as tunnels and underground shopping malls are of great importance for their significant role in economic and social development. In this paper, a new blastresistant method based on wave converters with spring oscillator for underground structures was put forward, so as to convert the shock wave with high frequency and high peak pressure to the periodic stress wave with low frequency and low peak pressure. The conception and calculation process of this new method were introduced. The mechanical characteristics and motion evolution law of wave converters were deduced theoretically. Based on the theoretical deduction results and finite difference software
Tunnels, subway stations, underground shopping malls, and so forth play an important role in the economic and social development, as well as the personal and property safety. In recent years, there have been a variety of researches on the blastresistant measures for underground structures, mainly focused on the design and optimization on structures and materials.
Blastresistant measures based on the structure optimization mainly contain the increase of stiffness or adoption of structures good for the reflection, diffraction, and scattering of the stress wave. Usually the arch structures and structures containing holes have better performance of wave dissipation than rectangular structures or structures without holes, so these kinds of structures have attracted many attentions of scholars [
Blastresistant measures based on the material optimization are mainly concentrated on the development of porous or lightweight materials of low stiffness, and the materials are developing gradually from the traditional inorganic porous materials or lightweight materials to the polymer materials and porous metal materials currently. Materials such as the rigid polyurethane foam, polypropylene fiber concrete, rubber concrete, foam concrete, foamed aluminium, and steel fiber reinforced concrete are good choices for the blastresistant materials. Yakushin et al. investigated the properties of lowdensity rigid polyurethane foams with hollow glass microspheres. The tension and compression properties in relation to the content of microspheres were determined in their work [
In the blastresistant methods of the traditional structure optimization, the construction process is usually complex, and sometimes the function of structures may even be affected. The shock wave mainly consists of high frequency components. In the blastresistant methods of porous or lightweight materials, the materials are easy to get damaged unrecoverably and have large deformation under the blasting load because of the low elastic modulus and the existence of the holes. Thereby the overall stability of underground structure and surrounding rock may be affected by the large deformation.
In order to improve the traditional antiknock methods, a new blastresistant method based on wave converters with spring oscillator for underground structures is put forward in this paper. The new method mainly consists of an array of wave converters and a distribution layer. Firstly, the conception of the new method is introduced, including the formation of the wave converter and distribution layer. Secondly, the calculation process of dynamic responses for underground structures adopting the new blastresistant method is presented. Thirdly, the mechanical characteristics and motion evolution law of the wave converter are derived, including the static constitutive relation of the wave converter, dynamic response partitioning of the wave converter, differential equation of motion for the spring oscillator, displacement transfer coefficient of the wave converter, and the stress inversion of the wave converter. A case study is also conducted to verify the applicability and rationality of the new method by comparing with the traditional structure.
The new method mainly consists of an array of wave converters and a distribution layer, shown in Figure
Schematic of the new blastresistant method.
Schematic of the wave converter.
The new blastresistant method combines such mechanisms as the spring deformation, inertia, and periodic vibration of the mass block to provide the resistance against the dynamic load. The selfsupport capacity of the surrounding rock can also be fully utilized. Via the wave converter, the shock wave with high frequency and high peak pressure can be transformed to the periodic stress wave with low frequency and low peak pressure. Thereby the shock wave is dispersed and materials under the converter can be prevented from crushing. Under the blasting load, the deformation process of the wave converter can be divided into such 3 periods as rapid loading stage, rapid unloading stage, and slow unloading stage.
The above 3 stages are determined by the relative displacement
Typical curve of
Establish numerical models and acquire the data needed for the calculation of the wave converter’s displacements in the rapid loading stage and rapid unloading stage. Model 1 without wave converters for the finite element analysis is set up, shown in Figure
Solve the differential equation of motion for the spring oscillator and obtain the law of motion in the rapid loading stage and rapid unloading stage. In model 1, the vertical displacementtime curve
Solve the differential equation of motion for the spring oscillator and obtain the law of motion in the slow unloading stage. According to step (
Calculate the dynamic response of the underground structure. Firstly, the initial pressure
Schematic of model 1.
Schematic of model 2.
Deformation process of the wave converter.
As for the mass block, the balance equation is
Combining (
So the static constitutive relation of the wave converter yields
The rapid loading stage, rapid unloading stage, and slow unloading stage correspond to the rapid compression stage, rapid recovery stage, and slow recovery stage, respectively. In the slow recovery stage, the length of the wave converter can be deemed as a constant. The demarcation point of stage 1 and stage 2 is that the relative displacement reaches the maximum. The duration time of stage 2 can be determined by the relative displacementtime curve. When the rock masses above the structure are in a wide range of elasticity state, the rapid unloading stage can not be ignored, but if the rock masses are in a wide range of plastic state, the rapid unloading stage can be ignored.
The computing time of stage 3 is advisable for 1 or 2 vibration periods. Via a large amount of computations, it is concluded that the computing time of stage 3 can be taken as 1 vibration period if stage 2 can not be ignored; otherwise it can be taken as 2 vibration periods.
Initial state of the fixedlength vibration.
As is shown in Figure
The length of the upper spring is
The amount of the spring compression is
The length of the lower spring is
The amount of the spring compression is
The oscillator acceleration yields
Then the differential equation of motion for the spring oscillator is
The above equation is an ordinary differential equation of the second order, which can be solved by the RungeKutta method of the fourth order. This equation can be transformed to following forms:
Assuming that
According to the RungeKutta method of the fourth order [
In the slow unloading stage, the vertical displacements of the wave converter top and bottom have few changes over time, so the length of the wave converter can be considered as a constant.
If the oscillator is in static equilibrium under the converter length of
The balance equation for the mass block is
The active force consists of the gravity force
In the rapid loading stage and rapid unloading stage, based on the numerical calculations in model 1 and model 2, the following equation can be derived:
Combining (
Based on the solutions on differential equations of motion in 3 stages, if the gravity force of the wave converter shell is ignored, the stresstime curves of the wave converter top and bottom in 3 stages can be gotten.
In stage 1 and stage 2, the function of the stresstime curve of the wave converter top is
The function of the stresstime curve of the wave converter bottom is
In stage 3, the functions of the stresstime curves of the wave converter top and bottom are, respectively,
Based on the software of
Figure
Mechanical parameters for the rock.
Density ( 
Elastic modulus (GPa)  Poisson ratio  Cohesion (MPa)  Internal friction angle (°)  Tensile strength (MPa) 

2400  13  0.28  1.1  45  0.91 
Mechanical parameters for the structure.
Density ( 
Elastic modulus (GPa)  Poisson ratio  Cohesion (MPa)  Internal friction angle (°) 

2400  30  0.21  3.68  58.7 
Mechanical parameters for foam concrete.
Density ( 
Elastic modulus (GPa)  Poisson ratio  Cohesion (MPa)  Internal friction angle (°)  Tensile strength (MPa) 

799  0.342  0.1  0.17  29  0.2 
Structure size and monitoring sections for internal forces in case 1.
The width, height, and thickness of the numerical models are 35.2 m, 28.85 m, and 1 m, respectively. The blasting load is assumed as a triangle wave acting on the ground surface (in Figure
Curve of the blasting load.
Model 1 built up based on
According to the symmetry, the monitoring positions for displacements are suggested to adopt the tops and bottoms of 5 wave converters along the width direction of the structure, shown in Figure
The curves of the displacement transfer coefficient
Internal forces of the structure for monitoring sections 1–5.
Monitoring section  1  2  3  4  5  

Case  1  2  1  2  1  2  1  2  1  2 
Bending moment ( 

Minimum  733  −43847  640  −45141  365  −56001  243  −92795  −290760  −129805 
Maximum  221068  163402  224395  148737  222567  95049  118881  8908  348  2039 


Shear force ( 

Minimum  −27317  −15193  2360  −3988  1194  −1956  1177  0  3474  23 
Maximum  −426  1037  164052  64656  439202  138742  1041408  214362  2040240  282142 


Axial force ( 

Minimum  −41094  −83637  −42568  −81050  −57842  −81420  −104766  −83599  −348627  −91759 
Maximum  1666072  921440  1596560  921280  1320330  921600  788366  932400  409633  949376 
Internal forces of the structure for monitoring sections 6–10.
Monitoring section  6  7  8  9  10  

Case  1  2  1  2  1  2  1  2  1  2 
Bending moment ( 

Minimum  −246269  −74504  −60566  −34165  −20880  −16684  −20016  −12038  −30226  −27187 
Maximum  −62  59448  11794  32440  20808  19901  4810  3787  10929  8644 


Shear force ( 

Minimum  −837708  −99838  −371508  −85756  −165964  −58982  −78672  −57528  −100307  −77861 
Maximum  −1537  211056  −1517  90487  −953  68296  27172  105168  51451  117882 


Axial force ( 

Minimum  −3218640  −814500  −2655720  −846588  −2229240  −891240  −1992840  −936840  −1573560  −834600 
Maximum  −4890  −1196  −1170  −2138  −1175  −1223  −620  −1812  −5041  −983 
Internal forces of the structure for monitoring sections 11–15.
Monitoring section  11  12  13  14  15  

Case  1  2  1  2  1  2  1  2  1  2 
Bending moment ( 

Minimum  230  664  261  614  295  555  97  419  −54210  −45028 
Maximum  35258  23027  36594  23699  39747  24339  32743  15492  15725  14552 


Shear force ( 

Minimum  −421  −92  −19776  −16431  −65196  −51572  −212922  −153744  −533254  −356511 
Maximum  4128  3608  319  411  −428  450  2836  718  5536  6486 


Axial force ( 

Minimum  −3124  −4749  −3185  −5153  −2953  −18050  −32879  −115840  −192165  −250929 
Maximum  680160  426831  667594  419751  615300  394217  518211  354017  381111  332100 
Curves of the displacement transfer coefficient with time.
Curves of
Curves of
Curves of
Curves of
Stresstime curves of the wave converter top.
Stresstime curves of the wave converter bottom.
In Tables
The peak absolute value of the shear force for monitoring sections in case 2 is also generally lower than that of case 1. The maximum drop in the roof, side wall, and floor is 86.2%, 75.6%, and 33.1%, respectively. The peak absolute value of the axial force for monitoring sections in case 2 is remarkably lower than that of case 1. The maximum drop in the roof, side wall, and floor is 44.7%, 74.7%, and 37.2%, respectively. For the roof, the decrease of the axial tensile force near the midspan is obvious, and the maximum drop occurs to the span center. The decrease of the axial tensile stress presents that the tensile failure in the roof can be alleviated via the wave converters.
The curves of the vertical normal stresses for inner and outer elements in the span center with time are shown in Figure
Vertical normal stress of inner and outer elements in the span center.
Inner element
Outer element
The peak horizontal tensile stresses of monitoring sections for roof in 2 cases are shown in Figure
Peak horizontal tensile stress of monitoring sections for roof in 2 cases.
In this paper, a new blastresistant method based on wave converters with spring oscillator for underground structures is put forward. The conception and calculation process of this new method are introduced. The mechanical characteristics and motion evolution law of the wave converter are derived. The dynamic responses of the traditional underground structure and the new blastresistant one are also calculated to verify the blastresistant effect of the new method. The following conclusions can be drawn through the study.
After the deployment of wave converters, the peak absolute values of the bending moment, shear force, and axial force decrease generally. The decrease of the peak internal forces means that smaller size and less steel are needed in the design of the structure, which could help reduce the costs.
After the adoption of wave converters, the peak vertical tensile stress for inner element and the peak vertical compressive stress for outer element in the span center drop remarkably, which means that the possibility of spalling damage for roof is reduced, and the impact load acting on the roof is decreased.
With wave converters, the peak horizontal tensile stresses of inner elements for roof are generally lower than that of the traditional structure, which could reduce the amount of reinforcing bars.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors gratefully acknowledge the support from Chongqing Graduate Student Innovation Project under Grant no. CYB14103, Chongqing Research Program of Basic Research and Frontier Technology under Grants nos. cstc2014jcyjA30015, cstc2015 jcyjBX0073, cstc2014jcyjA30014, and cstc2015 jcyjA30005, and Science and Technology Project of Land Resources and Real Estate Management Bureau of Chongqing Government under Grant no. CQGTKJ2014052.