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.

Plastic Deformation of Semi-Infinite Beams Subject to Transverse Impact Loading at the Free End

1956 ◽  
Vol 23 (2) ◽  
pp. 239-243
Author(s):  
M. F. Conroy
Keyword(s):  
Plastic Deformation ◽  
Impact Loading ◽  
Constant Velocity ◽  
Bending Moment ◽  
Transverse Force ◽  
Square Root ◽  
Transverse Loading ◽  
Transverse Impact ◽  
Velocity Impact ◽  
Time Part

Abstract The object of this paper is to consider the plastic deformation of semi-infinite beams subject to dynamic transverse loading at the free end. The type of loading considered is that of a constant bending moment, together with a transverse force the magnitude of which is inversely proportional to the square root of time. Part 1 of the paper consists of a plastic-rigid analysis of the problem, based on the plastic-rigid analysis of infinite beams under transverse, constant velocity, impact loading developed by the author. Part 2 of the paper consists of an elastic-plastic solution of the problem, based on a theoretical analysis of the plastic deformation of infinite beams subject to transverse, constant-velocity impact loading developed by H. F. Bohnenblust. Specific problems are considered for which the deflection solutions obtained by elastic ideally plastic and rigid ideally plastic analyses are compared.

Plane-Strain Bending With Isotropic Strain Hardening at Large Strains

2010 ◽  
Vol 77 (6) ◽  
Author(s):  
Sergei Alexandrov ◽  
Yeong-Maw Hwang
Keyword(s):  
Strain Hardening ◽  
Plane Strain ◽  
Closed Form Solution ◽  
Bending Moment ◽  
Form Solution ◽  
Isotropic Hardening ◽  
Large Strains ◽  
Rigid Plastic ◽  
Elastic Plastic ◽  
Hardening Law

Finite deformation elastic-plastic analysis of plane-strain pure bending of a strain hardening sheet is presented. The general closed-form solution is proposed for an arbitrary isotropic hardening law assuming that the material is incompressible. Explicit relations are given for most popular conventional laws. The stage of unloading is included in the analysis to investigate the distribution of residual stresses and springback. The paper emphasizes the method of solution and the general qualitative features of elastic-plastic solutions rather than the study of the bending process for a specific material. In particular, it is shown that rigid-plastic solutions can be used to predict the bending moment at sufficiently large strains.

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.

On the Effect of Shear on Plastic Deformation of Beams Under Transverse Impact Loading

1960 ◽  
Vol 27 (1) ◽  
pp. 107-110 ◽  
Author(s):  
B. Karunes ◽  
E. T. Onat
Keyword(s):  
Impact Loading ◽  
Pure Shear ◽  
Bending Moment ◽  
Transverse Impact ◽  
Rigid Plastic ◽  
Inertia Effects ◽  
Interaction Diagram ◽  
Shear Effects ◽  
The Impact ◽  
Special Case

The impact problem for a rigid-plastic beam is formulated by using an interaction curve relating shearing force and bending moment for fully plastic action, and allowing for shear and rotary inertia effects. Using a simplified interaction diagram, the problem of point-impact loading is solved for a special case. The analysis shows that the shear effects are of considerable importance when the parameter μ0 = 2Q0l/M0 is less than 20 where Q0 and M0 are plastic-carrying capacities of the cross section for pure shear and bending, respectively, and 21 is the length of the beam.

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.

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.

scholarly journals Finite Pure Plane Strain Bending of Inhomogeneous Anisotropic Sheets

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 145
Author(s):  
Sergei Alexandrov ◽  
Elena Lyamina ◽  
Yeong-Maw Hwang
Keyword(s):  
Plane Strain ◽  
Yield Criterion ◽  
Large Strains ◽  
Rigid Plastic ◽  
Plastic Properties ◽  
Elastic Plastic ◽  
Starting Point ◽  
The Difference ◽  
Inside Radius ◽  
Elastic Unloading

The present paper concerns the general solution for finite plane strain pure bending of incompressible, orthotropic sheets. In contrast to available solutions, the new solution is valid for inhomogeneous distributions of plastic properties. The solution is semi-analytic. A numerical treatment is only necessary for solving transcendent equations and evaluating ordinary integrals. The solution’s starting point is a transformation between Eulerian and Lagrangian coordinates that is valid for a wide class of constitutive equations. The symmetric distribution relative to the center line of the sheet is separately treated where it is advantageous. It is shown that this type of symmetry simplifies the solution. Hill’s quadratic yield criterion is adopted. Both elastic/plastic and rigid/plastic solutions are derived. Elastic unloading is also considered, and it is shown that reverse plastic yielding occurs at a relatively large inside radius. An illustrative example uses real experimental data. The distribution of plastic properties is symmetric in this example. It is shown that the difference between the elastic/plastic and rigid/plastic solutions is negligible, except at the very beginning of the process. However, the rigid/plastic solution is much simpler and, therefore, can be recommended for practical use at large strains, including calculating the residual stresses.

Elastic-Plastic Analysis of an Elbow With Axial Part-Through Internal Crack at Crown Under In-Plane Bending

2008 ◽  
Author(s):  
K. M. Prabhakaran ◽  
S. R. Bhate ◽  
V. Bhasin ◽  
A. K. Ghosh
Keyword(s):  
Finite Element ◽  
Crack Depth ◽  
Bending Moment ◽  
Internal Crack ◽  
J Integral ◽  
Finite Element Analyses ◽  
Integrity Assessment ◽  
Elastic Plastic ◽  
Axial Part ◽  
Plane Bending

Piping elbows under bending moment are vulnerable to cracking at crown. The structural integrity assessment requires evaluation of J-integral. The J-integral values for elbows with axial part-through internal crack at crown under in-plane bending moment are limited in open literature. This paper presents the J-integral results of a thick and thin, 90-degree, long radius elbow subjected to in-plane opening bending moment based on number of finite element analyses covering different crack configurations. The non-linear elastic-plastic finite element analyses were performed using WARP3D software. Both geometrical and material nonlinearity were considered in the study. The geometry considered were for Rm/t = 5, and 12 with ratio of crack depth to wall thickness, a/t = 0.15, 0.25, 0.5 and 0.75 and ratio of crack length to crack depth, 2c/a = 6, 8, 10 and 12.

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