A.1 EXAMPL E: Walk in g Exc it ati on - Pedes tr ian Foo tb ri dg e (Steel Str uc tu re) Source: Example 4.1 from AISC Steel Design Guide Series 11. Purpose: To verify the AISC equation 4.1 for walking excitation using 2D SAP2000 modelling. Description:
I j (comp) w j β (damping)
2388 2388 x10 x10 mm 12.1 kN/m 1%
SAP2000 Modeling: File Name: Exp 1 Pedestrian Bridge
Figure 1.1 Extruded view of I Beam 6
4
I j = 2388 x10 mm (I.e. IW530x66 x 6.82 prop modifier)
Modeling Input:
Figures 1.2 and 1.3 illustrate the beam line information. Figure 1.2 shows the b eam section, and the moment of inertia property modifier used to account for c omposite stiffness. Figure 1.3 shows the line load information. Table 1.1 summarizes the first three modal frequencies of the system. Figure 1.2 Line Information – Assignments
Figure 1.3 Line Information – Loads
Table 1.1 Modal Frequencies Mode Frequency (Hz) 1 6.51 2 23.3 3 45.2
Figure 1.4 Time History Sine Function Definition After performing the modal analysis, the first fundamental period and frequency are extracted to form the Sine loading function definition (i.e. walking excitation function). Figure 1.4 illustrates the function input. Five cycles of continuous excitation is chosen with 20 steps per cycle. T he amplitude is calculated as per the numerator of equation 4.1 from AISC guideline 11, and assuming a unit point load P. st
1 Period: 1/ 6.51 sec 0.41 x e
-
x .
x1000 N
Figure 1.5 Linear Modal History Figure 1.5 summarizes the linear modal history. A unit point load P is applied at the point of maximum deflection corresponding to the fundamental mode. Load P is then combined in resonance with the Sine Function defined in Figure 1.4 to simulate walking excitation. Periodic motion type is chosen to observe the steady state behaviour. Modal damping is also defined here. The number of output time steps is the product of no. of cycles x no. of steps/cycle from Figure 1.4. Output time step size is the quotient of period divided by no. of steps/cycle from Figure 1.4. No. steps/ cycle x No. of cycles Modeling Output:
Period/ No. steps per cycle
Figure 1.6 Plot Function Traces In SAP, the plot function traces can be extracted from the menu bar: Display/ Show Plot Functions. Define plot function as joint disp/ forces and select UZ acceleration. In Etabs, the equivalent plot function traces are under the menu: Display/ Show Time History Traces.
Results: Table 1.2 summarizes the modal response and acceleration from the AISC Publication and SAP model.
Table 1.2 System Modal Response and Accelerations Po Period Frequency Joist ∆ j
AISC SAP2000
s 0.147 0.154
Hz 6.81 6.50
mm 6.84 7.43
kN 0.41 0.41
apeak 2 m/s 0.0294 0.0282
apeak %g 3.0 2.9
Discussion: Maximum accelerations are similar in both cases. The higher midspan deflection in the SAP2000 model may be attributed to the contribution of shear deflection. As a result, a lower frequency contributes to a lower stiffness. Overall, the AISC method and SAP2000 modeling give good comparative results.
A.2 EXAMPL E: Walk in g Exc it ati on – Off ic e Flo or (Steel Str uc tu re) Source: Riverside South, Canary Wharf, London England. Purpose: To demonstrate walking excitation anal ysis using 3D Etabs Finite Element modeling (concrete on metal deck application). Description:
Riverside South consists of a three building office complex situated in Canary Wharf. RS1 is a 40+ storey office tower with concrete on metal deck slab construction, reinforced concrete core walls, and column free floor area between the façade and core wall. The floor plate is approximately 75m long x 45m wide, while the core is 43m x 17m. The North and South ‘tongues’ of the typical trading floor are under review for floor vibrations. Table 2.1 summarizes the typical trading floor lo ading - strength design and vibrations probable loads. The probable percentage of design load is project specific, based on judgement, and by no means is the norm. It is important not to overestimate the actual/ probable loads that will be imposed. In general, an increase in mass will decrease the floor frequency and reduce the floor accelerations. Table 2.1 Typical Trading Floor Loading Strength Vibrations Design Load Probable Load Live load 6 kPa 0.25 KPa SDL 1.2 KPa 0.6 KPa Cladding 1.5 KPa 0.75 KPa Slab SW 2.04 KPa 2.04 KPa Steel SW in Etabs
% of Design Load 4% 50% 50% 100% 100%
Etabs Modeling: File Name: Exp 2 RS1 Vibrations Walking
Figure 2.1 illustrates the Etabs floor plate model. Refer to Appendix A for a description of the column connection, beam to beam fixity, beam to core wall fixity, and other modeling criteria. The deflected shape shown is that of the st fundamental mode (1 mode), which also corresponds to the floor region of interest here.
Figure 2.1 3D view of Etabs floor plate model.
Modeling Input:
Figure 2.2 summaries the slab material properties. The slab composition is 80 mm of lightweight concrete on 60 mm of composite steel deck and is represented as shell elements. The shell element thickness, concrete type, and stiffness modifier are shown. The slab thickness refers to the depth of slab above the flute; the flute stiffness is accounted for i n the moment of inertia modifier. The s lab self weight is defined as a separate load case. Table 2.1 summarizes the first three modal frequencies of the system. Figure 2.2 Slab Material Properties
Table 2.1 Modal Frequencies Mode Frequency (Hz) 1 3.17 2 3.53 3 3.97
Figure 2.3 Time History Sine Function Definition After performing the modal analysis, the mode shape of interest (location specific) is defined. Its period and frequency are extracted to form the Sine loading function definition (i.e. walking excitation function). Figure 2.3 illustrates the function input. Five cycles of continuous excitation is chosen. The amplitude is calculated as per the numerator of equation 4.1 from AISC guideline 11, and assuming a unit point load P. st
1 Period: 1/ 3.17 sec 0.29 x e
-
x .
x1000 N
Figure 2.4 Time History A unit point load P is applied at the point of maximum deflection corresponding to the mode representing the area of interest (see Figure 2.1). Load P is then combined in resonance with the Sine Function f rom Figure 2.3 to simulate walking excitation. Periodic motion type is chosen to vie w the steady state behaviour that is of interest. Modal damping is also defined here. Number of output time steps is the product of no. of cycles x no. of steps/cycle from Figure 2.3. Output time step size is the quotient of period divided by no. of steps/cycle from Figure 2.3.
Modeling Output:
Figure 2.5 Time History Function Traces In Etabs, the plot time history function traces can be extracted from: Display menu/ Show Time History Traces. Define plot function as joint disp/ forces and select UZ acceleration.
Results: Table 2.2 summarizes the modal response and accelerations. The load corresponds to the magnitude of P as per equation 4.1 from AISC guideline 11. T he acceleration is displa yed as a percentage of gravity for convention.
Table 2.2 System Modal Response and Accelerations Mode 1
Period s 0.315
Frequency Hz 3.17
Load N 95.6
apeak 2 m/s 0.059
apeak %g 0.60
Discussion: The first modal frequency of this floor is 3.17 Hz, which is greater than the 3 Hz required to prevent rogue jumping (AISC guideline). Generally, one should avoid having a frequency lower than 3 Hz, unless the floor beam/ truss is spanning a large distance and is inducing a large portion of the floor plate. For the latter case, there may be a large enough amount of mass induced in the vibration to make the issue obsolete.
The peak walking acceleration is 0.6% g. Using linear extrapolation from Figure 2.1 of AISC guideline 11, the maximum acceptable level of peak acceleration is 0.6% g; therefore, the region of the floor plate in question is acceptable for vibrations.
A.4 EXAMPL E: Walk in g Ex ci tati on – Flat Slab (Conc ret e Stru ct ur e) Source: “Hospital Floor Vibration Study: Comparison of Possible Hospital Floor Structures With Respect To NHS Vibration Criteria”. The Concrete Centre. Purpose: To assess the accuracy of SAP2000 at predicting the natural floor frequency of flat slabs, and assess floor vibrations due to walking excitation. Description: A fixed floor layout of 75mx15m floor plate made up of 7.5x7.5m bays has been considered. For the reinforced concrete flat slab (I.e. no drop panel) scenario, three sl ab thicknesses were analyzed - 300 mm, 330 mm, and 350 mm.
Figure 4.1 Floor plate dimensions of a typical hospital floor setting.
Table 4.1 Loading Summary and Floor Properties Strength Vibrations Design Load Probable Load Live load 4.0 kPa Services and Finishes 1.0 KPa Partition 1.0 KPa Vibration Mass Considered 1.0 kPa Slab SW Steel SW β (damping) Column Size f’c
% of Design Load
variable in SAP2000
16% of above 100% 100%
3.0% 300x300 mm square 32 MPa for slab; 40 MPa for column
Calculation o f Cracked Concrete Stiff ness: The vibration performance of reinforced concrete slabs is affected significantly by the stiffness assumed. The cracked concrete stiffness depends on several fac tors such as reinforcement ratio, construction load, and environmental exposure. A comprehensive side study was undertaken using SAFE (FE analysis and design for slabs). In general, 50% crack stiffness best represented the slab deflections (see Appendix B). Combined with a dynamic loading factor of 1.2 for reinforced concrete slabs used for vibrations analysis, the slab stiffness should be reduced to 60% (I.e. 50% x 1.2).
SAP2000 Modeling: File: Flat Slab
Figure 4.1 illustrates the SAP2000 floor plate model. Although the source article suggested that the edge of slab to be restraint by the façade, the author has taken the conservative approach to leave the edge of slab free to rotate and deflect.
Figure 4.1 3D view of SAP2000 floor plate model.
Modeling Input:
Figures 4.2 and 4.3 summarize the slab material properties for the 330 mm slab thickness example. Note the 60% slab stiffness reduction in the stiffness modification table. Table 4.2 summarizes the first three modal frequencies of the system. Figure 4.2 Shell Properties and Stiffness Modification Factors
Table 4.2 Modal Frequencies Mode Frequency (Hz) 1 7.27 2 7.30 3 7.46
Figure 4.2 Slab Material Properties
Results: The analysis procedure is similar to that of Example 2. The results are summarized in Table 4.3 below.
Table 4.3 System Modal Response and Accelerations Slab Thickness Journal SAP2000 (mm) Frequency apeak apeak Frequency 2 (Hz) m/s %g (Hz) 300 5.24 Hz 0.0197 0.20 6.70 Hz 330 7.43 Hz 0.0095 0.10 7.27 Hz 350 8.97 Hz 0.0066 0.07 7.64 Hz
apeak 2 m/s 0.0200 0.0100 0.0093
apeak %g 0.20 0.10 0.10
Discussion: The modeling procedure of 60% cracked and f ree edge condition yielded reasonable results compared to the journal findings. Although not summarized in Table 4.3, pinning the edge of the slab produced similar frequencies; however, the accelerations were much lower and erroneous for this case.
APPENDIX: Fin it e Elemen t Mod eli ng Pro ced ur es an d Tec hn iq ues Modeling Criterion
Modeling Procedure/ Techniq ue
Concrete on metal deck
Model as continuous shell elements, with Inertia stiffness modified to account for greater concrete depth in the flute direction.
Steel beam to beam connection
Model as per strength design. Beam to column connections can be assumed rigid.
Beam elevation
Shift beam down from slab centerline using insertion point option under Assign menu/ ‘Frame/Cable/Tendon’.
Reinforced concrete slab
Model as continuous shell elements. Assume 50% cracked properties (m11, m22, m12 = 0.5).
Column support
Model columns with rigid connections to the slab, and pinned at the infections points above and below the slab (assumed at column mid height).
Cladding support at slab edge
Slab edge is assumed free to deflect and rotate.
Core wall connection
Beams and slabs connected to core wall are assumed rigid.
Note: SAP2000 is preferred over Etabs for vibration analysis. SAP2000 has a more rigorous FE engine suitable for sensitive floor deflections, but Etabs is still acceptable.
