Stress analysis and applicability analysis of the elliptical head

As the main pressure components of pressure vessels, the mechanical performance of cylinders and heads affects the normal operation of pressure vessels. At present, no unified theoretical formula exists for the connection region between an elliptical head and the cylinder. Therefore, the authors consider the standard elliptical head as the research object. First, the theoretical stress calculation formula is deduced according to the deformation continuity equation. Second, the stress is experimentally measured using an internal-pressure thin-walled-vessel stress measurement device, and the theoretical and experimental stress values in the discontinuous region between the elliptical head and cylinder are analysed and compared to verify the accuracy and applicability of the theoretical stress calculation formula. The results show that the theoretical stress calculation formula in the discontinuous region between the elliptical head and cylinder is valid. By comparing and analysing the theoretical and experimental stress values, the accuracy and applicability of the theoretical stress calculation formula in the discontinuous region are verified. The findings can provide guidance for the stress measurement of internal-pressure vessels.

www.nature.com/scientificreports/ most of the existing studies, the moment theory has not been used to determine the stress. In addition, no unified theoretical calculation formula exists for the connection region between the elliptical head and cylinder [24][25][26][27] . Therefore, in this study, the standard elliptical head is considered the research object. First, the theoretical stress calculation formula in the discontinuous region between the elliptical head and cylinder is deduced according to the deformation continuity equation 28 . Second, the experimental stress is measured using an internalpressure thin-walled vessel stress measurement experimental device, and the theoretical and experimental stress values in the discontinuous region are compared and analysed to verify the accuracy and applicability of the theoretical stress calculation formula in the discontinuous region between the elliptical head and cylinder 19,29 . The findings are expected to have guiding significance for the stress measurement of internal-pressure vessels.

Theoretical stress calculation formula in the discontinuous region between the elliptical head and cylinder
First, the edge force Q 0 and edge moment M 0 are calculated according to the deformation continuity equation 30 , and subsequently, the internal force and internal moment are determined according to the edge force Q 0 and edge moment M 0 . Finally, the theoretical stress calculation formula in the discontinuous region between the elliptical head and cylinder can be obtained by substituting the internal pressure, internal force and internal moment into the calculation formula of the membrane stress and bending stress 31 . The specific analysis process is described in the following text.
Solution of the edge force Q 0 and edge moment M 0 in the discontinuous region of the elliptical head. From the perspective of stress and manufacturing, an elliptical head represents the ideal structural form and is widely used in the petroleum, chemical, medicine, machinery, construction, nuclear power and other industries. The connection structure between the elliptical head and cylinder is shown in Fig. 1.
The connection boundary deformation diagram is shown in Fig. 2, in which (a) shows the overall deformation diagram, (b) shows the membrane deformation caused by the internal pressure P, (c) shows the bending deformation caused by the edge force Q 0 , and (d) shows the bending deformation caused by the edge moment M 0 . According to the deformation continuity condition, after the superposition of the membrane deformation (b) and bending deformation (c) and (d), the total deformation values (a) of the elliptical head and cylinder at the connection boundary are equal.
Under the action of the edge force and edge moment, the rigid body motions of the elliptical head and cylinder are constrained, and they are viewed as rigid bodies. The deformation continuity equation of the elliptical head and the cylinder can be expressed as where subscripts 1 and 2 represent the elliptical head and cylinder, respectively. Superscripts P, Q 0 and M 0 represent the internal pressure, edge force and edge moment, respectively.  According to the literature 13 , the membrane deformation equations of the elliptical head can be expressed as (2) �   www.nature.com/scientificreports/ where t 1 is the thickness of the elliptical head, a is the length of the semi-major axis, and b is the length of the semi-minor axis. P is the internal pressure, μ is Poisson's ratio, and E is the elastic modulus. The bending deformation equations of the elliptical head can be expressed as where R is the second radius of curvature at the boundary. The membrane deformation equations of the cylinder can be expressed as where t 2 is the thickness of the cylinder. The bending deformation equations of the cylinder can be expressed as When t 1 = t 2 , substituting Eqs. (2), (3), (4) and (5) into Eq. (1) yields the edge force and edge moment as follows: www.nature.com/scientificreports/ Solution of the theoretical stress in the discontinuous region of the elliptical head. The connection boundary load diagram is shown in Fig. 3, in which (a) is the overall load diagram, (b) is the diagram of the load caused by the internal pressure P, (c) is the diagram of the load caused by the edge force Q 0 , and (d) is the diagram of the load caused by the edge moment M 0 . The internal pressure P produces the internal membrane force, which generates the membrane stress, and the edge force Q 0 and edge moment M 0 produce the internal bending force, which generates the bending stress. The total stress can be achieved by superposition of the membrane stress and bending stress. According to the literature [30][31][32] , the total stress in the discontinuous region can be divided into the membrane stress and bending stress, and the theoretical stress equation of the elliptical head can be expressed as where subscripts θ and ω correspond to the tangential direction of the parallel circle and longitude, respectively. σ ω and σ θ denote the total longitudinal stress and circumferential stress, respectively. σ P ω and σ P θ denote the longitudinal stress and circumferential stress under internal pressure P, respectively. N The internal force and internal torque equations of the elliptical head can be expressed as www.nature.com/scientificreports/ where R 1 is the radius of the first curvature, R 2 is the radius of the second curvature, x and y are the coordinates of the measurement point on the elliptical head, a is the long half shaft, and b is the short plate shaft. Substituting Eq. (8) into Eq. (7) yields the total theoretical stress in the edge region of the elliptical head as follows: The theoretical stress equation of the cylinder is expressed as where subscripts θ and x correspond to the tangential direction of the parallel circle and longitude, respectively. σ x and σ θ represent the total longitudinal stress and circumferential stress, respectively. σ P x and σ P θ represent the longitudinal stress and circumferential stress under internal pressure P, respectively. N

Stress analysis and verification in the discontinuous region between the elliptical head and cylinder
To verify the correctness of the theoretical stress in the discontinuous region between the elliptical head and cylinder, the experimental stress of the elliptical head under different pressures is investigated using an internal-pressure thin-walled vessel stress measurement experimental device. The accuracy of the theoretical and experimental stress values in the discontinuous region is verified by comparing the errors of the theoretical and experimental stress values to provide reference for the stress distribution of the elliptical head.
Measurement of experimental stress in the discontinuous region of the elliptical head. The material and mechanical properties of the prototype for the experiment are listed in Table 1.
The experimental device is shown in Fig. 4. First, water is added to the pressure pump, and by shaking the pressure pump longitudinally, the pipeline is pressurized. The pressure is transmitted to the internal pressure www.nature.com/scientificreports/ vessel to produce stress and strain. The stress is measured using a pressure gauge and transmitted to a computer through a pressure sensor. The strain is measured using a strain gauge. Ten strain gauge measurement points are set. The distance from each measurement point to the vertex of the elliptical head is shown in Table 2. Measurement points 1 to 9 are located on the elliptical head, measurement point 10 is located on the cylinder, and the distribution of the measurement points is shown in Fig. 5.
Considering constant geometric dimensions of the elliptical head, the internal pressure P value is set as 0.1 MPa, 0.2 MPa, 0.3 MPa, 0.4 MPa and 0.5 MPa. By changing the internal pressure P, the influence of the internal pressure on the experimental values of the longitudinal stress and circumferential stress at each measurement point is investigated, as shown in Figs. 6 and 7. When the internal pressure P is unchanged, the longitudinal stress value is always greater than zero, corresponding to tensile stress. Measurement point 1 is located at the vertex, and measurement point 10 is located at the equator. The longitudinal stress gradually decreases from measurement points 1 to 10. The longitudinal stress is the maximum at measurement point 1, and the longitudinal stress at measurement point 10 is half the stress value at measurement point 1. The reduction range from measurement   www.nature.com/scientificreports/ points 8 to 10 is less because these measurement points are located in the discontinuous region, which bears not only the membrane stress but also the discontinuous stress. In terms of the circumferential stress, the variation trend is more complex and can be divided into three stages 33 . In the first stage, from measurement points 1 to 6, the circumferential stress is greater than zero, corresponding to tensile stress; in the second stage, from measurement points 7 to 9, the circumferential stress is less than zero, corresponding to compressive stress; in the third stage, the circumferential stress is greater than zero at measurement point 10, corresponding to tensile stress. From measurement points 7 to 10, the stress first decreases and later increases, similar to the longitudinal stress, affected by the discontinuous stress 34 . At measurement point 1, the longitudinal stress and circumferential stress are approximately equal. At measurement point 6, the circumferential stress changes from tensile stress to compressive stress. This phenomenon occurs because the stress distribution of the elliptical head is complex. The value of the longitudinal stress is always greater than zero, corresponding to tensile stress, and it gradually decreases from the vertex to the equator. The circumferential stress is related to a/b. When a/b < 2 0.5 , the stress value is always greater than zero, corresponding to tensile stress, and this value gradually decreases from the vertex to the equator. When a/b = 2 0.5 , the value is zero at the equator, and when a/b > 2 0.5 , compressive stress occurs. In this study, the a/b value of the experimental device is 2, and thus, the longitudinal stress is always tensile stress, and the circumferential stress is compressive stress [35][36][37][38] .   www.nature.com/scientificreports/ Calculation of the theoretical stress in the discontinuous region of the elliptical head. According to the arc length formula, the meridian angles (radian system) of measurement points 1 to 9 starting from the boundary are 1.57, 1.43, 0.62, 0.60, 0.43, 0.33, 0.23, 0.11 and 0.06, respectively. Measurement point 10 is distributed on the cylinder. According to the conversion, the distance in the meridian direction from the boundary of measurement point 10 is 10 mm. The distances between the measurement points and edge of the elliptical head are listed in Table 3.
The stress enhancement coefficient of each measurement point, calculated using the abovementioned formula, is presented in Table 4: By substituting these data into the total stress calculation formula of the elliptical head and cylinder edge region, the theoretical longitudinal stress and circumferential stress at measurement points 1 to 10 can be obtained, as shown in Figs. 8 and 9. The findings are consistent with the variation law of the experimental stress: For the same internal pressure P, the theoretical longitudinal stress is always greater than zero, corresponding to tensile stress, and its value gradually decreases from measurement points 1 (vertex) to 10 (equator). The theoretical circumferential stress is compressive stress 39 . At the vertex, the theoretical longitudinal stress is approximately equal to the theoretical circumferential stress 40 .

Comparison of two analysis methods for the elliptical head.
To compare the accuracy of the two analysis methods of the elliptical head, the internal pressure load of 0.1 MPa is considered as an example, and the comparison of the theoretical stress and experimental stress is presented in Figs. 10 and 11. The variation law of the theoretical stress of the elliptical head is consistent with that of the experimental stress, which verifies the correctness and reliability of the theoretical method. For the longitudinal stress, a large error is noted between the theoretical and experimental values at measurement point 1, which can be attributed to the inaccurate location of the strain gauge or sensitivity of the temperature and test instrument.
Finite element analysis of the elliptical head. Considering the elliptical head as the research object, an FEM analysis is conducted using ANSYS software under an internal pressure load of 0.1 The model is shown in Fig. 12a, and the distributions of the longitudinal stress and circumferential stress are shown in Fig. 12b,c,    Figure 11. Circumferential stress obtained using the two analysis methods. www.nature.com/scientificreports/ respectively. The longitudinal stress is always greater than zero, corresponding to tensile stress. The circumferential stress decreases gradually from the apex to the equator, and compressive stress occurs.
To compare the accuracy of the FEM-obtained stress and experimental stress of the elliptical head, the internal pressure load of 0.1 MPa is considered as an example, and the comparison of the FEM-obtained stress and experimental stress is shown in Figs. 13 and 14. The FEM-obtained longitudinal stress first decreases, subsequently increases, and later gradually decreases from measurement points 1 to 10. This phenomenon is highly inconsistent with the variation trend of the experimental longitudinal stress, and the relative error percentage is up to 45%. The FEM circumferential stress first decreases and later increases from measurement points 1 to 10, and compressive stress occurs, consistent with the change trend of the experimental longitudinal stress. However, the relative error percentage is extremely large, up to even 600%. The comparison results show that the theoretical formula is superior to the FEM for solving this problem.

Conclusion
1. The theoretical stress calculation formula for the discontinuous region between the elliptical head and cylinder is obtained considering the deformation continuity equation, edge force, edge moment, internal force and internal moment. 2. When the internal pressure load is constant, the theoretical meridian stress is always greater than zero, corresponding to tensile stress, and this value decreases gradually from the vertex to the equator. The variation trend for the theoretical circumferential is relatively complex and can be divided into three stages; moreover, this stress is compressive. At the vertex, the longitudinal stress and circumferential stress are approximately equal. The change from measurement points 8 to 10 is influenced by the discontinuous stress, and the variation trend is abrupt. 3. By comparing and analysing the theoretical stress and experimental stress in the discontinuous region between the elliptical head and cylinder, the accuracy and applicability of the theoretical stress calculation formula in the discontinuous region are verified.  www.nature.com/scientificreports/