Road tankers are the most used means of transporting petroleum product to end users due to its cost effectiveness and energy-efficiency. The cylindrical tank has been well designed for by ASME VIII divisions 1 and 2 using analytical equations. Petrol tankers are not circular but elliptical probably for stability during transportation. This paper has used the finite element method to investigate in-plane displacements and Von-Mises stresses in both circular and elliptical cylindrical tanks under full loading. An elliptical OANDO
^{?} tanker of 66.78 m
^{3}volume and shell thickness of0.2 mmand an equivalent volume circular cylindrical tank was used for the simulation. MATLAB
^{?} was used to generate geometrical mesh model of the petroleum tankers, extract element coordinates and conduct the finite element analysis. Plane strain condition was used in analyzing a section of the petroleum tanker. It was observed that an equivalent volume circular cylindrical tank was under a higher internal pressure (16,858 N/m
^{2}) compared to the elliptical cylinder (14,480 N/m
^{2}). Von-Mises stress and in-plane displacements showed direct linear relationships with internal fluid pressure. Von-Mises stress in the elliptical tank was found to be lower (5.7 × 10
^{6} N/m
^{2}) than for the circular tank (8 × 10
^{6} N/m
^{2}). In plane displacements was zero in the longitudinal direction for both tanks and of the order of 10
^{-4} mm in the y-direction for both tanks with the circular larger by about 2.5 ×
^{ }10
^{-3 }cm. So in addition to tank stability on the lorry, the Von-Mises stresses were lower as well for the elliptical tank. It was also observed that Von-Mises stresses were far below the yield stress of the steel plate. However, the effect of weldment area on lowering of yield stress was not studied. Stress values were validated using analytical method and found to be insignificantly different (P > 0.05).

Road tankers are cylindrical or ellipsoidal pressure vessels used to convey liquids especially petroleum products. Petroleum tanker is widely used over other means of transporting petroleum products (pipeline, trains, badges and trucks) due to its cost effectiveness and energy-efficiency [

Road tankers constructed locallyare designed based on ASME standards for pressure vessels (ASME VIII divisions 1 and 2). The ASME standard presents design formulae that are simple to use, but limited to specific geometries and geometry details such as welded supports and openings. It does not put into consideration several actions or combination of actions such as local loads, seismic load, wind loads and external pressure in its design formula. A better approach is the design by analysis. According to Josef et al. [

The Finite Element Method is a numerical technique ideally suited to digital computers in which a continuous elastic structure (continuum) is divided (discretized) into smaller but finite well defined sub-structures (element) that can be represented by simple equations [

There are several publications on application of Finite element analysis of specific pressure vessels. Mirko et al., [

Most Finite element analysts develop own codes using MATLAB, C++, or FORTRAN. Meshing of geometrical models is done manually giving poor results. Other challenges faced by finite element analysts include: high cost of acquiring CAD software for numerical simulation, difficulty in managing read data from standard CAD programs such as dwg format (AutoCAD file) when using own code, and erroneous results, due to improper use of commercial software.

MATLAB is a computer language and the software has an interactive computing environment that enables numerical computation, analysis and data visualization [

The ellipsoidal tanker which is predominantly more common compared to the cylindrical ones has three inner segments (Figures 1 and 2). Each segment can be divided into four symmetrically similar sections (^{®}.

The common ellipse (

also was written as

where a represents half the horizontal axis, and b half the vertical axis as shown in

After generating the meshed model, an element was extrapolated and used in developing the stiffness matrix and force vector, needed to evaluate the displacement vector for plane strain elasticity of the model and VonMises stress.

The equation for the rectangular shell elements was derived by combining the linear rectangular element (membrane element) and the plate element (bending element).

The equilibrium equations for a three-dimensional linear elasticity express in terms of stress is given [17,18] as;

where are the stress components, are the body forces.

The above equilibrium equation expressed in terms of displacement for two dimensional analyses is given as follows. The equation was reduced to two dimensional since plane strain condition is being considered.

h_{x} and h_{y} are the in-plane displacement in the x and y directions respectively of the Cartesian coordinate system.

The finite element equation is expressed as

[K] is the stiffness matrix, {F} is the force vector and {U} is the displacement vector.

For plane strain condition, assuming homogeneity of material,

E and v are the material elastic constant and poison ratio respectively.

where is the in-plane displacement vector, are the weight functions. The weight function for the rectangular element was used in the analysis.

The standard equation governing plate bending [

B_{ij} is the bending stiffness matrix for the shell element. E, v and t are the elastic modulus, poison ratio and thickness respectively of the tanker being considered.

For the plate element, on application of the classical plate theory [

Every term retains their meaning as above except for which is used to represent one out-of-plane displacement in the thickness axis and two rotations. Mass matrix was neglected since dynamics of the system was not considered.

. Where represents one out-of-plane displacement in the thickness direction and two rotations and are the weight functions. B_{ij}’ are the material elasticity matrices for the bending element.

The stiffness matrices, and force vectors are basically a function of the side length of the rectangular element (Figures 8(a) and (b)) used to form the tanker model. From

x_{i}, y_{i} and z_{i} are the coordinate locations of each element. I = 1 to 4 represents the four nodes of a 4 nodal rectangular element.

where [K] is the 8 × 8 membrane stiffness matrix and [Kb] is the 12 × 12 bending stiffness matrix, both combining to give the total elemental stiffness matrix [KG] in the form shown below

The force vectors were combined in similar manner to form the total elemental force vector as follows

Each element has five degrees of freedom (two in-plane displacements u and v, one out-of-plane displacements w, and two rotations. The displacements and bending at the edges were given zero since they are fixed.

Code for evaluating the finite element equation {U} =, introducing boundary conditions and estimating the displacement was done using the flow algorithm (

A_{ij} represent the elasticity matrices for both the membrane element and bending element respectively.

[B] Represents the strain vectors and moment vector for membrane and bending element respectively.

[Stress] is the bending and membrane stress being computed.

2.7. Analytical Von-Mises Stress Calculation To compute the Von-Mises stress analytically, there are three principal stresses which are all functions of the pressure.

P1 = Principal stress 1 = Hoop stress = pressure × horizontal radii axis/t;

P2 = Principal stress 2 = Hoop stress/2;

P3 = Principal stress 3 = radial stress pressure;

Length of tanker = 485 cm;

Vertical axis of tanker = 180 cm;

Horizontal axis of tanker = 244 cm;

Thickness of tanker = 0.2 cm;

Poison ratio = 0.3;

Material of construction = A516M Grade 70;

Specified minimum yield stress = 25 × 10^{7} N/m^{2};^{}

Maximum allowable stress = 13.8 × 10^{7} N/m^{2};

Elastic modulus = 200 × 10^{9} N/m^{2}.^{}

^{6} N/m^{2} and this decreased linearly to zero as the tank is offloaded until there is an empty tank with zero VonMises stress.

^{−4 cm}. The circular cylindrical tank showed higher circumferential displacement than the elliptical with a difference of about 2.5 × 10^{−3 cm}.

^{−2 m shell thickness, Von-Mises stress increased rapidly till yield stress is reached.}

The direct linear relationships between internal loading pressure and Von-Mises stress (

The results also show that circular cylinderical tanks have higher internal pressures as well as higher VonMises stresses than elliptical tanks of the same loading volume (

The in-plane displacements (Figures 14 and 15) showed good tanker material integrity and nullifies any idea of failure by yielding but just as had been mentioned earlier future work on weldment defects coupled with stresscorrosion need to be done to investigate what is transpiring here which could lead to other types of failure over time.

The inverse and non-linear relationship between tank shell thickness and Von-Mises stress (

Von-Mises stresses in circular and elliptical petroleum tankers under full loading have been obtained using the finite element method. It was observed that an equivalent volume circular cylindrical has a higher internal pressure (16,858 N/m^{2}) compared to the elliptical cylinder (14,480 N/m^{2}). Von-Mises stresses showed a linear relationship with variation in internal fluid pressure. Von-Mises stresses in the elliptical tank were found to be lower (5.7 × 10^{6} N/m^{2}) than for the circular tank (8 × 10^{6} N/m^{2}. VonMises stresses were far below the yield stress of the steel plate (25 × 10^{7} N/m^{2}). However, the effect of weldment area and stress corrosion on lowering of yield stress was not studied. Values obtained for the circular cylindrical tank were compared with ASME VIII divisions 1 and 2 standard values and found not to be significantly different (P > 0.05).