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\begin_body

\begin_layout Title
Structural Design of AguaClara Plants
\end_layout

\begin_layout Author
Lily Siu
\end_layout

\begin_layout Abstract
Our main objective we wish to accomplish this summer is to analyze the reinforce
ment configuration and structural strength of the sedimentation and flocculation
 tank walls.
 In the previous semester, the structural design team analyzed the structural
 capabilities of the columns and walls for the Alauca plant using various
 assumptions and load cases.
 The previous team analyzed the walls as closely spaced concrete columns.
 By modeling the walls as columns the flexural support provided by the horizonta
l reinforcement is unaccounted for, but it allowed for the use of the same
 tools and procedures that is used for beam analysis.
 We seek to attempt to validate the previous team's calculations as well
 as suggesting methods to analyze the horizontal reinforcement in order
 to reduce over-designing.
 This report is meant to augment the Spring 2011 report.
\end_layout

\begin_layout Section
Introduction
\end_layout

\begin_layout Standard
AguaClara plants use a poured concrete slab and stacks of bricks (see Figure
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Poured-slab-reinforced"

\end_inset

) with both horizontal and vertical rebar for reinforcement.
 The relatively shallow tanks are approximately 1.6
\begin_inset space \thinspace{}
\end_inset

m to 2
\begin_inset space \thinspace{}
\end_inset

m deep and thus the hydrostatic forces are relatively small.
 This has made it possible to construct walls that are only 15
\begin_inset space \thinspace{}
\end_inset

cm thick when finished.
 
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename IMG_0044.JPG
	scale 40

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Poured-slab-reinforced"

\end_inset

Bricks with Both Horizontal and Vertical rebar at the Alauca water treatment
 plant.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Unreinforced masonry built using stacked bonds provide no flexural support,
 and therefore no structural capabilities.
 In order to analyze this complex system involving reinforced masonry, key
 assumptions need to be made.
 Under tension, it is very likely that the mortar will separate from the
 bricks.
 Because of this, the mortar and the bricks provide no support in tension,
 and the columns must be analyzed as cracked.
 It is useful to imagine the bricks simply as spacers for the rebars, to
 hold the rebars in place but not provide any structural support.
 It is then the function of the plaster to prevent seepage into the wall
 even if the interior mortar has separated from the bricks, so the wall
 must be reanalyzed to ensure that the plaster is not cracked.
 
\end_layout

\begin_layout Standard
By imaging the bricks as spacers, the whole wall can indeed be analyzed
 using traditional concrete techniques, given that the compressive stress
 is changed to the compressive stress of clay masonry and the allowable
 tension in the 
\begin_inset Quotes eld
\end_inset

concrete
\begin_inset Quotes erd
\end_inset

 set to 0 or cracked.
 With these assumptions, the wall can then also be modeled with two-dimensional
 stresses using traditional concrete techniques.
\end_layout

\begin_layout Section
Column and Wall Design
\end_layout

\begin_layout Standard
Following the Spring 2011 report, the column with the highest moment in
 the Alauca plant design was analyzed.
 The scenario is a free standing 1.74
\begin_inset space \thinspace{}
\end_inset

m wall without rubble or back-fill contributing to the support.
\end_layout

\begin_layout Standard
The flocculation and sedimentation tank columns involves #3 rebars and #2
 ties 20
\begin_inset space \thinspace{}
\end_inset

cm apart (see Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Column-Rebar-Configuration"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Wall-and-Column"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename column layout.JPG
	scale 50

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Column-Rebar-Configuration"

\end_inset

Column Rebar Configuration
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
The flocculation and sedimentation tank wall layout involves bricks laid
 with running joints, inter-spaced with #3 rebars (see Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Wall-Rebar-Configuration"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Wall-and-Column"

\end_inset

) .
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\begin_inset Float figure
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status open

\begin_layout Plain Layout
\align center
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	filename wall layout 1.JPG
	scale 50

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Wall Rebar Configuration up to 1m (in cm)
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename wall layout 2.JPG
	scale 50

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\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Wall Rebar Configuration from 1m to 1.74m (in cm)
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

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\begin_inset Caption

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LatexCommand label
name "fig:Wall-Rebar-Configuration"

\end_inset

Wall Rebar Configuration
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

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wide false
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	filename DSC06123.JPG
	scale 40

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\begin_layout Plain Layout
\begin_inset VSpace defskip
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\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename DSC06142.JPG
	scale 44.5

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Wall-and-Column"

\end_inset

Wall and Column Rebar Configuration at the Alauca Water Treatment Plant
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Design Shear and Moment Calculations
\end_layout

\begin_layout Standard
The first step in the analysis is to calculate the shear and moment loads
 which would act on the column.
 These loads were determined by the water depth and the tributary area which
 transferred a portion of the total water pressure against the wall to the
 column or wall element.
 For our analysis of the tank walls, we modeled each 0.28
\begin_inset space \thinspace{}
\end_inset

m (two halves of a brick plus a 2
\begin_inset space \thinspace{}
\end_inset

cm joint) width of wall as a column.
 This was to account for the vertical rebar in the wall which provides flexural
 support.
 Each stack of brick and vertical rebars could now be analyzed as a single
 column.
 In our analysis, we made several key assumptions:
\end_layout

\begin_layout Standard
• Columns and wall sections act as cantilever beams with tributary loads
\end_layout

\begin_layout Standard
• Columns will have tributary loads from half the span of the masonry walls
 on each side of the column
\end_layout

\begin_layout Standard
• Columns will act as t-beams, capitalizing on the reinforcement inside
 the wall section
\end_layout

\begin_layout Standard
• Wall sections will have a tributary load area of its own width (0.28
\begin_inset space \thinspace{}
\end_inset

m)
\end_layout

\begin_layout Standard
• The water level reaches the top of the tank height
\end_layout

\begin_layout Standard
• The height of the tank (H
\begin_inset script subscript

\begin_layout Plain Layout
Floc
\end_layout

\end_inset

) is 1.74
\begin_inset space \thinspace{}
\end_inset

m without support from the back-fill, 0.8
\begin_inset space \thinspace{}
\end_inset

m with support from the back-fill, and 1.17
\begin_inset space \thinspace{}
\end_inset

m with support from the rubble-work
\end_layout

\begin_layout Standard
The maximum load at the base of the column (P
\begin_inset script subscript

\begin_layout Plain Layout
Max
\end_layout

\end_inset

) with units (N/m) is a function of the density of water (ρ), the acceleration
 due to gravity (g), the height of the flocculator tank wall (H
\begin_inset script subscript

\begin_layout Plain Layout
Floc
\end_layout

\end_inset

), and the tributary width of load (W
\begin_inset script subscript

\begin_layout Plain Layout
Trib
\end_layout

\end_inset

) (Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Pmax"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
P_{Max}=\rho\cdot g\cdot H_{Floc}\cdot W_{Trib}\label{eq:Pmax}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
For the entire wall and column system, the tributary width of load (W
\begin_inset script subscript

\begin_layout Plain Layout
Trib
\end_layout

\end_inset

) was calculated by taking the length of the wall (L
\begin_inset script subscript

\begin_layout Plain Layout
FlocWithWalls
\end_layout

\end_inset

) and dividing by two (Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WTrib"

\end_inset

).
 The column with the largest tributary width and greatest height would experienc
e the worst case moment and shear.
 This value will change depending on the dimension for the wall, but for
 the Alauca plant, the tributary width for the column was 1.99
\begin_inset space \thinspace{}
\end_inset

m.
 For the wall sections, the tributary width was taken to be 0.28
\begin_inset space \thinspace{}
\end_inset

m.
 These walls would only have to support the moment from the water pressure
 affecting the section.
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
W_{Trib}=\frac{L_{FlocWithWalls}}{2}\label{eq:WTrib}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The shear in the column (V
\begin_inset script subscript

\begin_layout Plain Layout
y
\end_layout

\end_inset

) is a function of the maximum load at the base of the column (P
\begin_inset script subscript

\begin_layout Plain Layout
Max
\end_layout

\end_inset

), the height of the flocculator tank wall (H
\begin_inset script subscript

\begin_layout Plain Layout
Floc
\end_layout

\end_inset

), and the distance from the base of the column (y) (Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Shear"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
V_{y}=\frac{1}{2}\left(P_{Max}-\frac{P_{max}\cdot y}{H_{Floc}}\right)\cdot\left(H_{Floc}-y\right)\label{eq:Shear}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The moment in the column (M
\begin_inset script subscript

\begin_layout Plain Layout
y
\end_layout

\end_inset

) is a function of the maximum load at the base of the column (P
\begin_inset script subscript

\begin_layout Plain Layout
Max
\end_layout

\end_inset

), the height of the flocculator tank wall (H
\begin_inset script subscript

\begin_layout Plain Layout
Floc
\end_layout

\end_inset

), and the distance from the base of the column (y) (Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Moment"

\end_inset

).
\end_layout

\begin_layout LyX-Code
\begin_inset Formula 
\begin{equation}
M_{y}=\frac{1}{2}\left(P_{Max}-\frac{P_{Max}\cdot y}{H_{Floc}}\right)\left(H_{Floc}-y\right)\left(Y+\frac{H_{Floc}-y}{3}\right)\label{eq:Moment}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Shear-and-Moment"

\end_inset

 shows the shear and moment from the wall tributary section with respect
 to the distance from the base of the column when no support from the back-fill
 is assumed in the flocculator tank column.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename shearandmoment.png
	scale 70

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Shear-and-Moment"

\end_inset

Shear and Moment in the Column with respect to Distance from the Base
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The maximum shear and moment would occur at the base.
 We set y=0 and calculated the maximum shear and moment using equations
 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Shear"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Moment"

\end_inset

 respectively.
 The results for this example are as shown below in Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Maximum-Shear-and"

\end_inset

.
\end_layout

\begin_layout Standard
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wide false
sideways false
status open

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\begin_inset Float table
wide false
sideways false
status open

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\align center

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<column alignment="center" valignment="top" width="2cm">
<column alignment="center" valignment="top" width="2cm">
<column alignment="center" valignment="top" width="2cm">
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\end_layout

\end_inset
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\begin_layout Plain Layout
Tributary Width (m)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

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Load, P
\begin_inset script subscript

\begin_layout Plain Layout
Max
\end_layout

\end_inset


\begin_inset Newline newline
\end_inset

(kN/m)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Shear
\begin_inset Newline newline
\end_inset

(kN)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Moment 
\begin_inset Newline newline
\end_inset

(kN m)
\end_layout

\end_inset
</cell>
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\begin_inset Text

\begin_layout Plain Layout
Concrete Column
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.15
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.56
\end_layout

\end_inset
</cell>
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\begin_inset Text

\begin_layout Plain Layout
2.23
\end_layout

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</cell>
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\begin_inset Text

\begin_layout Plain Layout
1.29
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Masonry Wall Sections
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.28
\end_layout

\end_inset
</cell>
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\begin_inset Text

\begin_layout Plain Layout
4.78
\end_layout

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\begin_inset Text

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4.16
\end_layout

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2.41
\end_layout

\end_inset
</cell>
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<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Composite Concrete Column and Masonry Wall
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.99
\end_layout

\end_inset
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33.96
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\begin_layout Plain Layout
29.54
\end_layout

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17.13
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Caption

\begin_layout Plain Layout
Max Shear and Moment at Bottom of Tank
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

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\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Shear (kN)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Moment 
\begin_inset Newline newline
\end_inset

(kN m)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Masonry Wall Sections
\end_layout

\end_inset
</cell>
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\begin_inset Text

\begin_layout Plain Layout
0.75
\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.94
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Composite Concrete Column and Masonry Wall
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.34
\end_layout

\end_inset
</cell>
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\begin_inset Text

\begin_layout Plain Layout
6.66
\end_layout

\end_inset
</cell>
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</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Caption

\begin_layout Plain Layout
Max Shear and Moment at 1m from Bottom of Tank
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:Maximum-Shear-and"

\end_inset

Maximum Shear and Moment in Tank Walls and Columns
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Flexural Analysis
\end_layout

\begin_layout Standard
Since the columns and walls are loaded in flexure, a concrete beam analysis
 was used to calculated the flexural strength and shear strength.
 Three components of the tank walls were analyzed, the composite flexural
 strength of the concrete column with the walls, acting as a T-beam (henceforth
 referred to as T-beam), under the load of the entire tributary area (1.99
\begin_inset space \thinspace{}
\end_inset

m); the flexural strength of the concrete columns under the load of just
 its exposed area (0.15
\begin_inset space \thinspace{}
\end_inset

m); and the flexural strength of the masonry wall per analyzed column (0.28
\begin_inset space \thinspace{}
\end_inset

m).
\end_layout

\begin_layout Standard
The first step of the analysis is to determine whether the thickness of
 the wall is adequate to sustain deflections.
 This is done using table 9.5(a) from ACI Building code 318 (see Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:ACI-318-08-Table"

\end_inset

).
 The wall slab is considered to be one end continuous since it is a two
 span slab, so the thickness of the slab must be greater than
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
l/24=1.99m/24=0.083m
\]

\end_inset


\end_layout

\begin_layout Standard
The thickness of the wall slab is 15
\begin_inset space \thinspace{}
\end_inset

cm.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename ACI 318 Table 9.5a.JPG

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:ACI-318-08-Table"

\end_inset

ACI 318-08 Table 9.5(a)
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
For the T-beam analysis, the amount of wall reinforcement that can be included
 in the column reinforcement must be calculated.
 This distance is called the effective width, b
\begin_inset script subscript

\begin_layout Plain Layout
eff
\end_layout

\end_inset

.
 The effective width is a function of the span length, l, beam width, bw,
 span thickness, t and the beam spacing (Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:beff"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
b_{eff}=\min\begin{cases}
0.25\cdot l\\
bw+16\cdot t\\
12\cdot[beamspacing]
\end{cases}\label{eq:beff}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The effective width of this column-masonry wall system is 0.5
\begin_inset space \thinspace{}
\end_inset

m.
 Because the effective width of this column does not include any additional
 reinforcement from the masonry wall, the T-beam analysis is the same as
 the concrete column analysis except for a much greater shear and moment
 from the increased tributary width.
\end_layout

\begin_layout Standard
To calculate the moment resistance offered by the reinforcement you first
 determine the tension force, T, by multiplying steel area, A
\begin_inset script subscript

\begin_layout Plain Layout
s
\end_layout

\end_inset

 and the strength of the steel, f
\begin_inset script subscript

\begin_layout Plain Layout
y
\end_layout

\end_inset

 (Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:T"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
T=A_{s}\cdot f_{y}\label{eq:T}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Because the internal moment is radially symmetric along the neutral axis,
 the compression force, C, is equal to the tension force, T (Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:C"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
C=T\label{eq:C}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The stress experienced by the concrete in compression is calculated using
 the 
\begin_inset Quotes eld
\end_inset

Whitney
\begin_inset Quotes erd
\end_inset

 Stress Block.
 The 
\begin_inset Quotes eld
\end_inset

Whitney
\begin_inset Quotes erd
\end_inset

 Stress Block translates a non-linear stress distribution into an equivalent
 constant stress distribution.
 The height of the stress block, a, is a function of the concrete compressive
 stress, f'
\begin_inset script subscript

\begin_layout Plain Layout
c
\end_layout

\end_inset

, the compression force, C, and the width of the member (Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:a"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
a=\frac{C}{0.85\cdot f'_{c}\cdot bw}\label{eq:a}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The distance from the compression face to the location of the steel is the
 thickness of the beam, h, minus the cover, minus the stirrup diameter,
 minus the radius of the reinforcing bar (Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:d"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
d=h-\{cover\}-\{Diameter_{stirrup}\}-\{Radius_{rebar}\}\label{eq:d}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The internal moment, M
\begin_inset script subscript

\begin_layout Plain Layout
n
\end_layout

\end_inset

, is then calculated by multiplying the tension force, T, by the internal
 moment arm (Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Mn"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
M_{n}=T\cdot(d-\frac{a}{2})\label{eq:Mn}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Flexural-Analysis-Inputs"

\end_inset

 shows the various inputs that were used to calculate the internal moment,
 M
\begin_inset script subscript

\begin_layout Plain Layout
n
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="9" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="2cm">
<column alignment="center" valignment="top" width="2cm">
<column alignment="center" valignment="top" width="2cm">
<column alignment="center" valignment="top" width="2cm">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
T-beam
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Concrete Column
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Masonry Wall Sections up to 1
\begin_inset space \thinspace{}
\end_inset

m
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Wall Sections from 1
\begin_inset space \thinspace{}
\end_inset

m to 1.74
\begin_inset space \thinspace{}
\end_inset

m
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Thickness, h (cm)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
15
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
15
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
15
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
15
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Width, bw (cm)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
15
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
15
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
28
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
28
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Compressive Strength f'
\begin_inset script subscript

\begin_layout Plain Layout
c
\end_layout

\end_inset

 (psi)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3000
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3000
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8250
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8250
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Reinforcing Bar number
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Reinforcing Stirrups number
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
N/A
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
N/A
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Cover (cm)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.5
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.5
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.5
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.5
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Number of Tension-side Bars
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Yield Strength of Rebar, f
\begin_inset script subscript

\begin_layout Plain Layout
y
\end_layout

\end_inset

 (ksi)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
40
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
40
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
40
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
40
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:Flexural-Analysis-Inputs"

\end_inset

Flexural Analysis Inputs
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
The internal moment, M
\begin_inset script subscript

\begin_layout Plain Layout
n
\end_layout

\end_inset

 must then be multiplied by a strength reduction factor, Φ.
 The Strength reduction factor, Φ, is a function of the strain in the tension
 steel.
 The factor is used to dissuade a design that has a compression controlled
 failure because compression failures are much more dramatic than tension
 failures (Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:strength reduction factor"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\Phi=\begin{cases}
0.65 & \varepsilon_{s}\leq\varepsilon_{y}\\
0.65+\left(\varepsilon_{t}-\varepsilon_{y}\right)\left(\frac{0.250}{\left(-0.005-\varepsilon_{y}\right)}\right) & \varepsilon_{y}\leq\varepsilon_{s}\leq0.005\\
0.90 & \varepsilon_{s}>0.005
\end{cases}\label{eq:strength reduction factor}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Flexural-Analysis-Results"

\end_inset

 shows the calculated internal moments M
\begin_inset script subscript

\begin_layout Plain Layout
n
\end_layout

\end_inset

 for the various members.
 By comparing it to Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Maximum-Shear-and"

\end_inset

, we can see that the wall cannot function as a T-beam, but instead the
 masonry wall does contribute significantly to the strength of the wall.
 However, we can also see that the masonry wall itself is not enough to
 withstand the force.
\end_layout

\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="3cm">
<column alignment="center" valignment="top" width="2cm">
<column alignment="center" valignment="top" width="2cm">
<column alignment="center" valignment="top" width="2cm">
<column alignment="center" valignment="top" width="2cm">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
T-beam
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Concrete Column
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Masonry Wall Sections up to 1
\begin_inset space \thinspace{}
\end_inset

m
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Wall Sections from 1
\begin_inset space \thinspace{}
\end_inset

m to 1.74
\begin_inset space \thinspace{}
\end_inset

m
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Internal Moment, M
\begin_inset script subscript

\begin_layout Plain Layout
n
\end_layout

\end_inset

 (KN
\begin_inset space \thinspace{}
\end_inset

m)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.37
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.17
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.34
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.55
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Internal Moment with Strength Reduction Factor of 0.9 (KN
\begin_inset space \thinspace{}
\end_inset

m)
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3.93
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3.75
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.11
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.40
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:Flexural-Analysis-Results"

\end_inset

Flexural Analysis Results
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Section
Suggestions and Future Work
\end_layout

\begin_layout Standard
An alternate construction technique can be implemented to increase the flexural
 capability of the wall.
 Using perforated bricks instead of solid bricks would allow additional
 reinforcement to be placed in the wall, running bonds, and also mortar
 to be grouted into the bricks.
 This would provide increased flexural support from additional steel reinforceme
nt and a more continuous wall.
 The moment capacity, Mn*Φ, from adding one additional rebar to the masonry
 wall is 4.18
\begin_inset space \thinspace{}
\end_inset

KN
\begin_inset space \thinspace{}
\end_inset

m,.
 This is sufficient to withstand the hydrostatic force and may even negate
 the need for the concrete columns.
 Perforated bricks would also allow for a more adaptable design, depending
 on the tank depth and maximum moment.
\end_layout

\begin_layout Standard
In the following semester, the beam analysis calculations should be translated
 into a Mathcad document and laboratory testing for the strength of the
 wall should be performed.
\end_layout

\begin_layout Standard
The wall should be around 1
\begin_inset space \thinspace{}
\end_inset

m high by 0.8
\begin_inset space \thinspace{}
\end_inset

m wide.
 The clay bricks should be of comparable to those used in the water treatment
 plants.
 The mortar shall also be of comparable strength to that used in the water
 treatment plants.
 Type N masonry mortar can be made with:
\end_layout

\begin_layout Enumerate
2 cylinders (3 by 6 inch) Portland Cement
\end_layout

\begin_layout Enumerate
4 cylinders hydrated lime
\end_layout

\begin_layout Enumerate
24 cylinders sand
\end_layout

\begin_layout Enumerate
11 cylinders water
\end_layout

\begin_deeper
\begin_layout Enumerate
Add water incrementally until the desired consistency is achieved.
 State with no more that 50% of the target water content.
 More water may be needed to give the mortar the proper consistency, but
 adding water will lower the strength of the mortar.
\end_layout

\end_deeper
\begin_layout Standard
The first wall should be built as the current design with stacked bonds
 interspaced with rebar (see Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Test-Wall-Stacked"

\end_inset

).
\end_layout

\begin_layout Standard
The second wall should be built with perforated bricks and running bonds
 (use half bricks where necessary), the rebar shall be interspaced and mortar
 shall be grouted into the perforations (see Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Test-Wall-Running"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename Test Wall 1.JPG
	scale 80

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Test-Wall-Stacked"

\end_inset

Test Wall with Stacked Joints
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename Test Wall 2.JPG
	scale 80

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Test-Wall-Running"

\end_inset

Test Wall with Running Joints
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
The testing procedure shall be as follows:
\end_layout

\begin_layout Enumerate
Cut plastic sheet 4 feet wide and about 12 feet long
\end_layout

\begin_layout Enumerate
Place test frame on plastic
\end_layout

\begin_layout Enumerate
Lay coursing marks to correspond to the top surface of the brick for each
 brick, allowing 2
\begin_inset space \thinspace{}
\end_inset

cm for joints and rebar
\end_layout

\begin_layout Enumerate
Build wall with rebars and mortar
\end_layout

\begin_layout Enumerate
Perform one mortar flow test as soon as mortar is ready to use, and one
 more flow test as soon as the wall is finished
\end_layout

\begin_deeper
\begin_layout Enumerate
Wipe brass table and mold clean
\end_layout

\begin_layout Enumerate
Place mold on center of table
\end_layout

\begin_layout Enumerate
fill half depth with mortar
\end_layout

\begin_layout Enumerate
Tamp with rectangular prism 20 times
\end_layout

\begin_layout Enumerate
Fill full depth
\end_layout

\begin_layout Enumerate
Tamp 20 times
\end_layout

\begin_layout Enumerate
Smooth top with trowel using a sawing motion across the top of the mold
\end_layout

\begin_layout Enumerate
Wipe table top clean and dry
\end_layout

\begin_layout Enumerate
Remove mold and immediately turn handle 25 times in 15 seconds
\end_layout

\begin_layout Enumerate
Measure the major and minor diameters of the resulting 
\begin_inset Quotes eld
\end_inset

pancake
\begin_inset Quotes erd
\end_inset


\end_layout

\end_deeper
\begin_layout Enumerate
Wait 28 days
\end_layout

\begin_layout Enumerate
Perform strength test using a suction device on a fistula in the plastic
\end_layout

\begin_layout Enumerate
Measure:
\end_layout

\begin_deeper
\begin_layout Enumerate
Height difference in water column at instant of flexural failure of wall
\end_layout

\begin_layout Enumerate
Air pressure difference
\end_layout

\begin_layout Enumerate
Average width of wall (around 82
\begin_inset space \thinspace{}
\end_inset

cm)
\end_layout

\begin_layout Enumerate
Space of wall from center of lower support to center of upper support, l
 (around 94
\begin_inset space \thinspace{}
\end_inset

cm)
\end_layout

\begin_layout Enumerate
Distance from center of lower support to plane of failure, x
\end_layout

\begin_layout Enumerate
Bending moment at plane of failure
\begin_inset Newline linebreak
\end_inset


\begin_inset Formula 
\[
M=\frac{wl}{2}x-\frac{wx^{2}}{2}
\]

\end_inset

 where:
\begin_inset Newline newline
\end_inset

w is the Air pressure multiplied by the average width of the wall
\begin_inset Newline newline
\end_inset

l is the height of the wall (94
\begin_inset space \thinspace{}
\end_inset

cm)
\begin_inset Newline newline
\end_inset

x is the distance from center of lower support to plane of failure
\end_layout

\begin_layout Enumerate
Width of wall at failure plane, b (around 82
\begin_inset space \thinspace{}
\end_inset

cm)
\end_layout

\begin_layout Enumerate
Thickness of wall at failure plane, t (around 6
\begin_inset space \thinspace{}
\end_inset

cm)
\end_layout

\begin_layout Enumerate
Area of wall at failure plane, A (b*t)
\end_layout

\begin_layout Enumerate
Moment of inertia at failure plane
\begin_inset Formula 
\[
I=\frac{1}{12}bt^{3}
\]

\end_inset


\end_layout

\begin_layout Enumerate
Flexural stress at failure plane
\begin_inset Formula 
\[
\sigma_{flexural}=\frac{M\cdot\frac{t}{2}}{I}
\]

\end_inset


\end_layout

\begin_layout Enumerate
Weight of upper portion of failed wall, P
\end_layout

\begin_layout Enumerate
Axial compression stress due to self-weight at failure plane
\begin_inset Formula 
\[
\sigma_{axial}=\frac{P}{A}
\]

\end_inset


\end_layout

\begin_layout Enumerate
Net tension stress at failure plane
\begin_inset Formula 
\[
\sigma_{axial}-\sigma_{flexural}
\]

\end_inset


\end_layout

\begin_layout Enumerate
Net compression stress at failure plane
\begin_inset Formula 
\[
\sigma_{axial}+\sigma_{flexural}
\]

\end_inset


\end_layout

\end_deeper
\end_body
\end_document
