scholarly journals Deriving the Beam Equation using the Minimum Total Potential Energy Principle and Solving the Equation Numerically

2018 ◽  
Author(s):  
Magnus Komperød
Keyword(s):  
Potential Energy ◽  
Total Potential Energy ◽  
Beam Equation ◽  
Energy Principle ◽  
Potential Energy Principle ◽  
Total Potential ◽  
Minimum Total Potential Energy

scholarly journals Postbuckling Analysis Of A Rectangular Plate Loaded In Compression

2015 ◽  
Vol 15 (2) ◽  
Author(s):  
Jozef Havran ◽  
Martin Psotný
Keyword(s):  
Rectangular Plate ◽  
User Program ◽  
Nonlinear Fem ◽  
Buckling Mode ◽  
Energy Principle ◽  
Initial Imperfections ◽  
Potential Energy Principle ◽  
Total Potential ◽  
The Stability ◽  
Minimum Total Potential Energy

Abstract The stability analysis of a thin rectangular plate loaded in compression is presented. The nonlinear FEM equations are derived from the minimum total potential energy principle. The peculiarities of the effects of the initial imperfections are investigated using the user program. Special attention is paid to the influence of imperfections on the post-critical buckling mode. The FEM computer program using a 48 DOF element has been used for analysis. Full Newton-Raphson procedure has been applied.

Bending of Sandwich Plates of Variable Thickness

1988 ◽  
Vol 55 (2) ◽  
pp. 419-424 ◽  
Author(s):  
N. Paydar ◽  
C. Libove
Keyword(s):  
Potential Energy ◽  
Variable Thickness ◽  
Middle Surface ◽  
Total Potential Energy ◽  
Sandwich Plates ◽  
Energy Principle ◽  
Total Potential ◽  
Small Deflection ◽  
The Face ◽  
Deflection Theory

A small deflection theory, consisting of differential equations and a total potential energy expression, is presented for determining the stresses and deformations in variable thickness elastic sandwich plates symmetric about a middle surface. The theory takes into account the contribution of the face-sheet membrane forces (by virtue of their slopes) to the transverse shear. A finite-difference formulation of the stationary total potential energy principle is presented along with an illustrative application.

scholarly journals Accurate Element of Compressive Bar considering the Effect of Displacement

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Lifeng Tang ◽  
Jing Xu ◽  
Hongzhi Wang ◽  
Xinghua Chen
Keyword(s):  
Potential Energy ◽  
Shape Function ◽  
Variation Principle ◽  
Shape Functions ◽  
Governing Equations ◽  
Energy Principle ◽  
Potential Energy Principle ◽  
Total Potential ◽  
Stiffness Matrices ◽  
Function Element

By constructing the compressive bar element and developing the stiffness matrix, most issues about the compressive bar can be solved. In this paper, based on second derivative to the equilibrium differential governing equations, the displacement shape functions are got. And then the finite element formula of compressive bar element is developed by using the potential energy principle and analytical shape function. Based on the total potential energy variation principle, the static and geometrical stiffness matrices are proposed, in which the large deformation of compressive bar is considered. To verify the accurate and validity of the analytical trial function element proposed in this paper, a number of the numerical examples are presented. Comparisons show that the proposed element has high calculation efficiency and rapid speed of convergence.

scholarly journals Three-Point Bending of Seven Layer Beams – Theoretical and Experimental Studies

2016 ◽  
Vol 62 (2) ◽  
pp. 115-133 ◽  
Author(s):  
E. Magnucka-Blandzi ◽  
P. Paczos ◽  
P. Wasilewicz ◽  
A. Wypych
Keyword(s):  
Potential Energy ◽  
Experimental Studies ◽  
Middle Surface ◽  
Total Potential Energy ◽  
Three Point Bending ◽  
Simply Supported ◽  
Total Potential ◽  
The Subject ◽  
Minimum Total Potential Energy

Abstract The subject of the analytical and experimental studies therein is of two metal seven-layer beam - plate bands. The first beam - plate band is composed of a lengthwise trapezoidally corrugated main core and two crosswise trapezoidally corrugated cores of faces. The second beam - plate band is composed of a crosswise trapezoidally corrugated main core and two lengthwise trapezoidally corrugated cores of faces. The hypotheses of deformation of a normal to the middle surface of the beams after bending are formulated. Equations of equilibrium are derived based on the theorem of minimum total potential energy. Three-point bending of the simply supported beams is theoretically and experimentally studied. The deflections of the two beams are determined with two methods, compared and presented.

Consistent Derivations of Spring Rates for Helical Springs

2009 ◽  
Vol 131 (7) ◽  
Author(s):  
Clive L. Dym
Keyword(s):  
Potential Energy ◽  
Total Potential Energy ◽  
Helical Spring ◽  
Helical Springs ◽  
Total Potential ◽  
Coiled Spring ◽  
Displacement Based ◽  
Minimum Total Potential Energy ◽  
First Time ◽  
Consistent Application

The spring rates of a coiled helical spring under an axial force and an axially directed torque are derived by a consistent application of Castigliano’s second theorem, and it is shown that the coupling between the two loads may not always be neglected. The spring rate of an extensional spring is derived for the first time through the use of the displacement based principle of minimum total potential energy. The present results are also compared with available derivations of and expressions for the stiffness of a coiled spring.

The Study for the Critical Moment of Arches in the Action of Endmoment Considering Shear Lag

2012 ◽  
Vol 446-449 ◽  
pp. 1380-1383
Author(s):  
Y. J. Chen ◽  
Qi Zhi Luo
Keyword(s):  
Potential Energy ◽  
Deformation Behaviour ◽  
Power Method ◽  
Shear Lag ◽  
Energy Principle ◽  
Potential Energy Principle ◽  
Basic Assumptions ◽  
Total Potential ◽  
Out Of Plane ◽  
Basic Displacement Functions

Based on six basic assumptions and deformation behaviour, the total potential energy of arches are obtained. The basic displacement functions are expressed with spline function. According to minmum potential energy principle, the characteristic equation is deducted. The characteristic value is solved by multiplication power method. An exampleof arches in the action of endmoment is calculated. The critical positive and negtive moment of out-of-plane stability in arches with single sysmetry axis section are studied.

A simplified formulation of adhesion problems with elastic plates

2009 ◽  
Vol 465 (2107) ◽  
pp. 2217-2230 ◽  
Author(s):  
Carmel Majidi ◽  
George G. Adams
Keyword(s):  
Potential Energy ◽  
Contact Area ◽  
Contact Region ◽  
Bending Moment ◽  
Total Potential Energy ◽  
Work Of Adhesion ◽  
Contact Radius ◽  
Algebraic Complexity ◽  
Elastic Plates ◽  
Total Potential

The solution of adhesion problems with elastic plates generally involves solving a boundary-value problem with an assumed contact area. The contact region is then found by minimizing the total potential energy with respect to the contact area (i.e. the contact radius for the axisymmetric case). Such a procedure can be extremely long and tedious. Here, we show that the inclusion of adhesion is equivalent to specifying a discontinuous internal bending moment at the contact region boundary. The magnitude of this moment discontinuity is related to the work of adhesion and flexural rigidity of the plate. Such a formulation can greatly reduce the algebraic complexity of solving these problems. It is noted that the related plate contact problems without adhesion can also be solved by minimizing the total potential energy. However, it has long been recognized that it is mathematically more efficient to find the contact area by specifying a continuous internal bending moment at the boundary of the contact region. Thus, our moment discontinuity method can be considered to be a generalization of that procedure which is applicable for problems with adhesion.

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