Pressure Vessel Design Case Study
This case study considers the design of a cylindrical storage vessel typical
of those used in chemical and process industries to store liquids. Corrosion
resistance, strength and ease of fabrication make composite materials
particularly attractive for this sort of application. The installed cost of a
GRP vessel compares favourably with that of more traditional materials, such as
stainless steel and lined carbon steel vessels. The majority of such
vessels have diameters in the range 1 to 10 m, with wall thicknesses of between
5 and 50 mm.
In many respects, the process of designing a composite vessel is the same as
that facing the designer of metal vessels. The design must take into account the
design stress resulting from the pressure and size of the vessel in
question. However, the composite designer is faced with the additional
task of designing the material to be used. In so doing, they will
generally take the opportunity to use a variety of differing layers within the
laminate construction in order to achieve the most economical and desirable
combination of properties.
The design methodology used in this case study is that developed in BS4994.This requires that the design process is considered in three stages, assessment
of allowable strain, calculation of the applied unit loads and the selection of
an appropriate laminate configuration.
Case Study Parameters
The vessel considered in this case study is a cylindrical vessel, internal
diameter 1.75 m with an effective pressure of 2 bar (0.2 MPa). The
operating temperature for the vessel is 40°C. In service, the vessel
contents level will primarily be static, although on occasion, the vessel will
be emptied and refilled. The case study will follow the design process,
using the BS4994 methodology, to develop a suitable laminate
configuration.
Allowable Design Strain
BS4994 determines an allowable design strain through the use of a
number of part factors, which account for the effects of loading, environment
and manufacturing conditions on the longterm chemical and mechanical behaviour
of the GRP laminates.
These part factors are defined as follows:
 k_{1 }method of
manufacture (range 1.6 to 3.0)
 k_{2 }long term
behaviour (range 1.2 to 2.0)
 k_{3 }temperature (range
1.0 to 1.2)
 k_{4 }cyclic loading
(range 1.1 to 1.4)
 k_{5 }curing procedure
(range 1.1 to 1.5)
The product of these factors, and a further safety factor of 3.0 results in
an overall design factor, K, which is used to evaluate the allowable design
strain, e_{L}.
For the case considered here, these part factors are evaluated as follows:
 For hand layup, part factor k_{1} = 1.6
 For long term behaviour, part factor k_{2} = 2.0
 For temperature, assuming operation at 40°C, and use of a resin system
with a heat distortion temperature of 80°C or higher, part factor k_{3}
= 1.0
 For cyclic stressing, assuming occasional filling and emptying, part
factor k_{4} = 1.1
 For curing procedure, assuming post cure at elevated temperature, part
factor k_{5} = 1.1
Therefore, as
The "load limited" allowable limit loading u_{L} is
given by
where u is the ultimate tensile unit strength (UTUS is in N/mm per kg/m^{2} ) of
the material, and K is the design factor calculated above.
chopped strand mat (CSM) the UTuS is 200
N/mm/(kg/m^{2}), thus u_{L} = 17.2 N/mm/(kg/m^{2})
woven rovings (WR) the UTuS is 300 N/mm/(kg/m^{2}), thus u_{L} =
25.8 N/mm/(kg/m^{2})
The load limited allowable strain is given by
where u and K are as previously defined and X is the
laminate extensibility.
For CSM, the extensibility is 12 700 N/mm/(kg/m^{2}), giving e_{L}
= 0.14%
For WR, the extensibility is 16 200 N/mm/(kg/m^{2}), giving e_{L}
= 0.16%
There is a further overriding upper limit to the design strain of the lesser of
0.2% or 0.1 x e_{r} (where e_{r}
is the fracture strain of unreinforced resin in a simple tensile test.
Assuming a resin strain to failure of 3%, then, in this case, the design remains
load limited and the design unit loading u_{x} = u_{L},
i.e. 17.2 N/mm/(kg/m^{2}) and 25.8 N/mm/(kg/m^{2}) for CSM
and WR respectively.
Applied Loads
The applied loading on the vessel is then calculated using conventional
analysis techniques. In this case, assuming no significant axial loading,
the vessel wall circumferential unit stress is given by:
where P is the pressure, D is the vessel diameter and t is the
vessel wall thickness.
Laminate Construction
At this point, it is possible to design the laminate construction.
The total quantity of reinforcement, in this first case for a vessel constructed
simply from multiple CSM layers, is simply determined by:
where w_{x} is the weight of a single layer and nx is the number
of layers.
Therefore a total weight of 10.2 kg m^{2} of reinforcement is
required. The distribution of this would be selected according to
manufacturers' individual preferences, but one suitable configuration would be:
2 layers 300 g m^{2} (one at each surface) = 0.6 kg m^{2}
16 layers 600 g m^{2}
= 9.6 kg m^{2}
Total = 10.2 kg m^{2}
Assuming a glass content of 30% for CSM, the wall thickness would be 2.2 mm per
kg/m^{2} of glass, giving a total wall thickness of 22.4 mm.
A more efficient structure is obtained using a combination of CSM with WR,
in which case the laminate construction is determined as follows:
The design unit loading in the WR must be reduced such that the strain
does not exceed the design limit for CSM, hence
per kg/m^{2} of glass
The design of the laminate can then be determined from
Therefore a suitable design would be as follows:
Detail 
Calculation 
Total 
Reinforced gel coat 
 
 
1500 g/m^{2} CSM 
17.2 x 1.5 
25.80 
800 g/m^{2} WR 
x5 
22.6 x 0.8 
x5 
129.10 
450 g/m^{2} CSM 
17.2 x 0.45 
800 g/m^{2} WR 
22.6 x 0.8 
18.08 
300 g/m^{2} CSM 
17.2 x 0.30 
5.16 
Resin rich layer with binding tissue 
 
 
TOTAL 

178.14 
In this case, assuming a glass content of 30% for CSM with 2.2 mm per
kg/m^{2} of glass, and a glass content of 55% for CSM with 0.95
mm per kg/m^{2} of glass, the vessel wall thickness would be 13.5 mm.
Dished End Design
If a torispherical end is desired for such a vessel, a typical geometry would
be h_{i} /D_{i} = 0.25 and r_{i} /D_{i}
= 0.15 (Note that this is slightly deeper than would be used for a typical
metallic construction).
At these values, the shape factor K_{s} is approximately equal
to 1.78. The membrane unit load for a domed end subject to pressure is
given by
For the current case, that is
Assuming a construction of CSM mat and woven rovings, similar to that for the
vessel shell, gives a required weight of reinforcement is given by
Therefore a suitable design would be as follows:
Detail 
Calculation 
Total 
Reinforced gel coat 
 
 
1200 g/m^{2} CSM 
17.2 x 1.2 
20.64 
800 g/m^{2} WR 
x12 
22.6 x 0.8 
x12 
309.84 
450 g/m^{2} CSM 
17.2 x 0.45 
800 g/m^{2} WR 
22.6 x 0.8 
18.08 
300 g/m^{2} CSM 
17.2 x 0.30 
5.16 
Resin rich layer with binding tissue 
 
 
TOTAL 

353.72 
This gives an actual laminate thickness of 25.06, assuming a glass content of
30% for CSM with 2.2 mm per kg/m^{2} of glass, and a glass
content of 55% for CSM with 0.95 mm per kg/m^{2} of glass, as
previously.
For a laminate of this thickness,
and the assumed value of K_{s} = 1.78 is reasonable. If it
had been found that the value of K_{s} was not acceptable, then
the calculation would need to be repeated with a better estimate for the value
of K_{s} until convergence was achieved.
Reference: BS4994  Specification for Vessels and Tanks in Reinforced Plastics,
BSI 1973.
Keywords: BS4994, Design, Design strain, Part factors, Laminate, Code
