Design Project 1

“Design of a Wooden Pile-and-Plank Retaining Wall”

ENGR 0145

Matthew Barry

March 17, 2017

Jordan Gittleman

Alexander Houriet

Tanmoy Sarker

Desmond Zheng

AbstractThis report will analyze our calculations, data, and analysis used to design a wooden pile-

and-plank retaining wall. The retaining wall, made of pressure-treated Standard Structural

Timber instead of stone or reinforced concrete, will be capable of holding a high lateral load

greater than 500 lb/ft while also minimizing costs.

Our group initially approached this design by assuming that the square piles are evenly

spaced and extend into the ground 5 feet while being used to create a 5 feet high terrace. The

planks (horizontal boards connected to the piles) are the same size and equally spaced,

connecting at the middle of each pile. To calculate the optimal size and minimal cost, we used

mathematical data and defined an equation and force diagrams of how the pressure, shear force,

and moment relate alongside the section modulus. By using certain dimensional assumptions and

theoretical analysis, we managed to calculate a minimum required section modulus.

Our group utilized a spreadsheet to calculate the minimum cost based off combinations of

plank and spacing sizes to find the minimum total cost along with the materials used. By

utilizing conceptions of the wooden pile-and-plank retaining wall and mathematical data

analysis, this design report will show how we computed the minimum total cost while meeting

the given constraints.

2

IntroductionOur objective is to design a retaining wall capable of supporting high lateral backfill

loads from the soil with minimal cost. A retaining wall is a structure use to change topography

and mediate erosion from storm water runoff, as shown in Figure 1. Our retaining wall will be

made of wooden planks and piles with concrete footers. They will be composed of equally

spaced square piles that are evenly spaced and extend 5 feet into the ground while creating a 5

feet high terrace. The planks (horizontal boards connected to the piles) are the same size and

equally spaced, connecting at the middle of each pile. Each plank size, pile size, and spacing

between piles must be checked so that the section modulus is greater than the required section

modulus to support the load of the soil. This design is constrained between 2 and 41 piles and

pressure-treated Standard Structural Timber (US Customary Units).

To calculate the section modulus of piles and planks, our group analyzed the pressure

distribution along the distance of the center of each pile. Subsequently, the section modulus of

each plank is inversely proportional to the number of piles, while each plank and pile has their

own size requirements. Following this analysis, the cost of the planks and piles were calculated.

Standard structural timbers are available in 8, 10, and 12 ft. lengths and cost $14 per cubic foot.

Also, there is an additional cost of $40 per pile for the concrete footing needed to support each

pile. This data was inputted into an Excel spreadsheet and the optimal choice was computed.

Minimizing the cost of this is essential to this design of a retaining wall.

Our group intends to approach this problem with a hypothesis that as the piles increase in

number, the mandatory section modulus for each pile and plank will decrease and subsequently

smaller piles and planks can be used. Therefore, we hypothesize that the constraints of the cost

will restrain the ideal plank to a thickness of two inches.

Figure 1

3

Analysis & DesignTo effectively arrive at an optimal design, we will first perform mechanical analysis to

derive our design constraints and then apply the given cost parameters. We will approach the

design of straight pile-and-plank retaining wall by assuming the piles to behave as cantilevered

with a distributed load. We will also assume the planks to be simply-supported beams placed

between the piles. We have the following general free body diagram for the planks:

Figure 2

To simplify our analysis, we apply our given maximum pressure of 500 lbf t2 in our

calculations and let the distributed load be 500 t lbft :

4

Figure 3

Because the plank is of rectangular geometry, we have a concentrated load of 500 tL lbft at

the centroid located at L2 :

Figure 4

We have the following summation of forces:

5

∑ F y=R A+RB−500 tL=0

Given that our rectangular plank is symmetrical, we let RA=RB, and have the following

values for RA and RB:

RA+R A−500 tL=0

2 RA=500tL

RA=RB=250 tL

We have the following shear diagram, and have labeled the region that constitutes our

maximum moment, M max:

Figure 5

We calculate our M max accordingly:

M max=12

(250tL )( L2 )lb∗ft=62.5t L2lb∗ft

6

We have the following formula for the minimum section modulus, Smin :

Smin=M max

σmax

Using our value of M maxand the given maximum allowable flexural stress of 1200 psi, we

derive the minimum section modulus for the planks:

Sminplank=62.5 t L2

1200

lb∗ftpsi

∗12 inches

1 ft=0.625t L2i n3

We now examine the distributed load on the piles. Given that the distributed load is in the

shape of a right-angle trapezoid, we divided it into a rectangular distributed load and a triangular

distributed load as shown in Figure 6.

Figure 6

7

Let y1 and A1 be the centroid and area for the rectangular distributed load respectively,

and y2 and A2 be the centroid for the triangular distributed load, respecitvely. We have the

following formulas for the centroid C and area A of rectangular geometry and triangular

geometry, where b is the base and h is the height, and the base of the geometry is parallel to the

neutral-axis of the geometry:

yrectangle=b2

, A rectangle=bh

y triangle=b3

, Atriangle=bh2

We apply these formulas to derive values for y1, A1, y2, and A2 using values for b and h

from Figure 1. We disregard the units given from Figure 6 in our calculations because we are

applying them in a different context:

y1=52=2.5 , A1=(5 ) (100 )=500

y2=53=1.667 , A2=

(500−100)(5)2

=1000

Let yc be the centroid of the trapezoidal distributed load, which consists of the

rectangular distributed load and triangular distributed load together. We have the following the

formula for the centroid of the total geometry:

yc=y1 A1+ y2 A2

A1+ A2

We substitute our derived values for y1, A1, y2, and A2 into the formula:

yc=(2.5 ) (500 )+(1.667 ) (1000 )

500+1000=1.944

Thus, we have the following free body diagram for the pile:

8

Figure 7

We have the following summation of forces and moments:

∑ F y=R A−1500 L=0

∑ M x=M A−( yc ) ( R A )=0

RA=1500 Llb

M A−(1.944 ) (1500 L )=0

M A=2916 Llb∗ft

We calculate our Smin for the piles accordingly:

Sminpile=2916 L

1200

lb∗ftpsi

∗12inches

1 ft=29.16 L i n3

RA

MA

9

Now that we have determined the required Section Modulus, we can calculate the cost of

various sizes of retaining walls. The Base variable, n, is the number of piles in the wall. The

smallest possible n is 8 piles and the given maximum is 41. The length (L) of the plank in

between each pile is equal to 80

n−1 , 80 feet being the total length of the wall and n−1 being the

number of planks in that section. Sminpile

for this section is found using the previously derived

equation, 29.16 L. Another initial condition was that the piles must be square, so the only timbers

tested were those of symmetrical sizes.

Table 1

Any timber with a Sall greater than Smin is marked yes and the cost was then calculated.

Concrete cost (c ) has a flat cost of $40 per pile for the concrete footing.

c=40 n

Total pile cost (C) uses the following variables and constants: dressed width, wd, dressed height,

hd, pile length, 10 (ft), timber cost per cubic foot, 14, number of piles, n, and concrete cost, c:

C=( wd

12∗hd

12∗10)14 n+c=

35 wd hd n36

+c

10

Nominal Size Sall (in3)

4x4 7.94

6x6 27.7

8x8 70.3

10x10 143

12x12 253

# of piles length of plank Smin Concrete

costn L (feet) Smin (in^3) Sall 4x4 Sall 6x6 Sall 8x8 Sall 10x10 Sall 12x12 c ($) 4x4 6x6 8x8 10x10 12x128 11.42857143 333.2571429 NO NO NO NO NO 320.00$ 422.20$ 555.28$ 757.50$ 1,021.94$ 1,348.61$ 9 10 291.6 NO NO NO NO NO 360.00$ 474.98$ 624.69$ 852.19$ 1,149.69$ 1,517.19$ 10 8.888888889 259.2 NO NO NO NO NO 400.00$ 527.76$ 694.10$ 946.88$ 1,277.43$ 1,685.76$ 11 8 233.28 NO NO NO NO YES 440.00$ 580.53$ 763.51$ 1,041.56$ 1,405.17$ 1,854.34$ 12 7.272727273 212.0727273 NO NO NO NO YES 480.00$ 633.31$ 832.92$ 1,136.25$ 1,532.92$ 2,022.92$ 13 6.666666667 194.4 NO NO NO NO YES 520.00$ 686.08$ 902.33$ 1,230.94$ 1,660.66$ 2,191.49$ 14 6.153846154 179.4461538 NO NO NO NO YES 560.00$ 738.86$ 971.74$ 1,325.63$ 1,788.40$ 2,360.07$ 15 5.714285714 166.6285714 NO NO NO NO YES 600.00$ 791.63$ 1,041.15$ 1,420.31$ 1,916.15$ 2,528.65$ 16 5.333333333 155.52 NO NO NO NO YES 640.00$ 844.41$ 1,110.56$ 1,515.00$ 2,043.89$ 2,697.22$ 17 5 145.8 NO NO NO NO YES 680.00$ 897.19$ 1,179.97$ 1,609.69$ 2,171.63$ 2,865.80$ 18 4.705882353 137.2235294 NO NO NO YES YES 720.00$ 949.96$ 1,249.38$ 1,704.38$ 2,299.38$ 3,034.38$ 19 4.444444444 129.6 NO NO NO YES YES 760.00$ 1,002.74$ 1,318.78$ 1,799.06$ 2,427.12$ 3,202.95$ 20 4.210526316 122.7789474 NO NO NO YES YES 800.00$ 1,055.51$ 1,388.19$ 1,893.75$ 2,554.86$ 3,371.53$ 21 4 116.64 NO NO NO YES YES 840.00$ 1,108.29$ 1,457.60$ 1,988.44$ 2,682.60$ 3,540.10$ 22 3.80952381 111.0857143 NO NO NO YES YES 880.00$ 1,161.06$ 1,527.01$ 2,083.13$ 2,810.35$ 3,708.68$ 23 3.636363636 106.0363636 NO NO NO YES YES 920.00$ 1,213.84$ 1,596.42$ 2,177.81$ 2,938.09$ 3,877.26$ 24 3.47826087 101.426087 NO NO NO YES YES 960.00$ 1,266.61$ 1,665.83$ 2,272.50$ 3,065.83$ 4,045.83$ 25 3.333333333 97.2 NO NO NO YES YES 1,000.00$ 1,319.39$ 1,735.24$ 2,367.19$ 3,193.58$ 4,214.41$ 26 3.2 93.312 NO NO NO YES YES 1,040.00$ 1,372.17$ 1,804.65$ 2,461.88$ 3,321.32$ 4,382.99$ 27 3.076923077 89.72307692 NO NO NO YES YES 1,080.00$ 1,424.94$ 1,874.06$ 2,556.56$ 3,449.06$ 4,551.56$ 28 2.962962963 86.4 NO NO NO YES YES 1,120.00$ 1,477.72$ 1,943.47$ 2,651.25$ 3,576.81$ 4,720.14$ 29 2.857142857 83.31428571 NO NO NO YES YES 1,160.00$ 1,530.49$ 2,012.88$ 2,745.94$ 3,704.55$ 4,888.72$ 30 2.75862069 80.44137931 NO NO NO YES YES 1,200.00$ 1,583.27$ 2,082.29$ 2,840.63$ 3,832.29$ 5,057.29$ 31 2.666666667 77.76 NO NO NO YES YES 1,240.00$ 1,636.04$ 2,151.70$ 2,935.31$ 3,960.03$ 5,225.87$ 32 2.580645161 75.2516129 NO NO NO YES YES 1,280.00$ 1,688.82$ 2,221.11$ 3,030.00$ 4,087.78$ 5,394.44$ 33 2.5 72.9 NO NO NO YES YES 1,320.00$ 1,741.60$ 2,290.52$ 3,124.69$ 4,215.52$ 5,563.02$ 34 2.424242424 70.69090909 NO NO NO YES YES 1,360.00$ 1,794.37$ 2,359.93$ 3,219.38$ 4,343.26$ 5,731.60$ 35 2.352941176 68.61176471 NO NO YES YES YES 1,400.00$ 1,847.15$ 2,429.34$ 3,314.06$ 4,471.01$ 5,900.17$ 36 2.285714286 66.65142857 NO NO YES YES YES 1,440.00$ 1,899.92$ 2,498.75$ 3,408.75$ 4,598.75$ 6,068.75$ 37 2.222222222 64.8 NO NO YES YES YES 1,480.00$ 1,952.70$ 2,568.16$ 3,503.44$ 4,726.49$ 6,237.33$ 38 2.162162162 63.04864865 NO NO YES YES YES 1,520.00$ 2,005.47$ 2,637.57$ 3,598.13$ 4,854.24$ 6,405.90$ 39 2.105263158 61.38947368 NO NO YES YES YES 1,560.00$ 2,058.25$ 2,706.98$ 3,692.81$ 4,981.98$ 6,574.48$ 40 2.051282051 59.81538462 NO NO YES YES YES 1,600.00$ 2,111.02$ 2,776.39$ 3,787.50$ 5,109.72$ 6,743.06$ 41 2 58.32 NO NO YES YES YES 1,640.00$ 2,163.80$ 2,845.80$ 3,882.19$ 5,237.47$ 6,911.63$

Viablility

Initial Load Condition TestingWood Cost+Concrete Cost (Piles are 10ft long, nominal sizes

given but calculations done with dressed size)

Total Cost Analysis for Piles

Table 2

This begins the same as pile cost, with n and L calculated as they are above. Since the

wall is 5 feet high, we decided to only test planks with a dressed height hd=7.5. This is the only

height of the plank that will fit evenly in the wall. This decreases scrap, making it the most

efficient choice.

Table 3

11

Nominal Timber Size Sall (in3)

2x8 15.3

4x8 34

6x8 51.6

8x8 70.3

Dressed height, hd (in) Number of planks to

reach 60in

3.625 16.55

5.625 10.67

7.5 8

9.5 6.32

11.5 5.22

13.5 4.44

15.5 3.87

17.5 3.42

Table 4

The Sminplank requires the plank thickness to be calculated, therefore a new Smin

pile must be

calculated for each timber size tested. Those planks with passable Sall are marked in Table 5. The

total plank cost (C) uses the following variables and constants: dressed thickness, t d, plank

length rounded up to the next integer, LR, dressed height, 7.25, planks per section, 8, number of

planks, n−1, and cost per cubic foot, 14:

C=7.512

t d LR (8 ) (n−1 ) (14 )=70 t d LR (n−1 )

The total cost can be found by adding the cheapest option for piles and planks at any

given n.

12

# of Piles length of plank

n L (feet) Smin (in^3) Sall 2x8 Smin (in^3) Sall 4x8 Smin (in^3) Sall 6x8 Smin (in^3) Sall 8x8 2x8 4x8 6x8 8x88 11.42857143 132.6530612 NO 295.9183673 NO 448.9795918 NO 612.244898 NO9 10 101.5625 NO 226.5625 NO 343.75 NO 468.75 NO10 8.888888889 80.24691358 NO 179.0123457 NO 271.6049383 NO 370.37037 NO11 8 65 NO 145 NO 220 NO 300 NO12 7.272727273 53.71900826 NO 119.8347107 NO 181.8181818 NO 247.933884 NO13 6.666666667 45.13888889 NO 100.6944444 NO 152.7777778 NO 208.333333 NO14 6.153846154 38.46153846 NO 85.79881657 NO 130.1775148 NO 177.514793 NO15 5.714285714 33.16326531 NO 73.97959184 NO 112.244898 NO 153.061224 NO16 5.333333333 28.88888889 NO 64.44444444 NO 97.77777778 NO 133.333333 NO17 5 25.390625 NO 56.640625 NO 85.9375 NO 117.1875 NO18 4.705882353 22.49134948 NO 50.17301038 NO 76.12456747 NO 103.806228 NO19 4.444444444 20.0617284 NO 44.75308642 NO 67.90123457 NO 92.5925926 NO20 4.210526316 18.00554017 NO 40.16620499 NO 60.94182825 NO 83.1024931 NO21 4 16.25 NO 36.25 NO 55 NO 75 NO22 3.80952381 14.73922902 YES 32.87981859 YES 49.88662132 YES 68.0272109 YES 796.25$ 1,776.25$ 2,695.00$ 3,675.00$ 23 3.636363636 13.42975207 YES 29.95867769 YES 45.45454545 YES 61.9834711 YES 834.17$ 1,860.83$ 2,823.33$ 3,850.00$ 24 3.47826087 12.28733459 YES 27.41020794 YES 41.5879017 YES 56.710775 YES 872.08$ 1,945.42$ 2,951.67$ 4,025.00$ 25 3.333333333 11.28472222 YES 25.17361111 YES 38.19444444 YES 52.0833333 YES 910.00$ 2,030.00$ 3,080.00$ 4,200.00$ 26 3.2 10.4 YES 23.2 YES 35.2 YES 48 YES 947.92$ 2,114.58$ 3,208.33$ 4,375.00$ 27 3.076923077 9.615384615 YES 21.44970414 YES 32.5443787 YES 44.3786982 YES 985.83$ 2,199.17$ 3,336.67$ 4,550.00$ 28 2.962962963 8.916323731 YES 19.89026063 YES 30.17832647 YES 41.1522634 YES 767.81$ 1,712.81$ 2,598.75$ 3,543.75$ 29 2.857142857 8.290816327 YES 18.49489796 YES 28.06122449 YES 38.2653061 YES 796.25$ 1,776.25$ 2,695.00$ 3,675.00$ 30 2.75862069 7.728894174 YES 17.24137931 YES 26.15933413 YES 35.6718193 YES 824.69$ 1,839.69$ 2,791.25$ 3,806.25$ 31 2.666666667 7.222222222 YES 16.11111111 YES 24.44444444 YES 33.3333333 YES 853.13$ 1,903.13$ 2,887.50$ 3,937.50$ 32 2.580645161 6.763787721 YES 15.08844953 YES 22.89281998 YES 31.2174818 YES 881.56$ 1,966.56$ 2,983.75$ 4,068.75$ 33 2.5 6.34765625 YES 14.16015625 YES 21.484375 YES 29.296875 YES 910.00$ 2,030.00$ 3,080.00$ 4,200.00$ 34 2.424242424 5.968778696 YES 13.31496786 YES 20.2020202 YES 27.5482094 YES 938.44$ 2,093.44$ 3,176.25$ 4,331.25$ 35 2.352941176 5.62283737 YES 12.5432526 YES 19.03114187 YES 25.9515571 YES 966.88$ 2,156.88$ 3,272.50$ 4,462.50$ 36 2.285714286 5.306122449 YES 11.83673469 YES 17.95918367 YES 24.4897959 YES 995.31$ 2,220.31$ 3,368.75$ 4,593.75$ 37 2.222222222 5.015432099 YES 11.1882716 YES 16.97530864 YES 23.1481481 YES 1,023.75$ 2,283.75$ 3,465.00$ 4,725.00$ 38 2.162162162 4.747991234 YES 10.59167275 YES 16.07012418 YES 21.9138057 YES 1,052.19$ 2,347.19$ 3,561.25$ 4,856.25$ 39 2.105263158 4.501385042 YES 10.04155125 YES 15.23545706 YES 20.7756233 YES 1,080.63$ 2,410.63$ 3,657.50$ 4,987.50$ 40 2.051282051 4.273504274 YES 9.533201841 YES 14.46416831 YES 19.7238659 YES 1,109.06$ 2,474.06$ 3,753.75$ 5,118.75$ 41 2 4.0625 YES 9.0625 YES 13.75 YES 18.75 YES 758.33$ 1,691.67$ 2,566.67$ 3,500.00$

t=6 plank (dressed t=5.5) t=8 plank (dressed t=7.5) Costt=2 plank (dressed

t=1.625)t=4 plank (dressed

t=3.625)

Table 5

Cheapest Option

n 8x8 10x10 12x12 2x8 4x8 6x8 8x822 -$ 2,810.35$ 3,708.68$ 796.25$ 1,776.25$ 2,695.00$ 3,675.00$ 3,606.60$ 23 -$ 2,938.09$ 3,877.26$ 834.17$ 1,860.83$ 2,823.33$ 3,850.00$ 3,772.26$ 24 -$ 3,065.83$ 4,045.83$ 872.08$ 1,945.42$ 2,951.67$ 4,025.00$ 3,937.92$ 25 -$ 3,193.58$ 4,214.41$ 910.00$ 2,030.00$ 3,080.00$ 4,200.00$ 4,103.58$ 26 -$ 3,321.32$ 4,382.99$ 947.92$ 2,114.58$ 3,208.33$ 4,375.00$ 4,269.24$ 27 -$ 3,449.06$ 4,551.56$ 985.83$ 2,199.17$ 3,336.67$ 4,550.00$ 4,434.90$ 28 -$ 3,576.81$ 4,720.14$ 767.81$ 1,712.81$ 2,598.75$ 3,543.75$ 4,344.62$ 29 -$ 3,704.55$ 4,888.72$ 796.25$ 1,776.25$ 2,695.00$ 3,675.00$ 4,500.80$ 30 -$ 3,832.29$ 5,057.29$ 824.69$ 1,839.69$ 2,791.25$ 3,806.25$ 4,656.98$ 31 -$ 3,960.03$ 5,225.87$ 853.13$ 1,903.13$ 2,887.50$ 3,937.50$ 4,813.16$ 32 -$ 4,087.78$ 5,394.44$ 881.56$ 1,966.56$ 2,983.75$ 4,068.75$ 4,969.34$ 33 -$ 4,215.52$ 5,563.02$ 910.00$ 2,030.00$ 3,080.00$ 4,200.00$ 5,125.52$ 34 -$ 4,343.26$ 5,731.60$ 938.44$ 2,093.44$ 3,176.25$ 4,331.25$ 5,281.70$ 35 3,314.06$ 4,471.01$ 5,900.17$ 966.88$ 2,156.88$ 3,272.50$ 4,462.50$ 4,280.94$ 36 3,408.75$ 4,598.75$ 6,068.75$ 995.31$ 2,220.31$ 3,368.75$ 4,593.75$ 4,404.06$ 37 3,503.44$ 4,726.49$ 6,237.33$ 1,023.75$ 2,283.75$ 3,465.00$ 4,725.00$ 4,527.19$ 38 3,598.13$ 4,854.24$ 6,405.90$ 1,052.19$ 2,347.19$ 3,561.25$ 4,856.25$ 4,650.31$ 39 3,692.81$ 4,981.98$ 6,574.48$ 1,080.63$ 2,410.63$ 3,657.50$ 4,987.50$ 4,773.44$ 40 3,787.50$ 5,109.72$ 6,743.06$ 1,109.06$ 2,474.06$ 3,753.75$ 5,118.75$ 4,896.56$ 41 3,882.19$ 5,237.47$ 6,911.63$ 758.33$ 1,691.67$ 2,566.67$ 3,500.00$ 4,640.52$

Pile Cost Plank Cost

Table 6

13

ResultsWe have the following relationship between the number of piles and the cheapest option

as shown in Figure 8.

20 25 30 35 40 $-

$1,000.00

$2,000.00

$3,000.00

$4,000.00

$5,000.00

$6,000.00

Number of Piles vs Cost

Number of Piles

Tota

l Cos

t

Figure 8

From Figure 8, we conclude the cheapest option occurs with 22 10”x10”x10’ piles, using

2”x8”x12’ planks. We have the following bill of materials of our design in Figure 9.

Piece Dimensions Units Unit cost Total cost

Pile 10"x10"x10' 22 $87.74 $1,930.28

Plank 2"x8"x12' 56 $14.22 $796.32

Concrete Footing

N/A 22 $40 $880.00

Final Cost: $3,606.60

Figure 9

14

The parameters listed above represent the best possible configuration of this retaining

wall given the constraints at hand. This design is the most efficient in regards to not only cost,

but ordering of materials as all planks are based on a maximum pressure that only the bottom-

most plank would normally be under. This design also allows for the wall to have an in-built

factor of safety, ensuring it will withstand some additional pressure in irregular instances.

Additionally, the minimization of scrap both decreases waste product and time required to

construct the wall.

15