Nonlinear analysis and plastic deformation of pipe elbows subjected to in-plane bending
Introduction
Due to their specific deformational behavior when exposed beyond the elastic limit, pipe elbows are capable of plasticizing over large areas when a system is overstressed. Thus, they absorb considerably large thermal expansions and seismic movements in addition to cushioning transiently loaded systems by energy dissipation as a result of plastic material flow. However, clear understanding of the deformation and stress behavior beyond the elastic limit is important to assure a sufficient safety margin.
The pipe bend problem has been analyzed by several approaches either elastic or inelastic using small and large deformation solutions with both theoretical and numerical techniques, in addition to experimental investigations.
Dhalla [1]performed a detailed nonlinear analysis of two 16 in elbows (cold and hot) tested up to collapse. The detailed nonlinear shell analysis was performed using thin shell finite elements. The stiffening effects of straight pipe welded to the elbow were included in the analysis. For local deformation response, such as the hoop and axial strain, the overall analytical–experimental correlation was good up to about 60% of the collapse moment. At higher load levels, the discrepancy between prediction and measurement increased but differences were considered to be within acceptable engineering bands.
Dhalla [1]also compared his results with simplified analysis results obtained by Sobel and Newman [2]showing that shell analysis predictions are in better agreement with the experimental results than the corresponding simplified pipe bend analysis predictions since the latter neglects end stiffening and elbow ovalization.
Suzuki and Nasu [3]performed nonlinear finite element analysis of a 12 and 24 inch outside diameter butt welded elbow subjected to in-plane bending using four node flat shell elements. The differences between experimental and analytical results increased to 10% in the nonlinear analysis and good correlation appeared in the linear part. The circumferential strains in large deformations begin to concentrate at both sides of the elbow after yielding occurs, and the maximum strains grow rapidly with the increase in load. The absolute maximum strain appears in the inner surface of the elbow under in plane bending. They showed that the patterns of strain distribution were predicted successfully by the finite element solution.
Hilsenkopf et al. [4]performed two series of tests on 90° large-radius elbows with different D/t ratios under in-plane (closing and opening) and out of plane moments. Changes in elbow angular deflection and ovalization of the mid-section were recorded as a function of the applied moment. The influence of pressure, temperature and cyclic loading was also studied. Experimental limit moment was achieved for angular rotation values of 6–11° which is large compared to values normally encountered in practice.
Kussmaul et al. [5]performed six pipe bend tests to determine the quasi-static deformation behavior of pipe bends in the nonlinear range and to obtain the local and global failure behavior as a function of load history. The results showed that there was no difference in the equivalent stress distributions of bends subjected to in-plane opening mode loading and in-plane closing mode loading within the elastic range. It was shown that the location of initial yielding depends on geometry. The point of maximum strain in the thin-walled bends is on the inner surface at the bend crown due to the dominant cross-sectional ovalization. In contrast, the maximum strain of the thick-walled elbows is on the outer surface at the intrados because prismatic bending is dominant in this case and the bend behavior approaches that of a straight pipe. In the case of bends lying between thin and thick-walled regimes, yielding appeared at two zones with a comparable state of stress, one on the outer surface at the intrados and one on the inner surface at the bend crown.
Shalaby [6]and Shalaby and Younan 7, 8, 9determined the limit loads for pipe elbows under in-plane closing and opening bending moments with and without pressure for various pipe bend factors. Limit moments were determined for both opening and closing cases. In the opening case the limit moments were higher because of the geometric stiffening effects.
Section snippets
Scope of the work
In this work plastic strain distribution for a pipe bend is investigated. The development of the strain with increasing bending moment is studied till instability. The strain and stress distributions across the thickness and around the elbow cross-section are studied from the initiation of yielding till instability to study the development of through the thickness yielding (plastic hinges) under opening or closing moments.
Analysis
ABAQUS [10], which is a general nonlinear finite element analysis package, was used with its special elbow element that takes the ovalization deformation into consideration but assumes constant deformation along the element length.
From the finite element solution, the plastic strain and stress distributions across the thickness for different positions around the cross-section are determined.
Cross-section analysis
In order to investigate the deformation behavior of the pipe elbow, it is very important to study the cross-section deformations and the developed stresses that cause these deformations. This behavior may be represented and plotted in the form of stress or deformation history (i.e. variation with loading history) at a certain point on the cross-section, spatial variation of stress or deformation around the cross-section at a certain load (snap shot), and cross-section deformed shape at a
Conclusions
The behavior of pipe elbows in the elastic range is the same under in-plane closing and opening moments. The stress and strain distributions are the same in both cases but with opposite signs. The elbow starts to yield at the same value of applied load and end rotation in both cases of in-plane closing and opening moments. For thin elbows, the location of first yield is at the inside surface of the elbow crown in both cases.
The behavior of pipe elbows in the plastic regime is quite different
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Simplified, detailed, and isochronous analysis and test results for the in-plane elastic plastic and creep behavior of an elbow
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