# Wave equation analysis

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Wave equation analysis is a numerical method of analysis for the behavior of driven foundation piles. It predicts the pile capacity versus blow count relationship (bearing graph) and pile driving stress. The model mathematically represents the pile driving hammer and all its accessories (ram, cap, and cap block), as well as the pile, as a series of lumped masses and springs in a one-dimensional analysis. The soil response for each pile segment is modeled as viscoelastic-plastic. The method was first developed in the 1950s by E.A. Smith of the Raymond Pile Driving Company.

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.

A deep foundation is a type of foundation that transfers building loads to the earth farther down from the surface than a shallow foundation does to a subsurface layer or a range of depths.

A pile driver is a device used to drive piles (poles) into soil to provide foundation support for buildings or other structures. The term is also used in reference to members of the construction crew that work with pile-driving rigs.

Wave equation analysis of piles has seen many improvements since the 1950s such as including a thermodynamic diesel hammer model and residual stress. Commercial software packages (such as AllWave-PDP and GRLWEAP) are now available to perform the analysis.

One of the principal uses of this method is the performance of a driveability analysis to select the parameters for safe pile installation, including recommendations on cushion stiffness, hammer stroke and other driving system parameters that optimize blow counts and pile stresses during pile driving. For example, when a soft or hard layer causes excessive stresses or unacceptable blow counts.

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Geotechnical engineering is the branch of civil engineering concerned with the engineering behavior of earth materials. Geotechnical engineering is important in civil engineering, but also has applications in military, mining, petroleum and other engineering disciplines that are concerned with construction occurring on the surface or within the ground. Geotechnical engineering uses principles of soil mechanics and rock mechanics to investigate subsurface conditions and materials; determine the relevant physical/mechanical and chemical properties of these materials; evaluate stability of natural slopes and man-made soil deposits; assess risks posed by site conditions; design earthworks and structure foundations; and monitor site conditions, earthwork and foundation construction.

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Effective stress is a force that keeps a collection of particles rigid. Usually this applies to sand, soil, or gravel.

A pile is a slender element cast in the ground or driven into it. Since pile construction as well as the final product are mostly invisible, engineers have often questioned their integrity, i.e. their compliance with project drawings and specifications. In fact, experience has shown that in piles of all kinds flaws may occur. The purpose of integrity testing is to discover such flaws before they can cause any damage.

High strain dynamic testing is a method of testing deep foundations to obtain information about their capacity and integrity, and in some cases, to monitor their installation. It is codified by ASTM D4945-12 - Standard Test Method for High-Strain Dynamic Testing of Piles.

The Statnamic load test is a type of test for assessing the load carrying capacity of deep foundations which is faster and less expensive than the static load test. The Statnamic test was conceived in 1985, with the first prototype tests carried out in 1988 through collaboration between Berminghammer Foundation Equipment of Canada and TNO Building Research of the Netherlands. Guidance on rapid load pile testing can be found in: Methods for Axial Compressive Force Pulse (Rapid) Testing of Deep Foundations. Sanken D7383 - 08 Standard Test.

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The Franki piling system is a method used to drive expanded base cast-in-situ concrete (Franki) piles. It was developed by Belgian Engineer Edgard Frankignoul in 1909 and has achieved considerable worldwide success since.

The Kansa method is a computer method used to solve partial differential equations. Partial differential equations are mathematical models of things like stresses in a car's body, air flow around a wing, the shock wave in front of a supersonic airplane, quantum mechanical model of an atom, ocean waves, socio-economic models, digital image processing etc. The computer takes the known quantities such as pressure, temperature, air velocity, stress, and then uses the laws of physics to figure out what the rest of the quantities should be like a puzzle being fit together. Then, for example, the stresses in various parts of a car can be determined when that car hits a bump at 70 miles per hour.

## References

• Smith, E.A.L. (1960) Pile-Driving Analysis by the Wave Equation. Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers. Vol. 86, No. EM 4, August.