Division III (include assigned division number from I to X)
ALLOWABLE BUCKLING LOAD CALCULATION AS PER RCC-MR
ELASTIC AND INELASTIC ROUTE-A COMPARISON
Ashok Kumar, S. D Sajish, S. Jalaldeen, K. Velusamy, P. Puthiyavinayagam
Indira Gandhi Centre of Atomic Research, Kalpakkam-603102, India
ABSTRACT
RCC-MR appendix A-7 gives two alternate, independent methodologies, namely elastic and inelastic route, to predicts safe buckling load for thin shell structures. Buckling load is calculated as per both approaches for a cylinder under three different loading scenarios. For elastic route, charts given in RCC-MR appendix A-7 are used to calculate reduction factor, which are further multiplied to linear eigenvalue buckling load. For inelastic route, three different imperfection shapes are computed for each case and a nonlinear analysis is carried out on imperfect geometry. About each equilibrium state, an eigenvalue buckling analysis is carried out till eigenvalue buckling load is reduced to unity. Safe buckling loads predicted by both elastic and inelastic route for cylinder under axial and biaxial load are very close. In case of cylinder under external pressure, there is significant difference between buckling load predicted by both elastic and inelastic route, with elastic route being less conservative.
INTRODUCTION
Buckling in fast reactor components is difficult problem due to involved plasticity. Due to high operating temperature, yield stress of material reduces, and buckling generally occurs in plastic regime. RCC-MR appendix A-7 proposes two different approaches to design against buckling in fast reactor components. First approach is called elastic route, and uses simplified elastic analyses and given charts, which provides reduction factors to be multiplied with linear elastic buckling load. These reduction factors involve reduction in buckling strength due to imperfection and plasticity. Two set of charts are provided for both stable and post buckling behaviour. These charts have been developed with beam based calculation, with superimposed post buckling behaviour of axisymmetric cylindrical under axial load [1]. This method is simple and easy to use, and conservative.
The second approach follows inelastic route, which calls for carrying out combination of nonlinear stress and bifurcation buckling analyses. Nonlinear analysis includes, both with geometric and material nonlinearity on imperfect geometric model, and bifurcation buckling to search for equilibrium step and load state in nonlinear analysis which results in unity bifurcation buckling load. This approach is complicated, time consuming and often less conservative compared to first approach. In this work, we calculate and compare safe buckling load by both approaches for three cases. Three cases represent three stress states in a cylinder, namely compressive hoop stress, compressive longitudinal stress and a combination of equal compressive hoop and longitudinal stress.
SAFE BUCKLING LOAD CALCULATION METHODOLOGIES
Elastic Route Methodology [2]
From linear stress analysis, primary local membrane stress Pm is computed. In present work, only membrane
stress state considered, hence step for cases where bending stresses are dominating is not discussed. With λ as minimum buckling load factor computed from eigenvalue value buckling analysis, a parameter ξ is computed as per following equation,
where σy is initial yield stress of material. Another imperfection parameter defined as ratio of imperfection and thickness of shell h is,
d
h
With computed ξ and d, a chart with suitable post buckling behaviour is used to evaluate parameters X and Y, which are reduction factor for critical elastic buckling stress and yield stress.
Inelastic Route Methodology [2]
An Imperfection shape is computed by carrying out series of elastic and inelastic bifurcation buckling (buckling with pre-stress). If eigenvalues computed are very close, then several combination of buckling modes shall be considered.
Further imperfection shape is superimposed geometric model scaled by imperfection magnitude. On imperfection geometry, nonlinear stress strain analysis shall be carried out. For each equilibrium state in nonlinear analysis, an eigenvalue buckling analysis shall be carried with applied load and pre-stress effect of equilibrium state. The load state for which bifurcation buckling load becomes unity is called buckling load of structure.
FINITE ELEMENT MODELLING
Geometric Model
Three different loading scenarios considered are, a cylinder under external pressure, cylinder under axial load and a cylindrical under axial load and external pressure. Radius of cylinder for all three cases is 5 meter and height is 10 meter. For cylinder under external pressure and axial load, thickness is considered as 20 mm. For cylinder under combined loading axial loading, thickness has been considered 10 mm. Figure-2 shows geometry model of cylinder.
Material Properties
For linear elastic stress and buckling analysis, Young’s modulus and poison’s ratio are 200 GPa and 0.3 respectively. For nonlinear stress analysis, plastic strain-stress relation with initial yield stress 253 MPa is used. Figure-3 shows tensile stress-plastic strain curve used for elastic-plastic analysis.
Imperfection Shape
For choice of imperfection shape in nonlinear analysis, nonlinear analyses with combination of buckling modes is carried out. For cylinder under biaxial state of stress, imperfection shape is constructed by superimposing first mode, which exhibits buckling mode under external pressure and 17th mode which
shows influence of axial load on buckling. Both modes are scaled by magnitude of half of thickness. Imperfection shape for cylinder axial load assumed to be first buckling mode scaled by half of thickness. For cylinder under external pressure, imperfection shape is constructed by combining first three buckling modes, scaled in reducing order of half (h/2, h/4 and h/8), where first mode is scaled by half of thickness.
Loading and Boundary conditions
The load applied (design load) for all three cases is eigenvalue buckling load of structure under respective stress state. For cylinder under biaxial state of stress, this load is external pressure of 2.05x104 Pa and axial
pressure applied load is external pressure of 1.37x105 Pa. Both ends of cylindrical shell are fixed with
simply supported boundary condition.
RESULT AND DISCUSSION
All the load magnitudes described in this section are normalized with respect to linear elastic buckling load of respective case. This approach helps to express obtained safe buckling loads as reduction factors.
Cylinder under axial load
Figure-2 Cylinder geometric model
For cylinder axial loading, nonlinear analysis predicts that load carrying capacity is reduced by factor of approximately 5. Figure 4 shows load displacement plot of cylinder under axial load. As shown by load displacement plot, slope of load displacement curve is negative after attaining first peak, hence unstable post buckling charts are used for elastic route. Figure 5 shows first buckling mode of cylindrical shell, which is axis-symmetric. Figure 6 shows bifurcation load predicted by eigenvalue analysis about applied load states. Unity buckling load occurs at load state of magnitude 0.142, which hence as per inelastic route is normalized buckling load. For safe buckling load as per elastic route, elastic buckling stress in cylinder 510 MPa, which results in ξ= 2.0158. For d=0.5, buckling reduction factor from unstable post buckling charts found to be 0.1383. Since elastic buckling load is unity, normalized buckling load for structure is 0.1383. Buckling load predicted by both methods are very close.
Figure-4 Load displacement plot for cylinder under axial loading
Figure-5 First Buckling mode under axial loading
Cylinder under external pressure
For cylinder external pressure, post buckling behaviour is stable and structure retains its load carrying capacity. Figure 7 shows load displacement plot of cylinder external pressure. There is no significant reduction in load carrying capacity due to imperfection in structure. Figure 8 shows first buckling mode of cylinder under external pressure. Figure 9 shows the bifurcation load predicted by eigenvalue analysis about applied load states. Unity buckling load occurs at load state of magnitude 0. 495 which is normalized safe buckling load as per inelastic route. For safe buckling load as per elastic route, the elastic buckling stress in cylinder 34.24 MPa, which results in ξ=0.1354. For d=0.5, buckling reduction factor from stable post buckling charts is found to be 0.653. Normalized buckling load for cylinder external pressure is 0.653. In case of stable post buckling behaviour, the difference between buckling load predicted by elastic and inelastic route is significant. Elastic route appears to be less conservative compared inelastic route.
Figure-7 Load displacement plot for cylinder under
external pressure Figure-8 First Buckling mode under external pressure
Cylinder under biaxial load
Figure-10 shows load displacement plots for cylinder under biaxial load. The load axis is normalized with respect to linear elastic buckling load (Pc).
At peak load in load displacement plot, slope of load displacement curve is negative and cylindrical shell under equal biaxial stress exhibit unstable post buckling behaviour. Even though, under combined loading, the first buckling mode appears to be similar to a buckling mode of cylinder under external pressure. Figure 11 shows the first buckling mode of cylinder under biaxial loading. Figure 12 shows plot of linear buckling load predicted by eigenvalue analysis about applied load state and applied load. The horizontal orange line in plot corresponds to buckling load factor of magnitude unity, which cut buckling load factor curve at abscissa 0.486. Hence normalized buckling load factor as per inelastic route is 0.486.
Following elastic route, the equivalent elastic buckling stress in cylinder 10.2 MPa, which gives ξ=0.0403. For d=0.5, buckling reduction factor from unstable post buckling charts found to be 0.4913. Since elastic buckling load is unity, normalized buckling load for structure is 0.4913.
Figure-10 Load displacement plot for cylinder under
CONCLUSION
Two design methodologies given in RCC-MR appendix A-7 are compared for three different stress states. Cylinder axial load, with axisymmetric imperfection shows significant reduction in buckling load in riks analysis, however safe buckling loads predicted by both elastic and inelastic route for this case are very close. Cylinder under external pressure shows stable post buckling behaviour, where there is not significant reduction in load carry capacity in post buckling regime. There is significant difference between buckling load predicted by both elastic and inelastic route, with elastic route being less conservative. In case of compressive biaxial state of stress of equal magnitude, stress at which buckling occurs is small. Safe buckling load predicted by both route is very close if unstable post buckling behaviour is assumed.
ACKNOWLEDGEMENT
The authors gratefully acknowledge SRI-AERB for the computational tools and computing facilities utilized in the present study.
REFERENCES
Autrusson, B., Acker, D., Hoffmann, A., 1987. Discussion and validation of a simplified analysis
against buckling. Nuclear Engineering and Design. 98.3, 379-393.
RCC-MR, 2012. Analyses taking account of Buckling, Section III-tome-1-subsection Z-Appendix
A-7, AFCEN, Paris