A NEW BLAST-RESISTANT DESIGN METHOD OF RC MEMBERS AND ITS APPLICATION IN PERFORMANCE BASED BLAST-RESISTANT DESIGN PROCEDURE

Design method is very essential and important for engineers. Present study proposes a new blast-resistant design method. First, two common used blast-resistant design methods are discussed. By analyzing the disadvantage of the common procedures, a new blast-resistant design method is proposed. The new design method has less design loops, as well as good control of the maximum displacement and ductility. Then, a dimensionless P-I diagram, which is convenient for design, is proposed based on the new design method. Finally, the new blast-resistant design method is applied in the performance-based blast-resistant design (PBBD) procedure described using a detailed design example. The design example shows that the proposed design method could be easily applied in the PBBD procedure.


INTRODUCTION
Performance-Based Design (PBD) method is firstly proposed in earthquake engineering [1]. Recently, there is a trend in civil engineering community to use PBD method in other subfields of structural engineering [2][3][4][5]. Meanwhile, blast-resistant design is necessary with an increasing attention after September 11, 2001. Thus, the application of PBD method in the field of blastresistant design will be very meaningful [6,7].
Performance-Based Earthquake Design (PBED) is more mature and provides a certain reference for PBBD procedure. During the development process of PBED, two generations design procedures are proposed by engineers [8]. The first-generation procedure is a deterministic framework. It includes methods of defining performance, methodologies to calculate building dynamic response and structural response parameters to assess performance levels of structures [7,9]. The second-generation procedure is a full probabilistic framework. It considers the inherent uncertainties and variability in structural response and provides risk management decisions for engineers [9]. Similar to PBED method, studies about the PBBD procedure are also categorized into two kinds. The first one does not consider the uncertainties while the second one does. Studies of the first kind include uncertainties about blast loads [10,11], fragility curves of RC structures [12,13], loss estimation of buildings after explosion [14], etc. Studies of the second kind include determination of explosion scenarios [7,15], determination of damage criteria [16,17], optimization design methods [18,19], etc. However, few studies pay attention to the blast-resistant design procedure.
The design method is different from the dynamic response analysis method. The design method is to calculate the structural configuration based on the objective performance, while the dynamic response method is to calculate the structural performance based on the known configuration of structures. The dynamic response method could also be used as design method by the application of try-and-error method. For example, UFC 3-340-02 [20] directly uses the dynamic response analysis method single-degree-of-freedom (SDOF) to design the RC beam. The design procedure in UFC 3-340-02 for RC beams (named as Design procedure A) is shown in Figure 1. After the determination of blast load, geometric sizes, materials and objective performances, trial designs are conducted for many times until the objective performances are satisfied. The trial designs will add extra work to designers.  In order to reduce the design loops, dynamic increase factors are calculated using SDOF method. Then, the blast load is transformed to an equivalent static load. RC members are designed using the static design method. This method is called as equivalent static load design method [21], which has less design loops, as shown in Figure 2 (named as Design procedure B). This method is more convenient to designers and widely accepted in China.
In present study, a new blast-resistant design procedure was presented. The new procedure combines the advantages of Design procedures A and B. The maximum displacement and ductility ratio are simultaneously used as performance indexes in the new procedure. Then, a neat PBBD procedure is present based on the new design procedure.

THEORETICAL BASICS OF DESIGN PROCEDURE FOR RC MEMBERS
RC member subjected to blast load is simplified into a perfect elastic-plastic SDOF system shown in Figure 3.
Equation (1) is used to calculate the dynamic response ̈+ ( ) = ( ) (1) where is the span length, ( ) is the resistance function, and ( ) is the linear blast load.

Fig. 3 -SDOF system
The resistance is calculated by Equation (2) according to [22] (Bounds 2010) where is the ductility ratio, 0 is the peak pressure, is the blast load duration, is the natural frequency of vibration.
is calculated by Equation (3 is the equivalent elastic stiffness, is the equivalent mass. The resistance is also computed by Equation (4), shown as = / (4) where is the maximum mid-span displacement. Then, Equation (2) is re-arranged as Equation (5) If the blast load is impulse-controlled load, calculations of and are simplified as After the calculation of and , parameters of cross section could be designed [22]. Present study takes a simple supported RC beam as an example. The resistance is given by Equation (10) is the ultimate moment capacity at the mid-span, given by Equation (11), where is the dynamic yield stress of the longitudinal reinforcement, ′ is the dynamic concrete compressive strength, 1 is the longitudinal reinforcement ratio, b is the width of beam, is the distance from the extreme compression fiber to the centroid of the longitudinal tension reinforcement. is given by Equation (12) Equations (6) ~ (10) are used to compute the design variables of cross section. The stirrup reinforcement ratio is calculated by Equation (15) [23] where is the ultimate shear force, is the shear capacity of the concrete, is the dynamic yield strength for shear reinforcements, ∅ is the capacity reduction factor.

NEW DESIGN PROCEDURE FOR RC MEMBERS
Equations (3)~(9) show that the maximum displacement and the ductility ratio can be calculated using and . Therefore, the designers control the objective performances and by the design of and . The new design procedure is shown in Figure 4. Compared with Design procedure A and B, both maximum displacement and ductility ratio are controlled without increasing the design loops. For convenience of design, non-dimensional P-I diagrams are proposed in Figure 5. The black lines represent the ductility ratio . The red lines represent the maximum displacement . Drawing method of the non-dimensional P-I diagrams are described as follows.
A simple example is present here to describe the design procedure in detail. A simply support RC rectangular beam needs to be designed. The design variables is the width b, the height h and the longitudinal reinforcement 1 . The other parameters and corresponding values are listed in Table 1 Step 1 is to calculate ′ according to Equation (22).
Step 2 is to find the intersection point ( ′ , ′ ) of P-I curves according to the value of ′ and . From Figure 4, we find that the intersection point ( ′ , ′ ) ≈ (5.5, 1.4).
The dynamic response of the designed RC beam is shown in Figure 6. The calculated results, = 0.0336 and = 5.9, are quite close to the objective performances. This means that the new design procedure has good control of and .

APPLICATION OF NEW DESIGN PROCEDURE IN THE PBBD PROCEDURE
The object of PBBD procedure is to control the performances of structures under blast load to satisfy the request of building owners. However, it is hard to predict the magnitude of blast hazards [24]. The recommend method is to assume some blast scenarios which may be determined by building owners, decision maker or engineers [25]. Several blast scenarios are very necessary for design, because only one blast scenario may be unsafety, which will be illustrated in the following part.
The procedure of PBBD for RC members is shown in Figure 7. The new design procedure is used after the determination of blast load and the corresponding objective performances. The blast loads should represent the possible explosive scenarios and be determined using explosive The new design procedure is used to design the RC beams according to several blast loads and objective performances. The key point is to find the relations between ( , ) and ( , ).
Steps are listed as follows: Step 1 is to find the ( , ) (i=1, 2, 3…) corresponding to the objective performance using Figure 5; Step 2 is to calculate the ( , ) corresponding to ( , ); Step 3 is to find the ( , ) (j=1, 2, 3…) corresponding to the objective performance using Figure 5; Step 4 is to calculate the ( , ) corresponding to ( , ); Step 5 is to determine the ranges of ( , ) which satisfies the objective performance; Step 6 is to determine a design point ( , ); Step 7 is to design the RC members and check performances. If the objective performance is not satisfied, return to Step 6.
A simple design example is presented here. Parameters of a simply supported RC beam are shown in Table 2. After the explosive possibility analysis and loss risk analysis. We assume that three reverse triangle blast load are considered, listed in Table 3. and are both used as performance index. The assumed performance levels are listed in Table 4. The objective performance is shown in Table 5. The objective performance is that the RC beam should satisfy Performance level 1 under blast load 1, simultaneously satisfy performance level 2 under blast load 2 and satisfy performance level 3 under blast load 3 at the same time.  Using the proposed design method and the corresponding non-dimensional P-I diagrams, the contour lines of and are for blast loads 1~3 are shown in Figure 8.

Fig. 8 -Contour lines of and for different blast loads
According to the objective performance, the design ranges of ( , ) are shown in Figure 9. performances under blast load 1 but satisfy the objective performances under blast load 3. This indicates that using several typical blast scenarios to design the RC members is more safety.
The performances of designed RC beam are check out using SDOF method. Calculated results are listed in Table 6. It shows that the objective performances are well satisfied.

CONCLUSIONS
The common used design procedures of RC members have many design loops, which adds much work to engineers and limits the development of PBBD procedure. Presented study proposed a new design procedure of RC members which has less design loops. The new design procedure is based on the SDOF method, which is widely accepted by designers. This indicates that the new procedure is convenient to use because SDOF is familiar to designers.
The new design procedure controls both the maximum displacement and the ductility very well. This means that the new procedure controls the performance well. For the convenience of design, non-dimensional design chart is proposed corresponding to the new design procedure. It shows that the new design chart is very convenient to design as well as to control the performance well.
The new design method is very suitable for the PBBD procedure. The application of the new design method in the PBBD procedure is presented in detail and explained using a design example. The example shows the importance of PBBD procedure because the PBBD procedure gives more safety design results than the common design methods.