VERIFICATION EXAMPLE 1 - Footbridge – Footbridge – Footfall Footfall Analysis REFERENCE : Example taken from A Design Guide for Footfall Induced Vibration of Structures, M.R.Wilford, P.Young, The Concrete Society 2006, page 42.
SPECIFICATION: In this problem will be analysed a concrete footbridge, having constant cross-section and two continuous equal spans of 20 m, as shown in the figure below.
Assumed cross-section cross-section properties: properties:
Material:
E= 38 GPa, density = 2400 kg/m3
RSA Calculation Model The 2-D beam members, each divided into 8 finite elements. Pinned supports.
SOLUTION:
2
area Ax=0.77 Ax=0.77 m 4 second moment of inertia Iy=0.056m mass per unit length m=1848kg/m
Definition of the Footfall Analysis case in RSA 2010. Generally there are two excitation methods: Self excitation analyzes the response in the same node to which the force is applied, Full excitation analyzes the response in any node, to the effect of the force applied to another node. All the parameters are set by default for Concrete Center excitation force method: -
Walking frequency range from 1 to 2.8 Hz.
-
Number of footsteps – a parameter that is important only for the resonant response (harmonic) analysis; a lower number of footsteps reduces a possibility of resonance
-
Walker’s weight - a parameter that is important only for the resonant response (harmonic) analysis; for the transient analysis, the excitation force is an impulse from the force of 746 N
Damping is assumed as 1.5% of critical, so define 0.015 as constant damping for all modes.
Press Modal analysis parameters button in the main definition dialog window. Footfall analysis solves modal eigenvalue problem first and this dialog box is for settings the modal analysis parameters. The number of calculated eigenmodes is limited to the frequency limit defined for FootFall analysis. For this example change the Frequency limit to 20 Hz, in order to obtain the first three modes.
By default, the active mass direction is Z only, for vertical vibrations. Like for normal Modal analysis case, the Footfall analysis case can ignore the self-mass of the model elements, when Ignore density is checked. User can also convert any static load case to masses. Footfall analysis generates lumped with rotation mass matrix and is solved with subspace iteration algorithm. Close the windows by pressing OK buttons. Run calculations.
A. Modal analysis – natural frequencies
Open results table from menu Results\Advanced\Modal analysis. Mode Theoretical (Hz) f1 4,22 f2 6,59 f3 16,90 Good agreement of results.
RSA 2010 (Hz) 4,212 6,579 16,814
The response factor for Footfall analysis is calculated using 2 different types of analysis: resonant response, which occurs for low frequency structures – the walking excitation can induce harmonic resonant with the structure natural modes of vibration. transient response , which occurs for high frequency structures – harmonic resonant cannot occur, so the footfall response is induced by a single dynamic impulse with the maximum frequency from the walking frequency range. The criterion of selecting an analysis type is the first, basic frequency of structure natural vibrations and the maximum footfall frequency. It is assumed that the resonant response can be induced up to fourth multiplication of the walking frequency. When the usual maximum walking frequency is about 2.5 Hz, then the limit for resonance possibility is about 10 Hz of the basic natural frequency.
As the sample structure basic natural mode is 4.21 Hz, then it is obvious that the resonant response will have stronger influence than the transient response.
B. Footfall analysis Results of the analysis include only results in nodes. One of the most important results of the analysis is a Response factor (Rf) which specifies how many times calculated vibrations exceed the vibration perceptible to a human. Open results table from menu Results\Advanced\Footfall analysis- tables.
The table shows for each node the maximum Response factor in the frequency domain. There are always calculated response factors for resonant and transient algorithm. Rf-overall is the greater value from the resonant and transient response factors. Frequency column shows for which frequency the overall response factor is obtained.
We can see that the maximum Response factor is obtained for nodes 7 and 14, which are actually in the middle of the spans. The maximum Response factor is obtained for walking frequency equal 2,11 Hz. More exactly it is 2,106 Hz which second harmonics (multiplication) gives the natural frequency 4.212. Hz of the second mode. The simple check according to the reference guide, for walking at 2.106 Hz (second harmonic), gives: Dynamic force Fh = 686 [N] * (0.069+0.0056*2*2.106) = 63.5 N
(refer table 4.3)
Resonant acceleration for mode 2 of modal mass 33322 kg, at node 7 with normalized eigenvector 0,97 value, is given by: a = 63.5*0.97*0.97 / (2*0.015*33322) = 0.0598 m/s2
(see 4.1)
Response factor Rf = 0.0598 / 0.0071 = 8,42
(see 4.7)
This gives good agreement with exact FEM solution.
For further analysis we can examine detailed diagrams of Footfall analysis results. Open dialog box from menu Results\Advanced\Footfall analysis- diagram. Footfall analysis diagrams display values of different results in walking frequency domain. This is particularly interested for resonant analysis results.
For Transient analysis diagrams the maximum values will be always obtained for the end of walking frequency range. Interesting can be diagram of Velocity in Time domain after impulse load.
Let’s examine diagram of Resonant response factor in Frequency domain for node 7 excited with the load at node 7. The diagrams can be plotted for any selection of resultant and excitation nodes, even if the analysis was performed with self-excitation method. The plots are calculated on-line.
The diagram shows the response factor due to each harmonics and the total, Overall Response factor. It is shown that the maximum response factor is obtained at walking frequency 2.11 Hz, mainly by the second harmonics. BS 5400 provides guidance on the acceptable levels for footbridges. For external bridges the response factor should be R<64 for single person excitation at most critical footfall rate. Thus the footbridge vibration are satisfactory according to the code.
