Rev. Téc. Ing. Univ. Zulia. Vol. 44, Nº 2, May-August, 2021, 104-116
Rev. Téc. Ing. Univ. Zulia. Vol. 44, No. 2, May-August, 2021.
Technical-Economic Analysis of Industrial Buildings Using
Non-Linear Lagrange Interpolation
Diego Fernando Loachamin-Chano , Andrea Elizabeth Freire-Luna , David
Patricio Guerrero-Cuasapaz
*
, Milton Bolívar Guerrón-Figueroa
Department of Civil Engineering, Salesian Polytechnic University. Quito, Ecuador.
* Corresponding author: dguerrero@ups.edu.ec
https://doi.org/10.22209/rt.v44n2a05
Received: 08 September 2020 | Accepted: 12 Febrary 2021 | Available: 15 April 2021
Abstract
Due to the notable increase of industrial buildings in Ecuador, it is necessary to carry out a technical-
economic analysis to know the impact of the cost of materials on these structures. To do this, a sample of 40
buildings was prepared, establishing the relationship between clear span (L), frame height (H) and distance between
frames (B), in order to determine a metric. For this study, the weight of the structure was established in kg/m². A
mathematical model was applied using the results (Lagrange interpolation), a polynomial was obtained that describes
the model of each group of buildings and its graphic representation, thus determining an approximation of the weight
of the structures. The metrics that were found have a direct impact on the cost of the materials of the structures, these
were obtained by multiplying them by the cost of the kg of steel. A verification was carried out with a sample of 6
buildings, in which we found the weight using the load and resistance factor design and later with a polynomial;
obtaining an error: for buildings modulated at 5 m a mean of 0.03 % with a deviation of 0.03 and for buildings
modulated at 6 m, a mean of 0.07 % with a deviation of 0.03. Keywords: 1.25Cr0.5Mo steel; posweld heat treatment;
creep resistance.
Keywords: abacus; interpolation; Lagrange; industrial buildings; weights.
Análisis Técnico-Económico de Naves Industriales Mediante
Interpolación No Lineal de Lagrange
Resumen
Debido al notable incremento de construcciones de naves industriales en Ecuador, es necesario realizar un
análisis técnico-económico para conocer la incidencia del costo de materiales sobre ellas. Para ello, se elaboró una
muestra de 40 naves, fijando relación entre luz libre (L), altura del pórtico (H) y distancia entre pórticos (B), con el
propósito de determinar una métrica. Para este estudio se estableció el peso de la estructura en kg/m². Con los
resultados se aplicó un modelo matemático (interpolación de Lagrange), se obtuvo un polinomio que describe el
modelo de cada grupo de naves y su representación gráfica, por consiguiente, se determinó una aproximación del
peso de estructuras. Las métricas que se hallaron tienen una repercusión directa en el costo de los materiales de las
estructuras, estas se obtuvieron al multiplicarlas por el costo del kg de acero. Se realizó una comprobación con una
muestra de 6 naves, en las se halló el peso utilizando el diseño por factores de carga y resistencia y luego con el
polinomio; obteniéndose un error: para naves moduladas a 5 m una media de 0,03 % con una desviación de 0,03 y
para naves moduladas a 6 m, una media de 0,07 % con una desviación de 0,03.
Palabras clave: ábacos; interpolación; Lagrange; naves industriales; pesos.
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Technical-economic analysis of industrial buildings
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Introduction
Industrial buildings are those that allow not only the storage of supplies, but are also used as production
centers, car washes, offices, shopping centers, industries, etc. (Hernández, 2015), which enable the satisfaction of
humans in various spheres of society. An industrial building is a unit for industrial use that houses the production
and/or storage of industrial goods, together with the workers, the machines that generate them, the internal
transportation, the exit and entry of goods, among others (Arnal et al., 2014).
The design of industrial buildings in Ecuador is based on the Ecuadorian Construction Standard (NEC,
2014), which is based on regulations such as: ASCE (2017), AISC (2016) and AISI (2016), which are dedicated to
studying the characteristics and forms of behavior of steel and the development of standards for the structural
calculation that governs steel design, thus developing the limit state method and the load and resistance factor design
(LRFD) (NEC, 2014; AISC, 2016; AISI 2016; ASCE, 2017; Cano y Imanpour, 2020).
In civil engineering, and more specifically in the construction field, it is essential to have a well-structured
budget for the correct implementation of a structure. We know that it is not always possible to have work quantities
without first carrying out a seismic resistant analysis and design of a building, such is the case of industrial buildings
(Guerrero, 2019). Consequently, this research proposed to carry out a structural analysis and design of a group of
industrial buildings that have particular characteristics in terms of length, height and spacing between frames. Based
on this analysis, the weights over m² of each building were obtained as a reference, which in turn were used to apply
a non-linear Lagrange interpolation, which led to the determination of an equation with its respective graphic
representation (abaci). With these results, it was possible to determine in a fairly approximate way the weight per m²
of industrial buildings different from those that were taken at the beginning of the research, so that later, using these
results, we would be able to get the cost of the material for each structure.
Experimental
The main objective of this research was to carry out a technical analysis of a group of industrial buildings,
in order to find the costs of steel material using a non-linear Lagrange interpolation, which is represented through
abaci.
Pre-design of the structural elements of industrial buildings
One of the considerations that were used in this research was a pre-design of the structural elements that
make up industrial buildings, with the aim of starting with data that will be approximate to the expected results in the
behavior of a frame under gravitational stresses, which must meet the resistance requirements. Generally,
reinforcements are placed in the areas with greater stresses to ensure the stability and safety of the structure. (Élez,
2016). In this specific way, the research had the following parameters as its starting point: column free height (H),
clear span of the building (L), modulation between frames (B1 or B2), defined in Table 1 and illustrated in Figure 1.
With this, the geometry of the structure was defined, obtaining 40 combinations of industrial buildings, which were
analyzed and modeled.
Table 1. Variants and combinations of free column height, frame width and length of the industrial building.
Free column height (Hi)
(m)
Clear
span (m)
Modulation (m)
H1
H2
H3
H4
L
B1
B2
6
7
8
9
15
5
6
6
7
8
9
20
5
6
6
7
8
9
25
5
6
6
7
8
9
30
5
6
6
7
8
9
35
5
6
Loachamin-Chano et al . 106
Rev. Téc. Ing. Univ. Zulia. Vol. 44, No. 2, May-August, 2021.
Figure 1. Geometry and variants of a metallic industrial building.
The industrial buildings formed by the structural profiling in accordance with ASTM A 1011 (2018) are: C-
type channels, cold-formed G-type purlins, structural profiling (ASTM A36, 2018; A36M-08, 2018) such as: hot-
formed L-type angles (ASTM A1011, 2018; A1011M-18a, 2018). This article, regarding the structural design, was
carried out with the load and resistance factor design method called LRFD, which considers the resistance or failure
condition, where the applied service loads are factored and the theoretical resistance of the material is reduced. The
criterion that the factored load must satisfy must be less than or equal to the reduced resistance (Garibov y
Bashirzade, 2020). This approach is based on two fundamental concepts: the limit state of resistance and the limit
state of service. The first focuses on the safety or load-bearing capacity of structures and includes plastic,
overturning, fatigue, fracture, and buckling resistances; while service limit states refer to the behavior of structures
under normal service loads and have to do with aspects associated with use and occupation, such as excessive
deflections, slips, vibrations and cracking (McCormac, 2013). The LRFD method (Garibov and Bashirzade, 2020;
Crisafulli, 2018), recommends that service loads (Qi) are multiplied by certain load or safety factors (λi), which are
almost always greater than 1.0; to obtain the “factored loads” used for the design of the structure. The magnitudes of
the load factors vary, depending on the type of load combinations, as can be seen in Equations 1 to 6 (NEC, 2014):
(1)
(2)
(3)
(4)
(5)
(6)
Where: U
i
: load combination, D: dead load, L: live load, L
r
: roof live load, S: snow load, R
r
: rain load, except flooding, W:
wind load.
The structure is sized to have a sufficient ultimate design resistance able to resist the factored loads. This
resistance is considered equal to the nominal theoretical resistance (Rn) of the structure member, multiplied by a
resistance factor Ø which is normally less than 1.0 (Hernández, 2015), according to:
(7)
The design resistance of each structural component is greater than or equal to the required determined
resistance, according to the LRFD load combinations, according to:
(8)
Where: R
u
: resistance required by the LRFD method, R
n
: nominal resistance given by the material, Ø: resistance factor, ØR
n
:
design resistance.
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In this research, the analysis and design of the structural elements was carried out using computer packages,
which were subjected to load combinations for each group of industrial buildings modulated at every 5 and 6 m, each
of them with a column height that varied between 6 and 9 m and a clear span of 15 to 35 m, based on the Ecuadorian
construction standard (NEC, 2014).
The parameters that intervene in the calculation of the seismic loading percentage, which will constitute the
basal shear applied to each industrial building, are tabulated in Table 2.
Table 2. Parameters that intervene in the calculation of the basal shear of industrial buildings.
Parameter
Variable
Value
Seismic zone
V
-
Zone acceleration factor
Z
0.4
Soil type
D
-
F
a
site factor
F
a
1.20
F
d
site factor
F
d
1.19
Inelastic behavior factor of the soil
F
s
1.28
Factor associated with the geographic location of the project
r
1
Spectral amplification ratio
ƞ
2.48
Acceleration at T = T
o
S
a
[g]
1.19
Limit period at T = T
o
T
o
[s]
0.127
Limit period at T = T
c
T
c
[s]
0.698
Limit period at T = T
L
T
L
[s]
2.856
C
t
Coefficient
C
t
0.073
Period calculation Coefficient
α
0.75
Total element height
h
n
[m]
-
Theoretical period method 1
T
1
[s]
0.441
Theoretical period method 2
T
2
[s]
0.57
Seismic force reduction factor
R
3.00
The live load (CV) had a value equal to the hail load which, according to the NEC SE CG (2014) standard,
by default, has a specific hail weight value equal to 1 T/m³ on lower slopes at 15% and a minimum load of 0.5
kN/m². In this case, we considered the accumulation of hail in a short time and a hail layer thickness of 10 cm in
height and a specific weight of 0.75 T/m is assumed.
3
; obtaining a load of 75 kg / m
2
, that used in all models: The
dead load (CM) we used was 5.0 kg/m
2
for the facilities and 4.0 kg / m
2
for the roof, having a constant value for the
entire set of industrial buildings of CM = 9 kg/m
2
and e CV = 75 kg/m
2
, and are represented in Figure 2 (Redroban,
2015).
Figure 2. Live load and dead load of industrial buildings.
The graphical representation in Figures 3 and 4 of the wind load, corresponding to windward W1, W2 and
leeward W3 and W4, with modulations between frames at every 5 and 6 m, respectively (Hernández, 2015).
Loachamin-Chano et al . 108
Rev. Téc. Ing. Univ. Zulia. Vol. 44, No. 2, May-August, 2021.
Figure 3. Wind load of industrial buildings. W1: windward column, W2: windward beam, W3: leeward beam, W4:
leeward column.
Figure 4. Summary of wind loads (W1, W2, W3 and W4) for industrial buildings with modulation between frames B
= 5 and 6 m (windward).
Design of the structural elements of industrial buildings
For these designs, we verified that they met the controls stipulated both in the ASCE (2017), SEI 7-16
(2017) and the NEC (2014) standards. The optimization of the structures is the next step to continue with the
objective of guaranteeing the service and resistance conditions. For this, we used the computational tool for the
modeling and analysis of the industrial buildings, to determine the total weight of all the sections that make up the
industrial shed structure, in a systematic way (ASCE, 2017). In this way, the forces and other maximum
requirements needed were obtained, both for columns and beams, in order to carry out a design of the elements
subject to compression, verifying that the limit states of local buckling and global buckling were met. Similarly, a
design was made for members subject to bending, which were analyzed for the effects of creep and torsional lateral
buckling, according to the steel construction manual (SAP 2000, 2016; AISC, 2016).
Once the analysis was carried out, the structural elements were designed, where the demand-capacity ratio
was verified, which involves the compression capacity with the ability to resist bending moments, and should not be
greater than 1.0 (Cuichan, 2016).
In this research, most of the elements were found between 0 and 95% of their capacity, that is, they were
working in a very acceptable way, since in metallic structures it should be ensured that they work at their maximum
140
190
240
290
340
390
15 17 19 21 23 25 27 29 31 33 35
Wind load W1 W2 W3 W4 (kg/m)
Clear span (m)
W1(6)(5) W2(6)(5) W3(6)(5) W4(6)(5) W1(7)(5) W2(7)(5)
W3(7)(5) W4(7)(5) W1(8)(5) W2(8)(5) W3(8)(5) W4(8)(5)
W1(9)(5) W2(9)(5) W3(9)(5) W4(9)(5) W1(6)(6) W2(6)(6)
W
3
W
4
W
2
W
1
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Rev. Téc. Ing. Univ. Zulia. Vol. 44, No. 2, May-August, 2021.
capacity, leveraging on their large advantages (Acero, 2011). In order to check the stresses of the structural elements
of the set of industrial buildings, a manual check was carried out with the computer package, choosing the elements
with the highest demand/capacity ratio, both in columns and in beams, and these are tabulated in Tables 3 and 4.
Table 3. Demand/capacity ratio in columns and beams with modulation between frames
H = 9 m, L = 25 m and B = 5 m.
Column
Beam
Members
Profile (mm)
D/C
S
D/C
M
Members
Profile (mm)
D/C
S
D/C
M
PE
C150x5x4
0.849
0.838
PE
C150x5x3
0.647
0.639
PI
C150x5x4
0.775
0.770
PI
C150x5x3
0.702
0.693
CD
L40x4x4
0.906
0.893
CD
L40x4x5
0.796
0.786
CH
L40x4x4
0.782
0.779
CH
L40x4x4
0.090
0.120
PE: outer profile, PI: inner profile, CD: diagonal lattice, CH: horizontal lattice, D/C
s
: demand index for the capacity
obtained by software, D/C
M
: demand index for the capacity obtained manually.
Table 4. Demand/capacity ratio in columns and beams with modulation between frames
H = 9 m, L = 25 m and B = 6 m.
Column
Beam
Members
Profile (mm)
D/C
S
D/C
M
Members
Profile (mm)
D/C
S
D/C
M
PE
C150x5x4
0.914
0.908
CE
C150x5x3
0.898
0.885
PI
C150x5x4
0.825
0.815
CI
C150x5x3
0.817
0.804
CD
L40x4x4
0.522
0.547
CD
L40x4x3
0.866
0.859
CH
L40x4x4
0.862
0.856
CH
L40x4x3
0.506
0.493
PE: outer profile, PI: inner profile, CD: diagonal lattice, CH: horizontal lattice, D/C
s
: demand index for the capacity
obtained by software, D/C
M
: demand index for capacity obtained manually.
Polynomial interpolation
The Lagrange high order interpolation method (Pacheco et al., 2012). The numerical analysis that defines the
Lagrange interpolation polynomial, corresponding to the n+1 given values, as the degree polynomial function at the
most n that takes different numerical values on the n+l {x
o
, x
1
, ..., x
n
.}, the n+l given values {y
o
, y
1
, ..., y
n
}. An
interpolating polynomial is obtained as a formal expression f (x) = a
o
+ a
1
x + a
2
x
2
+ ... +a
o
.x
n
(Sandoval, 2019).
By assuming known values at n+1 points x
0
, x
1
,…, x
n
(dependent values) and f
0
, f
1
, ..., f
n
(independent values) of a
function; a P
n
(x) polynomial was constructed with a degree less than or equal to n, as shown in Equation 9,
according to (Sandoval, 2019).
(9)
The P
n
(x) polynomial thus constructed is called an interpolating polynomial or an interpolation polynomial. One
possible way to solve the problem was to propose the following degree polynomial, indicated in Equation 10:
(10)
With undetermined coefficients a
i
, i = 0, 1,…, n , and impose that Equation 10 verifies Equation 9. This means that
obtaining the interpolating polynomial is equivalent to solving the system of linear equations given in Equation 11:
(11)
Loachamin-Chano et al . 110
Rev. Téc. Ing. Univ. Zulia. Vol. 44, No. 2, May-August, 2021.
Lagrange polynomial
In this research we used the Lagrange interpolation polynomial, since it allows us to generate a function that passes
through all the points, thus facilitating modeling and obtaining fairly reliable approximations. Given the set of points
(x
0
.f
0
).(x
1
.f
1
)… .(x
n
.f
n
), we consider the polynomial, using Equations 12 to 14 (Suárez, 2012).
(12)
Where L
i
(x):
(13)
The second-degree Lagrange polynomial is:
(14)
In this case f
n
(x) is the "y" interpolated and the "x" is the x to interpolate. The more data in the table, the
higher degree polynomial can be used. In this section, the polynomial obtained was of the fourth degree. For this
research, the f (x
i
) value corresponds to the weights of each industrial building modulated every 5 and 6 m, while the
Li (x) value is the Lagrange polynomial, which depends on the clear span of the frame, which varies from 15 to 35
m.
Results and Discussion
The cost of the materials of an industrial building when preparing a referential budget for a specific project
is one of the important factors that influences the feasibility and award to develop the project for the company that
budgets an offer (Guerrero, 2019). The determination of this parameter was obtained in a direct way, that is, the
weight of an industrial building in kg/m² indicated in Figures 5 and 6, multiplied by its area. In this research the area
ranged between 225 and 525 m² for frames modulated every 5 m, and from 270 to 630 m² for frames modulated
every 6 m. With these values, the weight of material that is related to the properties of the elements that was
considered in the design was obtained.
Based on the requirements and complying with the ASCE (2017), SEI 7-16 (2017), NEC SE AC (2014) and
NEC SE DS (2014) standards, this section presents the results of kg/m² achieved by industrial buildings as shown in
Table 5. Likewise, the numerical analysis is indicated by the Lagrange polynomial, with which the equations that
govern each group of buildings with modulations at 5 and 6 m were determined.
Interpolation result using the Lagrange polynomial
Once the weights in Table 5 were known and applying the programming language, the equations were
formed based on the Lagrange polynomial for the 5 and 6 m modulations of industrial buildings, with spans that vary
every 5 m and ranging from 15 up to 35 m respectively, as indicated in Figures 5 and 6.
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Table 5. Total weights per square meter obtained for buildings with modulations at 5 and 6 m.
Clear
span
(m)
Column
height
(m)
Weight
(kg/m
2
)
modulated at
5 m
Weight
(kg/m
2
)
modulated at
6 m
Clear
span
(m)
Column
height
(m)
Weight
(kg/m
2
)
modulated
at 5 m
Weight
(kg/m
2
)
modulated at
6 m
15
6
17.42
18.62
15
8
21.01
21.63
20
6
18.12
19.26
20
8
21.64
22.11
25
6
19.41
20.18
25
8
22.48
22.96
30
6
20.84
21.64
30
8
23.79
24.37
35
6
22.65
23.09
35
8
25.49
25.92
15
7
19.67
20.38
15
9
22.60
23.24
20
7
20.37
20.99
20
9
22.94
23.74
25
7
21.39
21.82
25
9
23.70
24.37
30
7
22.62
23.30
30
9
24.84
25.49
35
7
24.35
25.03
35
9
26.21
26.96
Figure 5. Interpolation curves (abacus) for industrial sheds with 5 m modulations.
17
18
19
20
21
22
23
24
25
26
27
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Weight in kg/m² in industrial buildings
Clear span (m)
Industrial sheds H=6 m
Industrial sheds H=7 m
Industrial sheds H=8 m
Industrial sheds H=9 m
Loachamin-Chano et al . 112
Rev. Téc. Ing. Univ. Zulia. Vol. 44, No. 2, May-August, 2021.
Figure 6. Interpolation curves (abacus) for industrial sheds with 6 m modulations.
Figures 7 and 8 show the reference costs for each group of industrial buildings, according to 5 and 6 m
modulations, respectively. The reference price for hot rolled steel coils reached US $99 per metric ton in 2019.
According to estimates for the year 2022, this figure would decrease slightly to around 463 US dollars (Díaz, 2021;
Investing, 2021).
Figure 7. Benchmark cost graph of industrial buildings modulated at 5 m (Investing, 2021).
17
18
19
20
21
22
23
24
25
26
27
28
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Weight in kg/m² in industrial buildings
Clear span (m)
Industrial sheds H=6 m
Industrial sheds H=7 m
Industrial sheds H=8 m
Industrial sheds H=9 m
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Figure 8. Benchmark cost graph of industrial buildings modulated at 6 m (Investing, 2021).
Determination of the normalized percent error
In order to establish the real value, it is sometimes difficult to have the true value, for such cases, one option
is to normalize the error, using the best possible approximation of the true value. For this study, the normalized
percent error is calculated with the expression given in Equation 15:
(15)
In order to validate the results obtained in the abaci of Figures 5 and 6; regarding the computational tool,
when obtaining the weight in kg/m² of the industrial building, we assumed that the true value is the one obtained
from the program and the approximation value is the value that was determined by the Lagrange polynomial. For
verification purposes, 6 additional industrial buildings were evaluated with intermediate values that do not exist in
Table 5, consequently, the results presented in Tables 6 and 7 were obtained.
Table 6. Total weights per square meter for intermediate values obtained for industrial sheds with a modulation of B
= 5 m.
B
H
L
Weight
with
software
(kg/m²)
Weight
with
equation
(kg/m²)
(%)
Error
Weight
Cost
(m)
(m)
(m)
(kg)
($)
5
6
28.84
20.49
20.50
0.05
8868.30
11227.27
5
7
24.64
21.31
21.31
0.00
7876.18
9971.24
5
9
25.58
23.80
23.81
0.04
9135.90
11566.05
Mean
0.03
Deviation
0.03
L: clear span, H: frame height, B: distance between frames.
Loachamin-Chano et al . 114
Rev. Téc. Ing. Univ. Zulia. Vol. 44, No. 2, May-August, 2021.
Table 7. Total weights per square meter for intermediate values obtained for industrial sheds with a modulation of B
= 6 m.
B
H
L
Weight
with
software
(kg/m²)
Weight
with
equation
(kg/m²)
(%)
Error
Weight
Cost
(m)
(m)
(m)
(kg)
($)
6
6
33.42
22.71
22.69
0.09
13649.40
17280.14
6
7
26.16
22.08
22.10
0.09
10406.45
13174.56
6
8
21.83
22.33
22.34
0.04
8778.28
11113.30
Mean
0.07
Deviation
0.03
L: clear span, H: frame height, B: distance between frames.
Determination of the proportionality of the H, L and B parameters Vs. Weight
In order to establish the level of proportionality that exists between the aforementioned parameters (H, L and B) with respect
to the weight per square meter of the group of buildings, a radial graph was obtained, which shows that there is a
proportionality from L = 15 m to L = 35 m, which is evidenced in Figure 9.
Figure 9. Radial graph of column free height Vs. weight per square meter.
Conclusions
When carrying out a structural analysis and design using a computer package for a set of industrial buildings
with different parameters for length, height and spacing of frames, we obtained the values corresponding to the index
between the weight and area of the structures, and when comparing them with the values obtained from the Lagrange
polynomial, a fairly low mean error was found. Therefore, we can conclude that the interpolation indicated in this
research, with the parameters described above, can simply be used to directly find the cost of the steel material in
industrial buildings.
Regarding the cost of the industrial buildings, the corresponding value of the cost of steel should be taken
into account, depending only on the area in which the project will be implemented; consequently, the preparation of
a budget will be much more agile than carrying out a previous structural analysis and later find the costs.
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Technical-economic analysis of industrial buildings
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Acknowledgments
The authors thank the Salesian Polytechnic University (Quito-Ecuador), in particular the civil engineering
department.
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