Volumen total de superadobe Al calcular el volumen de superadobe se obtienen las longitudes necesarias de saco y alambre, así como el volumen de material de drenaje.
Para el cálculo del volumen de superadobe, se calcula la longitud de cada hilada —longitud de la circunferencia que describe el tubo— en el punto medio del saco —por similitud con el cálculo del volumen de un toro— y se multiplica por la sección del saco lleno y compactado. En cada hilada, la longitud de la circunferencia es función del radio del domo a la altura del saco. Por debajo de la línea de surgencia el radio es constante.
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
A
(
n
)
=
s
w
s
h
2
π
r
n
(
A
)
=
s
w
s
h
2
π
(
r
−
l
+
1
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
)
{\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
A
=
s
w
s
h
2
π
∑
n
=
1
N
(
r
−
l
+
1
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
)
=
s
w
s
h
2
π
[
N
(
r
−
l
+
1
2
s
w
)
+
∑
n
=
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
{\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}}}
Ecuaciones Sección del saco Error al representar (error de sintaxis): {\displaystyle \color{Green}{L_v} \text{— anchura del saco vacío} \\ \color{Green}{s_h} — \text{altura del saco lleno y compactado} \\ s_w — \text{anchura del saco lleno y compactado} }
Sección rectangular Sección rectangular Error al representar (error de sintaxis): {\displaystyle s_w = \color{Green}{L_v} - \color{Green}{s_h} \\ A_{saco} = s_w \times \color{Green}{s_h} = \left(\color{Green}{L_v} - \color{Green}{s_h} \right) \times \color{Green}{s_h}}
Sección con laterales semicurculares Sección con laterales semicirculares Error al representar (error de sintaxis): {\displaystyle s_w = \color{Green}{L_v} - \dfrac{\pi \color{Green}{s_h}}{2} + \color{Green}{s_h} = \color{Green}{L_v} + \color{Green}{s_h}\left(1 - \dfrac{\pi}{2}\right) \\ A_{saco} = \left(\color{Green}{L_v} - \dfrac{\pi \color{Green}{s_h}}{2} \right) \times \color{Green}{s_h} + \pi \left(\dfrac{\color{Green}{s_h}}{2}\right)^2 = \color{Green}{L_v} \color{Green}{s_h} - \dfrac{\pi \color{Green}{s_h}^2}{4} }
Sección con laterales como segmentos circulares Sección con laterales como segmentos circulares 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 s_w = \color{Green}{L_v} - \dfrac{\pi \color{Green}{s_h}}{3} + 2 \left(\color{Green}{s_h} - \sqrt{\color{Green}{s_h}^2 - \left(\dfrac{\color{Green}{s_h}}{2} \right)^2} \right) = \color{Green}{L_v} - \dfrac{\pi \color{Green}{s_h}}{3} + 2 \left(\color{Green}{s_h} - \color{Green}{s_h} \dfrac{\sqrt{3}}{2} \right) = \color{Green}{L_v} + \color{Green}{s_h} \left( 2 - \dfrac{\sqrt{3}}{2} - \dfrac{\pi}{3} \right) = \color{Green}{L_v} + \dfrac{\color{Green}{s_h}}{3} \left( 6 - 3\sqrt{3} - {\pi} \right) \\ \sin \alpha = \dfrac{\dfrac{\color{Green}{s_h}}{2}}{\color{Green}{s_h}} = \dfrac{1}{2}; \alpha = 30^\circ \\ \begin{align} A_{saco} & = \left(\color{Green}{L_v} - \dfrac{\pi \color{Green}{s_h}}{3} \right) \times \color{Green}{s_h} + 2 \left(\pi \color{Green}{s_h}^2 \times \dfrac{2\alpha}{360} - \dfrac{1}{2} \times \color{Green}{s_h} \sqrt{\color{Green}{s_h}^2 - \left(\dfrac{\color{Green}{s_h}}{2} \right)^2} \right) \\ & = \color{Green}{L_v} \color{Green}{s_h} - \dfrac{\pi \color{Green}{s_h}^2}{3} + 2 \left(\pi \color{Green}{s_h}^2 \times \dfrac{1}{6} - \dfrac{1}{2} \times \dfrac{\color{Green}{s_h}^2 \sqrt{3}}{2} \right) \\ & = \color{Green}{L_v} \color{Green}{s_h} - \dfrac{\pi \color{Green}{s_h}^2}{3} + \dfrac{\pi \color{Green}{s_h}^2}{3} - \dfrac{\color{Green}{s_h}^2 \sqrt{3}}{2} \\ & = \color{Green}{L_v} \color{Green}{s_h} - \color{Green}{s_h}^2 \dfrac{\sqrt{3}}{2} \end{align} }
Sección para cálculos Usar una sección u otra conlleva variaciones en los cálculos, ya que el área de cada una es distinta:
Altura del saco lleno y compactado
Sección rectangular
Sección con semicírculos
Sección con segmentos circulares
in
2,00
3,00
5,00
2,00
3,00
5,00
2,00
3,00
5,00
cm
5,08
7,62
12,70
5,08
7,62
12,70
5,08
7,62
12,70
Anchura del saco vacío (cm)
30
126,59
170,54
219,71
132,13
183,00
254,32
130,05
178,31
241,32
35
151,99
208,64
283,21
157,53
221,10
317,82
155,45
216,41
304,82
40
177,39
246,74
346,71
182,93
259,20
381,32
180,85
254,51
368,32
45
202,79
284,84
410,21
208,33
297,30
444,82
206,25
292,61
431,82
50
228,19
322,94
473,71
233,73
335,40
508,32
231,65
330,71
495,32
55
253,59
361,04
537,21
259,13
373,50
571,82
257,05
368,81
558,82
60
278,99
399,14
600,71
284,53
411,60
635,32
282,45
406,91
622,32
Valores de sección El área de la sección del saco,
A
s
a
c
o
{\displaystyle A_{saco}}
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} s_w} \text{ — para el volumen de C} \\ r_{D,E} = {\color{Green}{r} + {1 \over 2} 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
(
A
)
=
r
−
l
+
1
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
{\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
(
B
)
=
r
−
l
+
3
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
{\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
=
∑
x
=
A
E
V
x
=
V
A
+
V
B
+
V
C
+
V
C
+
V
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
A
(
n
)
=
s
w
s
h
2
π
r
n
(
A
)
=
s
w
s
h
2
π
(
r
−
l
+
1
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
)
{\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
A
=
s
w
s
h
2
π
∑
n
=
1
N
(
r
−
l
+
1
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
)
=
s
w
s
h
2
π
[
N
(
r
−
l
+
1
2
s
w
)
+
∑
n
=
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
{\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
)
=
s
w
s
h
2
π
r
n
(
B
)
=
s
w
s
h
2
π
(
r
−
l
+
3
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
)
{\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
(
r
−
l
+
3
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
)
=
s
w
s
h
2
π
[
C
(
r
−
l
+
3
2
s
w
)
+
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
{\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
(
s
w
s
h
)
(
2
π
r
c
)
=
n
c
s
w
s
h
2
π
(
r
+
3
2
s
w
)
{\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
(
s
w
s
h
)
(
2
π
r
D
)
=
n
D
s
w
s
h
2
π
(
r
+
1
2
s
w
)
{\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
(
s
w
s
h
)
(
2
π
r
E
)
=
n
E
s
w
s
h
2
π
(
r
+
1
2
s
w
)
{\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
=
s
w
s
h
2
π
[
N
(
r
−
l
+
1
2
s
w
)
+
∑
n
=
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
+
s
w
s
h
2
π
[
C
(
r
−
l
+
3
2
s
w
)
+
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
+
n
C
s
w
s
h
2
π
(
r
+
3
2
s
w
)
+
n
D
s
w
s
h
2
π
(
r
+
1
2
s
w
)
+
n
E
s
w
s
h
2
π
(
r
+
1
2
s
w
)
=
s
w
s
h
2
π
[
(
N
+
C
)
(
r
−
l
+
1
2
s
w
)
+
C
s
w
+
∑
n
=
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
+
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
s
h
]
2
+
(
n
C
+
n
E
+
n
D
)
(
r
+
1
2
s
w
)
+
n
C
s
w
]
=
s
w
s
h
2
π
[
(
N
+
C
)
(
r
−
l
+
1
2
s
w
)
+
(
n
C
+
n
E
+
n
D
)
(
r
+
1
2
s
w
)
+
(
C
+
n
C
)
s
w
+
∑
n
=
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
+
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
=
s
w
s
h
2
π
[
(
N
+
C
+
n
C
+
n
D
+
n
E
)
(
r
+
1
2
s
w
)
−
l
(
N
+
C
)
+
(
C
+
n
C
)
s
w
+
∑
n
=
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
+
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
=
s
w
s
h
2
π
[
(
N
+
C
+
n
C
+
n
D
+
n
E
)
(
r
+
1
2
s
w
)
−
l
(
N
+
C
)
+
(
C
+
n
C
)
s
w
+
2
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
s
h
]
2
+
∑
n
=
C
+
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
{\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
o
=
V
t
i
e
r
r
a
+
V
e
s
t
a
b
i
l
i
z
a
n
t
e
=
V
t
i
e
r
r
a
+
p
e
s
t
a
b
i
l
i
z
a
n
t
e
×
V
t
i
e
r
r
a
=
(
1
+
p
e
s
t
a
b
i
l
i
z
a
n
t
e
)
×
V
t
i
e
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
i
e
r
r
a
=
V
t
u
b
o
1
+
p
e
s
t
a
b
i
l
i
z
a
n
t
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
i
e
r
r
a
=
V
a
r
c
i
l
l
a
+
V
a
r
e
n
a
s
+
g
r
a
v
a
s
=
p
a
r
c
i
l
l
a
×
V
t
i
e
r
r
a
+
p
a
r
e
n
a
s
+
g
r
a
v
a
s
×
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
i
e
r
r
a
=
(
p
a
r
c
i
l
l
a
+
p
a
r
e
n
a
s
+
g
r
a
v
a
s
)
×
V
t
i
e
r
r
a
{\displaystyle V_{tierra}=(p_{arcilla}+p_{arenas+gravas})\times V_{tierra}}
p
a
r
c
i
l
l
a
=
1
−
p
a
r
e
n
a
s
+
g
r
a
v
a
s
{\displaystyle p_{arcilla}=1-p_{arenas+gravas}}
Estabilizante
p
e
s
t
a
b
i
l
i
z
a
n
t
e
{\displaystyle p_{estabilizante}}
V
e
s
t
a
b
i
l
i
z
a
n
t
e
=
p
e
s
t
a
b
i
l
i
z
a
n
t
e
×
V
t
i
e
r
r
a
{\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:
p
a
g
u
a
{\displaystyle p_{agua}}
V
a
g
u
a
=
p
a
g
u
a
×
V
m
o
r
t
e
r
o
=
p
a
g
u
a
×
V
t
u
b
o
{\displaystyle V_{agua}=p_{agua}\times V_{mortero}=p_{agua}\times V_{tubo}}
![{\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 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_{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)
