By Alexander N. Papusha

Introducing a brand new functional procedure in the box of utilized mechanics constructed to unravel beam energy and bending difficulties utilizing classical beam concept and beam modeling, this striking new quantity deals the engineer, scientist, or pupil a progressive new method of subsea pipeline layout. Integrating use of the Mathematica software into those versions and designs, the engineer can make the most of this special approach to construct more advantageous, extra effective and not more expensive subsea pipelines, a crucial part of the world's strength infrastructure.

Significant advances were completed in implementation of the utilized beam concept in a number of engineering layout applied sciences over the past few many years, and the implementation of this thought additionally takes an incredible position in the functional sector of re-qualification and reassessment for onshore and offshore pipeline engineering. A basic technique of utilizing beam conception into the layout approach of subsea pipelines has been constructed and already included into the ISO guidance for reliability-based restrict country layout of pipelines. This paintings is based on those major advances.

The goal of the e-book is to supply the speculation, learn, and useful functions that may be used for academic reasons via body of workers operating in offshore pipeline integrity and engineering scholars. a must have for the veteran engineer and pupil alike, this quantity is a crucial new development within the strength undefined, a powerful hyperlink within the chain of the world's strength production.

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Extra info for Beam Theory for Subsea Pipelines: Analysis and Practical Applications

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I=1 Then let us formulate ODE boundary problem to be solved through unknown reactions as the following. d2 y = M(x); dx2 y(0) = 0; y(L) = 0. 2) The boundary conditions below will be used later for finding the reactions of constrains. y (0) = 0; y (L) = 0. 3) Beam in Classical Evaluations 15 Finally, components of the system of equations for finding the unknown reactions ties. 5)). 4) i=1 y (0) = 0; y (L) = 0. We have four equations to find four unknown reactions. 2 The equations of beam equilibrium Application of general method described in the previous section to the beam (Fig.

2 General Euler – Bernoulli method: Traditional approach The Euler – Bernoulli equation for the bending of the beam describes the relation between the beam’s deflection and the applied load. This equation is written in general form as the following (E. A. Witmer (1991–1992)). d2 dx2 YJ d2 y(x) dx2 = −q(x). 1) one has to add a boundary conditions, so the problem has to be solved in symbolic or numeric form or to being solved numerical. Diverse appropriated boundary conditions for various kinds constrains of beam will be considered later.

D2 y = M(x). 6) Y J0 — stiffness of beam; d2 y — curvature of the beam deflection; dx2 M(x) — bending moment in x. Moment distribution in the cross section evaluated along a beam through unknown reactions MA and VA in code: M(x ) = MA + VAx − qx2 /2; So, the equation of beam bending is written as the following (see Gere, J. M. and Timoshenko, S. , (1997)), p. 135). eq3 = Y J0 D[y[x], {x, 2}] == M(x) J0Y y (x) = MA + xVA − qx2 2 Symbolic solution regarding the beam deflection is presented as below sol = DSolve[{eq3, y[L] == 0, y[0] == 0}, y[x], x] //Flatten//Simplify y(x) → x(L − x) −4VA(L + x) − 12MA + q L2 + Lx + x2 24J0Y contained unknown reactions VA and MA.

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