Displacements in a Wide Curved Bar Subjected to Pure Elastic-Plastic Bending

1957 ◽  
Vol 24 (3) ◽  
pp. 447-452
Author(s):  
Bernard W. Shaffer ◽  
Raymond N. House
Keyword(s):  
Applied Load ◽  
Bending Moment ◽  
Incompressible Material ◽  
Pure Bending ◽  
Plastic Bending ◽  
Elastic Case ◽  
Elastic Plastic ◽  
Material Thickness ◽  
Perfectly Plastic ◽  
Order Of Magnitude

Abstract Equations have been obtained for the displacements and strains within a wide curved bar made of a perfectly plastic, incompressible material subjected to a pure bending moment which is sufficiently large to cause elastic-plastic stresses. It is found that whenever the applied load is within 95 per cent of the fully plastic bending moment, displacements and strains in the elastic-plastic problem are of the order of magnitude of the corresponding elastic case. It is also found that when the bending moment reaches approximately 65 per cent of the fully plastic bending moment, the change in material thickness reaches a maximum. It decreases to zero when the bar becomes completely plastic.

Some theoretical considerations relating to strain concentration in elastic-plastic bending of beams

1968 ◽  
Vol 3 (4) ◽  
pp. 304-312 ◽  
Author(s):  
M Radomski ◽  
D J White
Keyword(s):  
Strain Hardening ◽  
Numerical Solutions ◽  
Bending Moment ◽  
Rate Of Change ◽  
Plastic Bending ◽  
Elastic Plastic ◽  
Perfectly Plastic ◽  
Rate Of Increase ◽  
Plastic Zones ◽  
Elastic Perfectly Plastic

Theoretical derivations are presented for the relations between maximum deflection and the corresponding maximum strain for some simple beams subject to elastic-plastic bending. Both elastic-perfectly plastic and arbitrary stress-strain relations are considered. Where possible, explicit analytical solutions are given, but where this is not possible numerical solutions are obtained by means of computer programmes. The calculations show that in elastic-perfectly plastic material short plastic zones may develop and cause large strains in the beam even though the deflection corresponding to first yield is not greatly exceeded. On the other hand, strain hardening elongates the plastic zones, so producing a more favourable strain distribution along the length of the beam than would exist without it. The more pronounced the strain-hardening characteristic, i.e. the greater the rate of increase of stress with strain, the less concentrated will be the strains. The mode of loading is important in that the higher the rate of change of bending moment, in the region of ihe maximum bending moment, the more concentrated will be the local strains.

Effects of Torsion on Equivalent Bending Moment for Limit Load and EPFM Circumferential Pipe Flaw Evaluations

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Phuong H. Hoang ◽  
Kunio Hasegawa ◽  
Bostjan Bezensek ◽  
Yinsheng Li
Keyword(s):  
Finite Element ◽  
Pressure Vessel ◽  
Large Strain ◽  
Bending Moment ◽  
Limit Load ◽  
Elastic Plastic ◽  
Perfectly Plastic ◽  
Plastic Materials ◽  
Weighted Factor ◽  
Elastic Perfectly Plastic

The circumferential flaw evaluation procedures in ASME Boiler and Pressure Vessel Code Section XI nonmandatory Appendix C are currently limited to straight pipes under pressure and bending loads without consideration of torsion loading. The Working Group on Pipe Flaw Evaluation of the ASME Boiler and Pressure Vessel Code is developing guidance for considering the effects of torsion by a mean of an equivalent bending moment, which is a square root of sum square combination of bending moment and torsion load with a weighted factor for torsion moment. A torsion weighted factor, Ce, is established in this paper using large strain finite element limit load analysis with elastic perfectly plastic materials. Planar flaws and nonplanar flaws in a 10.75 in. (273 mm) OD pipe are investigated. Additionally, a finite element J-integral calculation is performed for a planar through wall circumferential flaw with elastic plastic materials subjected to bending and torsion load combinations. The proposed Ce factor for planar flaws is intended for use with the ASME B&PV Code Section XI, Appendix C for limit load and Elastic Plastic Fracture Mechanics (EPFM) circumferential planar flaw evaluations.

Plastic-Rigid Analysis of Long Beams Under Transverse Impact Loading

1952 ◽  
Vol 19 (4) ◽  
pp. 465-470
Author(s):  
M. F. Conroy
Keyword(s):  
Impact Loading ◽  
Bending Moment ◽  
Large Strains ◽  
Transverse Impact ◽  
Plastic Bending ◽  
Elastic Plastic ◽  
Straight Line ◽  
Probable Form ◽  
Constant Yield ◽  
Elastic Strains

Abstract The object of this paper is to set forth the results of an investigation of the behavior of long beams under transverse, constant-velocity impact loading, when a plastic-rigid type of analysis is adopted. It was expected that such an analysis would be satisfactory for problems involving large strains, and easier to evaluate than the corresponding elastic-plastic solution. Consideration is first given to the case of ideal plasticity. Elastic strains are neglected and the material of the beam is assumed to flow plastically at a constant yield limit. In this case expressions for the bending moment, shear force, curvature and deflection distributions along the beam are obtained analytically for any given impact velocity. The manner in which the solution for a beam having an elastic-ideally plastic bending moment-curvature relationship converges to the plastic-rigid solution, as EI increases, is discussed. Consideration is next given to the case of work-hardening where the material is assumed to obey a plastic-rigid bending moment-curvature relationship consisting of a straight line with nonzero slope. Unfortunately, difficulty arises in finding a solution analytically in this case. However, by considering the solution for a beam having the corresponding elastic-plastic bending moment-curvature relationship and a large EI-value, some speculation as to the probable form of the solution may be made.

The Elastic, Plastic Bending of a Simply Supported Plate

1961 ◽  
Vol 28 (3) ◽  
pp. 395-401 ◽  
Author(s):  
G. Eason
Keyword(s):  
Yield Condition ◽  
Constant Load ◽  
Flow Rule ◽  
Plastic Bending ◽  
Simply Supported ◽  
Elastic Plastic ◽  
Tresca Yield Condition ◽  
Perfectly Plastic ◽  
Elastic Perfectly Plastic ◽  
Von Mises

In this paper the problem of the elastic, plastic bending of a circular plate which is simply supported at its edge and carries a constant load over a central circular area is considered. The von Mises yield condition and the associated flow rule are assumed and the material of the plate is assumed to be nonhardening, elastic, perfectly plastic, and compressible. Stress fields are obtained in all cases and a velocity field is presented for the case of point loading. Some numerical results are given comparing the results obtained here with those obtained when the Tresca yield condition is assumed.

Mathematical Modeling and Modifying of Turntable’s Angle for 5/12 Automatic Numerical Controlling Bending Hoop Machine

2011 ◽  
Vol 189-193 ◽  
pp. 1955-1959
Author(s):  
Xiao Hong Zhang ◽  
Guo Fu Yin
Keyword(s):  
Mathematical Model ◽  
Rotation Angle ◽  
Bending Moment ◽  
Numerical Control ◽  
Bending Test ◽  
Spring Back ◽  
Plastic Bending ◽  
Elastic Plastic ◽  
The Mathematical Model ◽  
Nc Bending

Aiming at 5/12 full-automatic numerical control (NC) bending hoop machine, the paper analyzed the elastic-plastic bending deformation with the knowledge of elastic-plastic bending principle, theoretically mechanics and spring-back, and deduced the relationship of bending moment and curvature ratio, and built the mathematical model between bending hoop turntable’s rotation angle and stirrups angle considering the spring-back deformation. Then the different spring-back angles under two types of stirrups diameter, Φ=8 mm and Φ=10 mm, are measured. The mathematical model of turntable’s rotation angle and stirrup’s angle for 5/12 automatic NC bending hoop machine is modified according to the analysis of bending test data.

Numerical Study of Defect Free Elbows Subjected to In-Plane Bending Moment

2011 ◽  
Vol 189-193 ◽  
pp. 1494-1497
Author(s):  
Wang Chen ◽  
Yin Pei Wang ◽  
Pei Ning Li ◽  
Chen Jin ◽  
Xiao Ming Sun
Keyword(s):  
Numerical Study ◽  
Plastic Material ◽  
Three Dimensional ◽  
Bending Moment ◽  
Plastic Behavior ◽  
Piping System ◽  
Elastic Plastic ◽  
Perfectly Plastic ◽  
Bending Radius ◽  
Elastic Perfectly Plastic

Elbow is a type of components widely used in a piping system, and so it is very important to know the plastic carrying capacity of elbow. In this study, the elastic-plastic behavior of elbows with various ratios of t/rm and relative bending radius R/rm were investigated in detail by using of three-dimensional (3D) non-linear finite element (FE) analyses, assuming elastic-perfectly-plastic material behaviour and taking geometric nonlinearity into account. The analyses indicated that elbow exhibited different behavior obviously at the elastic-plastic states subjected to In-Plane opening bending moment and closing bending moment. The closed form equations of elbow involving effect of tangent pipes were established.

A General Method of Calculating the Mϵmax Diagram in Plastic Bending of Beams

1951 ◽  
Vol 18 (4) ◽  
pp. 353-358
Author(s):  
Aris Phillips
Keyword(s):  
New Method ◽  
Pure Bending ◽  
Stress Strain ◽  
Plastic Bending ◽  
Elastic Plastic ◽  
Bending Of Beams ◽  
General Method ◽  
Plastic Stress

Abstract In this paper we give a new method for finding the Mϵmax curve in the case of symmetrical pure bending of bars in plasticity. This method is both accurate and comparatively easy to use. At the end of the paper we show how it is possible to express any σ-ϵ curve as the sum of a number of elastic plastic-stress strain lines.

Dynamic Fracture of a Beam or Plate in Plane Bending

1976 ◽  
Vol 43 (1) ◽  
pp. 112-116 ◽  
Author(s):  
L. B. Freund ◽  
G. Herrmann
Keyword(s):  
Crack Length ◽  
Dynamic Fracture ◽  
Crack Tip ◽  
Bending Moment ◽  
Fracture Initiation ◽  
Fracture Plane ◽  
Multiple Reflections ◽  
Pure Bending ◽  
Strain Fracture ◽  
Beam Thickness

The dynamic fracture response of a long beam of brittle elastic material subjected to pure bending is studied. If the magnitude of the applied bending moment is increased to a critical value, a crack will propagate from the tensile side of the beam across a cross section. An analysis is presented by means of which the crack length and bending moment at the fracturing section are determined as functions of time after fracture initiation. The main assumption on which the analysis rests is that, due to multiple reflections of stress waves across the thickness of the beam, the stress distribution on the prospective fracture plane ahead of the crack may be adequately approximated by the static distribution appropriate for the instantaneous crack length and net section bending moment. The results of numerical calculations are shown in graphs of crack length, crack tip speed, and fracturing section bending moment versus time. It is found that the crack tip accelerates very quickly to a speed near the characteristic terminal speed for the material, travels at this speed through most of the beam thickness, and then rapidly decelerates in the final stage of the process. The results also apply for plane strain fracture of a plate in pure bending provided that the value of the elastic modulus is appropriately modified.

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