Dynamics of Structures

Chapter 5. Nonlinear Dynamic Response of SDOF

Chapter 5
5.1 Nonlinear SDOF Model5.2 Newmark - β Method 5.3 Elastoplastic Behavior

5.2 Formulation of the Newmark-β Method

Newmark-β Method has time step methods in its formaultion similar to other numerical methods. It introduces two parameters, β and γ\gamma . For numerical stability of the solution, both β and γ\gamma should have certain values. γ\gamma is always taken as 1/2, while β can be taken as 1/4 or 1/6. If β is taken as 1/4, the method is called the Constant Acceleration Method and if β is taken as 1/6, the method is called the Linear Acceleration Method.

The incremental equations of the Newmark - β Method can be written as,

Δu˙i=u¨iΔt+γΔu¨iΔt \Delta \dot{u}_i = \ddot{u}_i \Delta t + \gamma \Delta \ddot{u}_i \Delta t

Eq. 5.2.1

Δui=u˙iΔt+12u¨i(Δt)2+βΔu¨i(Δt)2 \Delta u_i = \dot{u}_i \Delta t + \frac{1}{2} \ddot{u}_i (\Delta t)^2 + \beta \Delta \ddot{u}_i (\Delta t)^2

Eq. 5.2.2

After setting γ\gamma to 1/2,

Δu¨i=1β(Δt)2Δui−1βΔtu˙i−12βu¨i \Delta \ddot{u}_i = \frac{1}{\beta (\Delta t)^2} \Delta u_i - \frac{1}{\beta \Delta t} \dot{u}_i - \frac{1}{2 \beta} \ddot{u}_i

Eq. 5.2.3

Δu˙i=12βΔtΔui−12βu˙i−(1−14β)Δtu¨i \Delta \dot{u}_i = \frac{1}{2 \beta \Delta t} \Delta u_i - \frac{1}{2 \beta} \dot{u}_i - (1 - \frac{1}{4 \beta} ) \Delta t \ddot{u}_i

Eq. 5.2.4

Newmark - β Triangular Load Calculator

m = 0.2 kip.s²/in

k = 10 kip/in

ζ = 0.05

Fₒ = 20 kips

t₁ = 0.02 sec

td = 0.06 sec

Time t = 1.5 sec

Output
Maximum Displacement0.00 in
Maximum Velocity0.00 in/s
Maximum Acceleration0.00 in/s2