Optimization of framed structures subjected to blast loading using equivalent static loads method

In this study, the optimum design of a three-dimensional framed steel structure subjected to blast loading is considered. The main idea of this research is to develop a practical formulation for the design optimization problem and to study the effect of including blast loads in the design process. The optimization problem is formulated to minimize the total weight of the structure subjected to American Institution of Steel Construction (AISC) strength requirements and blast design displacement constraints. The design variables for beams and columns are the discrete values of the W-shapes selected from the AISC tables. A car carrying 250 lbs of trinitrotoluene with a 50 ft standoff distance from the front face is modeled as the source of the blast loading. Pressure–time histories are calculated on the front, sides, roof, and rear faces of the structure. Since the problem functions are not differentiable with respect to the design variables, the gradient-based optimization algorithms cannot be used to solve the problem. Therefore, metaheuristic algorithms are used to solve the optimization problem. Linear and nonlinear dynamic analyses are carried out in the optimization process. The problems are solved using metaheuristic optimization with the equivalent static loads method (MOESL). In MOESL, the dynamic load is transformed into equivalent static loads (ESLs) then the linear static analysis is carried out in the optimization process. The problems are 4-bay × 4-bay × 3-story frames under serviceability and blast loading. It is shown that a penalty on the optimum structural weight is substantial for designing structures to withstand blast loads.


Introduction
Interest in the behavior of structures subjected to blast loading has increased over the last few decades as terrorist attacks increase around the globe.The attacks on the World Tarde Center in New York City in 1993 and Murrah Federal Building in Oklahoma City in 1995 showed the great damage that could happen due to a blast.In both attacks, structural failure caused more casualties and injuries than the blast wave itself (Cormie et al., 2009).Normally, conventional structures (many are moment resistance frames) are not designed to tolerate blast loads which are very high compared with service loads.For instance, a 10 lb of TNT at a distance of about 50 ft causes roughly peak pressure of 2.5 psi (360 psf) in a very short time (less than a second) compared to natural periods of structures.In comparison, the design snow load in the Midwest ranges from 5 to 50 psf (Longinow & Alfawakhiri, 2003).Thus, a small charge explosion could cause a catastrophic local or global failure of the structure.
Blast-resistant analysis of structures has been pursued in the literature for many years.Most of the research has focused on dynamic analysis, progressive collapse situations, and members (such as columns, beams, or slabs) subjected to blast loading.However, no study is available that considers the optimum design of 3D structures subjected to blast loading.Stea et al. (1977) presented a report that provided criteria and procedure for the design of steel frame structures subjected to blast loading based on dynamic analysis.The inelastic behavior of the frame members and second-order effects were considered in the analysis.Numerical examples were discussed and solved using a FORTRAN computer program called dynamic nonlinear frame analysis (DYNFA). Lee et al. (2011) studied the dynamic collapse behavior of two steel moment frames using nonlinear finite element analysis.The first model represented a blast and post-blast scenario and the second frame was modeled with a missing column.The study showed that the strain rate should be considered to predict a more exact progressive collapse response.Jeyarajan et al. (2015) investigated the response of a 10-story steel building frame subjected to high blast pressure using ABAQUS.The source of the blast load was a charge of 500 kg TNT placed at a distance of 20 m from the building.Various lateral bracing systems were studied to show their contributions to progressive collapse analysis.The study showed that higher redundancy in frames could redistribute the damaged members' loads to other floor levels and the vertical displacement could be reduced.On the other hand, the unbraced frame needs rigid beam-column connections to avoid very large displacement due to members' loss.Coffield and Adeli (2014) studied the effectiveness of different steel framing systems (moment-resisting frame, concentrically braced frame, and eccentrically braced frame) subjected to blast loading.They also studied the influence of structural irregularity on the structural response (Coffield & Adeli, 2015).In both papers, the structures are modeled and analyzed using the Applied Element Method.Khaledy et al. (2018) study the optimum design of 2D steel moment frames under blast loading using three techniques: Nonlinear Programming by Quadratic Lagrangian (NLPQL), Particle Swarm Optimization (PSO), and Multi Island Genetic Algorithm (MIGA).The weight of the structure is considered as the objective function and design variables are members' cross-sectional areas.Design variables are considered continuous and other geometrical properties of members are formulated based on polynomial functions of the cross-sectional areas.Only displacement constraints are considered.
Metaheuristic or stochastic algorithms have been used in structural optimization.In contrast to gradient-based algorithms, they are nature inspired and based on the successful evolutionary behavior of natural systems.The benefits of many of these algorithms are their robustness and dependability in terms of global convergence.Kaveh and Talatahari () studied the use of Particle Swarm Ant Colony Optimization (DPSACO) and Improved Ant Colony Optimization (IACO) for the design of 2D frame structures.Charged System Search (CSS) optimization is used to propose a performance-based optimal seismic design for steel frames and design of steel frames (Kaveh et al. (2010), Kaveh and Talatahari (2012), and Kaveh and Nasrollahi (2014)).The Cuckoo Search (CS) algorithm is used to find the best two-dimensional steel frame design for discrete variables (Kaveh & Bakhshpoori, 2013).Kaveh et al. (2015) created a framework to address the issue of the best performancebased design while taking into account the cost of building and the seismic damage to steel moment-frame structures.Large-scale space steel frame optimization problems are solved using cascade-enhanced colliding body optimization.The numerical results of the tested problems showed that the proposed framework is an effective tool for this type of problem (Kaveh & BolandGerami, 2017).Colliding Bodies Optimization (CBO), Enhanced Colliding Bodies Optimization (ECBO), Vibrating Particles System (VPS), and a hybrid algorithm based on VPS, Multi-design Variable Configurations (MDVC) cascade optimization, and Upper-Bound Strategy (UBS) are utilized to find the optimal design of three-dimensional (3D) irregular steel frames (Kaveh & Ilchi Ghazaan, 2018).The results of all the research showed that metaheuristic algorithms can tackle structural optimization problems efficiently.
The main objective of this work is to present a practical formulation for the optimum design of 3D steel frames subjected to blast loading.To this end, design variables, cost function, and constraints are studied and explained.The design variables are frame members (beams and columns) which are considered to be discrete (specifically, W-shapes selected from the AISC tables (AISC, 2017)) and they are organized in groups based on structural symmetry.The objective function is the total weight of the structure which is a function of the discrete design variables.Constraints are the AISC code strength requirements and DoD (2008) displacement requirements.They are also functions of the discrete design variables.Thus, the gradient-based optimization algorithms are unsuitable for this application because the problem functions cannot be differentiated with respect to the design variables.Therefore, metaheuristics (stochastic) optimization algorithms are used.In these algorithms, gradients are not needed to find an optimum solution.Instead, they search the entire design space for the best solution based on some stochastic strategy.There are many metaheuristics algorithms; however, in this study, Hybrid Harmony Search-Colliding Bodies Optimization (HHC) is utilized to find the minimum weight structure ("Optimization algorithms" section).For nonlinear optimization problems, metaheuristic optimization with the equivalent static load method (MOESL) is used ("Discrete variable optimization of structures subjected to dynamic loads using equivalent static loads" section).
MATLAB® is used to implement the algorithms and to model design examples interfacing with the structural analysis program SAP2000 using its Open Application Programming Interface (OAPI).That is, algorithms start with random design vectors that are sent to SAP2000 to analyze, then SAP2000 sends back information needed (nodal displacement, interaction ratio, etc.) to evaluate problem functions.Following this, MATLAB® uses this data to arrange and update design vectors using optimization algorithms and then send them back to SAP2000 for re-evaluation.

Blast design
Similar to seismic design, it is expected that some components will experience substantial nonlinear responses because designing structures subjected to blast loading to remain elastic is usually uneconomical (ASCE, 2010).Therefore, in designing blast-resistance structures, the maximum dynamic deflection and rotation are the criterion to prevent component failure.

Material design strength
Material under high strain rate loadings such as blast loads behaves differently from low rate and static loads.Generally, materials become stiffer under high-rate loadings which means an improvement in their mechanical properties.In addition, it is allowed to use the expected actual strength of the material instead of the minimum specified values in blast design (Gilsanz et al., 2013).
The high strain rate effect on some mechanical properties of steel is summarized as follows: 1.The modulus of elasticity ( E s ) remains the same.2. The yield strength ( f y ) and ultimate tensile strength ( f u ) increase the dynamic yield strength (f dy ) and the dynamic ultimate strength (f du ) , respectively.
Dynamic increase factors, DIF , are used to modify the static strength due to high-rate dynamic loads (DoD, 2008).
The average yield stress of steel grades 50 ksi or less is about 10% higher than the stress value specified by ASTM.Thus, for a blast-resistant design, the yield stress is 1.1 times the minimum yield stress.This factor is called the strength increase factor ( SIF ) or the average strength factor ( ASF ).SIF should not be used with high-strength steel (Gilsanz et al., 2013).

Strength reduction factors and load combination
As mentioned above, plastic deformations are allowed in the design of structures subjected to blast loads because of the nature of the blast load and to achieve an economical design.Also, it is allowed to use the nominal strength without the strength reduction factor ( = 1 ) for all modes of failure (ASCE, 2011).Blast loads are not combined with the loads that are not expected to be present when the blast happens.That is, wind, earthquake, part or all the live loads are not combined with blast loads; the basic load combination for all constructions material is as follows (ASCE, 2010): where DL is the dead load, LL is the live load, and BL is the blast load.In the absence of other governing criteria, Gilsanz et al. (2013) allow the following load combination:

Criteria for responses
There are many sources for response limits such as UFC 3-340-02 (DoD, 2008), Design of Blast Resistant Buildings in Petrochemical Facilities (ASCE, 2010), FEMA 356 (FEMA, 2000), and New York City Building Code (NYCBC, 2008).In this study, design criteria for a structural system are used with a medium response design (ASCE, 2010).That is, the maximum member end rotation shall be 2 degrees and the maximum side-sway deflection (or interstory drift (ISD)) is limited to 1/25 of the story height.To prevent extended structural collapse, beams are allowed to develop plastic hinges when columns are designed to remain elastic (Gilsanz et al., 2013).

Blast loading
When a blast occurs in the air, it forces the surrounding air out of its volume it occupies and the air molecules pile-up.A blast wave happens after that and it carries a huge amount of energy (Cormie et al., 2009).The blast wave travels fast and its pressure decays exponentially until it falls to the atmospheric pressure (positive phase).After that, the front wave pressure decreases further to be less than the atmospheric pressure (negative phase) and finally back to ambient value as shown in Fig. 1.In Fig. 1, P so is the peak overpressure or the incident pressure, P o is the ambient pressure, and P − so is the minimum negative pressure, P r is the reflected pressure, P − r is the minimum negative reflected pressure, t a is the arrival time, t o is the positive phase duration, and t − o is the negative phase duration.
The most commonly used approach for blast wave scaling is Hopkinson-Cranz scaling (or cube-root scaling).It is expressed as follows (Cormie et al., 2009) where Z is the scaled distance, R is the distance from the detonation source center to the point of interest, and W is the charge mass expressed in pounds of TNT.There are many types of explosives.TNT was chosen to be the blast load source.If another explosive is used, an equivalent TNT weight needs to be computed in order to use Eq.(3); these (1) are provided in conversion tables for different explosives (Cormie et al., 2009).
A hemispherical burst is considered in this work.It happens when an explosive charge is close to the ground, so the incident wave reflects immediately from the ground and interacts with the blast wave.To find blast wave properties of a hemispherical burst for a given scaled distance (Z), one can use Fig. 2. In Fig. 2, P r is the reflected pressure, P − r is the minimum negative reflected pressure, t a is the arrival time, t o is the positive phase duration, i r is the positive reflected impulse, i s is the positive incident impulse, U is the wave speed, and L w is the wavelength.They are presented on the y-axis while the x-axis represents the scaled distance Z. Blast loading calculations used in this study follow the methods presented in DoD (2008).For simplicity, a triangular simplification of the pressure-time history profile is used and the negative phase is ignored as shown in "Numerical examples" section.

The formulation for discrete structural optimization problems
The problem is to find the American Institute of Steel Construction (AISC) standard W-shape for each member of a steel frame structure and optimize its performance.In general, the nonlinear undamped dynamic response optimization problem with discrete design variables can be expressed as: where X is the vector of design variables with nvar unknowns, D i is a set of discrete values for the i th design variable, f (X) is a cost function (in this study, f (X) is the total weight of the structure), and g k is a constraint function that needs to be imposed at all time points.
The constrained optimization problem is defined in Eqs. ( 4)-( 6) to be transformed into an unconstrained problem so that the metaheuristic algorithms can be used to solve the problem.This can be done by defining a modified cost function F(X) to account for the constraint violations, as follows: where G(X) is a constraint violation function, ≥ 1 is the exploration penalty coefficient (in this study, = 10 ),  > 1 is the penalty function exponent (in this study, = 2 ), and max(0, g k t i ) ≥ 0 is the violation value of the k th inequality constraint at the time point t i .The present problem has just inequality constraints.The linear dynamic response problem (4) is the same as the nonlinear dynamic response problem except that K is not a function of the displacement vector u.
A linear static response optimization formulation is used to solve problems discussed in "Spatial steel frame subjected to service loads only" section using the equivalent static loads method with metaheuristic optimization ("Discrete variable optimization of structures subjected to dynamic loads using equivalent static loads" section).The linear static response optimization problem subjected to loading conditions can be stated as:

Design variables
In this study, the AISC (2017) W-shapes available in the manufacturer's catalog are desired for beams and columns.Since all sections are chosen from AISC tables and the assignment of a section specifies a number of cross-sectional properties for the member, the design variables are classified as linked discrete variables (Arora, 2017).Table 1 shows a part of AISC (2017) wide-flange sections.
In this study, one design variable is assigned for each member (the AISC section number).Once the section number is known, all the cross-sectional properties are known from the tables (Huang & Arora, 1997).This way the design variables Eqs. ( 1) and (2) become: where S i is an AISC W-shape number, i ∈ [1, 2, … , nvar] , S imin and S imax are the lightest and the heaviest sections, respectively, W-shapes allowed from the AISC table after arranging sections in ascending order based on their weights.

Cost function
The cost function is the criterion that is used to compare feasible designs to find the optimum solution (Arora, 2017).( 9) In this study, the problem is to minimize the total weight of the structure (in kips).Thus, Eqs. ( 5) and ( 10) become: where W s is the total weight of the structure, X is the design vector, NG is the total number of member groups for the structure, w ng is the weight per unit length (kips/ft) of the members in the ng th group (available in AISC's tables), MK is the number of members in the ng th group, and L mk is the length of the mkth member (ft).

Constraints
Restrictions imposed on the structural members are the strength requirements given in the AISC manual, inter-story displacement constraints, and geometrical requirements.
According to the AISC (2017), symmetric members subjected to axial force and biaxial bending must satisfy the interaction ratio and shear force strength requirements: Here is the resistance factor ( c = 0.85 and t = 0.90 for compression and tension, respectively).b = 0.9 is the flexural resistance factor.P u and P n are the required and the nominal axial strengths (compression or tension) (kips), respectively.M ux and M uy are the required flexural strengths about the major and the minor axes (kip-ft), respectively.M nx and M ny are the nominal flexural strengths about the major and the minor axes (kip-ft), respectively.M u and M n are required and the nominal flexural strengths about major or minor axes.V u and V n are required and the nominal shear strengths (kips), respectively.v = 0.9 is the resistance factor for shear.
Evaluating P n , M nx and M ny in Eq. ( 17) is an involved pro- cess that requires checking several failure modes (i.e., several "if then else" statements).For example, to find P n , first one needs to find whether the member force is tensile or compressive.For tension members, P n is calculated based on whether the gross section yields or the net section ruptures.The nominal strength for flexure of major or minor axis bending ( M nx or M ny in Eq. ( 17)) depends on the categorization of the member as compact, non-compact, or slender.
Constraints in Eqs. ( 18), and (18) need to be imposed at each point along the axis of every member in the structure.Thus, each equation represents infinite constraints.In the numerical process, the constraints are evaluated at several points along the axis of the member and they are imposed at the point where they have a maximum value.These constraint values are then used to evaluate the penalty function defined later in the paper.Thus, the total number of interaction ratio constraints (Eq.( 18)) equals the total number of members.The same is true for shear force constraints (Eq.( 18)).Also notice that constraints in Eq. ( 18) have a discontinuity at In addition, the nominal strength calculations have several discontinuities as explained in the previous paragraph.That is, gradient-based optimization algorithms are not suitable for this class of optimization problems.

Displacement constraints
The maximum member end rotation shall be 2 degrees and the maximum side-sway deflection (or inter-story drift (ISD)) is limited to 1/25 of the story height (high response design (ASCE, 2010)).where r and r−1 are lateral displacements of two adjacent stories (in), ru is the allowable lateral displacement, and h r is the rth story height (in).At each node, SAP2000 evaluates displacements and rotations in 3 dimensions.Displacements in x and y directions are extracted to evaluate Eq. ( 20) to impose these constraints.

Optimization algorithms
Metaheuristic algorithms are based only on simulations and do not require gradient information, such as the well-known genetic algorithms (Goldberg & Holland, 1988) and ant colony methods (Dorigo, 1992).They use random search in the entire design space instead of just the neighborhood of the current design as in the gradient search techniques.Also, the discrete variables can be treated routinely.Therefore, they are suitable for both continuous and discrete design variables and differentiable and non-differentiable problem functions.
In this study, hybrid harmony search-colliding bodies optimization (HHC) is utilized to find an optimum design for every case of study (Al-Bazoon, 2019).HHC uses two phases: the first is the Improved Harmony Search (IHS) (Mahdavi et al., 2007) algorithm with a design domain reduction technique.The second phase uses Enhanced Colliding Bodies Optimization (ECBO) (Kaveh & Ilchi Ghazaan, 2014).ECBO receives final designs from the first phase to enhance them further.

Hybrid improved harmony search-enhanced colliding bodies algorithm (HHC)
Compared to other metaheuristic algorithms, IHS is easy to implement and it works with any kind of problem.ECBO requires just one algorithmic parameter and it performs well in terms of the quality of final designs.However, IHS and ECBO have some shortcomings that were observed while solving some problems.IHS requires the specification of several algorithmic parameters to turn in that can affect the performance of the algorithm.ECBO makes steady progress toward the final design whereas IHS makes quite rapid progress towards a similar neighborhood.Therefore, IHS needs fewer simulations compared to ECBO to reach a neighborhood of the final design.However, after reaching the neighborhood of the final design, the progress of IHS becomes slow to reach the final design whereas ECBO continues to make good progress toward the solution (Al-Bazoon, 2019;Al-Bazoon & Arora, 2022a, 2022b).

Discrete variable optimization of structures subjected to dynamic loads using equivalent static loads
The design of structures subjected to blast loads requires nonlinear dynamic analysis (as described in "Blast design" section).Depending on the size of the structure, the nonlinear dynamic analysis (numerical solving of a system of nonlinear differential equations) might need a very long time.Metaheuristic algorithms require many structural analyses to reach the final design.Using metaheuristic algorithms could be impractical for this type of problem.Therefore, optimization by transforming dynamic to static loads is more efficient.
One of the well-known dynamic to static loads transformation methods is based on the displacement field obtained using dynamic analysis of the structure (Park, 2011).That is, the dynamic load is transformed into multiple equivalent static load sets.Then the equivalent static loads (ESLs) are considered as multiple loading conditions in the linear static response optimization process.This is called an ESL cycle of the optimization process.These cycles are repeated until the final design is obtained.This method works fine for gradient-based algorithms (Kang et al., 2001;Kim & Park, 2010;Park & Kang, 2003).The equivalent static load method (ESLM) with metaheuristic algorithms is tested and used in ).It is shown that ESL with metaheuristic algorithms (called MOESL) can reduce the number of linear or nonlinear dynamic analyses.Optimum design of structures subjected to dynamic loads using the MOESL proceeds as follows: Step 1. Generation of an initial population.
A population of designs is randomly generated from the design domain.
Step 2. Evaluation of designs.In this step, all designs from Step 1 are analyzed using a transient solver.Using the simulation results and Eqs. ( 6) to ( 8), the merit function F(X) is calculated for each design.Then the designs are arranged in an ascending order based on their merit function values.The best design is used to generate ESLs as follows: where K L is the linear stiffness matrix, u is the dynamic displacements vector, p s is the external static load vector, and n is the total number of the time steps.
Step 3. Optimum design with the calculated ESLs.
Using p s from Step 2 and the linear static response opti- mization formulation (Eqs.( 9) to ( 13)), the optimum design is found.This completes a cycle of the MOESL.
Step 4. Transient analysis of final design(s).Perform transient analysis of the best design. ( Step 5. Updating CB ESL and CM ESL .
Step 6. Initialization for a new ESL cycle.In this step, new ESLs are re-calculated and a new population of designs is generated from the design domains.
If the stopping criteria for the ESL step are satisfied, the population of designs is passed to full transient analyses.

Numerical examples
In this section, HHC and MOESL are applied for the optimum design of 3D steel frames.As mentioned earlier design variables are W-shapes selected from the AISC tables.
Figure 3 shows a 3D view and the dimensions of the structure (slabs and external walls are not shown).The structure is a 3-story, 4 bays in x and y directions with 4 inches concrete slab, and consisting of 197 members modeled using SAP2000 and MATLAB.All ground supports are fixed.Steel properties are: Young's modulus, E s =29,000 ksi, yield stress, F y =50 ksi, ultimate strength, F u =65 ksi, and poison's ratio, v s =0.3.Concrete properties are: Young's modulus, E c =3605 ksi, f ′ c =4000 psi, and poison's ratio, v c =0.2.The frame members are organized into 9 groups as shown in Fig. 3 and Table 2.Each group is treated as a design variable.Gravity loads are assigned as uniformly distributed loads on the first and second-floor slabs consisting of a design dead load of 60 psf and a design live load of 50 psf and on the roof slab consisting of a design dead load of 60 psf and a design live load of 25 psf.Table 3 gives design load combinations.
Three independent optimization runs were performed for each case study.

Spatial steel frame subjected to service loads only
In this design example, only service loads (no blast loads) and strength constraints are considered (only Comb 1 and Comb 2 in Table 3).It is used as a reference to compare with other design examples and to study the penalty of designing steel frames to resist blast loads.
Columns and beams are selected from the first 100 lightest standard W-shape sections provided in AISC tables (AISC, 2017) after rearranging sections in an ascending order based on their weight.This example is solved using linear static analysis (direct stiffness method).
The final designs are reported in Table 4 along with total structural weight and maximum values of interaction and shear ratios.The second run gives the best design with a total structural weight of 60.549 kips.

Spatial steel frame subjected to blast load
This design example has the same dimensions, design variables, and material properties as the previous study case.However, in addition to gravity loads and load combinations described in the previous section, blast loading is considered.The source of the blast loads is an automobile carrying a large charge of 250 lb of TNT.The structure has a stand-off distance of 50 ft from the charge's center as shown in Fig. 4. The structure is considered to be isolated with no opening (conservative assumption).In real cases, however, there are   windows and door openings that (if not designed to resist blast loading) vent some of the blast waves inside the building depending on the size of those openings.The façade of the structure is divided into 12 panels and the blast reflected pressure is evaluated at the center of each panel and distributed uniformly on that panel as shown in Fig. 4 (Karlos & Solomos, 2013).Side, roof, and rear blast loads are calculated at the center of each face and distributed uniformly on the surface.Table 5 shows the pressure-time history on the front, sides, roof, and rear faces.Using SIF and DIF ("Material design strength" section), the new strength values are as follows:

Optimum design with linear dynamic analysis
In this study case, beams and columns are designed according to AISC (2017) strength requirements ("Displacement constraints" section).That is, all members are designed to remain elastic.The following examples are solved using the Hilber-Hughes-Taylor method (linear direct integration) and the total analysis time is 1 s with a time step of 0.0025.The analysis time was selected after different designs indicated that the maximum response occurs between 0 and 1 s.
The blast loads are transported to beams and columns as distributed loads.The common design approach is that the stiffness of the outer periphery wall is not added to the stiffness of the structure.In this study, two approaches are investigated.
No external walls In this example, the stiffness of the external walls is not considered.This conservative procedure is used in most blast design references such as Gilsanz et al. (2013).Columns and beams are selected from the first 173 heaviest standard W-shape sections provided in AISC tables.
The final designs are reported in Table 6 along with total structural weight and maximum values of interaction and shear ratios.It shows that the second run reaches the best ( 22)  design with a total structural weight of 853.469 kips.This is about 14 times heavier than the best design found for the same structure subjected to service loads only.
No external walls with mass In this example, the stiffness of the external walls is not considered.However, the mass of the outer periphery walls (thickness of 4 in) is added as a dead load on beams.Columns and beams are selected from the first 173 heaviest standard W-shape sections provided in AISC tables (AISC, 2017) after rearranging sections in a descending order based on their weight.The final designs for the three runs are reported in Table 7 along with the total structural weight and maximum values of the interaction and shear ratios.It shows that the first run reaches the best design with a total structural weight of 827.182 kips.This is about 13.7 times heavier than the best design found for the same structure subjected to gravity loads only.

With external walls
The stiffness of the external walls that are connected to the steel frame is added to the stiffness of the structure in this study case.The outer periphery wall is a concrete wall with a thickness of 4 in.The outer periphery wall is pinned to the ground and attached to external beams only which are attached to the roof system to transfer loads directly into floor diaphragms to reduce the risk of progres-sive collapse as recommended in ASCE (2011).Columns and beams are selected from the first 100 lightest standard W-shape sections provided in AISC tables.
Table 8 gives the final designs of three optimization runs.The best design has a total structural weight of 77.818 kips.This design is about 28.5% heavier than the best design found for the same structure subjected to gravity loads only.
The external walls add quite an amount of stiffness to the structure.They act as shear walls that resist the lateral blast loads.This is an unconservative design.Although, the structure weight is quite lighter than the previous case studies, adding walls stiffness makes the structure less robust to progressive collapse when a wall fails during the blast event.

Nonlinear dynamic analysis
In this study case, columns are designed according to AISC (2017) strength requirements ("Displacement constraints" section) but beams are allowed to develop plastic hinges and blast design requirements are applied ("Displacement constraints" section).Steel Columns-Flexure elastic-perfectly plastic hinges provided by SAP2000 v.20 is modeled near the ends of each beam (CSI, 2017).The following examples are solved using the Hilber-Hughes-Taylor method (Nonlinear direct integration with P-delta).Considering the blast load duration and peaks of the response, the time range from the analysis is set from 0 to 1 s with a time step of 0.0025.
Testing the nonlinear dynamic models ("No External Walls" and "No External Walls with Mass" sections) with different designs shows that in most cases there are either convergence problems or the nonlinear structural analysis takes a long time because of the material nonlinearity, geometrical nonlinearity, and the size of the structure.This makes metaheuristic algorithms inconvenient to use since No external walls This study case is similar to "No External Walls" section.Namely, the stiffness of the external walls is not considered.However, beams are allowed to develop plastic.Beams are selected from the first 100 lightest standard W-shape sections and columns are selected from the heaviest 173 standard W-shape sections provided in AISC tables (AISC, 2017).
The final designs are shown in Table 9 along with total structural weight and maximum values of interaction ratio, shear ratio, member end rotation, and inter-story drift.The first run reaches the best design with a total structural weight of kips.This is about 6 times heavier than the best design found for the same structure subjected to gravity loads only.

No external walls with mass
This study case is similar to "No External Walls" section.However, the mass of the external walls is considered but their stiffness is not considered.Columns and beams are selected from the first 173 lightest standard W-shape sections provided in AISC.
The final designs are shown in Table 10 along with total structural weight and maximum values of interaction ratio, shear ratio, member end rotation, and inter-story drift.The first run reaches the best design with a total structural weight of 399.215 kips.This design is about 6.6 times heavier than the best design found for the same structure subjected to gravity loads only.
With external walls This study case is similar to "With External Walls" section.Namely, the stiffness of the external walls is considered.Columns and beams are selected from the first 100 lightest standard W-shape sections provided in AISC tables.
The best designs are shown in Table 11 along with total structural weight and maximum values of interaction ratio, shear ratio, member end rotation, and inter-story drift.The third run obtains the best design with a total structural

Concluding remarks
Optimum design of 3D steel frame structures subjected to service and blast loads is studied using metaheuristic optimization algorithms.The problem is formulated to minimize the total weight of the structure subjected to AISC strength requirements and displacement constraints.Depending on the problem, three types of analyses are carried out in the optimization process: linear static analysis, linear dynamic analysis, and nonlinear dynamic analysis.Hybrid Harmony Search-Colliding Bodies Optimization (HHC) is used to find an optimum design for every case of study.Metaheuristic optimization with equivalent static load (MOESL) is used to find the optimum design of the nonlinear dynamic study cases.
Table 12 shows the final designs of the structure with and without blast loading.It is seen that when beams and columns are designed to remain elastic, the optimum structure is about 14 times heavier to withstand the blast loads compared to the optimum design without the consideration of blast loading.However, when columns are only designed to remain elastic and beams are allowed to develop plastic hinges (with displacement requirements), the optimum structure is about 6 times heavier to withstand the blast loads compared to the optimum design without the consideration of blast loading.
When the stiffness of the external walls is considered (added to the stiffness of the structure), the final designs for linear and nonlinear dynamic analyses are only slightly heavier than the design without the blast load considerations.

Fig. 4
Fig. 4 3D steel frame and charge location

Table 2
Design variables

Table 3
Load combinations (AISC, 2017;Gilsanz et al., 2013) DL is dead load, LL is live load, and BL is blast load

Table 4
Final designs for 3D frame under service load only

Table 5
Pressure-time history on faces

Table 6
Final designs for 3D framed steel structure under service and blast loads (linear dynamic analysis)

Table 7
Final designs for 3D framed steel structure under service and blast loads with the mass of external walls (linear dynamic analysis)

Table 9
Final designs for 3D frame under service and blast loads (nonlinear dynamic analysis)

Table 10
Final designs for 3D framed steel structure under service and blast loads with the mass of the external walls (nonlinear dynamic analysis)

Table 11
Final designs for 3D frame with external walls under service and blast loads (nonlinear dynamic analysis) .626kips.This design is about 28% heavier than the best design found for the reference case study.