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 long-term chemical and mechanical behaviour of the GRP laminates.

These part factors are defined as follows:

• k₁        method of manufacture (range 1.6 to 3.0)
• k₂        long term behaviour (range 1.2 to 2.0)
• k₃        temperature (range 1.0 to 1.2)
• k₄        cyclic loading (range 1.1 to 1.4)
• k₅        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 lay-up, part factor k₁ = 1.6
• For long term behaviour, part factor k₂ = 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 k₃ = 1.0
• For cyclic stressing, assuming occasional filling and emptying, part factor k₄ = 1.1
• For curing procedure, assuming post cure at elevated temperature, part factor k₅ = 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/m² ) of the material, and K is the design factor calculated above.
 
       chopped strand mat (CSM) the UTuS is 200 N/mm/(kg/m²), thus u_L = 17.2 N/mm/(kg/m²)
                  woven rovings (WR) the UTuS is 300 N/mm/(kg/m²), thus u_L = 25.8 N/mm/(kg/m²)

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/m²), giving e_L = 0.14%
For WR, the extensibility is 16 200 N/mm/(kg/m²), 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/m²) and 25.8 N/mm/(kg/m²) 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/m² 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/m² of glass

The design of the laminate can then be determined from



Therefore a suitable design would be as follows:

In this case, assuming a glass content of 30% for CSM with 2.2 mm per kg/m² of glass, and a glass content of 55% for CSM with 0.95 mm per kg/m² 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:

This gives an actual laminate thickness of 25.06, assuming a glass content of 30% for CSM with 2.2 mm per kg/m² of glass, and a glass content of 55% for CSM with 0.95 mm per kg/m² 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