STRUCTURAL RESPONSE CALCULATION

Newmark-Beta Method for Dynamic Response Calculation

Principle

The Newmark-beta method is a numerical integration method used to solve dynamic equations, especially for time history analysis in structural dynamics. The basic idea is to iteratively solve for the system's response by applying predictor-corrector formulas for acceleration, velocity, and displacement at each time step.

Dynamic Equation

For a multi-degree-of-freedom (MDOF) system, the equation of motion can be written in matrix form as:

$$M\ddot{u}(t)+C\dot{u}(t)+Ku(t)=F(t)$$

where: - $M$ is the mass matrix - $C$ is the damping matrix - $K$ is the stiffness matrix - $u(t)$ is the displacement vector - $\dot{u}(t)$ is the velocity vector - $\ddot{u}(t)$ is the acceleration vector - $F(t)$ is the external force vector

Newmark-beta Method

The Newmark-beta method uses the following formulas to update the displacement and velocity at each time step:

1. Velocity update formula:

$${\dot{u}}_{n+1}={\dot{u}}_n+(1-\gamma )\mathrm{\Delta }t{\ddot{u}}_n+\gamma \mathrm{\Delta }t{\ddot{u}}_{n+1}$$

1. Displacement update formula:

$$u_{n+1}=u_n+\mathrm{\Delta }t{\dot{u}}_n+(\frac{1}{2}-\beta )\mathrm{\Delta }{t}^2{\ddot{u}}_n+\beta \mathrm{\Delta }{t}^2{\ddot{u}}_{n+1}$$

where: - $\mathrm{\Delta }t$ is the time step size - $\beta$ and $\gamma$ are Newmark parameters, typically $\beta =0.25$ and $\gamma =0.5$, which correspond to the average acceleration method

Procedure

1. Initial Conditions: Given initial displacement $u_0$ and initial velocity ${\dot{u}}_0$, compute the initial acceleration ${\ddot{u}}_0$.

2. Effective Stiffness Matrix:

$$K_{\text{eff}}=K+\frac{\gamma }{\beta \mathrm{\Delta }t}C+\frac{1}{\beta \mathrm{\Delta }{t}^2}M$$

1. Time Stepping:

2. Compute the effective force:

$$F_{\text{eff}}=F_{n+1}+M(\frac{1}{\beta \mathrm{\Delta }{t}^2}{u}_n+\frac{1}{\beta \mathrm{\Delta }t}{\dot{u}}_n+(\frac{1}{2\beta }-1){\ddot{u}}_n)+C(\frac{\gamma }{\beta \mathrm{\Delta }t}{u}_n+(\frac{\gamma }{\beta }-1){\dot{u}}_n+\mathrm{\Delta }t(\frac{\gamma }{2\beta }-1){\ddot{u}}_n)$$

• Solve for the new displacement:

$$u_{n+1}=K_{\text{eff}}^{-1}{F}_{\text{eff}}$$

• Compute the new acceleration:

$${\ddot{u}}_{n+1}=\frac{1}{\beta \mathrm{\Delta }{t}^2}(u_{n+1}-u_n)-\frac{1}{\beta \mathrm{\Delta }t}{\dot{u}}_n-(\frac{1}{2\beta }-1){\ddot{u}}_n$$

• Compute the new velocity:

$${\dot{u}}_{n+1}={\dot{u}}_n+\mathrm{\Delta }t((1-\gamma ){\ddot{u}}_n+\gamma {\ddot{u}}_{n+1})$$