Harmonic Analysis (FRF) on a pedestrian bridge with Robot Structural Analysis according to SETRA reg

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We will redo with Robot Structural Analysis the example page n°51, of the methodological guide for pedestrian bridges of SETRA (CEREMA), evaluation of the vibratory behavior under the action of pedestrians.

We find a mixed structure that we will study for class III (Class 3).

Thus we find the following hypotheses:

g= 9.80665 m/s²
Deck inertia = 0.0292 m4
Linear mass of the deck = 1456 kg/m
Young's modulus of steel = 2.1E+11 N/m²
useful width = 2.5 m
carried = 38.85 m
Mass of a pedestrian Go= 70 Kg (70 to 80kg)
Depreciation ksi= 0.6% (Mixed)

We will do the calculation in Class 3 (III)

density d= 0.5 pedestrians/m²
Number of pedestrians Np = 48.6 pedestrians
Total mass of pedestrians = 3399 kg
Linear mass of pedestrians = 87.5 kg/m
Linear mass for pedestrians per m² ro_s= 1544 kg/m

Thus the Analytical calculation gives:


1st mode: (with a little more detail from Excel)

High mode 1 frequency fh1= 2.1358 Hz
Low frequency mode 1 fb1= 2.0744 Hz
2nd mode: (with a little more detail from Excel)

High frequency mode2 fh2= 8.5432 Hz
Low frequency mode 2 fb2= 8.2975 Hz
Psi factor, vertically, we have a level between 1.7 and 2.1, and for class III, we can neglect the second harmonics.


ksi= 0.6% Mixed
10.8*(ksi/Np) 1/2 = 0.12
psi = 1 for fb1 = 2.08 Hz less than 2.1 Hz

As in the document, we will only analyze the Vertical load (v), for fb1= 2.08 Hz:

Surface load f(t)= d*(280*1)*Neq*y*cos(2p*fb1 xt)
Surface load = 16.81 cos(2p 2.07 xt ) N/m²
Linear load = 42.02 cos(2p 2.07 xt) N/m
Linear load = 4.28 cos(2p 2.07 xt) Kg/m
f(t)=4.28* sin(2 3.141592654*2.074367017 xt)
We then find analytically under Excel, as in the guide, the following acceleration:

Acc_max_z (fb1= 2.08 Hz)= 2.888 m/s²
We will use Robot to perform this calculation with a harmonic analysis in a first article. Then, with a temporal analysis for the example, because with a single harmony/frequency, a harmonic analysis is easier to perform in a second post of this blog.

1- Static Analysis

Case 1: Linear mass of the deck = 1456 kg/m or:

or PZ=-14.278482400[kN/m]


Case 2: Operation with 2.5m of deck x 70 Kg/m² = 175 Kg/ml, i.e.

PZ=-1.716163750[kN/m]


2- Modal Analyses

We will do 2 modal analyses, a first definition of the modal analysis with only the Self Weight (PP) following:


And a second with 50% of the pedestrian load conventie in mass:


Thus, we add to the dynamic mass with a coefficient of 1.00 the case1 of self-weight, and only to the modal analysis “Modal d05 III” n°4, 50% of the case2 of exploitation:



We find the previously requested effort:
f(t)=4.28* sin(2 3.141592654*2.074367017 xt)

Thus we create the analysis on a frequency domain following the “modal d05 III” case with the following parameters, by checking “Take into account the natural frequencies of the given interval”, thus we can enter from 2.07 to 2.09 Hz.


Note: we can also do a simple harmonic analysis but then we must put the 12 digits after the decimal point of the natural frequency of the gateway, and always the damping of 0.006, or 0.6% (Mixed):


We then create the load requesting 4.28 kg/ml or 0.04202 kN/ml on case 6 of the harmonic analysis on frequency domain:




We launch the calculations under Robot, then we go to “ Results/Advanced/Frequency response functions (FRF) – tables ”


Under AZ in m/s², for the case “FRF d05 III”, at the center nodes No. 22, we have AZ=2.889 m/s²

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Simulate harmonic excitation using Harmonic (one excitation frequency required) or FRF (series of excitation frequencies due to different machinery speed regarded) dynamic analysis. To define Harmonic or FRF analysis follow the steps.

Follow steps below:
Go to menu Analysis - Analysis type
Click on New to create new load case.
Select the Modal analysis then OK.
Define modal analysis parameters (particularly number of normal modes considered while dynamic analysis).
In the dialog Analysis type click New button again.
Select Harmonic or FRF analysis then OK.
Define the excitation frequency (serice fo frequencies for FRF analysis) and damping settings
OK
Select defined Harmonic or FRF from Load case filter
Open Loads - Load definition
Define loads regarded as dynamic load amplitudes while Harmonic ar FRF analysis.

Note
Non zero loads should be defined in harmonic or FRF load cases to obtain non zero results