Design of low-carbon and cost-efficient concrete frame buildings: a hybrid optimization approach based on harmony search

Figures & data

Figure 1. Research program of the present study.

Figure 2. Analytical procedure of the hybrid approach.

Table 1. Definition of the optimization problem on structural members.

Classification	Description	Expression	Note
Variable	Cross-section	bmin≤b≤ bmax hmin≤h≤ hmax	b and h are the cross-sectional width and height, respectively. The lower and upper limits of sectional dimension are constrained by the building operation requirements and construction practices
	Strength grade	gc,gs	gc and gs are the strength grades of concrete and steel, respectively. Longitudinal bars and stirrups may have different strengths
	Reinforcement	ds,ns,ls	ds,ns, and ls are the diameter, number, and length of steel bars, respectively
Objective	Carbon emission	f1 = Min(Er)	Er is the total change in emissions, of which a negative value indicates a reduction in emissions
	Monetary cost	f2 = Min(Cr)	Cr is the total change in costs, of which a negative value indicates a reduction in costs
Constraint	Load-bearing capacity	R≥ S	R and S represent member resistance and load effect, respectively; structures and components may be subjected to either separate or combined actions of axial force N, bending M, shear V, and torsion T
	Reinforcement ratio	ρmin≤ρ≤ ρmax	The reinforcement ratio ρ is limited in a certain range to avoid brittle failure of structural components
	Serviceability limit state	ω≤ ωmax	ω represents potential crack width, ωmax represents the upper limit of crack width
	Detailing requirements	fc≥fclim c≥ clim la≥lalim sv≥svlim	fclim and clim are the lower limits of concrete strength and relevant cover depth, respectively, due to durability requirements; lalim and svlim are the limits of anchorage length and spacing of stirrups, respectively

Figure 3. Main processes of the harmony search algorithm.

Table 2. Assessment of the constraint violation (VI) for the optimization of bending beams.

Constraint violation	Expression	Note
VI of the bending resistance	$V{I}_M=max\left(0,\frac{A_{s,min}-A_s}{A_{s,min}},\frac{A_s-A_{s,max}}{A_{s,max}}\right)$	As is the amount of longitudinal reinforcement based on the values of variables in a solution vector, As,min and As,max are the lower and upper limits of reinforcement depending on the bending capacity and limits of reinforcement ratios
VI of the shear resistance	$V{I}_V=max\left(0,\frac{s_{v,lim}-s_v}{s_{v,lim}},\frac{V-V_{lim}}{V}\right)$	sv is calculated spacing of stirrups based on the values of variables in a solution vector; svlim is the minimum spacing of stirrups; Vlim is the maximum shear force according to the limit conditions of oblique compression failure
VI of the crack width	$V{I}_{\omega }=max\left(0,\frac{\omega -{\omega }_{max}}{{\omega }_{max}}\right)$	ω is the crack width estimated based on the values of variables in a solution vector, ωmax is the maximum crack width limited by the design code
Total VI	$VI=V{I}_M+V{I}_V+V{I}_{\omega }$	The total VI is taken as the linear combination of the above three components.

Table 3. Engineering quantities based on subprojects for the initial building scheme.

Subproject	Content	Engineering quantity
Foundation	Ground leveling	929.9 m^2
	Mechanical excavation	1054.2 m^2
	Manual excavation	105.4 m^2
	Backfilling	624.1 m^3
	Manufacturing, casting, and curing concrete cushion	64.7 m^3
	Manufacturing, casting, and curing concrete foundation	196.9 m^3
	Manufacturing and processing foundation reinforcement	15.3 t
	Formwork for concrete cushion	41.9 m^2
	Formwork for concrete foundation	486.5 m^2
Framed structure	Manufacturing, casting, and curing concrete column	100.8 m^3
	Manufacturing and processing column reinforcement	19.1 t
	Formwork for concrete column	862.1 m^2
	Manufacturing, casting, and curing concrete beam	151.0 m^3
	Manufacturing and processing beam reinforcement	48.2 t
	Formwork for concrete beam	1487.4 m^2
	Manufacturing, casting, and curing concrete slab	254.7 m^3
	Manufacturing and processing slab reinforcement	22.2 t
	Formwork for concrete slab	2192.2 m^2
	Manufacturing, casting, and curing concrete stair	62.0 m^3
	Manufacturing and processing stair reinforcement	3.9 t
	Formwork for concrete stair (projected area)	473.6 m^2
	Manufacturing, casting, and curing concrete for other components	41.5 m^3
	Manufacturing and processing reinforcement for other components	3.0 t
	Formwork for other concrete components	465.7 m^2
Masonry wall	Concrete block masonry wall (thickness 200 mm)	444.1 m^3
	Concrete block masonry wall (thickness 100 mm)	59.0 m^3
	Manufacturing, casting, and curing concrete structural column	25.2 m^3
	Manufacturing and processing wall reinforcement	6.4 t
	Formwork for concrete structural column	313.1 m^2
Insulation and decoration	Leveling and plastering of concrete floor	2693.9 m^2
	Floor insulation (Polystyrene foam)	188.6 m^3
	Plastering, whitening, and painting of concrete column	862.1 m^2
	Plastering, whitening, and painting of masonry wall	5486.1 m^2
	Plastering, whitening, and painting of concrete beam and wall	1004.1 m^2
	Plastering, whitening, and painting of ceiling	3664.24 m^2
	Leveling, plastering, and painting of exterior wall surface	1170.0 m^2
	Insulation of exterior wall (Polystyrene foam)	1170.0 m^2
	Manufacturing and installation of plastic-steel window	354.2 m^2
	Manufacturing and installation of fireproof door	160.7 m^2
	Manufacturing and installation of wood door	86.4 m^2
	Roofing leveling (two layers with thickness of 20 mm)	673.5 m^2
	Roofing insulation by polystyrene foam board	94.3 m^3
	Roofing insulation by expanded perlite	70.7 m^3
	SBS-modified asphalt waterproofing (three layers)	673.5 m^2
	Manufacturing and casting, and curing of fine aggregate concrete	27.2 m^3
	Isolation and protection layers	673.5 m^2
Temporal work	Transportation and installation of tower crane	once
	Manufacturing, casting, and curing concrete foundation for tower crane	36.0 m^3
	Vertical transportation estimated by building area	2693.9 m^2
	Scaffolding estimated by building area	2693.9 m^2
	Building closure and safety net	1640.7 m^2
	Temporal electricity for lighting and others	5.0 MWh

Table 4. Building embodied emissions and costs of the initial scheme.

Objective	Process	Subproject
		Foundation	Framed structure	Masonry wall	Insulation & decoration	Temporal work	The entire building
Emission (tCO2e)	Production	110.16	426.99	120.94	433.56	32.14	1123.80
	Transportation	5.72	14.78	14.85	14.42	0.92	50.69
	Construction	8.80	14.38	0.74	1.14	71.36	96.42
	Total	124.68	456.16	136.53	449.12	104.42	1270.91
Cost (10^3 USD)	Labor	7.52	25.29	11.34	79.77	2.82	126.74
	Material	26.14	99.48	41.48	143.42	0.06	310.57
	Machinery	4.15	2.58	0.15	0.38	4.57	11.85
	Measure cost	3.95	44.23	3.98	5.92	44.09	102.18
	Other fees	6.38	30.55	9.06	71.22	3.15	120.37
	Total	48.15	202.14	66.02	300.71	54.69	671.71

Figure 4. Comparison of the embodied emissions and relevant costs for structural components of the assessed building. “E” and “C” represent emissions and costs respectively.

Figure 5. Trend of the maximum emission and cost reduction of the optimization results when the number of improvisations increases.

Table 5. Comparison of the optimization results for ten independent trials.

Trial	Time cost per thousand improvisations (s)	Performance of result
		Maximum efficient improvisations^a	Number of feasible solutions	Maximum cost reduction (10^3 USD)	Maximum emission reduction (tCO2e)
01	2578.0	7231	83,347	8.075	29.446
02	2536.3	7616	83,487	8.071	29.446
03	2570.5	6160	84,643	8.075	29.446
04	2517.9	4215	84,356	8.068	29.446
05	2419.8	7938	78,687	7.962	29.376
06	2454.7	1984	82,650	8.074	29.446
07	2499.5	1611	84,928	8.066	29.446
08	2453.1	6519	80,388	7.978	29.376
09	2468.8	3935	80,582	7.952	29.376
10	2506.0	6183	85,381	8.075	29.446

^aThe number of improvisations after which both the maximum cost and emission reduction remained unchanged till the end of the HS simulation.

Figure 6. Optimization results of emission and cost reduction based on the harmony search algorithm. Figure 6b demonstrates a partial view of Figure 6a to make the Pareto fronts clear. I1 and I2 represent the first and the second stages of optimization; A, A1, B, C, and C1 are five representative Pareto-optimal solutions.

Table 6. Main design variables and results of representative solutions for beams optimization.

Variable and result		Initial scheme	Solution A1	Solution C1	Solution A	Solution C
Sectional dimension of beams (h, b)	L1 and L2	(600, 200)	(600, 200)	(600, 200)	(550, 200)	(550, 200)
	L3 and L4	(600, 200)	(600, 200)	(600, 200)	(600, 200)	(600, 200)
	L5	(500, 200)	(500, 200)	(500, 200)	(500, 200)	(500, 200)
Strength grade (MPa)	Concrete	30	30	30	30	30
	longitudinal bars	400	500	500	500	500
	stirrups	400	300	400	300	400
Quantity of longitudinal bar (t)		41.65	29.95	29.90	30.07	29.99
Quantity of stirrups (t)		6.56	6.06	5.42	5.95	5.32
Concrete consumption (m^3)		150.97	150.97	150.97	146.71	146.71
Emission reduction (tCO2e)		0	27.95	29.45	28.84	30.41
Cost reduction (10^3 USD)		0	0.81	0.70	0.86	0.75

Table

Abbreviations
BIM	building information modeling
BOQ	bill of quantities
HM	harmony memory
HMCR	Harmony memory considering rate
HMS	Harmony memory size
HS	harmony search
LCA	life cycle assessment
NI	maximum number of improvisations
PAR	pitch adjustment rate
USD	U.S. dollar
VI	constraint violation
Notations
Cemb	project cost
Cfees	other fees
Clabor	monetary costs from labor quantities
Cmac	monetary costs from machinery
Cmat	monetary costs from material
Cmea	measure expenses
Cr	total change in costs
Econs	emissions from the construction process Edpembodied emissions of a desired process
Eemb	embodied emissions
Eprod	emissions from the production process
Er	total change in emissions
Etran	emissions from the transportation process
EFdp	emission factor of a desired process
Qdp	quantity of the desired process
Parameters
As	amount of longitudinal reinforcement
As,max	upper limit amount of reinforcement
As,min	lower limit amount of reinforcement
CD	crowding distance
c	concrete cover depth
clim	lower limit of concrete cover depth
fc	concrete strength
fclim	lower limit of concrete strength
K	numbers of equality constraints
L	numbers of inequality constraints
la	anchorage length
lalim	limit of anchorage length
M	bending
N	axial force
Nv	number of variables
P	number of objectives
R	member resistance
RP	non-domination rank of Pareto front
S	load effect
sv	stirrup spacing
svlim	limit of stirrup spacing
T	torsion
V	shear
ρ	reinforcement ratio
ω	calculated crack width
ωmax	maximum crack width limited by the design code
Variables, vectors, and matrices
b	cross-sectional width
ds	diameter of steel bars
h	cross-sectional height
gc	strength grade of concrete
gs	strength grade of steel
ns	number of steel bars
ls	length of steel bars
xi	ith design variable
xij	ith design variable in jth solution vector
HM0	matrix of initial harmony memory
HMnew	new harmony memory
HMsum	combined harmony memory
X	solution vector containing all variables
X	jjth solution vector
Functions
fp(X)	pth objective function
gl(X)	lth inequality constraint
hk(X)	kth equality constraint

Data availability statement

Essential research data have been included in the published article. Further information is also available from the corresponding author on reasonable request.