scholarly journals Closure to “Discussion of ‘Concentrated-Force Problems in Plane Strain, Plane Stress, and Transverse Bending of Plates’” (1947, ASME J. Appl. Mech., 14, p. A164)

1947 ◽  
Vol 14 (2) ◽  
pp. A165
Author(s):  
P. S. Symonds
Keyword(s):  
Plane Strain ◽  
Plane Stress ◽  
Concentrated Force ◽  
Transverse Bending

Concentrated-Force Problems in Plane Strain, Plane Stress, and Transverse Bending of Plates

1946 ◽  
Vol 13 (3) ◽  
pp. A183-A197
Author(s):  
P. S. Symonds
Keyword(s):  
Boundary Conditions ◽  
Plane Stress ◽  
Biharmonic Equation ◽  
Infinite Plate ◽  
Outer Edge ◽  
Transverse Force ◽  
Concentrated Force ◽  
Transverse Bending ◽  
Complementary Function ◽  
Concentrated Forces

Abstract A general method is described for the solution of problems of transverse bending of thin plates acted on by concentrated normal forces, and of problems of plane stress or plane strain, in which concentrated forces are applied to the boundaries. The solution is taken in two parts: (a) The special functions which give the stresses or deflections in the neighborhood of the concentrated forces. (b) A complementary function, satisfying the appropriate biharmonic equation, such that the complete solution satisfies the boundary conditions of the problem. For certain types of boundaries, this complementary function can be determined by expanding the concentrated-force functions as infinite trigonometric series. Then by addition of general solutions of the appropriate biharmonic equation, the required boundary conditions may be satisfied. The method is first illustrated by solving the plate-bending problem, for which the solution is known, of a clamped circular disk loaded by a transverse force at any point. It is then applied to the problem of an infinite plate fixed at an inner circular boundary, with outer edge free, and loaded by a transverse force at any point. This solution is obtained in finite form, and typical curves of deflection, bending moments, and shear forces are given in Figs. 3 to 8, inclusive. Using this result, solutions are next obtained for ring-shaped plates of finite outer radius, with the force applied either at the outer edge or at any point between the inner clamped edge and the outer free edge. The former case was previously solved by H. Reissner. Curves comparing the maximum moments and shears in the infinite plate with those of the annular plate with force either at the outer edge, or inside the ring are given in Figs. 9 to 12, inclusive. Finally, a solution is given of the problem in plane stress of a large plate containing an elliptical hole, which is loaded by line forces at the ends of the minor axes of the ellipse. Curves showing results of this solution are given in Figs. 14 and 15.

A Concentrated Force Problem of Plane Strain or Plane Stress

1947 ◽  
Vol 14 (3) ◽  
pp. A246
Author(s):  
A. E. Green
Keyword(s):  
Closed Form ◽  
Plane Strain ◽  
Plane Stress ◽  
Complex Variable ◽  
Elliptical Hole ◽  
Minor Axis ◽  
Concentrated Force ◽  
Large Plate ◽  
Force Problem ◽  
Variable Analysis

Abstract The problem in plane strain or plane stress of a large plate containing an elliptical hole, which is loaded by line forces at the ends of the minor axis of the ellipse, is solved in closed form by using complex variable analysis.

Improved Compliance Solutions for C(T), SE(B) and Clamped SE(T) Specimens Including Side-Grooves, Varying Thicknesses and 3-D Effects

2014 ◽  
Author(s):  
Gustavo Henrique B. Donato ◽  
Felipe Cavalheiro Moreira
Keyword(s):  
Crack Growth ◽  
Plane Strain ◽  
Plane Stress ◽  
Potential Drop ◽  
Crack Size ◽  
Growth Conditions ◽  
Size Estimation ◽  
Unloading Compliance ◽  
Integrity Assessments ◽  
Elastic Unloading

Fracture toughness and Fatigue Crack Growth (FCG) experimental data represent the basis for accurate designs and integrity assessments of components containing crack-like defects. Considering ductile and high toughness structural materials, crack growing curves (e.g. J-R curves) and FCG data (in terms of da/dN vs. ΔK or ΔJ) assumed paramount relevance since characterize, respectively, ductile fracture and cyclic crack growth conditions. In common, these two types of mechanical properties severely depend on real-time and precise crack size estimations during laboratory testing. Optical, electric potential drop or (most commonly) elastic unloading compliance (C) techniques can be employed. In the latter method, crack size estimation derives from C using a dimensionless parameter (μ) which incorporates specimen’s thickness (B), elasticity (E) and compliance itself. Plane stress and plane strain solutions for μ are available in several standards regarding C(T), SE(B) and M(T) specimens, among others. Current challenges include: i) real specimens are in neither plane stress nor plane strain - modulus vary between E (plane stress) and E/(1-ν2) (plane strain), revealing effects of thickness and 3-D configurations; ii) furthermore, side-grooves affect specimen’s stiffness, leading to an “effective thickness”. Previous results from current authors revealed deviations larger than 10% in crack size estimations following existing practices, especially for shallow cracks and side-grooved samples. In addition, compliance solutions for the emerging clamped SE(T) specimens are not yet standardized. As a step in this direction, this work investigates 3-D, thickness and side-groove effects on compliance solutions applicable to C(T), SE(B) and clamped SE(T) specimens. Refined 3-D elastic FE-models provide Load-CMOD evolutions. The analysis matrix includes crack depths between a/W=0.1 and a/W=0.7 and varying thicknesses (W/B = 4, W/B = 2 and W/B = 1). Side-grooves of 5%, 10% and 20% are also considered. The results include compliance solutions incorporating all aforementioned effects to provide accurate crack size estimation during laboratory fracture and FCG testing. All proposals revealed reduced deviations if compared to existing solutions.

scholarly journals A method of finding stress solutions for a general plastic material under plane strain and plane stress conditions

2020 ◽  
Vol 37 ◽  
pp. 100-107
Author(s):  
Sergei Alexandrov ◽  
Yeau-Ren Jeng
Keyword(s):  
Coordinate System ◽  
Plane Strain ◽  
Principal Stress ◽  
Plane Stress ◽  
Plastic Material ◽  
Yield Criterion ◽  
Boundary Value ◽  
Equilibrium Equations ◽  
Geometric Problem ◽  
Principal Lines

Abstract A general plastic material under plane strain and plane stress is classified by a yield criterion that depends on both the first and second invariants of the stress tensor. The yield criterion together with the stress equilibrium equations forms a statically determinate system. This system is investigated in the principal lines coordinate system (i.e. the coordinate curves of this coordinate system coincide with trajectories of the principal stress directions). It is shown that the scale factors of the principal lines coordinate system satisfy a simple equation. Using this equation, a method for constructing the principal stress trajectories is developed. Therefore, the boundary value problem of plasticity theory reduces to a purely geometric problem. It is believed that the method developed is useful for solving a wide class of boundary value problems in plasticity.

Transmission of Tension From a Bar to a Plate

1954 ◽  
Vol 21 (2) ◽  
pp. 147-150
Author(s):  
J. N. Goodier ◽  
C. S. Hsu
Keyword(s):  
Fatigue Strength ◽  
Strain Gage ◽  
Plane Stress ◽  
Fair Agreement ◽  
Lap Joint ◽  
Concentrated Force ◽  
Considerable Fraction ◽  
Stress Calculation ◽  
Local Character

Abstract When a bar or strip is lap-jointed to a plate, and transmits tension to it, the transmission is not effected only by a smooth distribution of force along the lap joint; there is also a highly concentrated force, a considerable fraction of the total tension, where the bar meets the plate, and a second force at the end of the bar. These forces are investigated by strain-gage measurements for various lengths of lap, and by a plane-stress calculation, with fair agreement. The results suggest that the fatigue strength of such joints will depend on the detailed local character of the joint where the bar meets the plate, rather than on the length of the joint.

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