Design of Silos Introduction
Silos and bunkers are usually used for storing granular materials. Silo ( also called deep bins) and bunkers (shallow bins). The important difference between the two is in the behavior of the stored material. This behavior difference is influenced by both bin geometry and characteristics of stored material. Material pressure against the walls and bottom are usually derermined by one method for silos and by another for bunkers. (The present ACI 313 Silos standard, however, uses the same method for both silos and bunkers) Silos and bunkers are made from many different structure materials. Concrete is the most frequently used materials. Concrete can offer the necessary protection to the stored materials, requires little maintenance, is aesthetically pleasing, and is relatively free of certain structural hazards (such as buckling) that may be present in silos and bunkers of thinner materials. Silos failures have alerted design engineers to the danger of designing silos for only static pressures due to stored material at rest. Those failures have inspired wide-spread wide-spread research into the variations of pressures and flow of materials. The research thus far has established beyond doubt that pressures during withdrawal may be significantly higher 1-4 or significantly lower than those present when the material is at rest. The excess (above static pressure) is called “overpressure” and the shortfall is called “underpressure.” One of the causes of overpressure is the switch from active to passive conditions which occurs during material withdrawal. While overpressures and underpressures are generally important in deeper silos, impact is usually critical only for shallow ones (bunkers) in which large volumes are dumped suddenly. Obviously, to design with disregard for either overpressure, underpressure or impact could be dangerous. Unless he elects to use silo static pressure equations for each, the designer must classify the structure as either a silo s ilo or bunker. Empirical approximation are preferred by b y many engineers. Tow such approximation are: a) H b) H H
> 1.5 A > 1.5D for circular silos tangular sil silos > 1.5a for rectan
“Slipformed” silos are constructed using a typically 4 ft. (1.2 m) high continuously moving form. “Jumpformed” silos are constructed using three typically 4 ft. (1.2 m) high fixed forms. The bottom lift is jumped to the top position after the concrete hardens sufficiently. A “hopper” is the sloping, walled portion at the bottom of a silo.
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“Stave silos” are silos assembled from small precast concrete units called “staves,” usually tongued and grooved, and held together by exterior adjustable steel hoops.
Loads q=
Vertical Pressure
γ R − µ kY 1−e ' µ k '
/R
(Janssen Formula)
Where
γ = weight per unit volume for stored material µ` = coefficient of friction between stored material and wall or hopper surface φ = = Angle of internal friction
k = 1 − sin φ R = is the hydraulic radius (ratio of area to perimeter) of horizontal cross section of storage space
• •
For Circular silos R=D/4 For polygonal silos R=D/4 for a circular shape of equivalent area.
•
For square silos a or shorter wall of rectangular silos use R=a/4
•
For the long wall b of rectangular silos use R=a`/4 where a` is the length of side of an imaginary square silo a ' =
2ab
a +b
Horizontal Pressure
p = kq
Vertical friction per unit length of wall perimeter
V = ( γ Y − q ) R
Note:
γ, k vary, the following combinations shall shall be used with with maximum: (1) Minimum µ`and minimum k for maximum vertical pressure q. (2) Minimum µ`and maximum k for maximum lateral pressure p. (3) Maximum µ`and maximum k for maximum vertical friction force V. Load Factors Load factors for silo or stacking tube design shall conform to those specified in ACI 318. The weight of and pressures due to stored material material shall be considered as live load. For concrete cast in stationary forms, strength reduction factors, φ, shall be as given in ACI 318. For slipforming, unless continuous inspection is provided, strength reduction factors given in ACI 318 shall be multiplied by 0.95.
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Pressures and loads for hoppers
The initial vertical pressure at depth h y q y = q 0 + γ h y
where q o is the initial vertical pressure at the top of the hopper. The initial pressure normal to the hopper surface at depth h y below top of hopper shall be the larger of:
qα = p n =
q y tan θ
tan θ
and
+ tan φ '
V n = pn
tan φ '
or qα = p n = q y
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( sin
2
θ +k
cos
2
θ)
and
V n = qy
Dr. Mohammed Arafa
θ ⋅ cos θ (1 − k ) sin
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Square and rectangular silos Horizontal Forces Due to Stored Material
The cross section of a rectangular silo is a rigid frame subject to outward pressure, p des varying with depth. Each wall will have axial tension, bending and shear. The horizontal (unfactored) tensile force per unit height at depth Y are: Fa = p b ,des (b 2 )
for wall a
Fb = p a ,des ( a 2 )
for wall b
Regular Polygonal silos The principle of design and detail of single or grouped polygonal silos are similar to those of rectangular silos. Their walls subjected to similar loading and similar horizontal tension, shaer and bending moment and vertical forces. The horizontal tensile force in any single regular polygonal silos is: T
sin θ = pdes ( a 2 ) 1 − cos θ
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Sections with combined tension and bending Strength design of walls subject to combined axial tension and flexure shall be based on the stress and strain compatibility assumptions of ACI 318 and on the equilibrium between the forces acting on the cross-section at nominal strength. For small eccentricity, the required tensile reinforcement area per unit height:
Small eccentricity A s
=
Fu e '
φ f y (d
− d ')
e
=
M u F u
<
h
2
A 's
− d ''
=
Fu e ''
φ f y (d
− d '')
Overpressure c d
During initial filling and during discharge, even when both are concentric, overpressures occur because of imperfections in the cylindrical shape of the silo, non-uniformity in the distribution of particle sizes, and convergence at the top of hoppers or in flow channels. A minimum overpressure factor of 1.5 is recommended for concentric flow silos even when they are of a mass flow configuration. The recommended factor recognizes that even though higher and lower point pressures are measured in full size silos, they are distributed vertically through the stiffness of the silo wall and can be averaged over larger areas for structural ENGC6353
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design. The 1.5 overpressure factor is in addition to the load factor of 1.7 required by ACI318 design pressure = 1.7 x c d x initial filling pressure.
The following Table shows the different C d factors at different depth zones
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Earthquake forces Earthquake loads may affect stability and strength. The UBC or IBC may be used. Seismic forces are assumed to act in any horizontal direction, but vertical acceleration forces are usually neglected. In computing lateral seismic force due to the mass of the stored granular material, the silo is assumed to be full, but the lateral force is less than it would be for a solid mass. The reduction of lateral force is allowed because of energy loss through inter-granular movement and particle-to-particle friction in the stored material. ACI 313 use not less 80% of the weight of the stored material as an effective live load, from which to determine seismic forces.
Wind forces Wind may affect the stability of empty silos, particularly tall, narrow silos or silos group. Foundation pressure and column stresses, however, may be worse with wind acting on the full silo. Wind load reduction may be applied for cylindrical shape may be applied to single circular for cylindrical Wind forces on silos shall be considered generated by positive and negative pressures acting concurrently. The pressures shall be not less than required by the local building code for the locality and height zone in question. Wind pressure distributions shall take into account adjacent silos or structures.
Thermal Loads Temperature and shrinkage steel requirement of ACI 318 apply to silos. In addition, hot stored materials may cause thermal stresses too high to be ignored. The approximate method illustrated below was developed specifically for cement storage silos. Its principles, however, should apply also to silos for storage of other not granular materials. In this method: 1- Tensile strength of the concrete is neglected 2- Wall temperatures are assumed to vary only radially. In building, the usual practice is to ignore a certain amount of inside-outside temperature difference (80 oF or 27oC for silos). The thermal effects of hot (or cold) stored materials and hot (or cold) air shall be considered. For circular walls or wall areas with total restraint to warping (as at corners of rectangular silos), the thermal bending moment per unit of wall height or width shall be computed by:
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M t =
E c h 2α c 12 (1 −ν )
∆T
Computation of bending moments due to thermal effects requires determining the temperature differential through the wall. To determine this differential, the designer should consider the rates at which heat flows from the hot material to the inside surface of the wall, through the wall thickness and from the wall to the atmosphere. The temperature differential may be estimated b y:
∆T = (T i −T 0 − 80o F ) K t = (T i −T 0 − 27o C ) K K t
=
0.08h 4.09 + 0.08h
Minimum wall thickness
The thickness of silo or stacking tube walls shall be not less than 6 in. (150 mm) for cast-in place concrete, nor less than 2 in. (50 mm) for precast concrete. The following formula can also be used in service loading t=
ε sh E s
+ fs − nf ct
100fs f ct
T
Load factors and strength reduction factors
• Load factors for silo or stacking tube design shall conform to those specified in ACI 318. The weight of and pressures due to stored material shall be considered as live load.
• For concrete cast in stationary forms, strength reduction factors, φ, shall be as given in ACI 318. For slip forming, unless continuous inspection is provided, strength reduction factors given in ACI 318 shall be multiplied by 0.95.
Allowable ultimate Compressive load
The compressive axial load strength per unit area for walls in which buckling (including local buckling) does not control shall be computed by Pnw = 0.55φ f c'
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Addi ti on al Lo ad at Open in gs The edges of bottom slab openings usually receive added line loads comprising the weight of the hopper and equipment supported by it, the weight of stored material in t he hopper and the design vertical pressures of material above the top of the hopper. Flat Bottom
The simplest flat bottom is a slab of uniform thickness. The flat bottom may also be a ribbed slab or beam-slab system. For a slab without hopper-forming fill, the design loads are dead load and pressure, q des computed at the top of the slab. Wu
= 1.4DL + 1.7qdes
With earthquake vertical friction at the wall is assumed to be zero, so that the ultimate vertical pressure on the bottom is: W u = 0.75 (1.4DL + 1.7γ H
)
For a slab with hopper-forming fill, the weight of the fill itself and the material stored in the hopper are added as dead load,. Slab Shear stresses should be checked. Conical Hooper
The design pressure, q may be computed from the above equation. In computing q y engineers usually use the dimensions at the top of the hopper to obtain R for the Janssen formulas, ignoring the reductions of cross section within the hopper. The conical hopper shell is subject to two tensile membrane forces. The meridional force, F m , is parallel to the generator line of the cone. The tangential force, F t , is in the plane of the shell and horizontal. The meridional force per unit width at depth Y is computed from equilibrium of the loads on the cone below that depth. These loads, shown on Figure, are the resultant of vertical pressures, q (at depth Y) and W, the combined weights of the hopper itself and material stored below depth Y plus any equipment supported by the hopper.
q y D
F mu = 1.7
+
W g + 1.4 π π D sin α D sin α W L
4 sin α q D F tu = 1.7 α 2sin α qα = P sin 2 α + q cos 2 α
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Both forces are maximum at the upper edge of the hopper, and approach zero at the lower edge. A conical hopper is usually supported at its upper end by a ring beam. This is merely a thickened portion of the conical shell with a cross section to satisfy the supporting condition and the loading. The ring beam depth should not be less than one-tenth of the hopper diameter. Pyramidal Hopper
Loads for pyramidal hoppers are the same as for conical hopper, Pyramidal hopper walls, however, are subject to bending as well as tensile membrane forces. The bending always include two-way, plate type bending and may include significant in-plane bending. The angles of slope for walls a and b are
a and
b ,
respectively.
If some arbitrary factors c a and c b are selected to define the division of the load to area A a and A b (such that for symmetrical case 2 c a +2 c b =1.0) then the ultimate meridional forces per unit
width of wall are: F mau
=
F mbu
=
1.7 (c aW L
+ A aq a ,des ) + 1.4c bW g a sin α a
1.7 (cbW L
+ Ab q b ,des ) + 1.4c bW g b sin α b
The ultimate horizontal forces F ta u and F tbu per unit length at depth Y may be approximated
b qαb ,des sin αa 2
Ftau = 1.7
and
a qα a,des sin α b 2
Ftbu = 1.7
Moment from two-way plate-type bending may be computed for inclined triangular walls with fixed sides and subjected to either uniform or triangular loading.
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Details and placement of reinforcement
• Where slipforming is to be used, reinforcement arrangement and details shall be as simple as practical to facilitate placing and inspection during construction.
• Reinforcement shall be provided to resist all bending moments, including those due to continuity at wall intersections, alone or in combination with axial and shear forces.
• Horizontal ties shall be provided as required to resist forces that tend to separate adjoining silos of monolithically cast silo groups.
• Unless determined otherwise by analysis, horizontal reinforcement at the bottom of the pressure zone shall be continued at the same size and spacing for a distance below the pressure zone equal to at least four times the thickness h of the wall above. In no case shall the total horizontal reinforcement area be less than 0.0025 times the gross concrete area per unit height of wall.
• Vertical reinforcement in the silo wall shall be ( φ10 diameter) bars or larger, and the minimum ratio of vertical reinforcement to gross concrete area shall be not less than 0.0020. Horizontal spacing of vertical bars shall not exceed 18 in. (450 mm) for exterior walls nor 24 in. (600 mm) for interior walls of monolithically cast silo groups. Vertical steel shall be provided to resist wall bending moment at the junction of walls with silo roofs and bottoms. In slipform construction, jack rods, to the extent bond strength can be developed, may be considered as vertical reinforcement when left in place
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Reinforcement pattern at intersecting walls
Miscellaneous details ENGC6353
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Typical Hooper Reinforcement
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Circular Concrete Ring-Beam and Column System Supporting a Steel Hopper
A steel hopper can do little to stiffen a concrete ring beam, therefore, all of the loads it applies to the beam—torsion included must be considered. The tables and equations which follow cover the design of such ring beams and columns having an even number (4 to 12) of equallyspaced columns, fixed at their bases, and rigidly connected to the ring beam. (See Fig. 1.) The following Equations are for working design loads (WSD) rather than ultimate. In this type of structure live load is much larger than dead load. Consequently, it is simpler and sufficiently accurate to apply a common live load factor, to the final answers in order to determine the USD values. The external design loads acting on the ring beam Fmu cos α
F x =
Fy = w beam +
and
1.7
F mu sin α 1.7
where w beam ,. is the weight of the ring beam per unit length. If the vertical pressure (by stored material) acting on the face of the ring beam is significant, it should be added to F y The WSD uniform torsional moment is M t = F m e The Cross sectional Area of the ring Beam is A r = a1b1 −
b 2a2 2
Coordinate of the centroid measured from the origin O are: x =
a1b12 / 2 − ( a2b2 / 2 )(b1 − b2 A r 2
y =
)
/3
b1a1
/2−
(a b 2
2
/2
)( a − a 1
2
)
/3
A r
An equivalent rectangle of height a and b is substituted for the pentagon a = 2y b = A r a
The column shear HA and upper end moment MA are found by solving the following equations: F x r 2 A r E r
+
H A r 3K 2 2E r I ry
12 M t r
E r a 3 ln ( r2 r1 )
+
+
H A L3 3E col I c
+
M A L2 2E col I c
C 2 r ( M A − Rec ) E r I rx
+
=0
C 3F y r 3 E r I rx
+
H A L2 2 E col I c
+
M A L2 E col I c
=0
The WSD moment at the column base is then M B
= H A hc − M A
The maximum WSD values of vertical shear, thrust, torque, and bending moment at the supports, midspans, and points of maximum torsion, for ring beams on 4, 6, 8, 10, or 12 supports are tabulated in Table 16-9. ENGC6353
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For USD, the final forces and moments given by the preceding equations or Table 16-9 should be multiplied by the live load factor, In all of the above equations, consistent units must be used.
Silo-Bottom: Steel Hooper supported on concrete ring
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Ring-beam cross Section
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Appendix Tables for Triangular Slab
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