Discussion of “Generalization of Hardy Cross Method”

1966 ◽  
Vol 92 (4) ◽  
pp. 59-78
Author(s):  
Allen E. Crenshaw ◽  
Joseph G. Perri ◽  
Frederic T. Mavis ◽  
James W. Gillespie
Keyword(s):  
Hardy Cross

scholarly journals An improvement of Hardy Cross method applied on looped spatial natural gas distribution networks

2017 ◽  
Author(s):  
Dejan Brkic
Keyword(s):  
Gas Flow ◽  
Solution Method ◽  
Distribution Networks ◽  
Original Method ◽  
Gas Distribution ◽  
Gas Network ◽  
Hardy Cross ◽  
Successive Substitution ◽  
Newton Raphson ◽  
Looped Network

Hardy Cross method is common for calculation of loops-like gas distribution networks with known node gas consumptions. This method is given in two forms: original Hardy Cross method-successive substitution methods and improved-simultaneous solution method (Newton-Raphson group of methods). Problem of gas flow in looped network is nonlinear problem; i.e. relation between flow and pressure drop is not linear while relation between electric current and voltage is. Improvement of original method is done by introduction of influence of adjacent contours in Yacobian matrix which is used in calculation and which is in original method strictly diagonal with all zeros in non-diagonal terms. In that way necessary number of iteration in calculations is decreased. If during the design of gas network with loops is anticipated that some of conduits are crossing each other without connection, this sort of network became, so there has to be introduced corrections of third or higher order.

scholarly journals Two Efficient Methods for Gas Distributive Network Calculation

2018 ◽  
Author(s):  
Dejan Brkić
Keyword(s):  
Pressure Drop ◽  
Thermodynamic Properties ◽  
Gas Flow ◽  
Flow Distribution ◽  
Gas Pipeline ◽  
Gas Flow Rate ◽  
Closed Loops ◽  
Loop Method ◽  
Hardy Cross ◽  
Flow Through

Today, two very efficient methods for calculation of flow distribution per branches of a looped gas pipeline are available. Most common is improved Hardy Cross method, while the second one is so-called unified node-loop method. For gas pipeline, gas flow rate through a pipe can be determined using Colebrook equation modified by AGA (American Gas Association) for calculation of friction factor accompanied with Darcy-Weisbach equation for pressure drop and second approach is using Renouard equation adopted for gas pipeline calculation. For the development of Renouard equation for gas pipelines some additional thermodynamic properties are involved in comparisons with Colebrook and Darcy-Weisbach model. These differences will be explained. Both equations, the Colebrook’s (accompanied with Darcy-Weisbach scheme) and Renouard’s will be used for calculation of flow through the pipes of one gas pipeline with eight closed loops which are formed by pipes. Consequently four different cases will be examined because the network is calculated using improved Hardy Cross method and unified node-loop method. Some remarks on optimization in this area of engineering also will be mentioned.

Discussion of “Hardy Cross and the Rise of Numerical Methods”

1972 ◽  
Vol 98 (9) ◽  
pp. 2127-2128
Author(s):  
Linton E. Grinter
Keyword(s):  
Numerical Methods ◽  
Hardy Cross

Hardy Cross and the Rise of Numerical Methods

1972 ◽  
Vol 98 (3) ◽  
pp. 691-694
Author(s):  
James Michalos
Keyword(s):  
Numerical Methods ◽  
Hardy Cross

Closure to “Hardy Cross and the Rise of Numerical Methods”

1973 ◽  
Vol 99 (3) ◽  
pp. 579-580
Author(s):  
James Michalos
Keyword(s):  
Numerical Methods ◽  
Hardy Cross

Aplicação do método de Hardy Cross: estudo, análise e proposta de reestruturação de um sistema de distribuição de água na comunidade de Macundú – Rio Claro/RJ

2013 ◽  
Vol 8 (22) ◽  
pp. 17
Author(s):  
Élcio Nogueira ◽  
José Damião Justino
Keyword(s):  
Hardy Cross

A análise do equilíbrio hidráulico de sistemas de distribuição de água malhados é baseada nas duas leis fundamentais da mecânica dos fluidos, a da continuidade (Lei de Conservação da Massa) e a da Lei de Conservação da Energia e, além disso, numa relação entre a vazão (ou velocidade) e a perda de carga (ou variação de pressão), estabelecida através das equações de Darcy - Weisbach (fórmula universal da perda de carga) ou Hazen - Williams. Em termos gerais, esta análise requer a resolução de um sistema de equações não lineares, frequentemente com um elevado número de incógnitas, dependendo da dimensão e da complexidade do sistema de distribuição de água em jogo. O primeiro método de solução aproximada para este tipo de problema (correções de vazões operadas individualmente para cada malha) foi apresentado por HARDY CROSS, no ano de 1936, sendo o mais antigo e, provavelmente, aquele que maior divulgação teve até o momento. O método de Hardy Cross é o mais utilizado dentre os métodos de aproximações sucessivas para o cálculo de rede malhadas, por possibilitar o desenvolvimento dos cálculos, em sistemas simples, além de ser um método provido de significado físico, que facilita a análise dos resultados intermediários obtidos. No presente trabalho, para o dimensionamento do sistema de distribuição de água, em malhas, de MACUNDÚ, foi aplicado o método de HARDY CROSS através do programa computacional REDEM.EXE desenvolvido pela Escola de Engenharia de São Carlos (EESC-USP, São Carlos, SP), que permite o dimensionamento ou verificação de uma rede de distribuição com até cem (100) trechos com um ou mais reservatórios de alimentação. O programa aceita como equações de resistência a equação de Hazen-Williams, ou a fórmula universal, ambas foram aplicadas, para efeito de comparação e análise. O sistema de distribuição considera-se resolvido quando são conhecidas as cotas piezométricas nos nós e as vazões em todas as tubagens, com o rigor pretendido. Os resultados obtidos estão apresentados em formas de gráficos e tabelas, e mostram-se consistentes com as condições físicas impostas para determinação das vazões individuais de cada trecho da malha analisada. O trabalho apresenta uma metodologia de execução exequível e refinada para sistemas de malhas de médio e grande porte.

scholarly journals An Efficient Iterative Method for Looped Pipe Network Hydraulics Free of Flow-Corrections

Fluids ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 73 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Iterative Method ◽  
District Heating ◽  
Distribution Networks ◽  
Initial Result ◽  
Hot Water ◽  
Heating Systems ◽  
Loop Method ◽  
Hardy Cross ◽  
Flow Through ◽  
Previous Iteration

The original and improved versions of the Hardy Cross iterative method with related modifications are today widely used for the calculation of fluid flow through conduits in loop-like distribution networks of pipes with known node fluid consumptions. Fluid in these networks is usually natural gas for distribution in municipalities, water in waterworks or hot water in district heating systems, air in ventilation systems in buildings and mines, etc. Since the resistances in these networks depend on flow, the problem is not linear like in electrical circuits, and an iterative procedure must be used. In both versions of the Hardy Cross method, in the original and in the improved one, the initial result of calculations in the iteration procedure is not flow, but rather a correction of flow. Unfortunately, these corrections should be added to or subtracted from flow calculated in the previous iteration according to complicated algebraic rules. Unlike the Hardy Cross method, which requires complicated formulas for flow corrections, the new Node-loop method does not need these corrections, as flow is computed directly. This is the main advantage of the new Node-loop method, as the number of iterations is the same as in the modified Hardy Cross method. Consequently, a complex algebraic scheme for the sign of the flow correction is avoided, while the final results remain accurate.

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.

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.

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