BoucWen

The Bouc-Wen model is a phenomenological mathematical representation used to describe nonlinear hysteresis in structures and materials under cyclic loading. It was introduced by Bouc in 1967 and further developed with Wen in 1971. The model captures the characteristic loop behavior observed in force–displacement or moment–rotation responses of steel frames, soils, dampers, and other components.

In its common form, the Bouc-Wen model separates the restoring force into an elastic part and a hysteretic part. Let x denote displacement and F the restoring force. The total restoring force is often written as F = k x + α z, where k is the initial stiffness and α scales the hysteretic component z. The internal variable z evolves according to a first-order differential equation that depends on the loading rate and the current state:

dz/dt = (1/η) [dx/dt − β |dx/dt| z − γ dx/dt |z|]

Here η is a rate parameter, while β and γ are dimensionless shape parameters that control the curvature and pinching of the hysteresis loop. Different parameter values produce a wide range of loop shapes, including softening, pinching, and degraded stiffness.

Variants of the Bouc-Wen model may involve multiple internal variables to represent more complex degradation or to better fit experimental data. The model is widely used in nonlinear dynamic analysis and finite element implementations because it provides a simple yet flexible way to reproduce hysteretic behavior without detailed material microstructure.

Limitations include its phenomenological nature and potential parameter identification challenges. Nevertheless, it remains a standard tool for simulating energy dissipation and strength deterioration in cyclic loading scenarios.