taivumakäyrän

Taivumakäyrän (deflection curve) is the curve describing the lateral displacement of a structural element under loading. It is the shape traced by an elastic beam, plate, or arch as it deforms from its original geometry when subjected to forces. In Finnish engineering literature, taivumakäyrä is used to analyze serviceability and strength.

In beam theory, the vertical deflection w(x) of a uniform, prismatic beam is a function of the position x along the length. The slope is θ(x) = dw/dx and the curvature κ(x) ≈ d^2 w/dx^2 for small deflections. The relation between bending moment M(x) and the deflection is M(x) = EI κ(x) ≈ EI d^2 w/dx^2, where E is Young's modulus and I is the second moment of area. For a beam with distributed load q(x), the fourth derivative of deflection relates to load: E I d^4 w/dx^4 = q(x). Boundary conditions determine the shape of the taivumakäyrä.

Applications include estimating maximum deflection, ensuring serviceability limits, and informing design. The taivumakäyrä is often obtained analytically for simple loading or numerically via finite element analysis. In practice it provides a visual and quantitative measure of how design members deform under expected loads.

See also: deflection, bending moment, Euler-Bernoulli beam theory, finite element method.