ANALYTICAL APPROACH TO PREDICT NONLINEAR PARAMETERS FOR DYNAMIC ANALYSIS OF STRUCTURES APPLIED TO BLAST LOADS

In this study, the Performance Based Design PBD method, which has been used only in seismic design by several codes, has been expanded to be applied to structures exposed to blast loads. The plastic hinge models used in PBD, which currently available for earthquake loads do not represent real behavior under the blast load. An analytical approach was proposed to represent the plastic behavior of flexural response under blast loads. The proposed model considers the following essential phenomena: concrete cover crushing, concrete core crushing, bar buckling in compression reinforcement, strain hardening in tensile reinforcement and softening in reinforcement bar. The proposed analytical approach has been validated with two experimental results of columns applied to blast loads and reasonable results has been seen


INTRODUCTION
Over the last few decades, the increasing of terrorist attacks on civilian buildings, led to growing research interests on the protection of civil facilities. In the case of severe loads, the design is based on deformation capacity, which is called Performance Based Design (PBD), instead of using strength capacity. Hence, in the present study, the PBD method will be used to investigate the blast resistance of structural components. For seismic design, there are several codes use the nonlinear dynamic analysis such as ASCE/SEI 41-13, PEER 2010, ACI 374.3R-16, FEMA 2012and FEMA 2009(Haselton, C.B. et al. 2016. To the authors knowledge, there are no code or research provide plastic hinge model for nonlinear dynamic analysis of structures applied to blast loads.
The seismic plastic hinge model was used in the nonlinear dynamic analysis in such cases, where the validity is questionable and need to be investigated. The researches done by Draganic and Sigmund, 2012, Kulkarni and Sambireddy, 2014, Vinothini and S. Elavenil, 2014, Priyanka and Rajeeva, 2015, and Shinde et al, 2016 are good examples of using seismic plastic hinge in dynamic analysis of structures applied to blast loads. The main objectives of the present study' s are to apply the PBD philosophy to structural components subjected to external explosions. The ASCE 41-13 Plastic Hinge Model was applied to the column tested by Kadhom B., 2016. The column was tested up to flexural failure only without the influence of the axial force and its dimensions was 150×150×2230 mm. The column was constructed with seismically reinforcement details, which consist of Ø11.3 mm longitudinal reinforcement for each concrete column and ties reinforcement Ø6.3 mm spaced at 37.5mm center to center. Fig. 2 describes the details of reinforcing steel, and Table 1 shows its mechanical properties. Concrete cover from the outer edge of the ties was 10mm and the concrete compressive strength was 44MPa.
The support conditions for these columns was simply supported. The amount of blast loads carried through the loading device (i.e. shock tube) on this column was 35kPa at duration of the positive phase 20ms, and its reflected impulse equals to 345.2 kPa.ms. In the current case study, the values of MPs according to ASCE 41-13 will be a=0.035, b=0.06 and c=0.2. ASCE 41-13 specifies flexural rigidity for columns with compression caused by design gravity loads ≤ 0.1Ag f`c corresponding to 0.3 Ec Ig. E=concrete modulus (taken as 57,000√f`c , psi), Ag=gross area of column cross-section, and f`c=concrete compressive strength. The slope of line BC, which represent the increase in strength due to hardening of reinforcement, shall be taken between zero and 10% of the initial slope according to ASCE 41-13. For more accurate calculations, the slope of line BC represent the difference between nominal strength, which is calculated from Eq. (1), and the probable strength which represents that associated with strain hardening of the steel reinforcement, i.e. 1.25 fy, which is calculated from Eq. (2). So, the slope of line BC will be equal to 1.25.  (Elwood J. et. al., 2007). The MPs, ultimate strength and slope of line BC are defined as shown Fig. 3, and entered as the user defined plastic hinge in ETABS as shown in Fig. 4, and then the nonlinear dynamic analysis is performed.    Fig. 6 shows the behavior of ASCE 41-13 plastic hinge, where it can be seen clearly that all the plastic hinge capacity will be exhausted within short time (i.e. 13 msec.), which is different from the experimental results (i.e. 48 msec.).

PROPOSED PLASTIC HINGE MODEL
In order to produce an accurate and comprehensive plastic hinge model for the nonlinear dynamic analysis, understanding of important phenomena in the behavior of both materials concrete and reinforcement is required. The proposed model considers the following essential phenomena: concrete cover crushing, concrete core crushing, bar buckling in compression reinforcement, strain hardening in tensile reinforcement and softening in reinforcement bar.
The models of concrete and reinforcement in tension and compression will be discussed in the coming sections.

Material Models and Material Properties
To establish the moment-rotation model of a concrete section, this requires calculating section properties based on material constitutive relations of concrete and steel, strain compatibility, and equilibrium (Shayanfar J., and H.A. Bengar, 2017). Thus, in the present section, the properties of materials which represent all the pre-and post-yield behavior are described.

i) Stress-Strain Model for Concrete
The modified Kent and Park model will be adopted in the present study. This model proposed by (Park et. al.1982) and was a modified method of the Kent and Park model published in 1971. Fig. 7 shows the modified Kent and Park model (Sharma A. et al., 2012).  (4) post-ultimate stress region (Yu., W.,2006), as shown in Fig. 8.

Linear elastic region
The assumed stress-strain relation in this region is: The idealized stress-strain relationship in this region is: Strain hardening region The strain hardening zone ranges from the ideal coordinates that strain hardening begins, (εsh,

b) Reinforcement in Compression
The stress-strain relationship of reinforcing steel in compression is the same as that in tension only, if the reinforcement was prevented from buckling (Dodd, L.L., and Restrepo-Posada, J.I, 1995). The modified Ohi and Akiyama model for bar buckling will be adopted in the present study (Kato et al., 1973). For monotonic loading, the buckling is assumed to begin at a critical buckling strain, εlb, indicated by point A as shown in Fig. 9. Post-buckling softening occurs by assuming, firstly, the stress decrease by τlbEs until the strain reaches ps. After that, the slope decreases by -0.005Es as shown Fig. 9. The term τlbEs is calculated from Eq. (16). The value of ps is equal to Ib+0.01. (Nakatsuka et al., 1999) relationship will be adopted to calculate the strain buckling lb as shown in Eq. (15).
The relationship considers the effect of the following parameters: lateral reinforcement spacing to confined core diameter ratios; confining stress; yield strength of lateral reinforcement, fyh; the shape of reinforcement (circular, rectangular); and the compressive strength of plain concrete. The bar buckling initially resisted by the lateral restraint provided by the concrete cover as well as the transverse reinforcement (Bai Z.Z. and Au, F., 2011).

Analytical Approach for Moment Rotation Behavior
The moment-rotation behavior represents the plastic hinge model for flexural action in Reinforced Concrete RC components under the effect of blast loads. In the previous studies, the moment-rotation was conducted until concrete core crushing (i.e. loss of the compression strength). The existence of the compression reinforcement in the section may increase the deflection until the bar buckling or rupture in tensile reinforcement occurs. Therefore, in the present study, the compression reinforcement role will be considered in the moment-rotation behavior in both pre-and post-concrete core crushing stages.
The proposed analytical approach is capable of predicting the concrete cover crushing, concrete core crushing, bar buckling in compression reinforcement, strain hardening in tensile reinforcement and softening in reinforcement bar (post-yield behavior), which produce the additional mechanism to dissipate the energy of components.
Pre-crushing of concrete core stage In this stage, the plastic rotation until concrete core crushing is taken into account. It is known that the RC columns consist of unconfined cover concrete and confined concrete core. Under blast load effects, the concrete cover crushing occurs at the earlier stages, due to high rate in strain. Thus, the plastic behavior of the proposed model starts when the concrete cover reaches its ultimate strength (i.e. ultimate strain 0.003). So, the strength of concrete cover will be neglected and the symbols k and kd refer to the depth and neutral axis depth of confined concrete core, respectively. Fig. 10 shows the strain diagram during the stage of pre-crushing of concrete core. The analytical approach is summarized in the following steps: 1-Assume strain at the concrete cover cm-cover reaches the ultimate value 0.003.

2-
Assume a value of neutral axis depth kd, starting from kd ≥0.5d.

3-
Find the strain at extreme fiber of confined concrete cm-core by interpolation.

4-Determine the parameters of compressive stress block of concrete  and  in region BC in
the modified Kent and Park model which is shown in Fig. 7. Eqs. (17) and (18)   Where k and Zm were described in section 3.1.1.
Cconc = f`dc b kd 19 Where f`dc: dynamic strength of concrete, b: width of cross section, and kd: depth of neutral axis of concrete core.
6-Find strain of steel si at different levels of section by interpolation depending on kd and  cmcore values and calculate the corresponding stress in steel bars fsi using the stress-strain curve which discussed in section 3.1.2 part I.

7-Determine the tensile force in reinforcement Ts from Eq. (20).
Ts= fds As 20 Where fds: dynamic strength of reinforcement, and As: area of reinforcement. 8-Check the equilibrium between compressive and tensile forces. If they are equal go to step 9, and if Cconc > Ts, decrease kd and return back to step 2, and if Cconc < Ts go to step 13. 12-Repeat steps 1-11 by increasing the value of cm-core with a specific increment such as 0.001, until the tensile force exceeds the compressive force.

9-
At large compressive strain values, the tensile force exceeds the compressive force and the concrete core crushing occurs, which means losing the section ability to resist compression. This phenomenon occurs earlier under blast loads due to the high strain rate of the material. At this stage, the compressive strength is transferred to the reinforcing steel in the compression zone. Hence, in the blast-resistant structural components, concrete beams with tension reinforcement only are not permitted. Compression reinforcement, at least equal to one-half the required tension reinforcement, must be provided (Ho et al., 2005).
Post-crushing of concrete core stage The following additional steps will describe the post-crushing stage of concrete as seen in Fig.   11.
13-Determine the buckling strain of compression reinforcement from Eq. (15), which was referred to in section 3.1.2, part II. 14-If the strain of steel, which will be referred by  s-h in this stage, is smaller to the buckling strain  bu , then, the stress of steel increased duo to the hardening corresponding to the stressstrain curve for reinforcement. Otherwise the stress of steel decrease by τlbEs until the strain of steel reaches ps. After that, the slope decreases by 0.005Es. The values of ps, τlbEs, and bu were discussed in section 3.1.2, part II. Where M is the moment at the post-crushing of concrete core stage, As is the total area of tension reinforcement within the beam, fds is the dynamic design stress of reinforcement, and d is the distance between the compression and tension reinforcement. Where Øs-h: curvature of reinforcement in post-crushing of concrete core stage, Øu: ultimate curvature at which concrete core crushing occur, and Lh: hardening hinge length. A part of the plastic hinge zone is assumed as the region over which the longitudinal reinforcement hardening, which is called Lh as shown in Fig. 12. Hardening hinge length will be assumed to be equal to H/2, which its validity will be checked later.

15-
Where H is the depth of section.  19-Repeat steps 14-18 by increasing the value of  s-h with a specific increment such as 0.001 until reaches the dynamic ultimate stress of reinforcement fdu.

Validation of the Proposed Analytical Approach
The proposed analytical approach was validation with the experimental program of Kadhom B., 2016. Fig. 13 displays the complete moment-rotation relationship conducted using the analytical approach mentioned in section 2.2. The figure covers all strains, starting with the strain of concrete cover crushing until the ultimate strain of steel, using element geometry and reinforcement details of Kadhom's column.  Engineering, Vol. 10, No. 3, July 2019 15 procedure are found. Point (B`) refers to the concrete core crushing, which occurs when the tensile force exceeds the compressive force and concrete loses its strength. The compression strength in this stage transferred to compression reinforcement. At the same time, the strain hardening begins and the strength will be increased with the increase in deflection. Point (C) refers to the end of the reinforcement hardening due to the absence of buckling in compression reinforcement in this element. As mentioned earlier the compression reinforcement behaves similarly to tension reinforcement in the absence of bar buckling. At the end of the hardening, sudden strength loss occurs and the curve reaches point (D).
The moment-rotation behavior will be approximated to find a plastic hinge model that can capture the important plastic response phenomena, as shown in Fig. 14

CONCLUSION
The following points are concluded from the present research: 1. The seismic plastic hinge characteristics of ASCE 41-13 are not valid in nonlinear dynamic analysis of structural components applied to blast loads.
2. An analytical approach was proposed that simulates real behavior under explosive loads.
The proposed model considers the following essential phenomena: concrete cover crushing,