Approximate Displacements of Exact Membrane Actions in a Shell Triangular Element

1983 ◽  
Vol 34 (4) ◽  
pp. 282-302 ◽  
Author(s):  
L.S.D. Morley
Keyword(s):  
Finite Element ◽  
Potential Energy ◽  
Positive Zero ◽  
Triangular Element ◽  
Minimum Potential Energy ◽  
Stress Functions ◽  
Minimum Potential ◽  
Triangular Elements ◽  
Physical Quantities ◽  
Geometric Analogue

SummaryExact stress functions which satisfy the homogeneous differential equations of equilibrium for membrane actions are available from the static geometric analogue of previously derived exact displacements of inextensional bending. For finite element evaluation it is necessary to know the displacements (and rotations) caused by these membrane actions. A method of calculating approximate displacements is described which uses the principle of minimum potential energy. Results are given for specimen triangular elements with positive, zero and negative Gaussian curvatures. A listing is appended of a Fortran computer program which allows calculation of these approximate displacements, rotations and other physical quantities for other element shapes.

Numerical study on contact force of paralleled beveloid gears using minimum potential energy theory

2020 ◽  
pp. 030932472093689
Author(s):  
Ruihua Sun ◽  
Chaosheng Song ◽  
Caichao Zhu ◽  
Yawen Wang ◽  
Kunming Liu
Keyword(s):  
Finite Element ◽  
Potential Energy ◽  
Contact Force ◽  
Force Distribution ◽  
Force Model ◽  
Gear Pair ◽  
Minimum Potential Energy ◽  
Contact Force Distribution ◽  
Minimum Potential ◽  
Energy Theory

Using minimum potential energy theory and slicing method, a computational approach to calculate the magnitude and distribution of contact force for paralleled beveloid gear pair was proposed in this article. The theoretical tooth contact model was built based on spatial gearing theory to calculate the mesh parameters including the coordinates, normal vectors, and equivalent radius for meshing points. Then, the analytical contact force model of paralleled beveloid gear pair was derived based on minimum potential energy theory. Finite element contact analysis was conducted to verify the proposed model. Finally, the influences of macro-geometry design parameters on the contact force distribution were investigated. Results show that the pressure angle has a limited influence on the contact force distribution. The increase in helix and cone angles will observably increase the asymmetry of contact force distribution as well as the fluctuation of contact force distribution for a single tooth. A good correlation was obtained between the proposed analytical model and the finite element model for the distribution and magnitudes of contact force.

Optimization of Finite Element Grids Based on Minimum Potential Energy

1973 ◽  
Vol 95 (1) ◽  
pp. 186-190 ◽  
Author(s):  
G. M. McNeice ◽  
P. V. Marcal
Keyword(s):  
Finite Element ◽  
Potential Energy ◽  
Axial Load ◽  
Necessary Conditions ◽  
Nodal Point ◽  
Initial Study ◽  
Minimum Potential Energy ◽  
One Dimensional ◽  
Independent Variables ◽  
Minimum Potential

An initial study has been made of a method for optimizing finite element grids. This method is based on the minimum potential energy where the nodal point positions are also treated as independent variables. Necessary conditions have been obtained for the optimized grids. Case studies demonstrate the procedure for a one-dimensional tapered bar under axial load and for a two-dimensional square membrane subjected to a parabolic tensile stress. The optimized grids were observed to give improved stress estimates.

Principles of minimum potential energy and complementary energy

2020 ◽  
pp. 228-248
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Boundary Condition ◽  
Potential Energy ◽  
Displacement Field ◽  
Mixed Boundary ◽  
Energy Functional ◽  
The Body ◽  
Minimum Potential Energy ◽  
Complementary Energy ◽  
Minimum Potential ◽  
Potential Energy Functional

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

Optimization for Minimum Potential Energy, Maximum Rigidity and Minimum Deformability

1984 ◽  
pp. 9-87
Author(s):  
Andrej M. Brandt ◽  
Wojciech Dzieniszewski ◽  
Stefan Jendo ◽  
Wojciech Marks ◽  
Stefan Owczarek ◽  
...  
Keyword(s):  
Potential Energy ◽  
Minimum Potential Energy ◽  
Minimum Potential ◽  
Energy Maximum

A Global Extremum Principle in Mixed Form for Equilibrium Analysis With Elastic/Stiffening Materials (a Generalized Minimum Potential Energy Principle)

1994 ◽  
Vol 61 (4) ◽  
pp. 914-918 ◽  
Author(s):  
J. E. Taylor
Keyword(s):  
Potential Energy ◽  
Problem Formulation ◽  
Extremum Problem ◽  
Equilibrium Analysis ◽  
Minimum Potential Energy ◽  
Problem Statement ◽  
Energy Principle ◽  
Potential Energy Principle ◽  
Minimum Potential ◽  
Minimum Potential Energy Principle

An extremum problem formulation is presented for the equilibrium mechanics of continuum systems made of a generalized form of elastic/stiffening material. Properties of the material are represented via a series composition of elastic/locking constituents. This construction provides a means to incorporate a general model for nonlinear composites of stiffening type into a convex problem statement for the global equilibrium analysis. The problem statement is expressed in mixed “stress and deformation” form. Narrower statements such as the classical minimum potential energy principle, and the earlier (Prager) model for elastic/locking material are imbedded within the general formulation. An extremum problem formulation in mixed form for linearly elastic structures is available as a special case as well.

Sign in / Sign up

Export Citation Format

Share Document