**
Powerhouse Design- Natick Dam
Web Page**

**Design of Powerhouse
Substructure W. Fay P.E.**

**Natick Hydroelectric
Project 3/28/90**

**West Warwick, Rhode Island**

**FERC License Project No.
3013-RI **

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Stamp**

**I) Methodology:**

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**A) Floor slabs to be designed
as one-way floor slabs using the**

**"ACI-Ultimate Strength Design
Methods"**

** **

**B) Walls to be designed as
one-way floor slabs using the "ACI-Ultimate**

**Strength Design Methods" and
assuming hydrostatic load distributions for**

**saturated soils.**

** **

**C) Beams to be designed using
ASI Code.**

** **

** **

**II) Design Approach:**

** **

**A) Design assumes waterbox
floor slab is supported on three sides by **

**reinforced concrete walls and
by a steel beam on the downstream, **

**tailrace side. Additionally,
the discharge pit will be subdivided into **

**two equal sized compartments
by a wall running in the streamwise **

**direction and this reinforced
wall will support the midspan of the **

**waterbox floor.**

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**Sketch:**

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**B) Waterbox floor slab loads
will consist of uniformly distributed **

**dead loads of water, rebar and
concrete and the point loads of the **

**turbine gate cases. The design
will incorporate four (4) transverse **

**beams, one on each side of the
gate cases to support the gate case **

**point load.**

** **

**C) The discharge pit walls
will be designed for external loads for when **

**the pit may occasionally be
dewatered by a coffer dam to maintain the **

**draft tubes.**

** **

**
(1)**

**D) Waterbox walls will be
designed for the internal pressure due to the **

**headwater and for external
pressure due to dewatering and emergency **

**repair work during highwater.**

** **

**E) Generator floor slabs will
be designed for a uniform floor loading of**

**300 lb/sq.ft and for the point
loads of the generator, turbine shaft,**

**turbine runner and hydraulic
thrust. This point load will be supported **

**by transverse beams similiar
to B) above. These beams will be designed **

**according to ASI Code to have
proper strength, with minimal deflection.**

** **

**F) The upstream side of the
generating floor will be supported by a **

**steel beam spanning the
waterbox inlets.**

** **

**G) The waterbox will be
divided into two (2) seperate chambers by a **

**structural wall.**

** **

**H) The main steel beam
supporting the waterbox floor slab will be**

**designed conservatively to
support the weight of the waterbox rear wall,**

**powerhouse rear wall and one
half (1/2) the weights of the waterbox**

**floor slab, the generating
room floor slab, the turbine/generator**

**hydraulic thrust point load
and the overhead crane point load (assumed**

**to be in the most compromising
position).**

** **

**I) The main steel beam
supporting the generating room floor slab and**

**spanning the waterbox inlets
will be designed conservatively to support**

**the weight of the generating
room upstream wall and one half (1/2) the**

**weights of the generating room
floor slab, the turbine/generator**

**hydraulic thrust point load
and the overhead crane point load (assumed**

**to be in the most compromising
position).**

** **

**J) The following elevations
will be assumed:**

** **

******(Datum is USGS-MSL)******

** **

**1) Top of discharge pit floor
slab - 21.0 msl**

** **

**2) Mean tailwater elevation -
31.5 msl**

** **

**3) Waterbox floor - 40.0 msl**

** **

**4) Mean headwater elevation -
49.1 msl**

** **

**5) Top of generating room
floor slab - 54.0 msl**

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**Assume main beams forming the
discharge pit/tailrace and water box inlet**

**arches are WF-30"x 124 #/ft.**

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**
(2)**

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**III) Design of the Waterbox
Floor Slab:**

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**A) Dimensions- from Fay
Engineering Services layout dated 12/27/89**

**entitled "Powerhouse
Longitudinal Section". Assume these dimensions are**

**correct.**

** **

**1) External dimensions:**

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**a) length = 40.0'**

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**b) width = 18'**

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**2) Internal dimensions:**

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**a) length = 36.0'**

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**b) width = 16.0'**

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**B) Loading:**

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**1) Water, total load = 62.4
#/ft3 * 16' * (38'* (50.5-40.0))**

** **

** = 398,361
lbs**

** **

**2) Water, load per foot width
of box = 10,483 lbs/ft-width of flume**

** **

**3) Water, load per foot width
of box per foot width of beam = 655 lb/ft**

** **

**4) Concrete, total load = 150
#/ft3 * 16' * 38.0'***

** **

**
(9"/(12"/ft)) = 68,400 lbf**

** **

**5) Concrete, load per foot
width of box = 68,400 lbf/38.0' = 1800 lbf/ft**

**width of box**

** **

**6) Concrete, load per foot
width of box per foot width of beam = **

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** 1800 lbf/ft-width/16' = 113
lbf/ ft**

** **

**7) Since the waterbox will be
periodically dewatered, treat the water **

**as a live distributed load and
use the larger live load overload factor **

**ACI ultimate strength
multipliers.**

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**C) Sketch of typical section
through proposed waterbox floor:**

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**
(3)**

**D) Design Calculations: -
reference, "Design of Concrete Structures", **

**9th ed., Winter & Nilson.**

** **

**1) Assume the yield strength
of the steel reinforcing is 30,000 psi and **

**that the compressive strength
of the concrete is 3000 psi.**

** **

**2) Select the trial thickness
of the slab, use L/20 from Table 5.1,**

**p.206 in Winter & Nilson.**

** **

**T= (12 "/ft * 18')/20 = 10.8"
approximately = 11"**

** **

**3) The slab weight is 150
#/ft3 * (11/12) = 138 PSF**

** **

**4) Apply the ACI load
multipliers and obtain the factored load:**

** **

**Dead Load = 113 PSF * 1.4 =
158.2 PSF**

** **

**Live Load = 655 PSF * 1.7 =
1113.5 PSF**

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**Total Factored Load = 1272 PSF**

** **

**5) Use the ACI moment
coefficients to determine the design moments at**

**the critical sections:**

** **

**a) Since the floor slab is
being designed as a one-way slab in the**

**short direction (ie: from the
inlet end to the tailrace end), the slab**

**will be resting on the main
30" beam which acts as the arch at the**

**rear of the poerhouse over the
tailrace and will be built into the top**

**of the rear (upstream wall) of
the discharge pit. At the tailrace end,**

**the floor slab is simply
supported and the beam is free to twist and**

**cannot be assumed to be rigid,
so use:**

** **

**(1/11)*Wu*ln^2 >>>>>>>>from
Table 8.1 W&N**

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**Sketch:**

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**b) At the upstream end, the
slab is to be built into the discharge pit**

**wall and can be assumed to be
rigid, so use:**

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**(1/14)*Wu*ln^2>>>>>>>>from
Table 8.1 W&N**

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**Sketch:**

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**
(4)**

**c) At the interior span use:**

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**(1/14)*Wu*ln^2>>>>>>>>from
Table 8.1 W&N**

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**d) At the tailrace: -M =
1/11*1.27 KSF *18'^2 = 37.4 ft-kips**

** **

**e) At the upstream end: -M =
1/14*1.27 KSF *18'^2 = 29.4 ft-kips**

** **

**d) At the midspan: -M =
1/16*1.27 KSF *18'^2 = 25.7 ft-kips**

** **

**5) Determine the maximum steel
ratio permitted by the ACI Code:**

** **

**Pmax = 0.75*Pbalanced =
0.75*0.85*B1*(fc'/fy)*(87,000/(87,000+fc'))**

** **

**this formula for fc'<4000 psi
and B1=0.85**

** **

**Pmax=0.75*0.85*0.85*(3000
psi/30,000 psi)*(87,000/(87,000 + 30,000))**

** **

** = 0.04**

** **

**6) Determine the minimum
required effective depth: (This is controlled**

**by the largest moment at the
tailrace)**

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**d^2 =
Mu/(phi*p*fy*b*(1-(0.59*p*fy/fc))) note: phi=0.9 for bending**

** **

** = (37.4
ft-kips*(12"\ft))/(0.9*.04*30*12*(1-.59*0.04*(30,000/3000)))**

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** = 45.3 in^2**

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**Therefore, d = 6.7 inchs**

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**7) Determine the minimum
effective depth using code restrictions:**

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**dm= 11"- 1" = 10"**

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**8) Since the calculated value
of 6.7 inches is less then the coded**

**effective depth, use d= 10
inches**

** **

**9) At the tailrace end, assume
the stress block depth a = 1.00 inch.**

**Then the area of steel
required per foot width in the top of the slab**

**is:**

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**As= Mu/(phi*fy*(d-a/2))= (37.4
ft-kips*12"/ft)/0.9*30*(10-1/2)=**

** **

** 1.75
in^2**

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**10) Check the assumed depth:**

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**a= As*fy/(0.85*fc'*b)= 1.75
in^2 *30,000/(0.85*3000*12"/ft)= 1.72 in**

** **

**11) The assumed area of steel
and the calculated area of steel are**

**reasonably close so use 1.72
in^2 of rebar per foot width of floor slab.**

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**
(5)**

**12) At the other critical
sections use the same lever arm to determine**

**the required cross sectional
areas of steel rebar:**

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**a) at the midspan: As= 25.7
ft-kips*12/(0.9*30*(11-1.72/2))= 1.13 in^2**

** **

**b) at the upstream wall: As=
29.4 ft-kips*12/(0.9*30*(11-1.72/2))= 1.29**

**in^2**

** **

**13) The minimum reinforcement
required to control shrinkage is: see p. **

**207, W&N.**

** **

**As= 0.002*12*11= 0.26 in^2/
12" with strip**

** **

**The required steel necessary
for shrinkage is met by the steel required **

**to meet the externally applied
loads.**

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**14) Determine the factored
shear force:**

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**Vu= 1.15 *
(1272*18/2)-1272*(11/12)= 13,165-1166 = 11,999 lbs**

** **

**15) The nominal shear strength
of the slab is:**

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**Vn= Vc= 2*b*d*fc'^0.5**

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** = 2*12*11*(3000 psi^0.5)**

** **

** = 14,460 lbs**

** **

**16) The design shear strength
is:**

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** phi*Vc= 0.85*14,460 lbs=
12,291 lbs**

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**17) Since the design shear
strength is above the required shear**

**strength, no additional steel
is necessary to resist the internal shear**

**forces.**

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** **

**IV) Design the Steel Main
Support Beam at the Tailrace**

** **

**A) Data for proposed Beam:**

** **

**1) height= 30"**

** weight= 124 lbs/ft-width**

** length= 55 feet**

** section modulus= 355 in^3**

** moment of inertia= 5360
in^4**

** depth= 30.16"**

** width= 10.521"**

** flange thickness= 0.93"**

** web= 0.585"**

** **

**2) Flat of flange=
(10.521"-0.585")/2= 4.97"**

** 90 % of the flat is 4.5"**

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**
(6)**

**B) Determine the Beam Loading:**

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**1) Point Loads:**

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**a) generator= 225 rpm, 32
pole, 197 kva= 15,800 lbs**

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**b) exciter= 1400 lbs**

**c) Shipping weight of turbine=
14,000 lbs**

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**d) Hydraulic thrust= 21,636 #**

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**2) Crane Load - assume a point
load in the middle of the beam of 5 tons**

**for a five ton bridge crane,
this gives a 2.0 safety factor on the crane**

**load.**

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**3) Total point load is:**

** **

**(15,800 # + 1400 # + 14,000 #
+ 21636 #)/2 + 5000 #= 31,418 lbs**

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**4) Dead Loads:**

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**a) Powerhouse backwall:**

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**(40'*((69.0-54.0)*0.67) +
38*((53.0-40.0)*1.0))*150 lbs/ft^3= 134,400 #**

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**b) Waterbox floor:**

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**(18'*40'*1')*150 lbs/ft^3=
108,000 #**

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**c) Generating room floor:**

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**(18'*40'*1')*150 lbs/ft^3=
108,000 #**

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**d) Water:**

** **

**18'*34'*(50.5-31.5)*62.4
lbs/ft^3= 725,587 lbs**

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**5) Total Distributed Load:**

** **

**(134,400 # + 31,418 # +
216,000 #/2 + 725,587 #/2)/40'= 15,915**

**lbs/ft-width beam**

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**6) Sketch the beam and the
loading diagram:**

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**
(7)**

**C) Determine the extreme fiber
stress and the maximum deflection:**

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**1) Even though the beam will
be embedded into the walls on either side, **

**the weight of the wall
immediately above the beam will not counter act **

**the loading and one should
assume that the beam is simply supported. **

**This is a conservative
assumption and will result in the largest **

**moment.**

** **

**2) Maximun Moment:**

** **

**15,915 lb/ft * (18.0'^2)/8 =
644,569 ft-lbs**

** **

**3) The extreme fiber stress
is:**

** **

**Sigma = M/Z = (644,569 ft-lbs
* 12"/ft)/355 in^3 = 21,788 psi**

** **

**4) From the AISC "Manual of
Steel Construction", 7th edition, P. 5-124, **

**section 1.5.1.4.1, the
allowable bending stress for W shapes is **

**0.66*Fy, where Fy is 36,000
psi.**

** **

**Therefore, the recommended
maximum load is:**

** **

**Sigma max= 0.66*36,000 psi=
23,760 psi <<<<<<<<*******

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**15) Since the calculated
stress is less then the maximum recommended by **

**AISC Code, the 30" beam is
alright for strength.**

** **

**16) Determine the maximum
deflection:**

** **

**a) For uniformly distributed
load with built in ends:**

** **

**delta max= w*l^4/(384*E*I)**

** **

** = 15,915 lb/ft *
(18.0'*12")^4/(384*29,000,000 psi* 5360 **

** in^4*12)= 0.58 inch**

** **

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**17) The calculated deflection
is negligible and should not effect the**

**concrete resting upon the
beam, especially since the major deflection**

**will take place when the
concrete is wet, before it dries.**

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**
(8)**

**V) Design the turbine gate
case support beams:**

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**A) Sketch the design:**

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**
Plan View**

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**B) Sketch the beam loading:**

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**C) Determine the beam size:
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**1) Size the beam at 80 % of
the AISC Codes maximum allowable stress or**

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**0.8 * 21,600 psi= 17,280 psi**

**2) Maximum moment is 8' * 3500
# = 28,000 ft-lbs**

** **

**3) Z= M/sigma= (28,000 ft-lbs
* 12 in/ft)/17,280 psi= 19.5 in^3**

** **

**D) From the AISC Handbook, 7th
ed., P. 1-42, tentatively choose a**

**W8"x24 lbs/ft**

** **

**E) Beam Properties:**

** **

** height= 7.93"**

** weight= 24 lbs/ft-width**

** length= 18 feet**

** section modulus= 20.8 in^3**

** moment of inertia= 82.5
in^4**

** depth= 7.93"**

** width= 6.50"**

** flange thickness= 0.398"**

** web= 0.245"**

**
(9)**

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**F) Check the deflection:**

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**Sigma max= P*L^3/(48*E*I)=
3500# * (15.5'*(12"/ft)^3)/(48 * 29,000,000**

**psi * 82.5 in^3= 0.00006 inch
<<<< okay**

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**The choosen beam will meet the
design requirements.**

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**VI) Design the generator
support beams:**

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**A) Sketch the design:**

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**
Plan View**

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**B) Sketch the beam loading:**

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**C) Determine the beam size:**

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**1) Size the beam at 80 % of
the AISC Codes maximum allowable stress or**

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**0.8 * 21,600 psi= 17,280 psi**

** **

**2) Maximum moment is 9.0' *
(21,636 #+ 17,600 #)/2 = 176,562 ft-lbs**

** **

**3) Z= M/sigma= (176,562 ft-lbs
* 12 in/ft)/17,280 psi= 123 in^3**

** **

**D) From the AISC Handbook, 7th
ed., P. 1-42, tentatively choose a**

**W12"x92 lbs/ft**

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**
(10)**

**E) Beam Properties:**

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** height= 12.62"**

** weight= 92 lbs/ft-width**

** length= 18 feet**

** section modulus= 125 in^3**

** moment of inertia= 789 in^4**

** depth= 12.62"**

** width= 12.155"**

** flange thickness= 0.856"**

** web= 0.545"**

**
**

**F) Check the deflection:**

** **

**Sigma max= P*L^3/(48*E*I)=
19618# * (18'*(12"/ft)^3)/(48 * 29,000,000**

**psi * 125 in^3= 1.1 inch <<<<
okay**

** **

**The chosen beam will meet the
design requirements. The deflection is **

**high, but in reality, the
beams will be supported along their edges by **

**the rigid generating room
floor**

**
**

**VII) Design the generator
floor:**

** **

**A) Dimensions- - from Fay
Engineering Services layout dated 12/27/89**

**entitled "Powerhouse
Longitudinal Section". Assume these dimensions are**

**correct.**

** **

**1) External dimensions:**

** **

**a) length = 40.0'**

** **

**b) width = 18'**

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**2) Internal dimensions:**

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**a) length = 36.0'**

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**b) width = 16.0'**

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**B) Loading:**

** **

**1) Concrete, total load = 150
#/ft3 * 16' * 36'***

** **

**
(12"/(12"/ft)) = 86,400 lbf**

** **

**2) Concrete, load per foot
width of box = 86,400 lbf/36' = 2400 lbf/ft**

**width of box**

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**3) Concrete, load per foot
width of box per foot width of beam =**

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** 2400 lbf/ft-width/16' = 150
lbf/ ft**

**
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**
(11)**

**C) Sketch of typical section
through proposed generator floor:**

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**
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**D) Design Calculations: -
reference, "Design of Concrete Structures",**

**9th ed., Winter & Nilson.**

** **

**1) Assume the yield strength
of the steel reinforcing is 30,000 psi and**

**that the compressive strength
of the concrete is 3000 psi.**

** **

**2) Select the trial thickness
of the slab, use L/20 from Table 5.1,**

**p.206 in Winter & Nilson.**

** **

**T= (12 "/ft * 16')/20 = 9.6"
approximately = 12"**

** **

**3) The slab weight is 150
#/ft3 * (12/12) = 150 PSF**

** **

**4) Apply the ACI load
multipliers and obtain the factored load:**

** **

**Dead Load = 150 PSF * 1.4 =
210.0 PSF**

** **

**Live Load = 300 PSF * 1.7 =
510.0 PSF**

** **

**Total Factored Load = 720 PSF**

** **

**5) Use the ACI moment
coefficients to determine the design moments at**

**the critical sections:**

** **

** a) Since the floor slab is
being designed as a one-way slab in the**

**short direction (ie: from the
inlet end to the tailrace end), the slab**

**will be resting on the main
30" beam which acts as the arch at the**

**front of the powerhouse, over
the inlets and will be built into the top**

**of the rear (downstream wall)
of the water boxes. At the inlet end, the**

**floor slab is simply supported
and the beam is free to twist and cannot**

**be assumed to be rigid, so
use:**

** **

** (1/11)*Wu*ln^2 >>>>>>>>from
Table 8.1 W&N**

** **

**Sketch: **

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**
(12)**

** **

**b) At the upstream end, the
slab is to be built into the discharge pit**

**wall and can be assumed to be
rigid, so use:**

** **

**(1/14)*Wu*ln^2>>>>>>>>from
Table 8.1 W&N**

** **

**Sketch:**

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**c) At the interior span use:**

** **

**(1/16)*Wu*ln^2>>>>>>>>from
Table 8.1 W&N**

**d) At the inlets: -M =
1/11*0.720 KSF *14'^2 = 12.8 ft-kips**

** **

**e) At the downstream end: -M =
1/14*0.720 KSF *14'^2 = 10.1 ft-kips**

** **

**f) At the midspan: -M =
1/16*0.720 KSF *14'^2 = 8.8 ft-kips**

** **

**5) Determine the maximum steel
ratio permitted by the ACI Code:**

** **

**Pmax = 0.75*Pbalanced =
0.75*0.85*B1*(fc'/fy)*(87,000/(87,000+fc'))**

** **

**this formula for fc'<4000 psi
and B1=0.85**

** **

**Pmax=0.75*0.85*0.85*(3000
psi/30,000 psi)*(87,000/(87,000 + 30,000))**

** **

** = 0.04**

** **

**6) Determine the minimum
required effective depth: (This is controlled**

**by the largest moment at the
inlet end)**

** **

**d^2 =
Mu/(phi*p*fy*b*(1-(0.59*p*fy/fc))) note: phi=0.9 for bending**

** **

** =
(12.8ft-kips*(12"\ft))/(0.9*0.04*30*12*(1-.59*0.04*(30,000/3000)))**

** **

** = 15.51 in^2**

** **

**Therefore, d = 3.9 inchs**

** **

**7) Determine the minimum
effective depth using code restrictions:**

** **

**dm= 12"- 1" = 11"**

** **

**8) Since the calculated value
of 4.7 inches is less then the coded**

**effective depth, use d= 11
inches**

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**
(13)**

**9) At the tailrace end, assume
the stress block depth a = 1.00 inch.**

**Then the area of steel
required per foot width in the top of the slab**

**is:**

** **

**As= Mu/(phi*fy*(d-a/2))= (12.8
ft-kips*12"/ft)/.9*30*(11-1/2)= 0.54 in^2**

** **

**10) Check the assumed depth:**

** **

**a= As*fy/(0.85*fc'*b)= 0.54
in^2 *30,000/(0.85*3000*12"/ft)= 0.53 in**

** **

**11) Reiterate assuming a=0.53
in:**

** **

**As=Mu/(phi*f*(d-a/2))=(12.8
ft-kips*12"/ft)/0.9*30*(11-.53/2)= 0.52in^2**

** **

**12) Reiterate assuming As=0.52
in^2:**

** **

**a= As*fy/(0.85*fc'*b)= 0.52
in^2 *30,000/(0.85*3000*12"/ft)= 0.51 in**

** **

**13) Reiterate assuming a=0.51
in:**

** **

**As=Mu/(phi*f*(d-a/2))=(12.8
ft-kips*12"/ft)/0.9*30*(11-0.51/2)= 0.53**

**in^2**

** **

** **

**14) The assumed area of steel
and the calculated area of steel are**

**reasonably close so use 0.53
in^2 of rebar per foot width of floor slab.**

** **

**15) At the other critical
sections use the same lever arm to determine**

**the required cross sectional
areas of steel rebar:**

** **

**a) at the midspan: As= 10.1
ksi*12/(0.9*30*(11-0.51/2))= 0.42 in^2**

** **

**b) at the upstream wall: As=
8.8 ksi*12/(0.9*30*(11-0.51/2))= 0.36 in^2**

** **

**16) The minimum reinforcement
required to control shrinkage is: see p.**

**207, W&N.**

** **

**As= 0.002*12*12= 0.288 in^2/
12" with strip**

** **

**The required steel necessary
for shrinkage is met by the steel required**

**to meet the externally applied
loads.**

** **

**17) Determine the factored
shear force:**

** **

**Vu=
1.15*(720*18/2)-720*(12/12)= 7452-720 = 6372 lbs**

** **

**15) The nominal shear strength
of the slab is:**

** **

**Vn= Vc= 2*b*d*fc'^0.5**

** **

** = 2*12*12*(3000 psi^0.5)**

** **

** = 15,774 lbs**

** **

**
(14)**

** **

**16) The design shear strength
is:**

** **

** phi*Vc= 0.85*15,774 lbs=
13,408 lbs**

** **

**17) Since the design shear
strength is above the required shear strength**

**by 15 %, no additional steel
is necessary to resist the internal shear**

**forces.**

** **

** **

**VIII Design Waterbox Walls:**

** **

**A) Dimensions- - from Fay
Engineering Services layout dated 12/27/89**

**entitled "Powerhouse
Longitudinal Section". Assume these dimensions are**

**correct.**

**1) External dimensions:**

** **

**a) height= 14'**

** **

**b) width = 18'**

** **

**2) Internal dimensions:**

** **

**a) length = 14'**

** **

**b) width = 18'**

** **

**B) Loading:**

** **

**1) Concrete, total load = 150
#/ft3 * 14' * 18'* (12"/(12"/ft)) = 37,800**

**lbf**

** **

**2) Concrete, load per sq.
ft. = 37,800 lbf/40' = 150 lbf/ft^2**

** **

**3) Water Load= 62.4 lbf/ft^3 *
14' wide * 18' deep * 1' thick/ 14'wide**

** = 1123 PSF. This
is the maximum load at the base of the**

** hydrostatic
load.**

** **

**
**

**C) Sketch of typical section
through proposed waterbox wall:**

** **

** **

** **

** **

** **

** **

** **

** **

** **

**
**

** **

** **

** **

** **

** **

**
(15)**

**D) Design Calculations: -
reference, "Design of Concrete Structures",**

**9th ed., Winter & Nilson.**

** **

**1) Assume the yield strength
of the steel reinforcing is 30,000 psi and**

**that the compressive strength
of the concrete is 3000 psi.**

** **

**2) Select the trial thickness
of the slab, use L/20 from Table 5.1,**

**p.206 in Winter & Nilson.**

** **

**T= (12 "/ft * 18')/20 = 10.8"
approximately = 11"**

** **

**3) The slab weight is 150
#/ft3 * (12/12) = 150 PSF**

** **

**4) Apply the ACI load
multipliers and obtain the factored load:**

** **

**Dead Load = 150 PSF * 1.4 =
210 PSF**

** **

**Live Load = 1123 PSF * 1.7 =
1910.0 PSF**

** **

**Total Factored Load = 2120 PSF**

** **

**5) Use the ACI moment
coefficients to determine the design moments at**

**the critical sections:**

** **

** a)The wall is rigidly built
into the floor slabs at the top and **

**bottom, so use:**

** **

** (1/14)*Wu*ln^2 >>>>>>>>from
Table 8.1 W&N**

** **

**Sketch:**

** **

** **

** **

** **

** **

** **

** **

** **

**
**

** **

**b) At the interior span use:**

** **

**(1/16)*Wu*ln^2>>>>>>>>from
Table 8.1 W&N**

**c) At the top: -M = 1/14*2.2
KSF *14'^2 = 30.8 ft-kips**

** **

**d) At the bottom: -M =
1/14*2.2 KSF *14'^2 = 30.8 ft-kips**

** **

**e) At the midspan: -M =
1/16*2.2 KSF *14'^2 = 27.0 ft-kips**

** **

**5) Determine the maximum steel
ratio permitted by the ACI Code:**

** **

**Pmax = 0.75*Pbalanced =
0.75*0.85*B1*(fc'/fy)*(87,000/(87,000+fc'))**

** **

**this formula for fc'<4000 psi
and B1=0.85**

**
(16)**

**Pmax=0.75*0.85*0.85*(3000
psi/30,000 psi)*(87,000/(87,000 + 30,000))**

** **

** = 0.04**

** **

**6) Determine the minimum
required effective depth: (This is controlled**

**by the largest moment at
either the top or the bottom)**

** **

**d^2 =
Mu/(phi*p*fy*b*(1-(0.59*p*fy/fc))) note: phi=0.9 for bending**

** **

** =
(30.8ft-kips*(12"\ft))/(0.9*0.04*30*12*(1-.59*0.04*(30,000/3000)))**

** **

** = 35.2 in^2**

** **

**Therefore, d = 5.9 inchs**

** **

**7) Determine the minimum
effective depth using code restrictions:**

** **

**dm= 12"- 1" = 11"**

** **

**8) Since the calculated value
of 5.9 inches is less then the coded**

**effective depth, use d= 11
inches**

** **

**9) At the tailrace end, assume
the stress block depth a = 1.00 inch.**

**Then the area of steel
required per foot width in the top of the slab**

**is:**

** **

**As= Mu/(phi*fy*(d-a/2))= (30.8
ft-kips*12"/ft)/.9*30*(11-1/2)= 1.3 in^2**

** **

**10) Check the assumed depth:**

** **

**a= As*fy/(0.85*fc'*b)= 1.3
in^2 *30,000/(0.85*3000*12"/ft)= 1.27 in**

** **

**11) Reiterate assuming a=1.27
in:**

** **

**As=Mu/(phi*f*(d-a/2))=(30.8
ft-kips*12"/ft)/0.9*30*(12-1.27/2)= 1.2 in^2**

** **

**12) Reiterate assuming As=1.20
in^2:**

** **

**a= As*fy/(0.85*fc'*b)= 1.20
in^2 *30,000/(0.85*3000*12"/ft)= 1.18 in**

** **

**13) Reiterate assuming a=1.18
in:**

** **

**As=Mu/(phi*f*(d-a/2))=(30.9
ft-kips*12"/ft)/.9*30*(12-1.18/2)= 1.20in^2**

** **

**14) The assumed area of steel
and the calculated area of steel are**

**reasonably close so use 1.20
in^2 of rebar per foot width of floor slab.**

** **

**15) At the other critical
sections use the same lever arm to determine**

**the required cross sectional
areas of steel rebar:**

** **

**a) at the midspan: As= 30.8
ksi*12/(0.9*30*(12-1.18/2))= 1.20 in^2**

** **

**b) at the bottom: As= 27
ksi*12/(0.9*30*(12-1.18/2))= 1.05 in^2**

** **

** **

**
(17)**

**16) The minimum reinforcement
required to control shrinkage is: see p.**

**207, W&N.**

** **

**As= 0.002*12*12= 0.288 in^2/
12" with strip**

** **

**The required steel necessary
for shrinkage is met by the steel required**

**to meet the externally applied
loads.**

** **

**17) Determine the factored
shear force: Note that the dead weight of **

**the vertical concrete does not
add to the shear component.**

** **

**
Vu=1.15*1910*14/2-2120*(12/12)= 15,375-2120 = 13,255 lbs**

** **

**15) The nominal shear strength
of the slab is:**

** **

**Vn= Vc= 2*b*d*fc'^0.5**

** **

** = 2*12*12*(3000 psi^0.5)**

** **

** = 15,774 lbs**

** **

**16) The design shear strength
is:**

** **

** phi*Vc= 0.85*15,774 lbs=
12,897 lbs**

** **

**17) The design shear strength
is slightly less then the required shear**

**strength. However, the
differential is small and no additional steel is**

**necessary to resist the
internal shear forces.**

** **

**18) Use 1.2 in^2 of steel for
the vertical reinforcement.**

** **

**IX) Design the discharge pit
walls:**

** **

**A) Assume the pit is dewatered
and drained externally. The height of**

**water is 31.5-20.5=11'**

** **

**B) This design is similiar to
the waterbox walls.**

** **

**C) Use 1.2 in^2 of steel per
foot width of wall.**

** **

** **

**X) Design main 30" beam
support column:**

** **

**A) Determine the load on the
column:**

** **

**1) Sketch the freebody
diagram:**

** **

** **

** **

** **

** **

** **

** **

** **

** **

**
(18)**

**Ra=((20,600 lbs +
(16,000*18'))/2 = 154,300 lbf**

** **

**2) the factored ACI Code load
is:**

** **

**Pu=((1.4*(16,000 lbf * 18')/2)
+ (1.7 * 20,600/2)**

** **

** = 201,600 lbf + 17,510 lbf =
219,110 lbf**

** **

**B) Determine the nominal axial
load strength of the column, Po, **

**assuming minimal eccentricity:**

** **

**1) Po= 0.85 * fc' * Aconc + fy
* Ast**

** **

**2) By ACI Code, the ratio of
the longitudinal steel area to the gross **

**column area must be:**

** **

**0.01 <= Pg <= 0.08. Try 0.025
to start.**

** **

**3) Assume the column is 12"
square, built into the discharge pit walls **

**and has a steel plate between
the WF beam flange and the concrete to **

**transmit the load.**

** **

**4) The gross area of the
column, Ag = 144 sq. in.**

** **

**5) Areas of steel are:**

** **

**Ast = Pg * Ag = 0.025 * 144
in. sq. = 3.6 in. sq.**

** **

**6) Po = 144 in. sq. *
(0.85*(1-0.025) * 3000 psi + 30,000 psi * 0.025**

** **

** = 466,020 lbf**

** **

**C) Determine the ACI factored
design strength:**

** **

**Pdesign = phi*Po = 0.80 * 0.70
* Po**

** **

** = 0.8 * 0.7 * 466,020
lbf**

** **

** = 260,971 lbf**

** **

**D) Since the factored ACI
design strength is greater then the factored **

**ACI load, this design will
work**

** **

**phi * Po > Pu >>>>>>>>>>>>
260,971 lbf > 219,110 lbf**

** **

**E) Determine the number of
bars and their size:**

** **

**1) Assume six bar design**

** **

**2) Area of single bar = Ast/#
bars**

** **

** = 3.6
in^2/6 = 0.60 sq. in.**

** **

**3) From ASTM rebar, table #7
bars are 0.6 in. sq.**

** **

**
(19)**

**F) Sketch design:**

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

**1) minimum spacing is 1.5 *
0.875 = 1.31"**

** **

**2) minimum tie wire size is #4**

** **

**3) concrete cover must be 1
1/2" thick**

** **

**4) every corner to be
supported by a tie wire**

** **

**5) tie wire spacing shall be
every 14" of column rise**

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

**G) Check the slenderness ratio
of the column:**

** **

**1) SR = K*Lu/r =
K*Lu/0.3*W K = 1 for unbraced columns**

**
Lu = 39 - 21 = 18'**

**
w = 12"**

** SR = 1 * 18'/(0.3*1') = 60**

** **

**Since 60>22 this design is
slender**

** **

**2) Determine w for SR= 22
minimum**

** **

**W = KLu/(0.3*SR) =
18'/(0.3*22) = 2.73' = 33"**

** **

**H) Retrofit design so that
wall is 33" thick for the 12" length of the **

**wall which the column is
embedded into.**

** **

** **

** **

** **

** **

** **

** **

**
(20)**

**I) Sketch design:**

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

**XI) Check the rock foundation
bearing capacity:**

** **

**A) Total weight of the
powerhouse and equipment:**

** **

**Water + 2 floors + machinery +
back & front wall + side walls =**

** **

**400,982 lbf + 100,000 lbf +
36,200 lbf + 432,000 lbf + 388,000 lbf =**

** **

**1,357,182 lbf**

** **

**B) Total surface bearing area
is:**

** **

**(40' + 2*18') * 1' = 76 sq.
ft.**

** **

** **

**C) The stress on the rock is:**

** **

**1,357,182 lbf/76 sq. ft. =
17,858 lbf/ft^2 = 8.92 tons/sq. ft.**

** **

**D) "American Civil Engineer's
Handbook", Merriman & Wiggin, 5th edition,**

**P. 711, table lists the
allowable soil pressures in short tons per sq.**

**ft. for very hard native
bedrock at 9 tons/sq.ft.. A short ton is 2000**

**lbs. Since the calculated
pressure of 4.5 tons/sq.ft. is much less then**

**15 tons/sq.ft. This design
should be all right if the walls are poured**

**directly on the rock
excavation. The footing should be chipped square**

**and level before the forms are
set up.**

** **

**XII) Rebar size and spacing
selection:**

** **

**A) Waterbox Floor:**

** **

**1) To obtain 1.72 in. sq./ft.
width slab, use one #8 bar every six**

**inches.**

** **

**2) Use #6 bar at 18 inch
spacing in the longitudinal spacing.**

** **

**B) Generating Room Floor:**

** **

**1) To obtain 0.53 in. sq./ft.
width slab, use one #7 bar every 12**

**inches.**

** **

** **

**
(21)**

** **

**2) Use #5 bar at 18 inch
spacing in the longitudinal spacing.**

** **

**C) Walls:**

** **

**1) To obtain 1.20 in. sq./ft.
width slab, use one #7 bar every 6**

**inches.**

** **

**2) Use #5 bar at 12 inch
spacing in the longitudinal spacing.**

** **

** **

** **

**
**

**ADDITIONAL CALCULATIONS
**

**July 19, 1990**

** **

** **

**A) Lap Splice Lengths -**

** **

**In tension**

** **

**- Bars are not larger than #11**

**- Minimum length of lap for
tension lap splices shall be as **

** required: Class A = 1.0 L,
Class B = 1.3 L and Class C = 1.7 L**

** **

**where: L > 0.0004*D*Fy*1.25¯D
= nominal diameter of bar**

**Fy = specified yield strength**

** = 30000 psi**

** L = 13.1 inches**

** **

** and L >
0.04*Ab*Fy/(Fc)^.5¯¯Ab = Area of individual ba**

** **

**Fc = specified compression**

**stength of concrete**

** = 4000 psi **

** **

**L = 14.2 inches, Class C = 24
inches**

** **

**In Compression **

** **

**L = 0.02*D*Fy/(Fc)^.5**

** **

** = 9.5 inches**

** **

**Thus, the longest the lap
splice would need to be in any condition **

**would be 24 inches long.**

** **

** **

**B) Make sure that the extra
circular reinforcement around holes in**

** the generator room floor
and water box floor is equivalent or **

** greater than the
reinforcement cut to create the holes.**

** **

** Maximum area cut = 2 - #7
Bars**

**
= 2[3.14*(0.875)^2]/4**

**
= 1.20 sq. in.**

** **

**
(22)**

** **

** Area added = 4 - #5 Bars**

**= 4[3.14*(0.625)^2]/4**

**= 1.22 sq. in.**

** **

** In the other case 4 - #8
bars are cut and 4 - #8 bars are added for**

**reinforcement.**

** **

**Therefore, in each case there
is enough reinforcement added to be**

**equivalent or greater than the
cross-sectional area cut to create the**

**holes.**

** **

** **

**C) Local Buckling Calculations
- addition of ed stiffeners**

** **

**Midsection of 30" I-beam**

** **

**Nmin = [reaction/(0.75)*Fy*tw]
- k; where Fy = yield strength**

** of steel**

**tw = web thickness**

**k = flange-to-web toe**

**fillet distance**

**Nmin = [(2)(154300
lbs)/(0.75)(30000 psi)(0.563)] - 1.06**

**= 22.7 inches **

** **

**Two ends of 30" I-beam**

** **

**Nmin = [(154300)/(0.75)(30000
psi)(0.563)] - 1.06**

**= 11.1 inches**

** **

**Turbine gate case support
beams**

** **

**Nmin = [(3500
lbs)/(0.75)(30000)(.25)] - 0.875**

**= 0 inches**

** **

**Generator support beams**

** **

**Nmin =
[(17600)/(0.75)(30000)(0.563)] - 1.563**

**= 0 inches**

** **

**D. Check for plate bearings
on end of beams in concrete wall**

** **

**Design for Fp = 0.35 fc
(Civil Eng Ref Manual, 4th Ed.)**

** = 1050 psi**

** **

**Turbine gate case support
beams**

** **

**Fp = 1050 psi > 1750
lbs/(6.5")(24") = 11 psi (no plate needed)**

** **

**Generator support beams**

** **

** Fp = 1050 psi >
17600/(12.13)(24) = 30 psi (no plate needed)**

** **

** **

** **

**
-23-**

**Main 30" support beam**

** **

**A = P/F = 310,000 lbs/1050 psi
= 298 sq in**

** **

**Need to spread load out onto
298 sq in, at present we have**

**a 24 inch X 30 inch steel
plate**

** **

**Area of steel plate =
(24")(30") = 720 sq in**

** **

**So the load is spread out
enough**

** **

**F = 310000 lbs/720 sq in = 431
psi < 1050 psi**

** **

** **

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**
-24-**