Jun 17, 2009 - with theoretical and numerical data for the cylinder under investigation. ... mounted on the inner and outer surface of the thick-walled cylinder and results were ... The thick-walled cylinder was tested as shown in Figure 1. An ... the cylinder is represented by ri, the outside radius by ro, the internal pressure by ...
Proceedings of the ASME 2011 International Mechanical Engineering Congress & Exhibition IMECE2011 November 11-17, 2011, Denver, Colorado, USA
Experimental Verification of Model Pressurized Thick-Walled Cylinder with Numerical and Theoretical Methods Sunilbhai P. Macwan Department of Mechanical Engineering South Dakota State University Brookings, SD, 57007, USA
Zhong Hu Department of Mechanical Engineering South Dakota State University Brookings, SD, 57007, USA
ABSTRACT Pressurized thick-walled cylinders undergo repeated cycles of high stress and temperatures that may severely shorten the life of the component. Testing pressurized cylinder can help to evaluate the strength of the cylinder. This research seeks to determine the pressure to which the component is subjected by instrumenting the outside of the cylinder, and to evaluate hoop strain and hoop stress of the internal and external surface of the pressurized thick-walled cylinder. This study provides experimental results and then compares them with theoretical and numerical data for the cylinder under investigation. Using the experimental method, an axial load up to 15,000 lb is applied to the cylinder using a Landmark 370 MTS unit to generate pressure inside the cylinder wall. Lamé equations are used to calculate hoop stress theoretically. The numerical data is obtained using finite element simulation (ANSYS) to calculate hoop stress and hoop strain at the internal and external surfaces of the cylinder. This work provides useful information for evaluating the strength of thick-walled cylindrical structures in a laboratory setting. INTRODUCTION Thick-walled cylinders have been widely used in various industries for many years. In practice, a cylinder is identified by its radius to thickness (r/t) ratio. A cylinder must have a radius-tothickness ratio of less than 10 to be considered a thick-walled cylinder [1]. By definition, a thick-walled cylinder is classified as a pressure vessel, i.e. a hydraulic cylinder, a gun barrel, a high pressure pipe, a boiler drum and a nuclear reactor. Safe design and operation of thick-walled cylinder is of primary concern for the designers and operators because unsafe design and faulty operation can lead to sometimes fatal accidents and financial losses. Because safety is the prime concern, all thick-walled cylinders are tested at 1.25 to 1.5 times greater pressure than that for which it was designed before being put into operation [2]. During the testing, visual inspection is carried out to detect any leakage and crack on the surface as well on weld joints. The pressure reading is also continuously monitored and pressure is recorded after some duration. Fatigue testing is also performed on thick-walled cylinder depending on the requirements. Extensive research has been carried out to determine appropriate methods to test and verify design of thick-walled cylinder. Hameed [3] compared the external expansion of thick-walled cylinder by measuring hoop strain on outer surface of cylinder. Elasto-plastic analysis of a thick-walled cylinder was performed by Zhao, Seshadri and Dubey [4]. Darijani, Kargarnovin and Naghdabadi [5] designed thick-walled cylinders under internal pressure. An experimental study
Fereidoon Delfanian Department of Mechanical Engineering South Dakota State University Brookings, SD, 57007, USA
to measure the maximum internal pressure on steel and aluminum material of thick walled cylinder was developed by Roach and Priddy [6]. Analytical and numerical methods are used to develop reliable design of thick-walled cylinders. The analytical expressions for the elasto-plastic stress, strain and displacement components of an internally pressurized closed end thick-walled cylinder of strain hardening material with large strains are derived by Gao [7]. In recent years, Finite Element Analysis (FEA) has become a valuable tool to design and analyze thick-walled cylinder at various pressures. Kihiu, Mutuli and Rading [8] characterized the stress in thick-walled cylinder by Finite Element Method. In a majority of cases, the design of thick-walled cylinder is carried out by considering hoop stress as the primary stress for failure of thick-walled cylinder, as it is the maximum stress which can be developed on the inside surface of the thick-walled cylinder. Research suggests that a crack starts from the inside surface of the thick-walled cylinder and propagates towards the outer surface of the cylinder [8] [9]. This research seeks to determine hoop and radial stress on the thick-walled cylinder at very high internal pressure for elastic region. To do so, three methods – theoretical, numerical and experimental-are selected and comparisons are made to validate results from all three methods. Details of the methods and procedures used will be described, and the results obtained from these methods will be discussed. Finally, comparisons will be done of the three methods and the validity of the results will be considered. In order to verify FEA results with the design stress, the experimental test is carried out by pressurizing the thick-walled cylinder beyond its design limit and determining stresses developed on it. To do so, strain gages are mounted on the inner and outer surface of the thick-walled cylinder and results were obtained at various internal pressures. EXPERIMENTAL METHOD The thick-walled cylinder was tested as shown in Figure 1. An axial compression load was applied on the thick-walled cylinder to generate pressure in the cylinder. First, the cylinder was filled with oil (SAE68) and then tested two times applying different loads each time – 10,000 lb and 15,000 lb at the 500 lb increments. Due to safety concerns, the first load was applied incrementally up to 10,000 lb on the cylinder, and the hoop strain and axial strain measured through the rectangular rosette strain gages. After obtaining successful results, additional safety precautions were taken before loading the cylinder to 15,000 lb. Essentially, the specimen was placed in a safety enclosure because the testing exceeded the cylinder’s design parameters. For both phases of the experimental work, a data 1
acquisition system, a laptop, wheat stone bridge circuits, wires, two AC supply system and a digital camera were used, see Figure 2.
Figure 3 Internal pressure vs. hoop strain THEORETICAL METHOD
Figure 1 Model test cylinder in Landmark MTS machine
Thick-walled cylinders are commonly used in industry to carry both liquids and gases under pressure. Because of high internal pressures, both radial and hoop stresses develop on the cylinder wall with values that are dependent upon the radius of the element under consideration. In determining the radial stress (σr) and the tangential (hoop) stress (σt), it is assumed that the longitudinal elongation is constant around the circumference of the cylinder [1]. When a thick-walled cylinder is exposed to internal or external pressure, the material that comprises the cylinder is subjected to pressure loading and, hence, stresses from all directions. Three types of stresses develop on the thick-walled cylinder under pressure -radial stress, hoop stress, and axial stress. First, the stresses being applied to the test specimen must be calculated. The inside radius of the cylinder is represented by ri, the outside radius by ro, the internal pressure by pi, and the external pressure by po in Figure 4[1]. The hoop stress (σθ), radial stress (σr) and axial stress (σz) are shown on the cylindrical element of thickness dz in Figure 4[1]. The hoop stress is acting on the tangential direction of thick-walled cylinder which is considered in designing the thick-walled cylinder. The radial and axial stresses act on the radial and axial directions of thick-walled cylinder, respectively.
Figure 2 Test set up The internal pressure vs. hoop strain graph (Figure 3) shows that the hoop strain increases linearly within the elastic region of the cylinder for all three strain gages. The plot from strain gage 1 and 2 once again overlaps with the exception of a few locations. The hoop strains from strain gage 1 and 2 deviate from each other around 4,000 psi to 5,000 psi as well as 7,000 psi to 8,000 psi internal pressure, but as the pressure increases, the hoop strain from both strain gages follows the same curves. The hoop strain from strain gage 3 deviates from hoop strain from strain gages 1 and 2. Strain gage 3 follows the strain gage 2 in parallel from beginning of the test and continues until the end of test.
Figure 4 Cylinder subjected to both internal and external pressure [1] 2
The Lamé equations for hoop stress, radial stress and axial stress for the internal and external pressure can be derived from cylindrical element thickness dz by considering stresses acting on it [1]. Consequently, it can be shown that tangential and radial stresses exist whose magnitudes are: 2
r r p p r r r
pr p r
t
i
i
o
r
2
2
i
o
2
2
o
i
o 2
i
(1)
r r p p pr p r r r 2
2
o
i
i
ro 2
o
2
2
i
o
2
2
o
i
o 2
i
(2)
The positive value indicates tension and negative value, compression. The special case of po = 0 gives:
r p r r
2 1 r o 2 2 r i
(3)
2
2 1 r o 2 2 r i
(4)
2
t
i
i
2
o
r p r r i
r
2
o
i
the cross-section area of the thick-walled cylinder was constructed and meshed using the PLANE42 elements, a four-node 2D axisymmetric structure element, an area was generated inside the cylinder which simulated fluid inside the cylinder by using element FLUID79, a four-node 2D axi-symmetric contained fluid element. The contact (CONTA171) and target (TARGE169) elements were applied on the contact surface of oil and inner radius of thick-walled cylinder in order to slide into the cylinder to create high pressure on the cylinder wall. The results of radial and hoop stresses obtained from ANSYS simulation are shown in Figures 5 and 6. The color bar indicates the development of the radial and hoop stresses on cylinder wall at different location. In the enlarged inset portion, the colors represent the range of radial and hoop stress at various radii. The range is very large for this particular radius of thick-walled cylinder. For the accuracy of the results, the values of radial and hoop stresses are carefully collected at the particular nodes.
The theoretical values for hoop and radial stress are evaluated at various radii for 500 psi to 15,000 psi internal pressure. The dimensions and various radii are mentioned in Table 1. Table 1 Dimensions of thick-walled cylinder Cylinder Dimensions Bore
1.13"
Inner Radius (ri)
0.565"
Outside Diameter
1.5"
Radius at point 1 (r1)
0.61125"
Thickness
0.185"
Radius at point 2 (r2)
0.6575"
Collapsed Height
6.5"
Radius at point 3 (r3)
0.70375"
Area
1 in2
Outer Radius (ro)
0.75"
Figure 5 Radial stress distributions at 15,000 psi
The radial stress and hoop stress are calculated from 500 psi to 15,000 psi of internal pressure in 500 psi increments. It is observed that the hoop stress and radial stresses increase as the internal pressure increases; however, the radial stress on the outer surface of the cylinder remains as zero, satisfying the boundary condition. In addition, the hoop and radial stresses at 15,000 psi internal pressure are calculated at various radii and depicts that the absolute values of hoop and radial stresses decrease from the inner to outer radius of the cylinder wall for the same internal pressure. NUMERICAL METHOD By using commercial software, finite element analysis was performed to evaluate hoop stress and hoop strain at the various radii of the pressurized thick-walled cylinder and then these results were compared with the theoretical and experimental methods. Because the thick-walled cylinder is symmetric about the center axis of the bore of the cylinder, an axisymmetric model was constructed. Once
Figure 6 Hoop stress distributions at 15,000 psi 3
The absolute value of radial stress developed on the outer surface of the thick-walled cylinder was 253 psi, while maximum radial stress developed on the inner surface of the thick-walled cylinder was 14,676 psi. Similarly, the maximum hoop stress of 53,815 psi occurred on the inner surface of the thick-walled cylinder, while minimum hoop stress of 38,866 psi developed on the outer surface of the thick-walled cylinder. As observed in theoretical method, absolute values of hoop and radial stresses decrease from the inner to outer radius of the cylinder wall for the same internal pressure in numerical method. In addition, hoop strain distribution on cylinder wall is evaluated from ANSYS simulation and presented in Figure 7. The hoop strain distribution on cylinder wall at ri is 1899.8 µε, at r1 is 1690.9 µε, at r2 is 1526.3 µε, at r3 is 1391.8 µε and at ro is 1280.2 µε. This depicts the hoop strain gradually decreasing from the inner surface to the outer surface of the thick-walled cylinder.
Table 2 Comparison between theoretical and numerical method Cylinder radius Inch
Theoretical Method
Numerical Method
Percent Error
Hoop Stress
Radial Stress
Hoop Stress
Radial Stress
Hoop Stress
Radial Stress
psi
psi
psi
psi
%
%
ri
0.565
54365.94
-15000
53815
-14676
1.013
2.160
r1
0.611
49315.96
-9950
48751
-9746
1.146
2.044
r2
0.658
45293.69
-5927
44822
-5840
1.041
1.465
r3
0.704
42038.06
-2672
41582
-2617
1.085
2.048
ro
0.75
39365.94
0.00
38866
-253
1.270
N/A
In comparing all three methods, it must be noted due to the limited access to inner surface of the thick-walled cylinder, the hoop strain was obtained for only the outer surface of the thick-walled cylinder and compared with the theoretical and numerical methods. Hoop stress for the internal pressure from 500 psi to 15,000 psi increment of 500 psi is presented in Figure 8. Looking at the plot, the results for hoop strain are linear as predicted for the elastic portion of thick-walled cylinder and the strain increases as the internal pressure on cylinder wall increases. For 5,000 psi internal pressure, the hoop strain is 430.82 µε for theoretical method, 425.87 µε for numerical method and 409.34 µε for experimental method. Consequently, the percent error between the theoretical and numerical methods is 1.149%, theoretical and experimental, 4.99% and finally, numerical and experimental, 3.88%. As the internal pressure increases up to 10,000 psi, the hoop strain is 861.65 µε for theoretical method, 853.74 µε for numerical method and 818.61 µε for experimental method. The percent error is within 0.91% for theoretical and numerical method and within 5% for the other two methods. Moreover, when the internal pressure reaches its final value, in this case 15,000 psi, the percent error is 0.949% for theoretical and numerical method, but it is 3.33% for theoretical and experimental methods and 2.40% for numerical and experimental methods.
Figure 7 Hoop strain distribution at 15,000 psi
COMPARISONS To determine the reliability of this research work, the three methods are compared graphically and statistically. Percent error method is used in comparing results from experimental and numerical methods with theoretical method which can be considered correct value. The hoop and radial stresses on various thicknesses of the cylinder wall are close to each other for 15,000 psi internal pressure with the percent error for hoop stress within 1.3% and radial stress within 2.2%. At the cylinder inner surface which is a critical point, the percent error in hoop stress is less than that of the outer surface. Hence, from this analysis, the strength of the thick-walled cylinder can successfully be predicted from Finite Element Analysis using ANSYS software. For the validity of the results, the elastic stress distribution was compared with the elastic response presented by J. M. Kihiu [8]. He observed that at the cylinder bore, the percentage error in hoop stress is 1.109 % and a slight discrepancy exists between the FEM and analytical values of stress at the outside and inside surfaces of the cylinder for the hoop and radial stresses. In this research work, the percentage error for hoop stress at the inner surface of cylinder is 1.013 %, which is very close the error discussed by Kihiu.
Figure 8 Hoop strain comparison at 15,000 psi CONCLUSIONS In this work, the results for radial stress and hoop stress obtained from the theoretical and numerical methods are in good agreement 4
with each other. The results for hoop strain are very close for all three methods on the outer surface of the thick-walled cylinder. It is observed from the study that the percent error between theoretical and numerical is within 1.0 %, between theoretical and experimental method is within 3.33 % and between numerical and experimental is within 2.4 % at 15,000 psi internal pressure. Hence, the percent error for all three methods is within 4 %. ACKNOWLEDGEMENT
[12]. ASTM standard D 5449/D 5449M – 93 (Reapproved 2006) “Standard test method for transverse compressive properties of hoop wound polymer matrix composite cylinders”, ASTM Int’l, Wed Jun 17 12:04:30 EDT 2009. [13]. G. A. Wedgwood, “The elastic properties of thick walled cylindrical shells under internal pressure,” 1929, pp 366-386. [14]. R. J. Eggert, “Design variation simulation of thick-walled cylinders”, Journal of mechanical design, ASME, June 1995, Vol. 117, pp. 221-228.
This research was sponsored by Benet Laboratories on behalf of the US Army Contracting Command Joint Munitions & Lethality Contracting Center and was accomplished under Cooperative agreement Number W15QKN-09-2-0002. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of Benet Laboratories or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation hereon. REFERENCE [1]. Joseph Shigley and Charles R. Mischke. Mechanical Engineering Design, 6th edition, McGraw-Hill, Inc, 1221 Avenue of the Americas, New York, NY, 10020, (2001). [2]. http://en.wikipedia.org/wiki/Hydrostatic_test [3]. Amer Hameed, Robert D. Brown and John G. Hetherington, “Comparison of the external expansion of a pressurized thick walled cylinder with predictions from a finite element model,” Journal of Battlefield Technology, Vol. 1, No. 2, pp. 10-14 (July 1998) [4]. W. Zhao, R. Seshadri and R. N. Dubey, “On thick-walled cylinder under internal pressure,” Journal of Pressure Vessel Technology, Vol. 125, pp. 267-273 (August 2003) [5]. Darijani , H., M. H. Kargarnovin and R. Naghdabadi, “Design of thick walled cylindrical vessels under internal pressure based on elasto-plastic approach,” Materials and Design Vol. 30 (2009): 3537 – 3544. [6]. Roach and Priddy, “Effect of Material Properties on the Strain to Failure of Thick-Walled Cylinders Subjected to Internal Pressure”, Journal of pressure vessel technology, May 1994, Vol. 116, Issue 2, pp. 96-105. [7]. Gao Xin-Lin, “An exact Elasto-Plastic solution for a closed-end thick-walled cylinder of elastic linear-hardening material with large strains”, International general of pressure vessel and piping 56, 1993, pp. 331-350, Elsevier science publishers ltd, England. [8]. J. M. Kihiu, S. M. Mutuli and G. O. Rading, “Stress characterization of autofrettaged thick walled cylinders,” International Journal of Mechanical Engineering Education, Vol. 31, No. 4 pp. 370-389 [9]. Joseph Perry and Jacob Aboudi, “Elasto-plastic stresses in thickwalled cylinder,” Journal of Pressure Vessel Technology, Vol. 125, pp. 248-252 (August 2003) [10]. Tony D. Andrews and Fred E. Brine, “Hydraulic testing of ordnance components,” Journal of Pressure Vessel Technology, Vol. 128, pp. 163-167 (May 2006) [11]. Omid Vakili, Zhong Hu and Fereidoon Delfanian, “Strength evaluation and fatigue prediction of a pressurized thick walled cylinder,” Proceedings of ASME International Mechanical Engineering Congress and Exposition, pp. 11-15 November 2007, Seattle, Washington. 5
