# Embodied Energy and Cost Optimization of RC Beam under Blast Load

Authors: Runqing Yu; Diandian Zhang; Haichun Yan
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Reinforced concrete (RC) structures not only consume a lot of resources but also cause continuing pollution. However, sustainable design could make RC structures more environmental-friendly. One important index for environmental impact assessment is embodied energy. The aim of the present study is to optimize the embodied energy and the cost of RC beam subjected to the blast loads. First, a general optimization procedure was described. Then, the optimization procedure was used to optimize the embodied energy and the cost of RC beams. Optimization results of the cost and the embodied energy were compared. It was found that the optimization results were influenced by the cost rationC (ratio of price of steel to price of concrete per unit volume) and the embodied energy ratio nE (ratio of embodied energy of steel to embodied energy of concrete per unit volume). An optimal design that minimized both embodied energy and cost simultaneously was obtained if values of nC and nE were very close.


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## Body

## 1. Introduction

Worldwide, RC structures consume a lot of energy [1] and give off large amounts of greenhouse gases [2]. In addition, the maintenance, repair, and refurbishment during the life of buildings also consume a lot of energy. Even when the building is abandoned, quantities of solid construction waste are difficult to recycle. All of those burden the environment.In order to make RC structures more friendly to environment, a lot of efforts have been made to reduce the influence of RC buildings on environment during the service life [3]. Recently, many researchers focused on the optimization of RC structures considering environment factors [4–7]. Paya-Zaforteza et al. [4] minimized the carbon dioxide (CO2) emissions and the cost using simulated annealing. Their research showed that the CO2 emissions optimization and the cost optimization were highly related. Yeo and Gabbai [5] explored the implications of using the embodied energy as the objective function to be minimized from the point of view of cost. Their results showed that a reduction of 10% in the embodied energy leads to an increase of 10% in the cost. In their study, certain values of the embodied energy were used, which was not true in practice because the exact value of embodied energy was not known [8, 9]. Medeiros and Kripka [6] proposed an optimal method to minimize the monetary and environmental costs of RC column. In their study, four environmental impact assessment indices—CO2, global warming potential (GWP), energy consumption, and Eco-indicator—were taken into consideration, but the cost was not considered. Yepes et al. [7] suggested a design method to optimize the cost and the CO2 emission of precast-prestressed concrete U-beam road bridges. Their analysis showed that a reduction of costs by 1 Euro can save up to 1.75 kg in CO2 emissions. Park et al. [10] provided structural design guidelines that reduced the CO2 emission and the cost of RC columns. Parametric study was conducted to investigate the influences of design factors on the CO2 emission and the cost.In conclusion, those researches optimized RC structures in normal operation condition considering environment factors. Extreme loading conditions, such as explosion, earthquake, and hurricane, were not considered. In the last century, the design that used to resist the extreme loads was not common. With the rapid development of economy, the design to resist the extreme loads was widely accepted by engineering field. An increasing number of buildings were designed considering the extreme loading conditions. Therefore, studies on the sustainable design of RC structures in extreme loading conditions are necessary.In this study, a sustainable design of RC beam under blast loads was discussed through minimizing the embodied energy. Meanwhile, the cost was used as objective function for comparison. First, a general optimal design procedure was presented. Then, parametric studies were conducted to analyze the influences of the material strength and sectional geometry on the optimal design. The cost and embodied energy of particular building materials (reinforcements and concrete) were taken as variables and their influences on the optimal design were presented. The optimization results of cost and the embodied energy were compared and the correlation between them was discussed, which may be helpful for sustainable deign of RC structural members against blast.


## 2. Methodology

### 2.1. Problem Description

In this study, an interior beam subjected to air blast load was considered. This problem was an illustrative example of UFC 3-340-02 [11]. Both ends of the beam were fixed. The beam was assumed to deform in a flexural shape. The cross section is shown in Figure 1. A uniform blast load is applied on the upper surface of the beam. The beam was designed according to the restrictions and guidelines found in the UFC 3-340-02 code.Figure 1
Cross section.There are two response indices that are usually used to define the damage levels of blast-loaded structural components in a flexural response regime: the support rotation angle (θ) and the ductility ratio (μ) [12]. In the present study, both of the two indices were used as design constraints.Four design parameters were considered in this study. The four parameters were the height of the beamh, the widthb, the ratio of longitudinal reinforcement ρ1, and the ratio of stirrup ρ2.


### 2.2. Objective Functions

The total cost and the total embodied energy were taken as objective function to be minimized. The total cost is defined as [13](1)C′=CCVC+CSVS1+VS2,where C′ is the total cost of beam, CC is the cost of concrete per unit volume, VC is the volume of concrete, CS is the cost of reinforcing steel per unit volume, and VS1 and VS2 are the volume of longitudinal reinforcement and shear reinforcement, respectively.It should be noted that the optimal values were affected by the relative values of the objective function only. Equation (1) divided by CS is (2)C′′=VCnC+VS1+VS2,where nC=Cs/Cc, C′′=C′/CS.The value ofnC varies from country to country; thus, (2) was more convenient to study the minimum cost of RC beam than (1) [13].Similarly to the cost objective function, the embodied energy objective function is written as follows:(3)E′′=VCnE+VS1+VS2,where E′′=E′/ES, E′ is the total energy, ES is the embodied energy of steel per unit volume, nE=ES/EC is the energy ratio, and EC is the embodied energy of concrete per unit volume.Comparing (2) with (3), a similarity can be found. The form of the energy objective function and the cost objective function is very similar. Thus, the cost objective function and the energy objective function are represented by (4) for the convenience of optimization calculation.(4)W=VCn+VS1+VS2,where n denotes nC and nE and W denotes C′′ and E′′.In the following analysis, the optimal designs for the costC′′ and the embodied energy E′′ were unified to the optimal design for W.


### 2.3. Design Constraints

The RC beam under blast load is simplified to SDOF system (see Figure2). The design constraints were given in mathematical expressions and the optimal design was transform into a constrained optimization problem.Figure 2
The simplified SDOF model of RC beam under blast load.
(a)
Equivalent system
(b)
Resistance curve
(c)
Blast loadThe design constraints were defined in accordance with the UFC 3-340-02 code [11]. The constraints for RC rectangular section beam under blast load were expressed.(1)
Performance limits are as follows:(5)θ≤θa,μ≤μa.(2)
Constraints of the ratio of longitudinal reinforcementρ1 and the ratio of stirrup ρ2 are as follows:(6)0.750.85K1fdc′fdy8700087000+fdy≥ρ1≥max⁡3fc′fy,200fy,ρ2≥0.0015.(3)
Limit of the direct shear stress is as follows:(7)0.18fdc′bd≤Rm2l.(4)
Limit of the diagonal tension stress is as follows:(8)l/2-dRmbd≤10fdc′0.5.(5)
Limit of the unreinforced web shear capacity is as follows:(9)1.9fdc′0.5+2500ρ1≤3.5fdc′0.5,where θa is the allowable support rotation angle, μa is the allowable ductility ratio, fc′ is the concrete compressive strength, fdc′ is the dynamic concrete compressive strength, fy is the yield stress of steel, fdy is the dynamic yield stress of steel, K1 is a coefficient associated with fc′, b is the width of beam, d is the distance from the extreme compression fiber to the centroid of the longitudinal tension reinforcement, Rm is the bending resistance of RC beam, and l is the span length.Some explanations of design constraints are presented here. Equations (5) are the deformation requirements. The value of θa and μa can be found in many literatures [11, 14, 15]. Equations (7)~(9) are the requirements that ensure shear failure does not occur.It should be noticed the parameters shown in (5)~(9) were interrelated. This interrelated relationship was reflected in the calculation of ρ2, θ, Rm, and μ. Calculations of ρ2, θ, Rm, and μ were present.(1)
ρ
2. ρ2 was computed by (10)ρ2=max⁡vu-vc,vc0.85fdy,where vc and vu are calculated by (11) and (12), respectively.(11)vc=1.9fc′0.5+2500ρ1,(12)vu=l/2-dRmbd.(2)
θ. θ is defined in (13)θ=arctan⁡2yml,where ym=μye, ym is the maximum midspan displacement, and ye is the elastic midspan displacement.y
e is calculated by (14)ye=RmK,where K is the equivalent elastic stiffness.K is given by (15) considering clamped boundary.(15)K=307ECIal4,where EC is the concrete modulus of elasticity and Ia is the average moment of inertia of the beams.I
a is given by (16)Ia=0.5bh312+Gbd3.The coefficientG in (16) is evaluated by [16](17)G=3320.3ρ13-181.98ρ12+5.8624ρ1ES7EC0.7,where ES is the steel modulus of elasticity.(3)
R
m. The resistance Rm is given by [11](18)Rm=8MN+MPl2,where MN is ultimate negative moment capacity at the support and MP is ultimate positive moment capacity at the midspan.Values ofMN and MP are calculated by (19) considering symmetrical reinforced concrete beam.(19)MN=MP=ρ1fdybd21-ρ1fdy1.7ρ1fc′.(4)
μ. μ is calculated by [17](20)2ωtd2μ-1+2μ-12μ1+1.4π/ωtd=P0Rm,where ω is the natural frequency, td is the duration time, and P0 is the peak pressure.ω is calculated by (21)ω=KKLMm,where m is the mass of the beam plus 20% of the slabs span perpendicular to the beam and KLM is the load-mass factor.


### 2.4. Optimization Technique and Verification

Many methods were used successfully in optimal design of RC structures, such as Generalized Reduced Gradient (GRG) method [13], heuristic optimization methods [18, 19], discretized continuum-type optimality criteria (DCOC) [20], and Lagrange multiplier method [21]. In this study, the sequential quadratic programing (SQP) method was used since SQP was efficient proved by an extensive comparative study done by Schittkowski [22]. Besides, SQP was easy to realize by the constrained nonlinear optimization solver “fmincon” in MATLAB.In order to get the global optimum, many random starting points were used. The random starting points were generated by Latin hypercube sampling (LHS) [23]. LHS was a matrix of i×j order. i was the number of sampling points to be examined. j was the number of design parameters. Each of j columns of matrix that contained sampling points 1,2,…,i was coupled to form the Latin hypercube. This generated random sample points, which ensured that all portions of design space were represented.The flowchart of optimum design method is shown in Figure3.Figure 3
Flowchart of optimization technique.


## 2.1. Problem Description

In this study, an interior beam subjected to air blast load was considered. This problem was an illustrative example of UFC 3-340-02 [11]. Both ends of the beam were fixed. The beam was assumed to deform in a flexural shape. The cross section is shown in Figure 1. A uniform blast load is applied on the upper surface of the beam. The beam was designed according to the restrictions and guidelines found in the UFC 3-340-02 code.Figure 1
Cross section.There are two response indices that are usually used to define the damage levels of blast-loaded structural components in a flexural response regime: the support rotation angle (θ) and the ductility ratio (μ) [12]. In the present study, both of the two indices were used as design constraints.Four design parameters were considered in this study. The four parameters were the height of the beamh, the widthb, the ratio of longitudinal reinforcement ρ1, and the ratio of stirrup ρ2.


## 2.2. Objective Functions

The total cost and the total embodied energy were taken as objective function to be minimized. The total cost is defined as [13](1)C′=CCVC+CSVS1+VS2,where C′ is the total cost of beam, CC is the cost of concrete per unit volume, VC is the volume of concrete, CS is the cost of reinforcing steel per unit volume, and VS1 and VS2 are the volume of longitudinal reinforcement and shear reinforcement, respectively.It should be noted that the optimal values were affected by the relative values of the objective function only. Equation (1) divided by CS is (2)C′′=VCnC+VS1+VS2,where nC=Cs/Cc, C′′=C′/CS.The value ofnC varies from country to country; thus, (2) was more convenient to study the minimum cost of RC beam than (1) [13].Similarly to the cost objective function, the embodied energy objective function is written as follows:(3)E′′=VCnE+VS1+VS2,where E′′=E′/ES, E′ is the total energy, ES is the embodied energy of steel per unit volume, nE=ES/EC is the energy ratio, and EC is the embodied energy of concrete per unit volume.Comparing (2) with (3), a similarity can be found. The form of the energy objective function and the cost objective function is very similar. Thus, the cost objective function and the energy objective function are represented by (4) for the convenience of optimization calculation.(4)W=VCn+VS1+VS2,where n denotes nC and nE and W denotes C′′ and E′′.In the following analysis, the optimal designs for the costC′′ and the embodied energy E′′ were unified to the optimal design for W.


## 2.3. Design Constraints

The RC beam under blast load is simplified to SDOF system (see Figure2). The design constraints were given in mathematical expressions and the optimal design was transform into a constrained optimization problem.Figure 2
The simplified SDOF model of RC beam under blast load.
(a)
Equivalent system
(b)
Resistance curve
(c)
Blast loadThe design constraints were defined in accordance with the UFC 3-340-02 code [11]. The constraints for RC rectangular section beam under blast load were expressed.(1)
Performance limits are as follows:(5)θ≤θa,μ≤μa.(2)
Constraints of the ratio of longitudinal reinforcementρ1 and the ratio of stirrup ρ2 are as follows:(6)0.750.85K1fdc′fdy8700087000+fdy≥ρ1≥max⁡3fc′fy,200fy,ρ2≥0.0015.(3)
Limit of the direct shear stress is as follows:(7)0.18fdc′bd≤Rm2l.(4)
Limit of the diagonal tension stress is as follows:(8)l/2-dRmbd≤10fdc′0.5.(5)
Limit of the unreinforced web shear capacity is as follows:(9)1.9fdc′0.5+2500ρ1≤3.5fdc′0.5,where θa is the allowable support rotation angle, μa is the allowable ductility ratio, fc′ is the concrete compressive strength, fdc′ is the dynamic concrete compressive strength, fy is the yield stress of steel, fdy is the dynamic yield stress of steel, K1 is a coefficient associated with fc′, b is the width of beam, d is the distance from the extreme compression fiber to the centroid of the longitudinal tension reinforcement, Rm is the bending resistance of RC beam, and l is the span length.Some explanations of design constraints are presented here. Equations (5) are the deformation requirements. The value of θa and μa can be found in many literatures [11, 14, 15]. Equations (7)~(9) are the requirements that ensure shear failure does not occur.It should be noticed the parameters shown in (5)~(9) were interrelated. This interrelated relationship was reflected in the calculation of ρ2, θ, Rm, and μ. Calculations of ρ2, θ, Rm, and μ were present.(1)
ρ
2. ρ2 was computed by (10)ρ2=max⁡vu-vc,vc0.85fdy,where vc and vu are calculated by (11) and (12), respectively.(11)vc=1.9fc′0.5+2500ρ1,(12)vu=l/2-dRmbd.(2)
θ. θ is defined in (13)θ=arctan⁡2yml,where ym=μye, ym is the maximum midspan displacement, and ye is the elastic midspan displacement.y
e is calculated by (14)ye=RmK,where K is the equivalent elastic stiffness.K is given by (15) considering clamped boundary.(15)K=307ECIal4,where EC is the concrete modulus of elasticity and Ia is the average moment of inertia of the beams.I
a is given by (16)Ia=0.5bh312+Gbd3.The coefficientG in (16) is evaluated by [16](17)G=3320.3ρ13-181.98ρ12+5.8624ρ1ES7EC0.7,where ES is the steel modulus of elasticity.(3)
R
m. The resistance Rm is given by [11](18)Rm=8MN+MPl2,where MN is ultimate negative moment capacity at the support and MP is ultimate positive moment capacity at the midspan.Values ofMN and MP are calculated by (19) considering symmetrical reinforced concrete beam.(19)MN=MP=ρ1fdybd21-ρ1fdy1.7ρ1fc′.(4)
μ. μ is calculated by [17](20)2ωtd2μ-1+2μ-12μ1+1.4π/ωtd=P0Rm,where ω is the natural frequency, td is the duration time, and P0 is the peak pressure.ω is calculated by (21)ω=KKLMm,where m is the mass of the beam plus 20% of the slabs span perpendicular to the beam and KLM is the load-mass factor.


## 2.4. Optimization Technique and Verification

Many methods were used successfully in optimal design of RC structures, such as Generalized Reduced Gradient (GRG) method [13], heuristic optimization methods [18, 19], discretized continuum-type optimality criteria (DCOC) [20], and Lagrange multiplier method [21]. In this study, the sequential quadratic programing (SQP) method was used since SQP was efficient proved by an extensive comparative study done by Schittkowski [22]. Besides, SQP was easy to realize by the constrained nonlinear optimization solver “fmincon” in MATLAB.In order to get the global optimum, many random starting points were used. The random starting points were generated by Latin hypercube sampling (LHS) [23]. LHS was a matrix of i×j order. i was the number of sampling points to be examined. j was the number of design parameters. Each of j columns of matrix that contained sampling points 1,2,…,i was coupled to form the Latin hypercube. This generated random sample points, which ensured that all portions of design space were represented.The flowchart of optimum design method is shown in Figure3.Figure 3
Flowchart of optimization technique.


## 3. Numerical Examples and Discussions

### 3.1. Example  1: Verify the Effectiveness of Optimum Design Method

Values of design parameters in this example are listed in Table1, where ρ is density of concrete.Table 1
Values of design parameters.
Design parameters
Values
L(in)
240
f
c
′/(psi)
4,000
E
C
/(psi)
3.8 × 106
ρ (lbs/ft3)
150
f
d
y (psi)
66,000
E
S (psi)
29 × 106
P
0 (psi)
7.2
t
d (ms)
60.7
θ
a
1°
μ
a
15
n
50The ranges ofb and h are given as follows in this example:(22)2.0≤hb≤3.5,h≤40in.The design variables obtained from the standard design approach solution and the optimal design solution are shown in Table2.Table 2
Optimal design solution.
Standard solution
Optimal solution
b (in)
18
12.5
h (in)
30
40.0
ρ
1
0.0045
0.00318
ρ
2
0.00235
0.00228
μ
9.0
15.0
ω
0.248
0.322
M
u (in·lbs)
4.45 × 106
4.29 × 106
W
4127.3
3429.7From Table2, it is found that h and μ are larger after optimization. This is explained that an increase of h and μ increases the bending resistance capacity of RC beam. By optimal design,W is smaller by 16.9%. If W is taken as the cost, it will save 16.9% of the cost after optimal design, which is very economical. The optimal design result is shown in Table 2, which proves the effectiveness of the optimal design method mentioned before.


### 3.2. Example  2: Effect of Section Size(b,h) on the Optimal Design

Energy ratio and cost ratio vary from country to country. Then, ranges of the energy rationE and the cost ratio nC should be gotten before parameter analysis. Considering the uncertainties of energy ratio and cost ratio, it is reasonable to extend the ranges that are shown in Tables 3 and 4. Therefore, 60≤nC≤200 and 10≤nE≤150 were used in this study.Table 3
Energy ratio.
Reference
[24]
[25]
[26]
Concrete
1.11 MJ/kg
3180 MJ/m3
1.3 MJ/kg
Reinforcing bar
35.3 MJ/kg
8.9 MJ/kg
11.1 MJ/kg
n
E
101
21.8
27.2Table 4
Cost ratio.
Reference
[27]
[28]
[6]
Concrete
55 £/m3
43.93 EUR/m3
65.65 USD/m3
Reinforcing bar
0.5 £/kg
534.53 EUR/ton
5826.30 USD/m3
n
C
70.9
94.9
88.9Setθ = 2° and μm≤10 and other design parameters are the same as presented in Example  1. When analyzing the effect of b on optimal design, set h=20 in and 10 in ≤b ≤ 20 in (Example  2(a)). When analyzing the effect of h on optimal design, setb = 15 in and 18 in ≤h ≤ 25 in (Example  2(b)).In order to compare the optimal results expediently, optimal design results forb=10 and h=10 were selected as the reference results. Then, optimal results were normalized by (23), where WR represents the reference results. Thus, the smaller the value of ΔW, the more effective the optimization. Normalized optimal results are shown in Figures 4 and 5. (23)ΔW=W-WRWR×100%.Figure 4
Effect ofb on ΔW (Example  2(a): h=20 in, 10 in ≤b ≤ 20 in).Figure 5
Effect ofh on ΔW (Example  2(b): b=15 in, 18 in ≤h ≤ 25 in).From Figures4 and 5, it is observed that the effect of cross section on optimization effectiveness ΔW varies with the value of n. When the value of n is small (n=10), ΔW decreases with the increase of values of b and h. As the value of n increases, effects of b and h change. Ranges of nE and nC are also indicated in Figures 4 and 5. When the difference between the value of nE and the value of nC is big, an increase in costs results in a reduction in embodied energy. However, when the value of nE is close to the value of nC, the cost optimal results reduce the embodied energy simultaneously.It shall be noticed that two parameters were used as deformation constraints:θ and μ. Only one parameter reached the allowed value in most cases. Take Example  2(a) as an example, the optimal values of μ/μa and θ/θa are shown in Figure 6. When the value of b is small, the allowed value θa is reached. As the value of b increases, the value of θ decreases while the value of μ increases until it reaches the allowed value μa. Two ranges were defined: θa range (in which θ=θa) and μa range (in which μ=μa). Trends are different in different ranges, shown in Figure 7. As the value of b increases, the resistance changes from curveA to curveE. CurveA to curveC is θa range and curveC to curveE is μa range. In θa range, the value of Rm decreases as the value of b increases. In μa range, the value of Rm increases as the value of b increases. Those show that the optimal results are quite different if different deformation constraints are adopted. In order to obtain a safe design, constraints of θ and μ should be considered simultaneously.Figure 6
Values ofμ/μa and θ/θa.Figure 7
Trends in different range.


### 3.3. Example  3: Effect of Material Strength on the Optimal Design

Values of design parameters were the same as those of Example  2(a). Optimal design results whenn=10 were chosen as the reference results. Then, optimal design results were normalized by (23). First, an investigation was performed to study the effect of concrete compression strength fc′ on optimal effectiveness ΔW (see Figure 8). It is clear to find that the optimal effectiveness closely relates to n. However, the relationship between fc′ and optimal effectiveness is not clear, which is different from the conclusion under conventional load [29].Figure 8
Effect offc′ on cost and embodied energy.Then, an investigation was performed to investigate the effects of the yield stress of steelfy on optimal effectiveness ΔW (see Figure 9). As the yield stress of steel increases, ΔW decreases regardless of n, which means that the optimal design is very effective if the yield stress of steel is small.Figure 9
Effect offy on cost and embodied energy.


## 3.1. Example  1: Verify the Effectiveness of Optimum Design Method

Values of design parameters in this example are listed in Table1, where ρ is density of concrete.Table 1
Values of design parameters.
Design parameters
Values
L(in)
240
f
c
′/(psi)
4,000
E
C
/(psi)
3.8 × 106
ρ (lbs/ft3)
150
f
d
y (psi)
66,000
E
S (psi)
29 × 106
P
0 (psi)
7.2
t
d (ms)
60.7
θ
a
1°
μ
a
15
n
50The ranges ofb and h are given as follows in this example:(22)2.0≤hb≤3.5,h≤40in.The design variables obtained from the standard design approach solution and the optimal design solution are shown in Table2.Table 2
Optimal design solution.
Standard solution
Optimal solution
b (in)
18
12.5
h (in)
30
40.0
ρ
1
0.0045
0.00318
ρ
2
0.00235
0.00228
μ
9.0
15.0
ω
0.248
0.322
M
u (in·lbs)
4.45 × 106
4.29 × 106
W
4127.3
3429.7From Table2, it is found that h and μ are larger after optimization. This is explained that an increase of h and μ increases the bending resistance capacity of RC beam. By optimal design,W is smaller by 16.9%. If W is taken as the cost, it will save 16.9% of the cost after optimal design, which is very economical. The optimal design result is shown in Table 2, which proves the effectiveness of the optimal design method mentioned before.


## 3.2. Example  2: Effect of Section Size(b,h) on the Optimal Design

Energy ratio and cost ratio vary from country to country. Then, ranges of the energy rationE and the cost ratio nC should be gotten before parameter analysis. Considering the uncertainties of energy ratio and cost ratio, it is reasonable to extend the ranges that are shown in Tables 3 and 4. Therefore, 60≤nC≤200 and 10≤nE≤150 were used in this study.Table 3
Energy ratio.
Reference
[24]
[25]
[26]
Concrete
1.11 MJ/kg
3180 MJ/m3
1.3 MJ/kg
Reinforcing bar
35.3 MJ/kg
8.9 MJ/kg
11.1 MJ/kg
n
E
101
21.8
27.2Table 4
Cost ratio.
Reference
[27]
[28]
[6]
Concrete
55 £/m3
43.93 EUR/m3
65.65 USD/m3
Reinforcing bar
0.5 £/kg
534.53 EUR/ton
5826.30 USD/m3
n
C
70.9
94.9
88.9Setθ = 2° and μm≤10 and other design parameters are the same as presented in Example  1. When analyzing the effect of b on optimal design, set h=20 in and 10 in ≤b ≤ 20 in (Example  2(a)). When analyzing the effect of h on optimal design, setb = 15 in and 18 in ≤h ≤ 25 in (Example  2(b)).In order to compare the optimal results expediently, optimal design results forb=10 and h=10 were selected as the reference results. Then, optimal results were normalized by (23), where WR represents the reference results. Thus, the smaller the value of ΔW, the more effective the optimization. Normalized optimal results are shown in Figures 4 and 5. (23)ΔW=W-WRWR×100%.Figure 4
Effect ofb on ΔW (Example  2(a): h=20 in, 10 in ≤b ≤ 20 in).Figure 5
Effect ofh on ΔW (Example  2(b): b=15 in, 18 in ≤h ≤ 25 in).From Figures4 and 5, it is observed that the effect of cross section on optimization effectiveness ΔW varies with the value of n. When the value of n is small (n=10), ΔW decreases with the increase of values of b and h. As the value of n increases, effects of b and h change. Ranges of nE and nC are also indicated in Figures 4 and 5. When the difference between the value of nE and the value of nC is big, an increase in costs results in a reduction in embodied energy. However, when the value of nE is close to the value of nC, the cost optimal results reduce the embodied energy simultaneously.It shall be noticed that two parameters were used as deformation constraints:θ and μ. Only one parameter reached the allowed value in most cases. Take Example  2(a) as an example, the optimal values of μ/μa and θ/θa are shown in Figure 6. When the value of b is small, the allowed value θa is reached. As the value of b increases, the value of θ decreases while the value of μ increases until it reaches the allowed value μa. Two ranges were defined: θa range (in which θ=θa) and μa range (in which μ=μa). Trends are different in different ranges, shown in Figure 7. As the value of b increases, the resistance changes from curveA to curveE. CurveA to curveC is θa range and curveC to curveE is μa range. In θa range, the value of Rm decreases as the value of b increases. In μa range, the value of Rm increases as the value of b increases. Those show that the optimal results are quite different if different deformation constraints are adopted. In order to obtain a safe design, constraints of θ and μ should be considered simultaneously.Figure 6
Values ofμ/μa and θ/θa.Figure 7
Trends in different range.


## 3.3. Example  3: Effect of Material Strength on the Optimal Design

Values of design parameters were the same as those of Example  2(a). Optimal design results whenn=10 were chosen as the reference results. Then, optimal design results were normalized by (23). First, an investigation was performed to study the effect of concrete compression strength fc′ on optimal effectiveness ΔW (see Figure 8). It is clear to find that the optimal effectiveness closely relates to n. However, the relationship between fc′ and optimal effectiveness is not clear, which is different from the conclusion under conventional load [29].Figure 8
Effect offc′ on cost and embodied energy.Then, an investigation was performed to investigate the effects of the yield stress of steelfy on optimal effectiveness ΔW (see Figure 9). As the yield stress of steel increases, ΔW decreases regardless of n, which means that the optimal design is very effective if the yield stress of steel is small.Figure 9
Effect offy on cost and embodied energy.


## 4. Conclusions

The sustainable design and the blast-resistance design were combined in this paper. A general optimization procedure to minimize the cost and the embodied energy of RC rectangular cross-section beam was present. The optimal design was turned into a nonlinearly constrained optimization. The optimal design was conducted using Latin hypercube sampling and the sequential quadratic programing (SQP) method. Several examples were present to investigate the optimization effectiveness. Several major conclusions are drawn as follows:(1)
The optimal results are different if different deformation constraints are adopted. In order to obtain a safe design, constraints ofθ and μ should be considered simultaneously.(2)
The optimization results are closely related to the cost rationC and the embodied energy ratio nE. When the value of nE is close to the value of nC, the cost optimal results reduce the embodied energy simultaneously.(3)
The optimal design is more effective if the yield stress of steel is small. In the present study, when the yield stress of steel is decreased (from 70000 psi to 60000 psi), the efficiency of optimal design will significantly increase for both cost (from 5.6% to 11.5%) and embodied energy (from 4.2% to 9.8%).



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*Source: 1907972-2017-03-22.xml*