INTRODUCITON
Prestressed concrete circular tanks are usually the best combination of structural form and material for the storage of liquids and solids. Their performance over the past half- century indicates that, when designed with reasonable skill and care, they can function for 50 years or more without significant maintenance problems. The first effort to introduce circumferential prestressing into circular structures was that of W. S. Hcwett, who applied the tie rod and turnbuckle principle in the early ea rly 1920s. 1920s . But the reinforcing steel available a vailable at that time had very low yield strength, limiting the applied tension to not more than 30,000 to 35,000 psi (206.9 to 241.3 MPa). Indeed, significant long-term losses due to concrete creep, shrinkage, and steel relaxation almost neutralized the prestressing force. As higher strength steel wires became available, J. M. Crom, Sr., in the 1940s, successfully developed the principle of winding high- tensile wires around the circular walls of prestressed tanks. Since that time, over 3,000 circular storage structures have been built of various dimensions up to diameters in excess of 300 feet (92 m).
The major advantage in performance and economy of using circular prestressing in concrete tanks over regular reinforcement is the requirement that no cracking be allowed. The circumferential “hugging” hoop stress in compression provided by external winding of the prestressing wires around the tank shell is the natural technique for eliminating cracking in the exterior walls due to the internal liquid, solid, or gaseous loads that the tank holds. Other techniques of circumferential prestressing using individual tendons which are anchored to buttresses have been more widely used in Europe than in North America for reasons of local economy and technological status.
Containment vessels utilizing circumferential prestressing, which can be either situ-cast or precast in segments, include water storage tanks, wastewater tanks and effluent clarifiers, silos, chemical and oil storage tanks, offshore oil platform structures, cryogenic vessels, vess els, and nuclear reactor pressure pr essure vessels. All these structures are considered thin shells because of the exceedingly small ratio of the container thickness to its diameter. Because no cracking at working-load levels is permitted, the shells are expected to behave elastically under workingload and overload conditions. When the prestressed members are curved, in the direction of prestressing, the prestressing is called circular prestressing. For example, circumferential
prestressing in pipes, tanks, silos, containment structures and similar structures is a type of circular prestressing. In these structures, there can be prestressing in the longitudinal direction (parallel to axis) as well. Circular prestressing is also applied in domes and shells.
The circumferential prestressing resists the hoop tension generated due to the internal pressure. The prestressing is done by wires or tendons placed spirally, or over sectors of the circumference of the member. The wires or tendons lay outside the concrete core. Hence, the centre of the prestressing steel (CGS) is outside the core concrete section.
The hoop compression generated is considered to be uniform across the thickness of a thin shell. Hence, the pressure line (or C-line) lies at the centre of the core concrete section (CGC). The following sketch shows the internal forces under service conditions. The analysis is done for a slice of unit length along the longitudinal direction (parallel to axis).
Internal forc es under service conditions
To reduce the loss of prestress due to friction, the prestressing can be done over sectors of the circumference. Buttresses are used for the anchorage of the tendons. The following sketch shows the buttresses along the circumference.
DESIGN PRINCIPLES AND PROCEDURES:
Considering the behaviour of circular tanks involves examining both the interior pressure due to the material contained therein acting on a thin-walled cylindrical shell cross section and the exterior radial and sometimes vertical
prestressing forces balancing the interior forces. The interior pressure is horizontally radial, but varies vertically depending on the type of material contained in the tank. If the material is water or a similar liquid, the vertical pressure distribution against the tank walls is triangular , with maximum intensity at the base of the wall. Other liquids which are accompanied by gas would give a constant horizontal pressure throughout the height of the wall. The vertical pressure distribution in tanks used for storage of granular material such as grain or coal would be essentially similar to the gas pressure distribution, with a constant value along most of the depth of the material contained. Figure shows the pressure distributions for these three cases of loading. The basic elastic theory of cylindrical shells applies to the analysis and design of the walls of prestressed tanks. A ring force causes ring tension in the thin cylindrical walls, assumed unrestrained at the ends at each horizontal section. The magnitude of the force is proportional to the internally applied pressure, and no vertical moment is produced along the height of the walls. If the wall ends are restrained, the magnitude of the ring force changes and a bending moment is induced in the vertical section of the tank wall. The magnitudes of the ring forces and vertical moments are thus a function of the degree of restraint of the cylindrical shell at its boundaries and are computed from the elastic shell theory and its simplifications and idealizations to be discussed subsequently. Liquid Load and Freely Sliding Base. From basic mechanics, the ring force is
and the ring stress is
Where d = diameter of cylinder, r = radius of cylinder, t = thickness of wall core, p = unit internal pressure at wall base = H, = unit weight of material contained in vessel.
Figure: Tank internal pressure diagrams, (a) Tank cross section, showing radial shear Q0 and restraining moment M0 at base for fixed-base walls, (b) Liquid pressure, triangular load, (c) Gaseous pressure, rectangular load, (d) Granular pressure, trapezoidal load.
The tensile ring stress at any point below the surface of the material contained in the vessel becomes
where H is the height of the liquid contained and y is the distance above the base. The corresponding ring force is
The maximum tensile ring stress at the base of the freely sliding tank wall for y = 0 becomes, Gaseous Load on Again from basic principles of ring stress is
Freely Sliding Base. mechanics, the constant tensile
Note that while theoretically the centreline diameter dimension is more accurate to use, the ratio t/d is so small that the use of the internal diameter d is appropriate. Liquid and Gaseous Load on a Restrained Wall Base. If the base of the
wall is fixed or pinned, the ring tension at the base vanishes. Because of the restraint imposed on the base, the simple membrane theory of shells is then no longer applicable, due to the imposed deformations of the restraining force at the wall base. Instead, bending modifications to the membrane stresses become necessary. And the deviation of the ring tension at intermediate planes along the wall height must be approximated.
If the vertical bending moment in the horizontal plane of the wall at an y height is My, the flexural stress in compression or tension in the concrete becomes
The distribution of the flexural stress across the thickness of the tank wall is shown in Figure
Ring tension and flexural stresses, (a) Ring tension internal force F in the horizontal section, (b) Flexural stress due to bending moment M in the wall thickness of the vertical section.
Ring Shear Q0 and Moment M 0 Gas Cont ainment If the edges of the shell are free at the wall base, the internal pressure produces only hoop stress f R = pr/t and the radius of the cylinder increases by the amount
Also, for full restraint at
And
Solving for Mo and Qo gives
And
the wall base,
CYLINDRICAL SHELL MEMBRANE COEFFICIENTS: The bending moment at any level along the height above the base of a cylindrical tank can be computed from the bending moment expression for a cantilever beam. This is accomplished by multiplying the cantilever moment values by coefficients whose magnitudes are functions of the geometrical dimensions of the tank and which are termed membrane coefficients. The basic moment expressions developed for the circular container can be rearranged into a factor H 2 /dt denoting geometry and a factor H 3 or pH 2 denoting cantilever effect, for liquid and gaseous loading, respectively. The tank constant is a function of rt or dt, where d is the tank diameter. Using poison’s ratio 0.2 for concrete, we have
⁄ 31 1. 3 0 1. 8 4 = ⁄ = ⁄ = ⁄ //⁄⁄ /
since =
The factor 1/H used in the basic bending expressions of 1.84/
⁄
. The product y con also be rewritten in terms of
using
y =H, where y is the height above the base.
Consequently, the moment M y in a wall section a distance y above the base can be represented in terms of the form factor or pH2 as follows:
and the cantilever factor H3
M y = numerical variant x form factor x cantilever factor Or
= × × / = =
the form factor is constant for the particular structure being designed. Hence, the product of the variant and the form factor produces the membrane coefficient C , so that
for a liquid load and
For a gaseous load.
/
The membrane coefficients C for various form factors and most expected boundary and load conditions. They significantly reduce the
computational efforts normally required in the design and analysis of shells, without loss of accuracy in the results. Using the membrane coefficients for the solution of the circular tank forces and moments should give results reasonably close.
PRESTRESSING EFFECTS ON WALL STRESSES FOR FULLY HINGED, PARTIALLY SLIDING AND HINGED, FULLY FIXED, AND PARTIALLY FIXED BASES: The liquid or gas contained in a cylindrical tank exerts outward radial pressure h or p on the tank walls, inducing ring tensions in each horizontal section of wall along its height. This ring tension in turn causes tensile stresses in the concrete at the outside extreme wall fiberes, resulting in impermissible cracking. To eliminate the cracking that causes leaks and structural deterioration, external horizontal prestressing is applied which induces inwards radial thrust that can balance the outward radial tension. Additionally in order to prevent the development of cracks in the inside walls when the tanks is empty, vertical prestressing is induced to reduce the residual tension within the range of the modulus of rupture of the concrete and with an adequate safety factor. In order to ensure against the development of cracking at the outside face of the tank wall, it is good practice to apply somewhat larger horizontal prestressing forces than are required to neutralize or balance the outward radial forces caused by the internal liquid or gas, thereby producing residual compression in the tank when it is full. Such an increase in circumferential prestressing forces through the use of additional horizontal prestressing steel, and sometimes mild vertical steel, also counteracts the effects of temperature and moisture gradients across the wall thickness in an adverse environment.
FREELY SLIDING WALL BASE: When the boundary condition is such that the wall at its base can freely slide when the tank is internally loaded, there is no moment in the vertical wall due either to liquid load or to prestressing when the tank is totally filled to height H. only a small nominal moment develops when the tank is partially filled, partially prestressed, or empty, and no vertical prestressing is necessary. The deflected shape of the freely sliding tank
While free sliding is an ideal condition that renders the structure statically determinate and hence most economical, it is difficult to achieve in practice. Frictional forces produced at the wall base after the tank becomes operational and the difficulty of achieving liquid tightness render this alternative essentially implementable.
Hinged Wall Base: For wall with a hinged connection to the base, the maximum radial forces due to the liquid retained and the prestressing at eh critical section a distance y above the base are almost equal to those in the freely sliding case at height y. but vertical moments are introduced, and vertical prestressing becomes necessary to reduce the tensile stresses in the concrete at the outer wall face.
In order to minimize the possibility of cracking, a residual ring compression of a minimum value of 200 psi (1.38 MPa) is necessary for wirewrapped prestressed tanks without diaphragms, and 100 psi (0.7MPa) for tank with a continuous metal diaphragm. The maximum tension at the inside face of the wall should not exceed
3√
at working load level. The deflected shape of
the tank walls and the stress variations in the concrete across the thickness of the section when the tank is empty and when it is full. For tank prestressed with pretension and post-tensioned tendons, the minimum residual compressive stress should be as stipulated.
RECOMMENDED PRACTICE FOR PRECAST PRESTRESSED CONCRETE CIRCULAR STORAGE TANKS Stresses General guidelines for precast prestressed concrete circular storage tanks are provided by the Prestressed Concrete Institute (Ref. 11.10, the American Concrete Institute (Refs. 11.7-11.9), and the Post Tensioning Institute (ref. 11 .10) for choosing the applicable allowable stresses, dimensioning, minimum wall thickness, and construction and erection procedure. The allowable stresses in concrete and shotcrete are given in table (Ref. 11.7), with modifications to accommodate the recommended stresses in Ref. 11.6. Allowable stresses in the reinforcement are given in table (Ref. 11.6). Table (Ref. 11.7) Type and limit of stress
Axial compression, f c
Concrete situ-cast and precast Temporarya Service load stresses f ci, stresses f c, psi psi 0.55f’ ci 0.45 f’ c
Shotcrete situ-cast Temporarya Service load stresses f gi, psi stresses f g, psi 0.38 f’ g 0.45f’ gi But not more
Axial tension Flexural compression, f c Maximum flexural tensionb, f t Minimum residual compression, f cv a
0
0
0.55 f’ ci
0.4 f’ c
≅ 3√
3√
200 psi
than 1600+40 tc psi 0 0.45 f’ gi
0
0.38 f’ g 200psi
Before creep and shrinkage losses. Fiber stress in precomposed tension zone.
b
Table Ref. 11.6
Type of stress Tendon jacking force Immediately after prestress transfer Post-tensioning tendons at anchorage and couplers, immediately after tendon anchorage Service load stress, f pe Nonprestressed mild steel at intital prestressing, f si 60 grade steel
Max allowable stress* 0.94 f py≤ 0.85 f pu 0.82 f py≤ 0.75 f pu 0.70 f pu
0.55 f pu f y/1.6 24,000
Corrosive storage
18,000
Dry storage
f y /1.8
*1,000 psi = 6,895
REQUIRED STRENGTH LOAD FACTORS The structure, together with its components and foundations, whould hace to be design strength exceeds the effect of factored load combinations specified by ACI 318, ANSI/ASCE 7-95, or as justified by the engineer based on rational analysis, with the following exceptions: Feature Initial liquid pressure
Load factor 1.3
Internal lateral pressire from dry material Prestressing forces: Final prestress after losses Strength reduction factor for both reinforcement and concrete, φ
1.7 1.7 0.9
The nominal moment strength equation M n is similar to the one used for linear prestressing, i.e.,
Or
= 2 = 2+ 2
When mild vertical steel Where
is used and
= vertical restressing steel per unit width of circumference, in 2.
f ps = stress in prestressed reinforcement at nominal strength, psi f y = yield strength of mild steel, psi
MINIMUM WALL-DESIGN REQUIREMENTS Circumferential forces Liquid Initial
= = +
per foot of wall
Backfill Initial
Where t is the total wall thickness.
Thickness and Stresses Core Wall Thickness
= = + .
Final Stress Due to Backfill and Initial Prestress
Deflection. The unrestrained initial elastic radial deflection of the wall due to initial prestressing is
Δ =
Where Ec = 57,00
r = tank inner radius, tco = thickness fo wall at top or bottom of wall,
√′
psi for both normal-weight concrete and shotcrete.
The final radial deflection f = 1.7i
Restraint Effects Maximum Vertical Wall Bending Due to Radial Shear
= 0.24√ = 0.68√
This moment occurs at a distance
From the base or top edge.
Radial shear for monolithic base details which may be assumed to provide hinged connection
= 0.38 This type of detail should be used only with situ-cast tanks which incorporate a diaphragm in their wall construction.
Mild Steel for Base Anchorge. If a diaphragm is used, extend the full area of the inside bars in a U-shape distance
= 1.4√ = 1.8√
Above the base. If no diaphragm is used, extend to
Above the base. Note that anchroge length has to be added to minimum area of nominal vertical steel at the base region is
= 0.003 = 0.75√
And should be extended above the base a distance of 3 ft or
Whichever is greater.
or
. The
Minimum Wall Thickness
Situ-Cast Walls Type of tank
Minimum wall thickness
Shotcrete-steel diaphragm tanks
3.5 in.
Tanks without vertical prestressing Tanks with vertical prestressing
8 in. 7 in.
Precast walls Types of tank Tanks with vertical pretensioning and external circumferential prestress
Minimum wall thickness 5 in.
Tanks with vertical pretensioning and internal circumferential prestress Tanks with vertical post-tensioning and internal circumferential prestress
6 in. 7 in.
It should be noted that for tanks prestressed with tendons a thickness not less than 9 in. is advisable for practical considerations.
CRACK CONTROL IN WALLS PRESTRESSED CONCRETE TANKS
OF
CIRCULAR
Vessey and Preston in Ref. 11.14 recommend the following expression based on Nawy’s work in Ref. 11.15 for the maximum crack width at the exterior surface of the prestressed tank wall:
ϵ
Where
= 4.1 ×10−ϵ√
= tensile surface strain in the concrete
I x = grid index =
= reinforcement spacing in direction “2”
= reinforcement spacing in perpendicular direction “1” (horizontal)
= concrete cover to centre of steel
= diameter of steel in main direction “1.”
The tensile strain can be computed from
Where
= stress parameter
= /
= actual stress in the prestressing steel = initial prestress before losses.
For liquid-retaining tanks, the maximum allowable crack width is 0.004 in.
STEP-BY-STEP PROCEDURE FOR THE DESIGN CIRCULAR PRESTRESSED CONCRETE TANKS
OF
The following trial and adjustment procedure is recommended for designing a prestressed concrete circular tank: Select the prestressing system, the type of prestressing wire, the concrete strength, and the type of restraint that can be accomplished under local conditions. Determine the contained material pressure on the wall: H for liquid and p for gas. Use the trapezoidal distribution for granular or solid containment. Find the unit ring force F = H for a completely sliding base, where r is the radius of the tank and y is the distance above the base. Choose, the applicable vertical moment coefficients for the particular load type and wall base restraint condition caused by liquid pressure
= + 1 + = +2 1 √121 = ( Δ) 61 Δ = + (+)
And determine the corresponding horizontal radial ring tensions
And
And
, where the offset
Where
≈ 0.20
3 1 = ⁄
for concrete.
Find the applicable membrane coefficients C. compute the appliocable ring force F = C Hr. Compute the critical vertical moments in the wall using the applicable membrane coefficient C . the equation for kmoment due to liquid is
Or
= + =
Due to gas load if applicable. Compute the moment at the base, where applicable, and at the critical y plane above the base. Choose the level of vertical prestressing force. Compute the concrete stresses across the thickness of the wall both for the condition when the tank is empty and for when it is totally full. Allow maximum residual axial compressive stress f cv = 200 psi at service and a
√′ = / = =0.9⁄2 = ⁄2+ ⁄2. maximum tensile stress f t = 3
.
Design both the horizontal and the vertical prestressing steel limiting stresses. Compute the factored moment using the applicable load factors. The required , where . compute the available nominal moment strength , or The available
has to be greater than or equal to the
required . Design the length L of the annular ring at the base of the wall from the equation
= + ⁄
Where t is the thickness of the wall and h the thickness of the base slab. Compute the percentage of prestress in the base to be transferred to the wall from the formula
Where S = 1.1 (h/t ) x
1 = 1+ /⁄ .
When only the outer rim of the slab ring is compressed by radial thrust at the rim, the value of S is modified to
⁄ 1 ℎ = + =
Where
In which
d o = outer diameter
d = inner slab ring diameter = d o - 2L. Check the minimum wall thickness requirements, and evaluate the unrestrained initial elastic radial deflection
Where
Δ =
r = tank inner radius, tco = thickness fo wall at top or bottom
√′ = 1.8√ = 0.005
of wall, Ec = 57,00 13.
psi for both normal-weight concrete and shotcrete.
The final radial deflection f = 1.7i Anchor the steel from the base to the wall such that the steel extends into
the wall a distance
or 3ft, whichever is greater. Also, ensure
that the minimum nominal vertical steel at the base region is
ϵ
Where
Verify the maximum crack width
= 4.1 ×10−ϵ√
= tensile surface strain in the concrete
I x = grid index =
= reinforcement spacing in direction “2”
= reinforcement spacing in perpendicular direction “1” (horizontal)
= concrete cover to centre of steel
= diameter of steel in main direction “1.”
The tensile strain can be computed from
Where
= stress parameter
= /
= actual stress in the prestressing steel = initial prestress before losses.
For liquid-retaining tanks, the maximum allowable crack width is 0.004 in. 15.
Design the roof cover dome after selecting the type of connection at the top of the tank wall. Limited the ratio of the rise h’ of the dome to its base d such that h’/d does not exceed 1/8.
Compute the required horizontal radial prestressing force P for the edge beam from the equation.
Where
And
= ℎ + ∅2∅ 1 = 2 ∅ [1+cos∅ cos∅] 4sin∅ cos2∅ ∅ = 1+cos∅ + 2
h = total depth of rim beam b = ring beam width WD = intensity of self-weight of shell per unit area (dead load) WL = intensity of live-load projection. 16.
Compute the ring-edge beam cross section
= ̅
Where Pi = initial prestressing force = P/
̅
= residual stress percentage
f c = allowable com pressive stress in the concrete, not to exceed 0.2 f′c but not more than 800-900 psi, in the edge beam. 17.
Compute the area of the edge beam prestressing tendon
= = t∅
Where f si is the allowable stress in the prestressing steel before losses, or
It is accurate analysis is not performed. In the latter, W is the total dead and live load on the dome due to W D + WL and f pe is the effective prestress after losses. 18.
Check the minimum dome thickness required to withstand buckling,
i.e., Min.
ℎ = ∅ .
Where
a = radius of dome shell
Pu = ultimate uniformly distributed design unit pressure due to dead load and live load = ( 1.4 D + 1.7L)/144 Ф = strength reduction factor for material variability = 0.7 βi = buckling reduction factor for deviations from true spherical surface
due to imperfections. βi = (a/r i)2 where r i
≤
1.4a
βc = buckling reduction factor for creep, material nonlinearity, and
cracking = 0.44 + 0.003 WL but not to exceed 0.53 Ec = initial modulus of concrete = 57,000
√ .
References; Prestressed-Concrete-A-Fundamental-Approach-5th-Ed-Nawy http://www.ogj.com/articles/print/volume-97/issue-14/in-this-issue/pipeline/decommissioningconcrete-c-3-tank-poses-safety-concerns.html http://www.abam.com/blog/2013/12/innovation-in-using-precast-prestressed-concrete-forliquefied-natural-gas-storage http://www.waterworld.com/articles/print/volume-31/issue-12/water-connections/saws-utilizesprestressed-concrete-storage-tank-for-durable-solution.html http://www.engineeringcivil.com/economics-of-r-c-c-water-tank-resting-over-firm-ground-vis-a-visprestessed-concrete-water-tank-resting-over-firm-ground.html http://www.wedotanks.com/precast-concrete-tanks.html