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I'm interested in learning what I might call "advanced statics" but am not sure what concepts this would include in engineering parlance.

Would it actually be called "matrix structural analysis"?

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The field of statics is based on analyzing finitely many (solid) members and discrete forces acting in equilibrium. It is a purely algebraic method of analysis (no calculus is necessary). To move to topics more advanced than this while still assuming equilibrium, you have to start eroding away at the "discreteness" assumption. In other words, you have you start considering infinitesimal contributions summed over an area or a volume (i.e. you really need vector calculus to do this for generalized 3d problems). Continuum Solid Mechanics, particularly steady state linear elasticity, is pretty much the next step up from statics if you're analyzing solids in equilibrium.

However, there is somewhat of a middle ground between the discrete nature of statics and continuum solid mechanics. The Finite Element Method makes additional assumptions to interpolate a continuous problem into discretely many pieces and resolves the degrees of freedom as a system of equations. But I would consider it more as a method to solve partial differential equations than an "advanced form of statics", because FEM is more general and applies not only to solid mechanics, but also fluid mechanics, heat transfer, electromagentics, and pretty much anything that can be described by a PDE.

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