4. Objectives of Research work

Transcription

1 4. Objectives of Research work 4.1 Objectives of Study: The design of bellows is challenging looking to varieties of applications and evaluation of stresses is further difficult to approximate due to its customized geometric parameters. The design of a bellows is complex as that involves an evaluation of pressure capacity, stress due to deflection, fatigue life, spring forces and stability. The determination of standard or series of geometry is not possible due to numerous variables and varieties of industrial applications. The geometric variables are inside diameter, material thickness, number of convolution, shape of convolution, height of convolution, pitch of convolution, number of plies, and method of reinforcement, manufacturing technique, material type and heat treatment. The research objectives are summarized below. 1. Analysis of Design Parameters: Design of the bellows for a specific requirement depends on several factors like, type of fluid, expected fluctuation of pressure, variations in temperature and probable axial, lateral or angular movements. Hence, the designer should be aware about all the variable conditions of the application. Each parameter results variations in design and performance of bellow, hence customized design approach is suggested by EJMA. Design of bellows in certain applications will involve a compromise of conflicting requirements. Hence, it is the responsibility of a design engineer that, bellow should be suitably designed for specific application or expected motions very precisely. It will be interesting to study about role of each individual design parameter on the performance of bellows. 2. Investigation of Stability of bellows: When internal pressure reaches excessive value, a bellow to become unstable and it deforms from the middle region. This kind of failure is termed as squirm failure. Squirm failure may be avoided, if the material thickness is on higher side, but this will greatly reduce the fatigue of bellows. Also if the material 117

2 thickness is considered on minimum criteria, its pressure withstanding capability is reduced. Hence, squirm is determining parameter to bellows performance in that it can greatly reduce both fatigue life and pressure capacity. Two most common forms of squirm are column squirm and inplane squirm. Determination of acceptable design with reference to stability criteria is the important task for the designers. 3. Investigation of Spring Rate of bellows: An important and additional feature required of the bellow is flexibility. The bellow should be flexible enough so as to take deflection and motion easily, and it should be strong enough to resist the pressure and temperature conditions. The flexibility may be measured in terms of stiffness or spring rate, i.e. force required to deflect bellow in the initial condition. It is a function of the dimensions of the bellows and the material from which it is made. It will be necessary to study about role of geometric features and resultant stiffness of bellow. Evaluation of actual spring rate is critical task for the designers. 4. Investigation of Fatigue Life expectancy: The total number of complete cycles which can be expected from the expansion joint based on data tabulated from tests performed at room temperature under simulated conditions is called fatigue life. A cycle is defined as one complete movement from initial positioning the piping system to the operating position and back to initial position just like a pendulum. Fatigue life is dependent upon the maximum stress range which the bellows is subjected, the maximum stress amplitude being the far less significant factor. It is also affected by various factors such as operating pressure, operating temperature, material of bellows, movement per convolution, the convolution pitch, the depth and shape of the convolutions and bellows heat treatment. Any change in these factors will result in a change of fatigue life of the expansion joint. 118

3 It is the responsibility of design engineers that the bellows should function as per the designed expected number of cycles. Also frequent testing of bellows will help designers to study the precise behavior of the bellows. 4.2 Research Methodology: The study about expansion joint should start with analysis of geometric parameters. It is necessary to understand influence of various geometric parameters of the bellows and their effect on performance of the bellow. There are three possibilities or research methods are considered for the present research work. They are as follows. 1) Analytical Approach: Geometry of bellows is directly affecting its performance. Hence, geometric relationships can be varied and their effect on the performance can be observed analytically. This approach will lead designers towards the proper understanding of each geometric parameter and precise design of bellows. Since, there are many design parameters; optimization technique may be implemented to improve the performance behavior of bellows. 2) Finite Element Analysis: Experimental testing is always time consuming, and costly affair to check the performance of bellows expansion joint. Also experimental measurement of stress is very difficult. Hence, stress analysis may be analyzed in the virtual environment, using Finite Element Analysis software. Once the results are validated in standard shape, any geometrical changes can be made in the component and performance can be evaluated from FEA results. Even prototype element can be made and results may be predicted for any newer expansion joint. This method is comparatively faster, reliable and economical compared to actual experimental methods. Only validation of result is required. 3) Experimental Testing Approach: In this methodology experimental performance testing of bellows can be carried out and results will indicate further understanding about the performance behavior of expansion joints. But, this method requires test rig facility along with instrumentation facilities in order to measure the results. The performance testing of bellows may be studied for the expected validation of pressure capacity (hydro test, pneumatic test), spring rate test, rupture test, squirm test, and fatigue life test. 119

4 Study of design parameters of bellows is carried out from the next paragraph. An analytical methodology is implemented to perform this study. Here, EJMA codes are utilized to study the effect of each geometric parameter on other factors. Results of parameters are analyzed using graphs. The FEA methodology is covered in chapter 5 and the testing of bellows are described in chapter Analysis of Design Parameters: The study of parameters is carried out analytically using EJMA mathematical relations. Due to large variations in geometry, and because of lack of standard manufacturing facility at various industries, many researchers have adopted the computerized technique to evaluate the stresses. Also instrumentation of stresses and strain is very much difficult in the experimentation work. Also by variation of geometry only there will be change in all stresses developed. Figure 4.1 shows important geometric parameters of bellows. Figure 4.1: Geometric parameters of Bellow Material thickness: Bellows are made from either thick or thin materials. Thick material bellows are used for high pressure applications. Bellows made from thin sheet metal used for comparatively lower pressure applications. Basic function of bellow is to take up the axial and lateral motions, occurring because of fluctuation in pressure and 120

5 temperature. Thin material bellows can undergo higher deformations. Hence, bellows should be flexible enough to take-up the variations and also strong enough to resist the pressure fluctuations. This conflicting need for thickness for pressure and thinness for flexibility is the unique design problem faced by the expansion joint designers. The kinds of stresses produced in bellows are circumferential membrane stresses and longitudinal membrane stress and longitudinal bending stresses. The circumferential membrane stress and longitudinal membrane stress is increasing with decrease in material thickness. But, the material thickness should be strong enough to withstand the inside pressure fluctuations. Bellows are supposed to take axial, lateral, and angular motions because of variations in pressure and temperature. Since this motion produces cyclic fluctuations, bellow material undergo low cycle fatigue. This phenomenon is observed by C Betch [3] in his study. In case of higher thickness bellow material, fatigue life is reduced, as it can resist less number of cycles. While thin material can undergo more number of cycles and hence their fatigue life is always higher. Selection of thickness should be optimum considering the pressure withstanding capacity and expected cycle life. However, thin material can be used in multiple plies is an alternate option for the designers, which will be discussed later on. Stress Analysis using Analytical Approach: The estimation of stresses is essential for the design of expansion joints for various applications. The steps are elaborating the design procedure of expansion joint. The procedure is based on design fundamentals and EJMA codes. [20] Thin cylinders, subjected to internal pressure, according to Barlow equation, Circumferential membrane stress = P D 2t (4.1) Modifying the equation for the bellows according to its geometries, Bellows tangent circumferential membrane Stress, S 1 = P Db nt k 2 nt (4.2) To consider the bellows convolutions geometry, Bellows circumferential membrane stress, S 2 = P D m 1 2nt w/ q (4.3) 121

6 The circumferential stress at convolution (S 1 ) and at tangent length S 2 should not exceed the permissible stress of the material. Thin cylinders, subjected to internal pressure, Longitudinal stress = P D 4t (4.4) Modifying the equation for bellows, taking mean diameter equivalent to (2 w) and adding number of plies; 2 P w Longitudinal membrane stress, S 3 = = 4 nt Longitudinal bending stress = S 4 = P D nt w t 2 p 2 P w 2nt C p (4.5) (4.6) The total longitudinal stress (S 3 +S 4 ) should not exceed the permissible stress of the material. For following set of dimensions, the stresses and other parameters are calculated using same program mentioned in Chapter 3 (page no. 109). Here to study effect of one parameter (like thickness), other dimensions are kept constant. Design Pressure, P = 50 N/cm 2 Material : SS 304 Inside Diameter, Db = 40.6 cm Yield stress = N/cm 2 Thickness, t = 0.08 cm Modulus Elasticity= N/cm 2 Height of Convolution, w = 2.3 cm Design temp. = 50 0 C Pitch of Convolution, q = 2.26 cm Elasticity at temp. = N/cm 2 Number of convolutions = 15 Number of ply = 1 Effect of Thickness: As the thickness increases, stresses decreases. Figure 4.2 show that circumferential stress and longitudinal stress decreases with increase in thickness. Also longitudinal stresses are always higher than circumferential stresses. Table 4.1: Effect of material thickness on stresses and Life cycles Thickness, t,cm Circumferential stresses,s 1 +S 2, N/cm Longitudinal Stresses, S 3 + S 4, N/cm Number of Life Cycles, Nc

7 Stresses, N/cm Material Thickness, cm Circumferential stress Longitudinal stress Figure 4.2: Effect of Material thickness Figure 4.3 shows that increase in bellow material thickness, which results in reduction in number of life cycles. So fatigue life is reducing nonlinearly. Number of cycles thickness of bellows, cm Life cycles Figure 4.3: Effect of thickness on Number of Life cycles Number of plies: Considering wall thickness and convolution size parameters, single wall thin bellows may have limited applications, considering stress or stability criterion. To overcome this limitation, multi-ply bellows can be made by telescoping two or more cylinders and forming them together. Multi-ply bellows may be advantageous for reducing the risk of sudden and complete failure. Also, in case of multi-ply the inner ply highly corrosion resistance material is used and as outer ply less costly high strength material can be used. Here, the fatigue resistance is limited by the inner ply. 123

8 In multiple plies material thickness is more (compare to single ply), so as strength of the bellows is more; hence they can be used for higher pressure applications. Also this will lower the stresses. Here the thickness of one ply and number of plies will give rise to total thickness of the material, e.g. instead of 0.08 cm thickness material, 2 plies of 0.04 cm thickness material can be employed. Multiple plies will limit the individual material thickness and increases the strength without reduction in number of cycles. Also there will be increase in factor of safety. In estimation of each stresses, thickness should be considered as the product of number of plies (n) and individual material thickness (t). (Refer equations 1 to 4) Table 4.2: Effect of plies on stresses and In-stability of bellow Number of Plies, n Circumferential stresses, S 1 +S 2,N/cm 2 Longitudinal Stresses, S 3 + S 4, N/cm 2 Column Buckling Pressure, Pcr, N/cm 2 In-plane buckling Pressure, Psi, N/cm 2 Figure 4.4 indicates that with increase in thickness, the induced stress values are reduced. As strength is increases, its stability is also increases Stresses, N/cm Number of Plies Circumferential Stresses Longitudinal Stresses Figure 4.4: Effect of Plies on Stresses Figure 4.5 indicates that the multiple plies increases the strength of bellows, which increases the buckling pressure values. 124

9 Buckling Pressure, N/cm Number of plies Pcr (column) Pcr (In-plane) Figure 4.5: Effect of Plies on Buckling pressure Number of Convolutions: Number of convolutions has no relation with stresses produced, since we evaluate maximum stress and that is going to be same in each convolution. The number of convolutions is directly affecting length of bellows. Length of bellow = Number of convolutions x Pitch of the bellow (4.7) The length of bellows has further effect on Lb/Db ratio. The column in-stability pressure of the bellow is depending on the Lb/Db ratio. Actually, number of convolution is the design variable, which can be selected according to expected movements of the bellows. If bellow is expected to take axial movement, x = 10 mm axial motion, and 4 convolutions are provided in the bellow. Then each convolution will have movement x 10 ex =, = = 2.5 mm deflection will occur per convolution. (4.8) N 4 The movement per convolution can be reduced if we increase the number of convolutions. The stresses due to deflection will depend on ex parameter, if higher the deflection, higher the stresses will be produced. Hence higher convolution number will control the deflection stresses. For lateral movement of bellows, ey = N 3D L b m y x (4.9) 125

10 For angular movements and multidirectional movements of the bellows, the deflection is non-linear and very unpredictable. Hence more number of convolutions is advisable for such cases. Figure 4.6 : Angular absorption of bellows Table 4.3: Effect of Number of convolutions on In-stability Pressure of bellow Number of Convolutions, N Buckling Pressure, Pcr, N/cm Figure 4.7 shows that as the number of convolutions increases, column stability pressure is decreases. Of course there is no affect on in-plane stability of convolutions. Column Buckling Pressure, N/cm Number of Convolutions Pcr (column) Figure 4.7: Effect of Number of Convolutions on Column Buckling pressure 126

11 Selection of pitch and height of convolutions makes different q/2w ratio in the bellow. Following figure shows four types of dimensions selected, which gives different values of q/2w ratio. Here an attempt is made to calculate circumferential stresses for each ratio. As the q/2w ratio increases, size of convolution increases x x x x 2 q/2w Ratio Figure 4.8.1: Size of convolutions, relating q/2w Table 4.4: Effect of Ratio, q/2w on Circumferential stresses of bellow Ratio, q/2w Circumferential stresses, S 1 +S 2, N/cm Figure 4.8 shows the effect of ratio on circumferential membrane stress. As the q/2w ratio is increases, circumferential membrane stress also increases. Hence smaller size convolution should be preferable in order to achieve axial motion. Circumferential Stress, N/m q/2w Ratio Figure 4.8.2: Effect of (q/2w) Ratio on Circumferential membrane stress Height of convolutions: The force required to deflect a bellow axially is a function of the dimensions of the bellow and the material from which it is made. Height of convolutions is one of the important parameter to increase the flexibility 127

12 or spring rate of bellows. The height of convolution (w) is around circular periphery of an axis. Figure 4.9 : Height of convolutions As the height of convolution increases, outer diameter (Do) and mean diameter (Dm) of bellows are increases. Figure 4.9 shows the geometric parameters. Mean diameter of bellow Dm = Db + (n x t) + (2 x w) (4.10) By increasing height of convolutions, mean diameter of bellows also increases. The bellow will become more flexible and it can take higher amount of axial deflections for higher values of height of convolutions. Generally U shape bellows can take higher axial motion compared to toroidal convoluted bellows. Figure 4.10 indicates that as height of convolutions is increases, circumferential membrane stress (S 2 ) decreases and longitudinal stress increases, as the stresses due to bending becomes significant. Figure 4.10 shows that 2 cm is the optimum value of height of convolution. Table 4.5: Effect of Height of Convolutions on Stresses of bellow Height of Convolutions, w, cm Circumferential stresses, S 1 +S 2, N/cm 2 Longitudinal Stresses, S 3 + S 4, N/cm 2 128

13 Stresses, N/cm Height of convolutions, cm Circumferential Stress Longitudinal Stress Figure 4.10: Effect of Height of Convolutions on Stresses Figure 4.11 indicates that as the height of convolutions increases, column instability pressure (Psc) increases and in-plane instability pressure (Psi) both increases up to certain limit (0.02 m), than beyond this value of convolution height, in-plane stability pressure decreases drastically compared to column instability pressure. This also indicates that, the both stability is maximum, when convolution shape is near to square. In this case height is 2.3 cm and width is 2.26 cm for a square shape of convolution. It can also conclude that minimum convolution height should be selected for equal stability strength. Table 4.6: Effect of Height of Convolutions on In-stability Pressure of Bellow Height of Convolutions, w Column Buckling Pressure, Pcr, Pa In-plane buckling Pressure, Psi, Pa In-stability Pressure, N/cm Height of convolutions, cm Pcr (column) Pcr (in-plane) Figure 4.11: Effect of Height of Convolutions on instability pressures 129

14 Pitch of Convolutions: Pitch is having direct relation with radius of convolutions for U shape convolution bellows. Pitch is four time the radius. As finer the pitch, lower the radius of convolution and convolution width is also decreases. For coarse pitch bellows, larger the radius of convolution and width of convolution is higher. Figure 4.12 : Coarse Pitch bellow Figure 4.13 : Fine Pitch bellow Fig : Squirm Also for higher pitch bellows, (keeping other parameters constant) length of bellows is increases. So bellows stability pressure value will decrease. Pitch of convolutions is critical parameter for the stability criteria. Pitch decides the effective length of bellow, and it affects on the length to diameter ratio (Lb/Db) of the bellow. Bellow becomes unstable because of excessive internal pressure. This phenomenon is known as squirm. The equation which is suggested by EJMA, indicates that number of convolutions and pitch are the two parameters which adversely affect the stability. Psc = 0.34 C f N 2 q iu (4.11) Figure 4.15 shows that the circumferential membrane stress is increasing with increase in pitch of bellows. Ratio (q/2w) is the dimensionless parameter to evaluate the stresses developed in the bellows. This ratio is depending on pitch and height of convolutions. It indicates that minimum size of convolution pitch will restrict the circumferential stresses in the bellow. 130

15 Table 4.7: Effect of Pitch of Convolutions on Stresses and In-stability of Bellow Pitch of Convolutions, q, cm Circumferential stresses, S 1 +S 2, N/cm Column instability Pressure, Psc, N/cm Circumferential Stress, N/cm Pitch of Convolutions, cm Circum. Stress Figure 4.15: Effect of Pitch on Circumferential membrane stress Figure 4.16 shows that column in-stability pressure is decreases with increase in pitch of bellows. Column Buckling Pressure, N/cm Pitch of Convolutions, cm Pcr (column) Figure 4.16: Effect of Pitch on Instability Pressure 131

16 Observations: Following observations are made from the study. 1. Circumferential stress and longitudinal stress decreases with increase in thickness of bellow material. Also longitudinal stresses are always higher than circumferential stresses. But for higher material thickness, numbers of life cycles are reduced due to low cycle fatigue. The minimum thickness should be with reference to strength pressure capacity consideration, while higher limit should be controlled by fatigue life. 2. Multiple plies of the bellow material, increase the overall thickness of the material, by which strength of bellow increases. Hence stresses are reduced with increase in thickness. Bellow can withstand higher pressure due to increase in overall thickness, and also stability of bellow increases. By keeping other parameters constant, the in-plane stability is always lower than column instability. 3. More number of convolutions can take higher deflection as deflection per convolution will be decreasing. Higher the number of convolutions, higher the length of bellow; and stresses due to deflection reduces in individual convolutions. But higher length bellows may become unstable (squirm) at critical pressure value. So, pressure capacity should be safe below critical pressure limit. 4. Ratio q/2w is important factor of the geometric dimensions considering the performance behavior of bellows. This ratio should be kept around 2. If the ratio is lesser the 2, circumferential stresses increases very drastically. 5. Bellow can take higher deflections if height of convolutions increases. But this will result in increase in the longitudinal stress of the bellow. Height of convolutions has implications on stability criteria of bellows. As height of convolutions increases, column stability increases, while in-plane stability decreases. Height of convolutions is restricted up to 25 or 30 mm. 6. As the pitch of bellows increases, circumferential membrane stress increases. But increase in pitch value will increase length of bellow and this will decrease the column in-stability. Squirm failure possibilities are always higher in case of higher pitch bellows. 132

17 4.2.2 Parametric Optimization: Optimization is mathematical technique of obtaining the best result under given circumstances. In design of engineering components, designers have to choose the correct dimensions of various parameters. The ultimate goal of all such decisions is either to minimize the effort or maximize the desired benefit. Since the effort required or the benefit desired in any practical situation can be expressed as a function of certain decision variables, optimization can be defined as the process of finding the conditions that gives the maximum or minimum value of a function. There are various methods developed to solve different optimization problems effectively. [B17] Optimization methods: 1. Single variable optimization 2. Multi variable optimization with no constraints 3. Multi variable optimization with equality constraints 4. Multi variable optimization with inequality constraints 5. Simplex method (LPP method) 6. Nonlinear Programming methods In the present study two-parameter graphical approach is utilized, as the objective is axial spring rate of bellow, which is mainly depends on thickness of material and height of convolutions. Both parameters develop adverse effect on the resultant axial spring rate, hence optimum results can be achieved. Objective function: The conventional design procedures aim at finding an acceptable or adequate design which merely satisfies the functional requirement of the problem. In general, there will be more than one acceptable design, and the purpose of optimization is to choose the best one of the acceptable design available. The criterion, with respect to which design is optimized, when expressed as a function of the design variables, is known as the criterion or objective function. Design constraints: In many practical problems, the design variables can not be chosen arbitrarily; rather, they have to satisfy certain specified functional and other requirements. The restrictions that must be satisfied to produce an acceptable design are collectively called design constraints. Constraints that represent limitations on behavior are called behavior constraints. 133

18 Constraints that represent physical limitations on design variables such as availability, transportability, etc. are known as side constraints. Statement of the problem: A bellow with following dimension is considered for analysis. Notations: Pressure = 1 kg/cm 2 = 10 N/ cm 2 (constant) Thickness of bellow material (t) = x 1 Diameter of bellow (Db) = x 2 Height of convolutions (w) = 3.5 cm (constant) Number of ply (n) = 1 Constant, C f = 1 Theoretical axial spring rate = ƒ iu Figure 4.17: Geometric parameters of a bellow Theoretical axial spring rate of bellows is an important parameter to be referred for the selection and design of expansion joint. This parameter can be calculated mathematically by using following equation as suggested by EJMA. Objective of the optimization is maximization of theoretical spring rate of bellows. Objective function: Theoretical axial spring rate, ƒ iu = Behavior constraints: Circumferential stress = x x 1.7 Dm Ebt ƒx = 3 w Cf = n = t Db = (x 1 3 ) (x 2 ) 3 x 1 x Circumferential stress Allowable stress Longitudinal stress Allowable stress P D 2t = (1) (2) x 2 x = 2 x1 2 x1 2 (4.12) (4.13) Longitudinal stress = P D 4t = (1) x 2 (4) x 1 = x 2 4 x 1 (4.14) x2 g 1 (x) = x 2 1 (4.15) 134

19 x2 g 2 (x) = x 4 1 (4.16) Side constraints: Thickness of bellow material = 0.01 to 0.1 cms Diameter of bellows = 100 to 1000 cms. g 3 (x) = - x (4.17) g 4 (x) = x (4.18) g 5 (x) = - x (4.19) g 6 (x) = - x (4.20) Now, the constrained surfaces are plotted in a two dimensional design space where the two axes represent the two design variables x 1 and x 1. First behavior constraint x 2 2 x (4.21) x 2 = 2540 x 1 Finding different variables x x Second behavior constraint x 2 4 x (4.22) x 2 = 5080 x 1 x x Now, the contours of the objective function are to be plotted before finding the optimum point. Optimized Region: ƒx = (x 3 1 ) (x 2 ) = constant = (x 3 1 ) (x 2 ) = 50 x x

20 ƒx = (x 1 3 ) (x 2 ) = 80 x x ƒx = (x 1 3 ) (x 2 ) = 200 x x ƒx = (x 1 3 ) (x 2 ) = 500 x x ƒx = (x 1 3 ) (x 2 ) = 1000 x x ƒx = (x 1 3 ) (x 2 ) = 2000 x x ƒx = (x 1 3 ) (x 2 ) = 3000 x x ƒx = (x 1 3 ) (x 2 ) = 5000 x x Figure 4.18 shows the plotting of all side constraints, behavior constraints and surface contours of the objective function. These contours are plotted for the different values of theoretical axial spring rate. The designer can select the thickness for given diameter of bellows for specific value of axial spring rate. The optimum region is triangle ABC. 136

21 For 100 mm diameter of bellow, 0.04 cm bellow thickness is the optimum point amongst the variables. At this point, the axial spring rate is 500 kg/cm. Figure 4.18: Graphical solution of the problem. 137

22 4.2.3 Finite Element Analysis: Many researchers have contributed their work through computation based analysis, in order to solve many differential equations quickly. [2] Even some researchers have used programming in order to get faster and reliable stress analysis. [B11] Finite Element Analysis is a numerical method of deconstructing a complex system into very small pieces (of user-designated size) called elements. The software implements equations that govern the behaviour of these elements and solves them all; creating a comprehensive explanation of how the system acts as a whole. These results then can be presented in tabular or graphical forms. This type of analysis is typically used for the design and optimization of a system far too complex to analyze by hand. Systems that may fit into this category are too complex due to their geometry, scale, or governing equations. More finite element case studies are made and they are described in fifth chapter Performance Testing of Bellows: Experimental performance testing of bellows with variable conditions, are always helpful to the designers for better confidence in the design. EJMA is also recommending frequent testing of bellows, so that the design and manufacturing methodology will be maintained qualitatively. [20] Also for several critical applications, proto type testing of bellows is safe for the manufacturer as well as the users. Using this as one of the research aspect, some performance tests are conducted and they are described in sixth chapter. 138