histats.com - statistiche web

Problems solving 1D: Equivalent Liner Analysis

A column of elastic material (single-lithology) resting on a rigid base has natural periods of vibration that depend on the mode of oscillation. The theoretical value for natural frequency of oscillation (f) is given by:

where:

  • H = column height of soil (for this verification H=100.0 m);
  • Vs = average velocity of shear waves
Layer Thickness Wave Vs
1 2.0 141
2 2.0 224
3 2.0 267
4 7.5 292
5 2.0 355
6 9.5 406
7 55.0 485
8 20.0 546

The calculated Natural Frequency is 1.063Hz equal to 0.94s of Period.

 

Geometry and Material Properties

The model is a column with dimension 100.0x10.0 meters and it make of thirty (30) elements and sixty-two (62) nodes. The elements have varying heights depending on the thickness of the layer (Height Min = 1.0m, Height Max = 10.0m).

Material Properties

Layer Density Wave Vs Shear Modulus Poisson Ratio Damping G/G0 - D%
1 19710 N/m2 141 m/s 2.796E7 N/m2 0.45 5.0 % Soil 1
2 19710 N/m2 224 m/s 7.057E7 N/m2 0.45 5.0 % Soil 1
3 19710 N/m2 267 m/s 1.003E8 N/m2 0.45 5.0 % Soil 1
4 18460 N/m2 292 m/s 1.123E8 N/m2 0.45 5.0 % Soil 2
5 18460 N/m2 355 m/s 1.660E8 N/m2 0.45 5.0 % Soil 2
6 18060 N/m2 406 m/s 2.124E8 N/m2 0.45 5.0 % Soil 3
7 19000 N/m2 485 m/s 3.189E8 N/m2 0.45 5.0 % Soil 4
8 20000 N/m2 567 m/s 4.254E8 N/m2 0.45 5.0 % Soil 4
    • Curve Soil 1
    • Curve Soil 2
    • Curve Soil 3
    • Curve Soil 4

 

Boudary Conditions

The column have a behavior in shear and the vertical motion (Y) is inhibited to eliminate bending modes. The loading is applied to the base (XY). You apply the load horizontal component only in the nodes with X-Direction locked:

  • Node 1, 1 --> Fix XY-Direction
  • Node 1, 2 --> Fix XY-Direction

Input Motion

The horizontal acceleration is Kobe FN SSE where:

  • Records = 8000
  • Time = 44.0 sec
  • Time step = 0.005 sec
  • PGA = 0.942 g

Results

To the first iteration Visual-Q4M calculate the fundamental Frequency of the model and it's show the value in run window.

The minimal Angular Frequency = 6.651 [s^(-1)]

The calculated Natural Frequency fN = 1.059 [Hz]

The value of natural frequency change if it's modified the stiffness of the model. During the calculation cycles (Max Run: 20) the value of the strain is calculated and for each element calculated the corresponding G/G0 and D% on the variation curve.

The figures show the horizontal transfer function at the top of the column (node 2,31) with a predominant frequency of 0.623 Hz.