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Performs thickness calculations and design for annular plate baserings, top rings, bolting, and gussets. These calculations are performed using industry standard calculation techniques.
Thickness of a Basering under Compression
The equation for the thickness of the basering is the equation for a simple cantilever beam. The beam is assumed to be supported at the skirt, and loaded with a uniform load caused by the compression of the concrete due to the combined weight of the vessel and bending moment on the down-wind / down-earthquake side of the vessel.
Thickness of a cantilever, t:
Where
fc = Bearing stress on the concrete
l = Cantilever length of basering
s = Allowable bending stress of basering (typically 1.5 times the code allowable)
There are two commonly accepted methods of determining the stress from the vessel and base-ring acting on the concrete. The simplified method calculates the compressive stress on the concrete assuming that the neutral axis for the vessel is at the centerline.
Stress acting on the concrete, fc:
Where:
W = Weight of the vessel together with the basering
M = Maximum bending moment on vessel
A = Cross-sectional area of basering on foundation
c = Distance from the center of the basering to the outer edge of the basering
I = Moment of inertia of the basering on the foundation
However, when a steel skirt and basering are supported on a concrete foundation, the behavior of the foundation is similar to that of a reinforced concrete beam. If there is a net bending moment on the foundation, then the force upward on the bolts must be balanced by the force downward on the concrete. Because these two materials have different elastic moduli, and because the strain in the concrete cross section must be equal to the strain in the basering at any specific location, the neutral axis of the combined bolt/concrete cross section will be in the direction of the concrete. Several authors, including Jawad and Farr (Structural Analysis and Design of Process Equipment, pg 428 - 433) and Megyesy (Pressure Vessel Handbook, pg 70 - 73), have analyzed this phenomenon. The software uses the formulation of Singh and Soler (Mechanical Design of Heat Exchangers and Pressure Vessel Components, pg 957 - 959). This formulation seems to be the most readily adaptable to computerization, as there are no tabulated constants. Singh and Soler provide the following description of their method:
In this case, the neutral axis is parallel to the Y-axis. The location of the neutral axis is identified by the angle alpha. The object is to determine the peak concrete pressure (p) and the angle alpha.
For narrow base plate rings, an approximate solution may be constructed using numerical iteration. It is assumed that the concrete annulus under the base plate may be treated as a thin ring of mean diameter c. Assuming that the foundation is linearly elastic and the base plate is relatively rigid, Brownell and Young have developed an approximate solution which can be cast in a form suitable for numerical solution. Let the total tensile stress area of all foundation bolts be A. Within the limits of accuracy sought, it is permissible to replace the bolts with a thin shell of thickness t and mean diameter equal to the bolt circle diameter c, such that:
Thickness, t:
Where:
A = Total cross-sectional area of all foundation bolts
P = Peak concrete pressure
l = Width of basering
c = Thin ring diameter
We assume that the discrete tensile bolt loads, acting around the ring, are replaced by a line load, varying in intensity with the distance from the neutral plane.
Let n be the ratio of Young's moduli of the bolt material to that of the concrete; n normally varies between 10 and 15. Assuming that the concrete can take only compression (non-adhesive surface) and that the bolts are effective only in tension (untapped holes in the base plate), an analysis, similar to that given above, yields the following results:
Where:
n = Ratio of elastic modulus of the bolt, Eb, to that of the concrete, Ec:
t3 = Width of the basering, similar to the cantilever length, l, in Jawad and Farr's
thickness equation previously mentioned
c = Bolt circle diameter
r1 - r4 = Four constants based on the neutral axis angle and defined in Singh and Soler's
equations 20.3.12 through 20.3.17, not reproduced here.
These equations give the required seven non-linear equations to solve for seven unknowns, namely p, c, a, and the ri (i = 1 - 4) parameters. The iterative solution starts with assumed values of s and p, so and po, taken from an approximate analysis performed first. Then a is determined using the above equation. Knowing a the dimensionless parameters r1, r2, r3, and r4 are computed. This enables computation of corrected values of p and s, named po' and so'). The next iteration is started with s1 and p1 where we choose:
This process is continued until the errors ei and Ei at the iteration stage are within specified tolerances --ei = Ei = 0.005 is a practical value,
Where:
After the new values of bolt stress and bearing pressure are calculated, the thickness of the basering is calculated again using the same formula given above for the approximate method.
Thickness of Basering under Tension
On the tensile side, if there is no top ring but there are gussets, then there is a discrepancy on how to do the analysis. For example, while Megyesy uses Table F (Pressure Vessel Handbook, pg 78) to calculate an equivalent bending moment, Dennis R. Moss uses the same approach but does not give a table (Pressure Vessel Design Manual, pg 126-129), and Jawad & Farr use a 'yield-line' theory (Structural Analysis and Design of Process Equipment, pg 435-436). Since the Jawad and Farr equation for thickness, t, is both accepted and explicit, the program uses their equation 12.13:
Thickness, t:
Where:
Bolt Load, Pt:
sy = Yield strength of the bolt
a = Distance between gussets
b = Width of base plate that is outside of skirt
l = Distance from skirt to bolt area
d = Diameter of bolt hole
Thickness of Top Ring under Tension
If there is a top ring or plate, its thickness is calculated using a simple beam formula. Taking the plate to be a beam supported between two gussets with a point load in the middle equal to the maximum bolt load, we derive the following equation:
Thickness, t:
Where:
Allowable stress, s:
Bending moment, M:
Where:
Cg = Center of gravity, depending on the geometry of the plate
Bolt Load, Pt:
Section Modulus, Z:
Width of Section, Wt:
Required Thickness of Gussets in Tension
If there are gussets, they must be analyzed for both tension and compression. The tensile stress, T, is the force divided by the area, where the force is taken to be the allowable bolt stress times the bolt area, and the area of the gusset is the thickness of the gusset, tgusset, times one half the width of the gusset, Wgusset (because gussets normally taper):
Where:
Required Thickness of Gussets in Compression
In compression (as a column) we must iteratively calculate the required thickness. Taking the actual thickness as the starting point, we perform the calculation in AISC 1.5.1.3. The radius of gyration for the gusset is taken as 0.289 t per Megyesy's Pressure Vessel Design Handbook, page 404. The actual compression is calculated as described above, and then compared to the allowed compression per AISC. The thickness is then modified and another calculation performed until the actual and allowed compressions are within one half of one percent of one another.
Basering Design
When you request a basering design, the software performs the following additional calculations to determine the design geometry:
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Selection of Number of Bolts
This selection is made on the basis of Megyesy's table in Pressure Vessel Handbook (Table C, page 67). Above the diameter shown, the selection is made to keep the anchor bolt spacing at about 24 inches.
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Calculation of Load per Bolt
This calculation of load, Pb, per bolt:
Where:
W = Weight of vessel
N = Number of bolts
R = Radius of bolt area
M = Bending moment
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Calculation of Required Area for Each Bolt
This is the load per bolt divided by the allowable stress:
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Selection of the Bolt Size
The software has a table of bolt areas and selects the smallest bolt with area greater than the area calculated above.
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Selection of Preliminary Basering Geometry
The table of bolt areas also contains the required clearances in order to successfully tighten the selected bolt (wrench clearances and edge clearances). The software selects a preliminary basering geometry based on these clearances. Values selected at this point are the bolt circle, basering outside diameter, and basering inside diameter.
Analysis of Preliminary Basering Geometry
Using the methods described previously for the analysis section, the software determines the approximate compressive stress in the concrete for the preliminary geometry.
Selection of Final Basering Geometry
If the compressive stress calculated above is acceptable then the preliminary geometry becomes the final geometry. If not, then the bolt circle and basering diameters are scaled up to the point where the compressive stresses are acceptable. These become the final basering geometry values.
Analysis of Basering Thicknesses
The analysis then continues through the thickness calculation described above, determining required thicknesses for the basering, top ring, and gussets.