Elastic foundation

Hetenyi, M. (1958) Beams on Elastic Foundations (University of Michigan Press). 

Biot MA (1937) Bending of an infinite beam on an elastic foundation. J Appl Mech 4 A-1... 

Features of the two-layer subpad response can be captured by considering the subpad as a plate on an elastic foundation. The governing equation for the deflection of a plate of bending stiffness D on a foundation of compression stiffness k is [S]... 

Compliance versus crack length for wood-wood TDCB specimen (Fig. 8). Exp., experi- FEA, finite element analysis TBEF, tapered beam on elastic foundation model. 

Figure 12 Modeling of TDCB specimen on an elastic foundation (after Ref. 13). 

Note These calculations are based on a beam on elastic-foundation methods. 

This issue was addressed in 1979 by McBride and Jacobs. Jacobs was from Fluor in Houston. The principle was to calculate stresses in two distinct areas, membrane and bending. Membrane stresses are based on pressure area times metal urea. Bending is based on AISC beam formulas. The neck-and-shell section (and sometimes the flange as well) is assumed as bent on the hard axis. This is not a beam-on-elastic-foundation calculation. It is more of a brute-force approach. 

Beam on elastic foundation methods where the elasHc foundation is the vessel shell. 

For local loads where a partial ring stiffener is to be used to reduce local stresses, the beam on elastic foundation method provides an ideal method for sizing the partial rings or stiffener plates. The stresses in the shell must then be analyzed by another local load procedure. Shell stresses can be checked by the beam-on-elastic-foundation method for continuous radial loads about the entire circumference of a vessel shell or ring. 

Figure 5-8. Dimensions and loading diagram for beam on elastic foundation anaiysis.

This procedure is based on the beam-on-elastic-foundation theory. The elastic foundation is the vessel shell and the beam is the partial ring stiffener. The stiffener must be designed to be stiff enough to transmit the load(s) uniformly over its full length. The flexibility of the vessel shell is taken into account. The length of the vessel must be at least 4.9v Rt to qualify for the infinitely long beam theory. 

Failure of a laminate under compressive load acting in the fibre direction is likely to involve microscopic buckling of individual fibres, which act as cylindrical beam-columns on an elastic foundation provided by the surrounding matrix. Compressive failure of glass FRP laminates is strongly influenced by imperfections, including ... 

Fig. 1. Beam on an elastic foundation (a) continuous beam shown on discrete elastic springs ...

The solution to the governing differential equation for the beam on elastic foundation problem yields the deflection of the beam of the form ... 

The characteristic beam on elastic foundation stress distribution, as shown in Fig. 2 for the applied moment case, is frequently encountered in a wide range of adhesive bond situations. In addition to lap joints mentioned earlier, peel tests, moisture-induced stresses, and curvature mismatch situations all tend to exhibit this characteristic distribution. Clearly, the beam on elastic foundation has important qualitative and, in many cases, quantitative applicability to a host of adhesively bonded joints. 

M. Hetenyi, Beams on Elastic Foundation Theory with Applications in the Fields of Civil and Mechanical Engineering, UnivCTsity of Michigan Press, Ann Arbor, 1946. 

Stress distribution bond thickness. Stress distribution Poisson s ratio. Stress distribution stress singularities. Stress distribution beam on elastic foundation and Stress distribution mode of failure are related articles. 

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