Discontinuous Maxwell–Rankine stress functions for space frames

2018 ◽  
Vol 33 (1) ◽  
pp. 35-47
Author(s):  
Allan McRobie ◽  
Chris Williams
Keyword(s):  
Stress Function ◽  
Three Dimensional ◽  
Shear Forces ◽  
Space Frames ◽  
Stress Functions

This article shows how bending and torsional moments in three-dimensional frames can be represented via a discontinuous Maxwell–Rankine stress function. The associated Rankine reciprocal contains polygonal faces whose areas represent forces. These faces are orthogonal to the member forces (which may include shear forces) and need not be orthogonal to the beams.

Notes on problems in hexagonal aeolotropic materials

1951 ◽  
Vol 47 (2) ◽  
pp. 401-409 ◽  
Author(s):  
R. T. Shield
Keyword(s):  
General Solution ◽  
Stress Function ◽  
Harmonic Functions ◽  
Present Note ◽  
Three Dimensional ◽  
Stress Distributions ◽  
Axially Symmetric ◽  
Isotropic Materials ◽  
Stress Functions ◽  
Single Stress

Three-dimensional stress distributions in hexagonal aeolotropic materials have recently been considered by Elliott(1, 2), who obtained a general solution of the elastic equations of equilibrium in terms of two ‘harmonic’ functions, or, in the case of axially symmetric stress distributions, in terms of a single stress function. These stress functions are analogous to the stress functions employed to define stress systems in isotropic materials, and in the present note further problems in hexagonal aeolotropic media are solved, the method in each case being similar to that used for the corresponding problem in isotropic materials. Because of this similarity detailed explanations are unnecessary and only the essential steps in the working are given below.

scholarly journals Displacements and Stress Functions of Straight Dislocations and Line Forces in Anisotropic Elasticity: A New Derivation and Its Relation to the Integral Formalism

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1721
Author(s):  
Markus Lazar
Keyword(s):  
Plane Strain ◽  
Stress Function ◽  
Function Fields ◽  
Three Dimensional ◽  
Anisotropic Elasticity ◽  
Green Tensor ◽  
Two Dimensional ◽  
Integral Formalism ◽  
Stress Functions ◽  
Generalized Plane Strain

The displacement and stress function fields of straight dislocations and lines forces are derived based on three-dimensional anisotropic incompatible elasticity. Using the two-dimensional anisotropic Green tensor of generalized plane strain, a Burgers-like formula for straight dislocations and body forces is derived and its relation to the solution of the displacement and stress function fields in the integral formalism is given. Moreover, the stress functions of a point force are calculated and the relation to the potential of a Dirac string is pointed out.

scholarly journals Analytical graphic statics

2021 ◽  
pp. 095605992110016
Author(s):  
Tamás Baranyai
Keyword(s):  
Stress Function ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Linear Functionals ◽  
The Past ◽  
Force Diagram ◽  
Graphic Statics ◽  
Reciprocal Diagrams ◽  
Force Systems ◽  
Stress Functions

Graphic statics is undergoing a renaissance, with computerized visual representation becoming both easier and more spectacular as time passes. While methods of the past are revived, little emphasis has been placed on studying the mathematics behind these methods. Due to the considerable advances of our mathematical understanding since the birth of graphic statics, we can learn a lot by examining these old methods from a more modern viewpoint. As such, this work shows the mathematical fabric joining different aspects of graphic statics, like dualities, reciprocal diagrams, and discontinuous stress functions. This is done by introducing a new, three dimensional force diagram (containing the old two dimensional force diagram) depicting the three dimensional equilibrium of planar force systems. A corresponding three dimensional “form diagram” (dual diagram) is introduced, in which forces are treated as linear functionals (dual vectors). It is shown that the polyhedral stress function introduced by Maxwell is in fact a linear combination of these functionals; and the projective dualities connecting these three dimensional diagrams are also explained.

scholarly journals Composite hollow truss with multiple vierendeel panels

2013 ◽  
Vol 66 (4) ◽  
pp. 431-438
Author(s):  
Augusto Ottoni Bueno da Silva ◽  
Newton de Oliveira Pinto Júnior ◽  
João Alberto Venegas Requena
Keyword(s):  
Bearing Capacity ◽  
Failure Modes ◽  
Three Dimensional ◽  
Analytical Calculation ◽  
Two Dimensional ◽  
Shear Forces ◽  
Vertical Displacements ◽  
New System ◽  
The Ideal

The aim of this study was to evaluate through analytical calculation, two-dimensional elastic modeling, and three-dimensional plastic modeling, the bearing capacity and failure modes of composite hollow trusses bi-supported with a 15 meter span, varying the number of central Vierendeel panels. The study found the proportion span/3 - span/3 - span/3, as the ideal relationship for the truss - Vierendeel - truss lengths, because by increasing the proportion of the length occupied by the central Vierendeel panels, the new system loses stiffness and no longer supports the load stipulated in the project. Furthermore, they can start presenting excessive vertical displacements and insufficient resistance to external shear forces acting on the panels.

Three-dimensional stress functions

1954 ◽  
Vol 258 (5) ◽  
pp. 371-382 ◽  
Author(s):  
H.L. Langhaar ◽  
M. Stippes
Keyword(s):  
Three Dimensional ◽  
Stress Functions

The Effect of Lifting vs. Lowering on Spinal Loading

1996 ◽  
Vol 40 (13) ◽  
pp. 599-603 ◽  
Author(s):  
Kermit G. Davis
Keyword(s):  
Muscle Activity ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Muscle Forces ◽  
Shear Forces ◽  
Spinal Loading ◽  
Spinal Loads ◽  
Trunk Strength ◽  
Anterior Posterior ◽  
Three Dimensional Space

In industry, workers perform tasks requiring both lifting and lowering. During concentric lifting, the muscles are shortening as the force is being generated. Conversely, the muscle lengthens while generating force during eccentric lowering. While research on various lifting tasks is extensive, there has been limited research performed to evaluate the lowering tasks. Most of the research that does exist on lowering has investigated muscle activity and trunk strength. None of these studies have investigated spinal loading. The current study estimated the effects of lifting and lowering on spinal loads and predicted moments imposed on the spine. Ten subjects performed both eccentric and concentric lifts under sagittally symmetric conditions. The tasks were performed under isokinetic trunk velocities of 5, 10, 20, 40, and 80 deg/s while holding a box with weights of 9.1, 18.2, and 27.3 kg. Spinal loads and predicted moments in three dimensional space were estimated by an EMG-assisted model which has been adjusted to incorporate the artifacts of eccentric lifting. Eccentric strength was found to be 56 percent greater than during concentric lifting. The lowering tasks produced significantly higher compression forces but lower anterior-posterior shear forces than the concentric lifting tasks. The differences in the spinal loads between the two lifting tasks were attributed to the internal muscle forces and unequal moments resulting from differences in the lifting path of the box. Thus, the differences between the lifting tasks resulted from different lifting styles associated with eccentric and concentric movements

scholarly journals Form-Finding of Shells Containing Both Tension and Compression Using the Airy Stress Function

2020 ◽  
Author(s):  
Masaaki Miki ◽  
Emil Adiels ◽  
William Baker ◽  
Toby Mitchell ◽  
Alexander Sehlstrom ◽  
...  
Keyword(s):  
Stress Function ◽  
Mixed Type ◽  
Airy Stress Function ◽  
Form Finding ◽  
Graphic Statics ◽  
Tension And Compression ◽  
Stress Functions ◽  
Complete Set ◽  
Pure Compression ◽  
Asymptotic Lines

Pure-compression shells have been the central topic in the form-finding of shells. This paper studies tension-compression mixed type shells by utilizing a NURBS-based isogeometric form-finding approach that analyzes Airy stress functions to expand the possible plan geometry. A complete set of smooth version graphic statics tools is provided to support the analyses. The method is validated using examples with known solutions, and a further example demonstrates the possible forms of shells that the proposed method permits. Additionally, a guideline to configure a proper set of boundary conditions is presented through the lens of asymptotic lines of the stress functions.

Analysis of Deep Beams

1951 ◽  
Vol 18 (2) ◽  
pp. 163-172
Author(s):  
H. D. Conway ◽  
L. Chow ◽  
G. W. Morgan
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Stress Distribution ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Normal Stress ◽  
Stress Function ◽  
Beam Theory ◽  
Deep Beam ◽  
Stress Functions

Abstract This paper presents a method of analyzing the stress distribution in a deep beam of finite length by superimposing two stress functions. The first stress function is chosen in the form of a trigonometric series which satisfies all but one of the boundary conditions—that of zero normal stress on the ends of the beam. The principle of least work is then used to obtain a second stress function giving the distribution of normal stress on the ends which is left by the first stress function. By superimposing the two solutions, all the boundary conditions are satisfied. Two particular cases of a given type of loading are solved in this way to investigate the stresses in a deep beam and their deviation from the ordinary beam theory. In addition, an approximate solution by the numerical method of finite difference is worked out for one of the two cases. Results from the two methods are compared and discussed. A method of obtaining an exact solution to the problem is given in an Appendix.

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