# Difference between revisions of "Retaining wall"

author Retaining wall Leif Roschier File:Ex retaining wall.pdf

## Theory and background

Classic example of nomograph found in Ref [1] is the formula connecting the proportions of retaining walls according to picture.

The picture can be scaled horizontally and vertically so that the wall cross-section is in general a rectangle as illustrated. The nomograph calculates equal retaining walls: one having straight faces (texture in figure) and one with on sloping faces (blue color). The problem follows equation

$(1+L)h^2-Lh(1+p)-\frac{1}{3}(1-L)(1+2p)=0$

that can be written as

$\begin{vmatrix} 2(L^2-1) & 3L(L+1) & -L(L-1) \\ 2(2p+1) & 3(p+1) & -(p+1)(2p+1) \\ h & 1 & -h^2 \end{vmatrix} =0,$

where p is defines as (density of earth)/(density of masonry). Similar nomograph is also found in Ref.[2]

#### References

1. H.J. Allcock, J.R. Jones and J.G.L. Michel, The Nomogram. Sir Isaac Pitman & Sons, Ltd., 4nd Edition, 1952.
2. d'Ocagne, Traité Nomographie, Gauthier-Villars (1899)

## Construction of the nomograph

The underlying equation is expressed as determinant and type 9 has to be used. This equation has to be transformed so we use

'transform_ini':True

Note that we have set the points at scale to be transformed with

'u_min_trafo':0.5,
'u_max_trafo':1.0,

in scale definitions.

## Generated nomograph

Retaining wall
Generated portable document file (pdf): File:Ex retaining wall.pdf

## Source code

"""
ex_retaining_wall.py

Retaining wall.

-----------------------------------------
| 2*(u*u-1) | 3*u*(u+1) | -u*(u-1.0)    |
-----------------------------------------
|      v    |     1     |    -v*v       | = 0
-----------------------------------------
| 2*(2*w+1) | 3*(w+1)   |-(w+1)*(2*w+1) |
-----------------------------------------

This program is free software: you can redistribute it and/or modify
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <http://www.gnu.org/licenses/>.
"""
from pynomo.nomographer import *
N_params_1={
'u_min':0.5,
'u_max':1.0,
'u_min_trafo':0.5,
'u_max_trafo':1.0,
'f':lambda u:2*(u*u-1.0),
'g':lambda u:3*u*(u+1.0),
'h':lambda u:(-u*(u-1.0)),
'title':'L',
'tick_side':'left',
'tick_levels':4,
'tick_text_levels':2,
}
N_params_2={
'u_min':1.0,
'u_max':0.75,
'f':lambda v:v,
'g':lambda v:1.0,
'h':lambda v:(-v*v),
'title':'h',
'tick_side':'right',
'tick_levels':3,
'tick_text_levels':2
}
N_params_3={
'u_min':1.0,
'u_max':0.5,
'u_min_trafo':1.0,
'u_max_trafo':0.5,
'f':lambda w:2.0*(2.0*w+1.0),
'g':lambda w:3.0*(w+1.0),
'h':lambda w:(-(w+1.0)*(2.0*w+1.0)),
'title':'p',
'tick_side':'left',
'tick_levels':4,
'tick_text_levels':2
}
block_params={
'block_type':'type_9',
'f1_params':N_params_1,
'f2_params':N_params_2,
'f3_params':N_params_3,
'transform_ini':True,
'isopleth_values':[[0.8,'x',0.7]]
}

main_params={
'filename':'ex_retaining_wall.pdf',
'paper_height':10.0,
'paper_width':10.0,
'block_params':[block_params],
'transformations':[('scale paper',)]
}
Nomographer(main_params)