IJSRD - International Journal for Scientific Research & Development Vol., Issue 05, 015 ISSN (online): 1-061 Post Buckling Analysis of Silos and Optimization of Additional Stiffeners Ms. S. H. Gujar 1 Prof. N. S. Biradar Prof. M. Chamnalli Mr. Manan Panchal 1,, Department of Mechanical Engineering 1 Dhole Patil College of Engineering Pune JSPM Imperial Wagholi Pune, Vaftsy CAE Pvt Ltd. Pune, Abstract Silos are probably the common form of large engineering shell structure in service. The high rate of structural failure in these structures must be considered by the designer and the complexity of their behavior. The paper presents buckling and post-buckling of silos under axial compression using a time-integrated finite element model as well as the analysis and Experimental results. Since stiffeners have the vital role to support the pressure vessel and to maintain its stability, they should be designed for load and internal pressure of the vessel. A model of vertical pressure vessel and stiffeners are created and stresses are evaluated using mathematical approach and ANSYS software. The analysis reveals the zone of high localized ω L R T u v z Vp P 0 P R D n N nozzles Dr nozzle Dr location V s V Vb H India stress at the junction of the pressure vessel and stiffeners due to operating conditions. The results obtained by both the methods are compared. The analyses provide significant new insight into the mechanisms underpinning collapse behavior of the shells. The tests which are carried out on cylindrical silo models, the measurement of the initial imperfections and the numerical analysis of the shells are presented. Key words: Post Buckling Analysis of Silos, Optimization of Additional Stiffeners Deflection of plate Length Radius Thickness Displacements along x axis Displacements along y axis. Half of the thickness of plate Process Volume External Pressure Self-weight/ Internal pressure Main Nozzle to Nozzle Centre Distance (Inlet/Outlet) Vessel Internal Radius Nozzle diameter Number of nozzles Drain Nozzle diameter Drain location Expected Stagnant Volume Stagnant Volume Buffer Volume Requirement Buffer Stock Cylindrical Length Main Stock Cylindrical Length Total Height of silo Density of water NOTATION Ca Vp A A1 A A A5 Nca C De S E Σ F r fr1 Allowable strength of material Welding efficiency corrosion allowance Expected Stagnant Volume Total cross-sectional area of reinforcementrequired in the plane under consideration. area in excess thickness in the vessel wall available for reinforcement Area in excess thickness in the nozzle wall availablefor reinforcement. area available for reinforcement when the nozzle extends inside the vessel wall. Cross-sectional area of material added as reinforcement Nozzle corrosion allowance Nozzle thickness Design constant, nominal plate diameter maximum allowable stress Joint efficiency Normal stress Strength reduction factor, not greater than 1.0 [seeug-1(a)] All rights reserved by www.ijsrd.com 877
fr fr fr Te Rn N wall Rp G Post Buckling Analysis of Silos and Optimization of Additional Stiffeners (IJSRD/Vol. /Issue 05/015/0) 1.0 for nozzle wall abutting the vessel wall and for nozzles shown in Fig. UG-0, sketch (j), (k),(n) and (o). Sn /Sv (lesser of Sn or Sp) /Sv Sp /Sv Thickness of shell Thickness or height of reinforcing element Inside radius of the nozzle Nominal wall thickness of nozzle Radius ofcurvature of plate Gravitational acceleration Nozzle diameter Nominal thickness of internal projection of nozzle wall I. INTRODUCTION Large steel silos are typical kind of thin walled structures which are widely used for storing huge quantities of granular solids in industry and agriculture. The sizes of silos vary as per their storage capacities. These steel silos are usually with large diameter to thickness ratios, which is particularly vulnerable to buckling due to internal pressure, wind pressure. The critical load capacity of cylindrical shells, subjected to pressure, depends on two geometric slender ratios of length to radius (L/R) and Radius to thickness (R/T). Therefore, the geometric imperfections would influence the buckling and post-buckling behaviour of these structures. The experimental studies show that the buckling strength of ideal shells without geometric imperfections is much more than that of imperfect shells. The imperfection sensitivity of these structures must be considered carefully and properly. A. Objective of Project: Objective of the project is to study the buckling and postbuckling behaviour of silo and verify its safety against buckling failure to validate the design. Additionally, to check whether addition of stiffeners may improve the buckling characteristics of the silo is also the aim of the thesis. Thus following are objectives of our project: To do linear buckling analysis of silo To do nonlinear buckling analysis of silo To experiment with stiffeners (sizes and location) to check for improvement. II. LITERATURE SURVEY The stability of conical shells under axial compression has been studied in the past both theoretically and experimentally by several investigators. The first shell buckling problem solved was cylindrical shells under axial compression. S Aghajari (1) has studied the Buckling and postbuckling behaviour of thin walled cylindrical steel shells with varying thickness subjected to uniform external pressure. He found that the buckling capacity of the thin walled cylindrical steel shells, with thickness with variation in height will increase. Chawalit Thinvongpituk (), have studied the Buckling of axially compressed conical shells of linearly variable thickness using structural model. He concluded that the reduction of the buckling resistance of cylindrical and conical shells due to their small variable thickness is proportional to the thickness reduction parameter and can be expressed in a simple linear equation. A Spagnoli (), have studied the elastic buckling and post buckling behaviour of widely stiffened conical shells under axial compression. Critical load increases as the panel width decreases in line with the behaviour of the cylindrical panel. M Shariati(), have studied the experimental results on buckling and post buckling behaviour of cylindrical panels with clamped and simply supported ends and come to the conclusion that by increasing the length of the panels, the buckling load decreases slightly. III. BUCKLING AND POST BUCKLING A. Buckling Buckling is that mode of failure when the structure experiences sudden failure when subjected to compressive stress. When a slender structure is loaded in compression, for small loads it deforms with hardly any noticeable change in the geometry and load carrying capacity. At the point of critical load value, the structure suddenly experiences a large deformation and may lose its ability to carry load. This stage is the buckling stage. The structural instability of buckling can be categorized as; Bifurcation buckling Limit load buckling In Bifurcation buckling, the deflection when subjected to compressive load, changes from one direction to a different one. The load at which bifurcation occurs is the Critical Buckling Load. The deflection path that occurs prior to the bifurcation is called as the Primary Path and that after bifurcation is called as secondary or post buckling path. In Limit Load Buckling, the structure attains a maximum load without any previous bifurcation, i.e. with only a single mode of detection. B. Post Buckling Post buckling stage is a continuation of the buckling stage. After the load reaches its critical value the load value may not change or it may start decreasing, while deformation continues to increase. In some cases the structure continues to take more loads after certain amount of deformation, to continue increasing deformation which eventually results in a second buckling cycle. Post buckling analysis being nonlinear, we obtain far more information than we obtain from linear Eigen-value analysis. The non-linear load-displacement relationship, which can be a result of the stress strain relationship with a nonlinear function of stress, strain and time. The changes in geometry due to large displacements; irreversible structural behaviour upon removal of external loads; change in the All rights reserved by www.ijsrd.com 878
(IJSRD/Vol. /Issue 05/015/0) boundary conditions such as change in the contact area and the influence of loading sequence on the behaviour of the structure, requires a nonlinear structural analysis. The nonlinear structural analysis depends upon the various structural non linearities. The structural nonlinearities can be classified as, a geometric nonlinearity, a material nonlinearity and a contact or a boundary nonlinearity. IV. PLATE BENDING THEORY Expression for stresses induced in rectangular plate subjected to Uniformly Distributed Load. Fig..1: Plate definition Fig..: Deflection of plate in the x-z direction. Let a rectangular plate subjected to UDL and z be the half of the thickness of plate. Let ω be the deflection of plate, u and v be the displacements along x and y axes. Hence, the displacement along x direction, u z 1 x Similarly, the displacement along y direction, v z y As per strain displacement relations, strains are given by- u x x v y y v u γ x y Put 1 and in equation, x z x y z y γ z z From plane stress condition, x 1 0 x E y 1 0 y 1 1 0 0 E x x y 1 E y y x 1 E 1 1 5 6 Substituting equation 5 in equation 6 Ez x 1 Ez y 1 y x Ez 1 1 7b 7a Fig..: Variation in moments and forces Let, M be bending moment and M be shear force. According to beam theory M x, M y, M are given by the following relations: t/ M x x z dz (8) t/ t/ M y yz dz t/ t/ M z dz t/ 9 10 All rights reserved by www.ijsrd.com 879
(IJSRD/Vol. /Issue 05/015/0) Substituting equation (7a), (7b) and (7x) in equation (8), (9) and (10) Let, t/ Ez M x z dz t/1 x y Ez t M x 1 x y 1 D 1 1 Where, D=Flexural rigidity of plate. Mx D x y Similarly, we have My D y x 11 1c M D 1 Comparing 7a and 1a x 1 Mx 1 1 Ez x y Comparing 7b and 1b ) y 1 1M y (1 Ez y x Comparing 7c and 1c 1 1 M1 1 Ez1 1 1a 1b (15) Now, the maximum and minimum stresses occur at z t/ is given by- 6Mx x max,min t 16 6M y y max,min t 17 6M max,min t In simple beam theory, moments are easily determined from equilibrium. However, for plate this is not easy method. Consider the tdxdy element. Where now the lateral load P and the changes in the forces and moments (, ) per unit lengths are shown. Also we have utilized the fact that M M. Summing forces in the z direction, yx 18 V x y Dividing by dxdy gives- Vx y Fz dx dy dydx px, ydxdy 0 19 V V x y px, y 0 x y Summing moments about an axis parallel to the axis through the element center gives- M My 1 Or 0, 0 M dx dy dy dx Vydx dy 0 x x y M My Vy x y Substituting an equation 1 and M M M x y Px, y x x y y Finally substituting equation simplifying results in P x,y x x y y D in equation 1 into and Where, P, Pressure applied on the surface of the plate in x yplane. Tabulated solutions of Uniformly Loaded rectangular plates- 1) The terms a and b are the dimensions of the sides of the plate. ) The load is a uniform pressure applied along the entire plate. ) The maximum stress and deflection are given by the formulas. () (5) Where, K 1 and K are numerical constants values. A. Mathematical Modelling A water tank.60m deep and.70m square is to be made of structural steel plate. The sides of the tank are divided into nine panels by two vertical supports (or stiffeners) and two horizontal supports that are, each panel is 0.90m wide and 1.0m high, thickness 6.79mm and average head of water on a lower panel.00m figure 1.alculate the maximum deflection of the panel. Fig. 1: Modelling D=.60m, Pb 0 L=.70m, max K max 5.79mm E= 00*109 Pa All rights reserved by www.ijsrd.com 880
(IJSRD/Vol. /Issue 05/015/0) a = 1.0m b = 0.90m Table 1: Parameters Ansys Result= ω max 6.869mm V. DESIGN CALCULATIONS FOR SILOS AND ITS STIFFENERS WITH OPTIMIZED RESULTS Sr. No. For V s < 0.1 V p, For high pressure applications P>1. mpa Head type = Flat head 1 Support = Direct to ground V p = 500 m, P 0 =1015 Pa=Atm, D n =0.m, N nozzles =6, Dr nozzle =0.1m, Dr location, = Head V s =0 m, V b = 5m, R=.6m, ) 5 H = L 0 + L 1 H=1.15 m 6, L 0 = NTD + D p D p = 1.8 m Determining internal pressure H = 19.6 mm, P 0 = 1015 Pa, For Low Pressure Applications (P < 1. mpa ) Sr. No. 1 V p = 500 m V s =0 m, V b = 5m, R=.6m P=, R=.6m,,, ca= + ca 9.76 mm,, P=, R n = 150 mm, nca = mm + nca 5 6 d=0.m,, F r1 =1 n t = 1, F r1 = 1, d=0.m, F r1 =1, 7 8 9 F r =1,,,, n t =1,,, F r =1,,, 10 D p = 19.6, R p = 66. mm + corrosion allowance 5 Sr. No. 6, D p = 19.6, d=0.m,, F r =1,,, R p = mm,,, + ca Nominal 7 D p = 68 mm, NTD= 8, All rights reserved by www.ijsrd.com 881
(IJSRD/Vol. /Issue 05/015/0) 9 10 11 1 1,,,,, E=1, ca=,, R n =150mm, E=1, ca=,, R p = mm, E=1, ca= DESIGN OF FLAT HEAD : d = Diameter of shell = 700 mm, C = 0.0, P=, S=, E=1 + ca + ca + ca Table : Calculations for Silos and its Stiffeners with Optimized Results Diameter of reinforcing pad = 68 mm Thickness of flat head = 1.7 mm Internal radius of flat head = 10 mm The reduced value of A a by reducing the value of D p. Sr. No. Dp A5 Aa Ar 1 19.6 5919.6 6717. 8.8 1100 76 5561.7 8.8 1000 16 961.7 8.8 900 56 61.7 8.8 5 800 96 761.7 8.8 6 700 6 161.7 8.8 7 650 06 861.7 8.8 8 68 05 89.7 8.8 A. Design of Flat Head: B. Final Dimensions of Silo: Fig. : Design of Flat Head Fig. : Final Dimensions of Silo Thickness of shell = 1 mm Diameter of shell = 700 mm Thickness of nozzle = 6 mm Diameter of nozzle = 00 mm Thickness of reinforcing pad = 6 mm VI. CONCLUSION The theoretical & ANSYS values are very close to each other. The results obtained after linear and nonlinear analysis, is calculated as critical buckling capacity of silo equal to 16 MPa, which is very higher value than actual internal pressure value 0.98 MPa for which silo is designed. Hence, silo is safe against buckling failure. ACKNOWLEDGMENT As a student of Post graduate M.E. (Design Engineering) at Dhole Patil College of Engineering, Wagholi, Pune. I have a great pleasure to present my project "Post Buckling Analysis of Silos and Optimization of additional stiffeners". I express my thanks to my Guide Prof. N. S.Biradar and Co-Guide Prof. Maheshwar C., PG Coordinator, Prof. Anantharama, Prof. Abhijeet Dandavate (Head of Department, Mechanical Engineering), Mr. Vinaay Patil,Mr. Manan panchal (Vaftsy CAE, Pune) and team members for their valuable support. REFERENCES [1] S. Aghajari and H. Showkati, (006) Buckling and post-buckling behaviour of thin-walled cylindrical steel shells with varying thickness subjected to uniform external pressure," Thin-Walled Structures, vol., pp. 90-909. [] A. Spagnoli and M. Chryssanthopoulos,(1999) Elastic buckling and post-buckling behaviour of widelystiffened conical shells under axial compression," Engineering Structures, vol. 1, pp. 85-855. [] A. B. Sabir and M. S. Djoudi, (1995) Shallow shell finite element for the large deflection geometrically nonlinear analysis of shells and plates," International Journal of Solids and Structures, vol. 1, pp. 5-67. [] P. Mandal and C. Calladine, (000) Buckling of thin cylindrical shells under axial compression," International Journal of Solids and Structures, vol. 7, pp. 509-55. [5] Mohamed Thabit Abdel-Fattah, (000) Non-linear behaviour of cylindrical shells containing elastic solids, pp. 1-1. [6] J. G. Teng, (1996) Buckling of thin shells: Recent advances and trends," ASME Journal of Applied Mechanics, vol. 9, pp. 6-7. All rights reserved by www.ijsrd.com 88
(IJSRD/Vol. /Issue 05/015/0) [7] P. Boresi, Richard J. Schmidt.(009) ''Advanced Mechanics Of Materials''. [8] S. Timoshenko, Theory of plates and shells. [9] Kelvin Tengende, TawandaMushiri and Talon Garikayi, (01) Finite element analysis ROM silo subjected to 5000 tons Monotonic loads at an anonymous mine in Zimbabwe, International Conference on challenges in IT, Engineering and Technology (ICCIET 01) July 17-18. All rights reserved by www.ijsrd.com 88