Stresses by Analysis and Experiment

1946 ◽  
Vol 155 (1) ◽  
pp. 20-40
Author(s):  
A. J. Sutton Pippard
Keyword(s):  
Relaxation Methods ◽  
Distribution Method ◽  
Moment Distribution ◽  
Stress Diagram ◽  
Hardy Cross ◽  
Moment Distribution Method ◽  
Fundamental Theorems ◽  
Elementary Space ◽  
Plane Stress Analysis ◽  
Made In

This is an attempt to outline in a single paper the bases of structural analysis when stresses do not exceed the limit of proportionality. The fundamental theorems are stated and the general problems in the analysis of just-stiff and redundant frameworks are discussed. The solution of a just-stiff frame is obtained when a compatible system of stresses is determined, but this is insufficient if the structure be redundant; the strains must then also be compatible. Passing reference is made to the best known methods of stressing just-stiff frames; the stress diagram and method of sections, but the method of tension coefficients is described in rather more detail with reference to the elementary space frame. The classical methods of dealing with redundant frames are next outlined. After a reference to slope-deflexion analysis, of which an early example was Clapeyron's treatment of the continuous beam (the theorem of three moments), the work of Castigliano is considered and in illustration his second theorem is used to obtain the stresses in a flywheel due to the heating of the rim by a brake. A most important contribution to the treatment of redundant structures was made in 1930 when Professor Hardy Cross described his moment distribution method. This is illustrated by an example and reference is made to the powerful relaxation methods developed by Professor R. V. Southwell. The paper concludes with a short description of an experimental method for obtaining influence lines and of another for the solution of certain problems of plane stress analysis.

Stress Analysis of Aircraft Frameworks

1941 ◽  
Vol 45 (367) ◽  
pp. 241-262 ◽  
Author(s):  
N. J. Hoff
Keyword(s):  
Critical Loads ◽  
Bending Moments ◽  
Lateral Loads ◽  
Distribution Method ◽  
Numerical Examples ◽  
Beam Columns ◽  
Moment Distribution ◽  
Hardy Cross ◽  
Moment Distribution Method ◽  
The Stability

SummaryIt is shown that the calculation of the critical loads of a plane framework is superfluous if the bending moments in the bars due to external moments and to lateral loads are determined by the Hardy Cross moment distribution method as extended by James. Convergence of this method is a proof of the stability of the framework. In Section 1 methods of determining stresses and critical loads in frameworks are discussed. Section 2 deals with the distortion patterns of beam columns on several supports below and above the critical loads. In Section 3 the method of proof of the convergence is outlined, and regular and particular cases are discussed with the aid of numerical examples. The final proof is given in Section 4.

The Analysis of Shear Distribution for Multi-Cell Beams in Flexure by means of Successive Numerical Approximations

1954 ◽  
Vol 58 (518) ◽  
pp. 122-127 ◽  
Author(s):  
Derek R. Samson
Keyword(s):  
Shear Flow ◽  
Unit Length ◽  
Design Stage ◽  
List Type ◽  
Numerical Approximations ◽  
Distribution Method ◽  
Moment Distribution ◽  
Hardy Cross ◽  
Moment Distribution Method

The determination of shear distributions for multi-cell sections can conveniently be considered as two separate problems.(i)the shear flow arising from a pure torsional couple applied in the plane of the section, and(ii)the shear flow arising from a resultant shear force acting on the section so as to produce no resultant twist per unit length of that section,i.e. through the shear centre of that section.Both problems involve finding the value of one unknown shear flow for each cell, so that in multi-cell construction a set of simultaneous equations is obtained, which must be solved to obtain the shear pattern.The solution of problem (i) by successive numerical approximations has been dealt with completely by Benscoter. The present paper gives a method for problem (ii).The usefulness of the method becomes more apparent with increasing numbers of spar webs as may be envisaged for future wing sections of small thickness chord ratios. It is analogous in the procedure used to the Hardy Cross moment distribution method for continuous beams, and has similar advantages. An approximate solution of any required degree of accuracy is quickly obtainable at the design stage and the method is readily adaptable for use by a computer with no knowledge of the structural or mathematical theory involved.

Stresses in Ring Structures

1945 ◽  
Vol 49 (418) ◽  
pp. 646-651
Author(s):  
L. G. Whitehead
Keyword(s):  
Simultaneous Equations ◽  
Principle Of Superposition ◽  
Distribution Method ◽  
Direct Extension ◽  
Moment Distribution ◽  
Hardy Cross ◽  
Ring Structures ◽  
Moment Distribution Method ◽  
Relaxation Procedure ◽  
Large Aircraft

The simple ring frames which are a common feature of metal fuselage construction are being superseded in some large aircraft by more complicated frames in which one or more cross members are added to stiffen the frame. The stress distribution in simple rings is readily analysed by the standard methods for redundant structures, but the direct extension of these methods to more complicated cases leads normally to the solution of a number of simultaneous equations. The method now proposed avoids this difficulty by breaking down the calculation into a number of simple steps and then applying the principle of superposition to combine the results so obtained. In this respect it is equivalent to the Hardy Cross moment distribution method (1) or to Southwell's method of the systematic relaxation of constraints (2) though in many applications the final superposition of loadings is so simple that the full moment distribution or relaxation procedure is not required.

scholarly journals A Matrix Formulation for the Moment Distribution Method Applied to Continuous Beams

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Arlindo Pires Lopes ◽  
Adriana Alencar Santos ◽  
Rogério Coelho Lopes
Keyword(s):  
Structural Analysis ◽  
Algebraic Equations ◽  
Displacement Method ◽  
Matrix Formulation ◽  
Distribution Method ◽  
Continuous Beams ◽  
Moment Distribution ◽  
Moment Distribution Method ◽  
Carry Over ◽  
The Moment

The Moment Distribution Method is a quite powerful hand method of structural analysis, in which the solution is obtained iteratively without even formulating the equations for the unknowns. It was formulated by Professor Cross in an era where computer facilities were not available to solve frame problems that normally require the solution of simultaneous algebraic equations. Its relevance today, in the era of personal computers, is in its insight on how a structure reacts to applied loads by rotating its nodes and thus distributing the loads in the form of member-end moments. Such an insight is the foundation of the modern displacement method. This work has a main objective to present an exact solution for the Moment Distribution Method through a matrix formulation using only one equation. The initial moments at the ends of the members and the distribution and carry-over factors are calculated from the elementary procedures of structural analysis. Four continuous beams are investigated to illustrate the applicability and accuracy of the proposed formulation. The use of a matrix formulation yields excellent results when compared with those in the literature or with a commercial structural program.

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