Cálculos aproximados. Se considera:
el volumen del tubo lleno, compactado y fraguado lo ocupan la tierra y el estabilizante; el agua no se considera aditiva para el volumen porque ocupa los espacios entre las partículas de los materiales;
el volumen del tubo se calcula con la anchura del mismo vacío para contar con margen de sobra de los materiales. Volumen total de superadobe Ecuaciones Sección del saco Error al representar (error de sintaxis): {\displaystyle \color{Green}{L_v} \text{— anchura del saco vacío} \\ \color{Green}{h} — \text{altura del saco lleno y compactado} }
A
s
a
c
o
=
L
v
×
h
{\displaystyle {A_{saco}=L_{v}\times h}}
A
s
a
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o
=
(
L
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−
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h
2
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×
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+
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2
)
2
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h
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h
2
4
{\displaystyle A_{saco}=\left(L_{v}-{\dfrac {\pi h}{2}}\right)\times h+\pi \left({\dfrac {h}{2}}\right)^{2}=L_{v}h-{\dfrac {\pi h^{2}}{4}}}
Error al representar (error de sintaxis): {\displaystyle \sin \alpha = \dfrac{\dfrac{h}{2}}{h} = \dfrac{1}{2}; \alpha = 30^\circ \\ A_{saco} = \left(L_v - \dfrac{\pi h}{3} \right) \times h + 2 \left(\pi \left(\dfrac{h}{2}\right)^2 \times \dfrac{2\alpha}{360} - \dfrac{1}{2}bh \right) }
Para el cálculo de todos los volúmenes, se considera la sección del saco como el producto
s
w
×
s
h
{\displaystyle \color {Green}{s_{w}}\times \color {Green}{s_{h}}}
Error al representar (error de sintaxis): {\displaystyle \color{Green}{s_w} \text{ — anchura del saco lleno y compactado} \\ \color{Green}{s_h} \text{ — altura del saco lleno y compactado}}
Error al representar (SVG (MathML puede ser habilitado mediante un plugin de navegador): respuesta no válida («Math extension cannot connect to Restbase.») del servidor «https://es.wikipedia.org/api/rest_v1/»:): {\displaystyle r_C = {\color{Green}{r} + {3 \over 2} \color{Green}{s_w}} \text{ — para el volumen de C} \\ r_{D,E} = {\color{Green}{r} + {1 \over 2} \color{Green}{s_w}} \text{ — para los volúmenes de D y E} }
r
n
{\displaystyle r_{n}}
l
{\displaystyle l}
h
n
{\displaystyle h_{n}}
Error al representar (error de sintaxis): {\displaystyle n − {1 \over 2}}
veces la altura del saco lleno
s
h
{\displaystyle s_{h}}
l
2
=
h
n
2
+
l
n
2
{\displaystyle \color {Green}{l}^{2}=h_{n}^{2}+l_{n}^{2}}
Error al representar (error de sintaxis): {\displaystyle \color{Green}{l}^2 = h_n^2 + l_n^2 \\ l_n = \color{Green}{l} - \color{Green}{r} + r_n \\ h_n = \left(n - {1 \over 2} \right)\color{Green}{s_h} }
Error al representar (error de sintaxis): {\displaystyle \color{Green}{l}^2 = \left[\left(n - {1 \over 2} \right)\color{Green}{s_h} \right]^2 + \left(\color{Green}{l} -\color{Green}{r} + r_n \right)^2 \\ \color{Green}{l} -\color{Green}{r} + r_n = \sqrt{\color{Green}{l}^2 - \left[\left(n - {1 \over 2} \right)\color{Green}{s_h} \right]^2} \\ r_n = \color{Green}{r} - \color{Green}{l} + \sqrt{\color{Green}{l}^2 - \left[\left(n - {1 \over 2} \right)\color{Green}{s_h} \right]^2} }
r
n
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=
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s
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{\displaystyle r_{n(A)}=\color {Green}{r}-\color {Green}{l}+{1 \over 2}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{1 \over 2}\right)\color {Green}{s_{h}}\right]^{2}}}}
r
n
(
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)
=
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3
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s
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+
l
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−
[
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{\displaystyle r_{n(B)}=\color {Green}{r}-\color {Green}{l}+{3 \over 2}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{1 \over 2}\right)\color {Green}{s_{h}}\right]^{2}}}}
V
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∑
x
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E
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=
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+
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+
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E
{\displaystyle V=\sum _{x=A}^{E}V_{x}=V_{A}+V_{B}+V_{C}+V_{C}+V_{E}}
V
A
{\displaystyle V_{A}}
N
— número de hiladas por encima de la línea de surgencia
{\displaystyle N{\text{ — número de hiladas por encima de la línea de surgencia}}}
Error al representar (error de sintaxis): {\displaystyle \color{Green}{l}^2 = h^2+(\color{Green}{l}-\color{Green}{r})^2 \\ h=\sqrt{\color{Green}{l}^2 - \left(\color{Green}{l} - \color{Green}{r} \right)^2} \\ N = \dfrac{h}{\color{Green}{s_h}} \\ N = \dfrac{ \sqrt{ \color{Green}{l}^2 - \left(\color{Green}{l} - \color{Green}{r} \right)^2 } }{\color{Green}{s_h}} }
V
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{\displaystyle {\begin{aligned}V_{A}(n)&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi r_{n(A)}\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right)\end{aligned}}}
V
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=
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{\displaystyle {\begin{aligned}V_{A}&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \sum _{n=1}^{N}\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right)\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[N\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\end{aligned}}}
V
B
{\displaystyle V_{B}}
Error al representar (error de sintaxis): {\displaystyle \color{Green}{h_c} \text{ — altura del contrafuerte por encima de la línea de surgencia (m)} \\ C \text{ — número de hiladas del contrafuerte por encima de la línea de surgencia} \\ C = \dfrac{\color{Green}{h_c}}{\color{Green}{s_h}} }
V
B
(
n
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s
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{\displaystyle {\begin{aligned}V_{B}(n)&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi r_{n(B)}\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}-\color {Green}{l}+{\frac {3}{2}}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right)\end{aligned}}}
V
B
=
s
w
s
h
2
π
∑
n
=
1
C
(
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{\displaystyle {\begin{aligned}V_{B}&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \sum _{n=1}^{C}\left(\color {Green}{r}-\color {Green}{l}+{\frac {3}{2}}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right)\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[C\left(\color {Green}{r}-\color {Green}{l}+{\frac {3}{2}}\color {Green}{s_{w}}\right)+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\end{aligned}}}
V
C
{\displaystyle V_{C}}
n
C
— número de hiladas hasta la línea de surgencia del contrafuerte
{\displaystyle \color {Green}{n_{C}}{\text{ — número de hiladas hasta la línea de surgencia del contrafuerte}}}
V
C
=
n
c
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{\displaystyle {\begin{aligned}V_{C}&=\color {Green}{n_{c}}(\color {Green}{s_{w}}\color {Green}{s_{h}})(2\pi r_{c})\\&=\color {Green}{n_{c}}\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}+{\frac {3}{2}}\color {Green}{s_{w}}\right)\end{aligned}}}
V
D
{\displaystyle V_{D}}
n
D
— número de hiladas hasta la línea de surgencia
{\displaystyle \color {Green}{n_{D}}{\text{ — número de hiladas hasta la línea de surgencia}}}
V
D
=
n
D
(
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)
(
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=
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{\displaystyle {\begin{aligned}V_{D}&=\color {Green}{n_{D}}(\color {Green}{s_{w}}\color {Green}{s_{h}})(2\pi r_{D})\\&=\color {Green}{n_{D}}\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)\end{aligned}}}
V
E
{\displaystyle V_{E}}
n
E
— número de hiladas en los cimientos
{\displaystyle \color {Green}{n_{E}}{\text{ — número de hiladas en los cimientos}}}
V
E
=
n
E
(
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)
(
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{\displaystyle {\begin{aligned}V_{E}&=\color {Green}{n_{E}}(\color {Green}{s_{w}}\color {Green}{s_{h}})(2\pi r_{E})\\&=\color {Green}{n_{E}}\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)\end{aligned}}}
Volumen total de superadobe
V
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n
D
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[
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=
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[
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{\displaystyle {\begin{aligned}V&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[N\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]+\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[C\left(\color {Green}{r}-\color {Green}{l}+{\frac {3}{2}}\color {Green}{s_{w}}\right)+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]+\color {Green}{n_{C}}\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}+{\frac {3}{2}}\color {Green}{s_{w}}\right)+\color {Green}{n_{D}}\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\color {Green}{n_{E}}\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[\left(N+C\right)\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+C\color {Green}{s_{w}}+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\left(\color {Green}{n_{C}}+\color {Green}{n_{E}}+\color {Green}{n_{D}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\color {Green}{n_{C}}\color {Green}{s_{w}}\right]\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[\left(N+C\right)\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\left(\color {Green}{n_{C}}+\color {Green}{n_{E}}+\color {Green}{n_{D}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\left(C+\color {Green}{n_{C}}\right)\color {Green}{s_{w}}+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[\left(N+C+\color {Green}{n_{C}}+\color {Green}{n_{D}}+\color {Green}{n_{E}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)-\color {Green}{l}\left(N+C\right)+\left(C+\color {Green}{n_{C}}\right)\color {Green}{s_{w}}+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[\left(N+C+\color {Green}{n_{C}}+\color {Green}{n_{D}}+\color {Green}{n_{E}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)-\color {Green}{l}\left(N+C\right)+\left(C+\color {Green}{n_{C}}\right)\color {Green}{s_{w}}+2\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\sum _{n=C+1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\end{aligned}}}
Mortero —tierra (arcilla más arenas y gravas) más estabilizante—
V
m
o
r
t
e
r
o
=
V
t
u
b
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=
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t
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e
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=
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t
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×
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×
V
t
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r
r
a
{\displaystyle V_{mortero}=V_{tubo}=V_{tierra}+V_{estabilizante}=V_{tierra}+p_{estabilizante}\times V_{tierra}=(1+p_{estabilizante})\times V_{tierra}}
V
t
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e
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a
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a
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e
{\displaystyle V_{tierra}={\dfrac {V_{tubo}}{1+p_{estabilizante}}}}
Tierra: arcilla más arenas y gravas
p
a
r
c
i
l
l
a
{\displaystyle p_{arcilla}}
p
a
r
e
n
a
s
+
g
r
a
v
a
s
{\displaystyle p_{arenas+gravas}}
V
t
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=
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a
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c
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×
V
t
i
e
r
r
a
{\displaystyle V_{tierra}=V_{arcilla}+V_{arenas+gravas}=p_{arcilla}\times V_{tierra}+p_{arenas+gravas}\times V_{tierra}}
V
t
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×
V
t
i
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r
r
a
{\displaystyle V_{tierra}=(p_{arcilla}+p_{arenas+gravas})\times V_{tierra}}
p
a
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a
=
1
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a
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n
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{\displaystyle p_{arcilla}=1-p_{arenas+gravas}}
Estabilizante
p
e
s
t
a
b
i
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a
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t
e
{\displaystyle p_{estabilizante}}
V
e
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{\displaystyle V_{estabilizante}=p_{estabilizante}\times V_{tierra}}
Agua El volumen de agua necesario se calcula en función del volumen de mortero, pero no se tiene en cuenta en la suma de volúmenes que intervienen en el resultado final:
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{\displaystyle p_{agua}}
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{\displaystyle V_{agua}=p_{agua}\times V_{mortero}=p_{agua}\times V_{tubo}}
![{\displaystyle r_{n(A)}=\color {Green}{r}-\color {Green}{l}+{1 \over 2}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{1 \over 2}\right)\color {Green}{s_{h}}\right]^{2}}}}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/49ae68cfb834b686089baf27472b4805d2a6c60d)
![{\displaystyle r_{n(B)}=\color {Green}{r}-\color {Green}{l}+{3 \over 2}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{1 \over 2}\right)\color {Green}{s_{h}}\right]^{2}}}}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/7e274f3f544683cf55588b51c633568df8ee6c52)
![{\displaystyle {\begin{aligned}V_{A}(n)&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi r_{n(A)}\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right)\end{aligned}}}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/7641a64d7fd3243007452599dac9adda6943a2e6)
![{\displaystyle {\begin{aligned}V_{A}&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \sum _{n=1}^{N}\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right)\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[N\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\end{aligned}}}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/d03902bba3ffc3098673b9640d7b2b6495b67501)
![{\displaystyle {\begin{aligned}V_{B}(n)&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi r_{n(B)}\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}-\color {Green}{l}+{\frac {3}{2}}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right)\end{aligned}}}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/cc1f3759d5e641858be1408650a7273dfc6b043c)
![{\displaystyle {\begin{aligned}V_{B}&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \sum _{n=1}^{C}\left(\color {Green}{r}-\color {Green}{l}+{\frac {3}{2}}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right)\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[C\left(\color {Green}{r}-\color {Green}{l}+{\frac {3}{2}}\color {Green}{s_{w}}\right)+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\end{aligned}}}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/19fddc0309a8dfc32668e5326466e7168e0e6712)
![{\displaystyle {\begin{aligned}V&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[N\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]+\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[C\left(\color {Green}{r}-\color {Green}{l}+{\frac {3}{2}}\color {Green}{s_{w}}\right)+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]+\color {Green}{n_{C}}\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}+{\frac {3}{2}}\color {Green}{s_{w}}\right)+\color {Green}{n_{D}}\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\color {Green}{n_{E}}\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[\left(N+C\right)\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+C\color {Green}{s_{w}}+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\left(\color {Green}{n_{C}}+\color {Green}{n_{E}}+\color {Green}{n_{D}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\color {Green}{n_{C}}\color {Green}{s_{w}}\right]\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[\left(N+C\right)\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\left(\color {Green}{n_{C}}+\color {Green}{n_{E}}+\color {Green}{n_{D}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\left(C+\color {Green}{n_{C}}\right)\color {Green}{s_{w}}+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[\left(N+C+\color {Green}{n_{C}}+\color {Green}{n_{D}}+\color {Green}{n_{E}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)-\color {Green}{l}\left(N+C\right)+\left(C+\color {Green}{n_{C}}\right)\color {Green}{s_{w}}+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\\&=\color {Green}{s_{w}}\color {Green}{s_{h}}2\pi \left[\left(N+C+\color {Green}{n_{C}}+\color {Green}{n_{D}}+\color {Green}{n_{E}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)-\color {Green}{l}\left(N+C\right)+\left(C+\color {Green}{n_{C}}\right)\color {Green}{s_{w}}+2\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\sum _{n=C+1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\end{aligned}}}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/3ea36395bebe9b872f6efd60785859be6a041301)
