Vibration Testing and Analysis of a Monumental Stair

Proceedings of the IMAC XXXII A Conference and Exposition on Structural Dynamics Rosen Plaza Hotel, Orlando, Florida February 3-6, 2014

Vibration Testing and Analysis of A Monumental Stair Mehdi Setareh, Ph.D., P.E., and Xiaoyao Wang Virginia Tech Blacksburg, Virginia 24061 ABSTRACT Excessive and annoying vibrations in buildings, stadiums, and footbridges have become more common in the past two decades due to several reasons including the tendency to optimize the use of building materials, use of higher strength and lighter structural properties, etc. Stair vibrations have also become an important design issue mainly due to the architects’ desire to create more innovative, slender, light, and flexible monumental structures. Large vibrations and movements of stairs can become serious safety and serviceability problems, as they have the potential to disrupt the evacuation during an emergency and also people may feel unsafe during the normal use. Even though over the past two decades a large number of research studies on the floor vibrations have been conducted, very little information is available for stairs. This paper presents experimental and analytical studies of a large monumental stair. Using an electrodynamic shaker and a series of seismic accelerometers, a set of modal tests was performed on the structure. This was done to estimate the dynamic properties of the stair. A computer model of the structure was created using the common practices used in structural engineering design offices.

Comparison of the measured stair responses and the results of computer analysis showed reasonable agreement. A Cumulative Modal Assurance Criterion (CMAC) was introduced, and was used to identify the degrees of freedom with low quality measured responses. It was found that CMAC performs better than Enhanced Coordinate Modal Assurance Criterion (ECOMAC) and Coordinate Modal Assurance Criterion (COMAC) when the response is due to the excitation of a single mode.

Several walking and running tests were conducted on the structure, which showed the structure may not be susceptible to large levels of vibrations for everyday use.

Keywords: Staircase; Vibration Serviceability; Human Vibrations; Cumulative Modal Assurance Criterion (CMAC); Coordinate Modal Assurance Criterion (COMAC); Enhanced Coordinate Modal Assurance Criterion (ECOMAC).

Introduction In recent years, a trend in the architecture community to design long-span monumental stairs has started. These structures are generally slender, light, and flexible, which result in low natural frequencies. They are often susceptible to excessive vibrations that can become annoying to people. Serviceability problems as related to excessive vibrations of stairs have been studied by researchers since the 1970s.

Nilsson [1] measured the magnitude of the footfall force of fifteen people descending a stair at 2 steps/sec (Hz) and found that it was up to four times the individuals’ body weights. Alcock and Lander [2] measured the average forcing function for a single footfall on a stair and concluded that descending at a normal rate generates forces up to two times the body weight.

Bishop, et al. [3;4] measured the forcing functions on stairs, and studied the effects of group loading, in addition to the acceptable vibration levels for stairs. They found that the presence of people could increase the damping ratio with negligible effects on the natural frequency. Kerr and Bishop [5;6] studied the difference between the floor and stair loadings due to people movements. With the help of twenty-five human subjects, an instrumented platform, and a staircase, they conducted various tests. They concluded that the most comfortable paces for ascent were near 2 Hz, and below 2.3 Hz for descents. When running, step frequencies of about 3.3 Hz for ascents and above 3.3 Hz for descents were the most comfortable speeds. Bishop, et al.

[3;4] studied stair vibration levels that can be acceptable to humans by introducing a non-dimensional factor, R, to be applied to the acceptable r.m.s. of acceleration based on the limits recommended by BS6472 [7]. They recommended R=32 for the light-use stairs (for example in office buildings), R=24 for heavy-use stairs (for example in public buildings), and R=64 for very lightly-used stairs and for cases where group effects are expected. These recommended limits are based on the vibrations

resulting from one person ascending or descending a staircase. However, the authors have not provided much detail on the basis and origin of these recommendations.

Kerr [8] used two stair mock-ups in the laboratory and conducted a number of tests with the help of twenty-five human subjects.

He concluded that ascending at 2 Hz was most comfortable for walking and 3.3 Hz for running up the stair. When ascending the stair, the subjects’ maximum speed was about 4.5 Hz. The subjects commented that they could easily descend at any speed below 4 Hz. The maximum step frequency for descents was about 5.5 Hz. Kim, et al. [9] used six steel and cast-in-place concrete stair mock-ups with relatively high natural frequencies, and found that, in general, the cast-in-place concrete stairs had better vibration serviceability performance than their steel structure counterparts. They also reported that the normal walking speed on the stairs was about 1.8 Hz. Davis and Murray [10] studied the vibration performance of a monumental stair using measurements and computer modeling. They measured the first mode natural frequency and damping ratio to be 7.3 Hz and 1.1% respectively. They recommended an acceptable peak acceleration of 1.7% g for walking by an individual, and 4.6% g for when a person runs or for a group walk. These recommendations were based on Bishop, et al. [3;4], and the authors did not independently verify them.

Arbitrio [11] presented a study of the vibration analysis of a stair using the SAP2000 structural analysis software. He didn’t conduct any vibration measurements on the structure. Huntington and Mooney [12] also conducted a Finite Element Analysis of a 39 ft long monumental stair made of steel. Without providing many details, they suggested the application of tuned mass dampers (TMD) to reduce the vibrations. Similar studies were conducted by Howes and Gordon [13] on two sets of stairs at the Art Gallery of Ontario, Toronto, Canada, which were designed by the renowned architect, Frank Gehry. They also installed a TMD in the landing area to reduce the vibrations; however, they did not provide any information on the level of stair vibration or the effectiveness of the TMD. Howes, et al. [14] studied the structural performance of a stair connecting three levels of a store in Las Vegas, Nevada. Their study was also limited to an analytical investigation using the SAP2000 structural analysis software. Eid, et al. [15] conducted a simplified vibration analysis of a stair located at a Canadian University. Even though, they measured a damping ratio of 4% based on the results of heel drop tests on a similar stair, they adopted a value of 1% for their computer analysis.

Belver, et al. [16] conducted a vibration analysis and testing of a steel staircase located at a British University, which had excessive vibrations when people ascended or descended. In addition to the structural vibration analysis using the ABAQUS computer software, they conducted an Operational Modal Analysis (OMA) of the ambient response of the stair. The measured first mode natural frequency was at 6.3 Hz with damping ratios between 0.4% and 0.6%.

Kasperski and Czwikla [17] studied the effects of stair geometry on step frequency and dynamic load factor. They found that the slowest step frequency range was about 0.8-1.9 Hz when skipping a step during ascents. From the results of their tests they concluded that the normal walks occur at 1.2-1.5 Hz when ascending and 1.6-3.6 Hz for descents. However, the reported range by the authors for ascents seems to be low and the upper limit for descents to be high.

Cappellini, et al. [18] conducted an Experimental Modal Analysis (EMA) of a stair using a shaker and a series of accelerometers to measure the dynamic properties of the structure. They also conducted an OMA of the same staircase. From these measurements, they estimated the first two mode natural frequencies and damping ratios to be: f1= 4.7 Hz, ξ1=0.4%; and f2=8.8 Hz, ξ2=0.3%. The computed natural frequencies from their Finite Element Model were greatly overestimated (f1=7.7 Hz and f2=13.7 Hz).

From the presented brief summary of the literature available on the vibration serviceability of stairs, it is clear that there are still many issues related to stair vibrations that require further studies. Almost all structural designs of monumental stairs are carried out by structural engineers using various Finite Element software without conducting any field testing to verify their designs. Structural engineers generally use simple models considering frame element for linear elements (beams, stringers, etc.) and shell elements for surface elements (decking, in-fill steel plates, etc.). As indicated above, there are very few studies available on the comparison of such analytical models with measured data and appropriateness of these models for predicting stair vibrations. Therefore, this paper presents the results of the modal testing conducted on a monumental stair after installation and before the addition of non-structural elements, and an analytical study of the structure using a structural analysis software.

The measured and computed dynamic properties of the structure are compared to each other. A Cumulative Modal Assurance Criterion (CMAC) is introduced and used to identify the degrees of freedom with low quality measured response of the structure. A brief evaluation of the measured vibration when an individual or group of people ascended or descended the stair at different speeds is presented.

Description of the Staircase The staircase, which has a steel structure, is located at the Broad Art Museum in Lansing, Michigan. It connects the first and second levels of the building. Details of the initial static design of the structure and subsequent modifications to meet the dynamic requirements have been discussed by Setareh [19].

Since this paper presents the results of the dynamic analysis and testing of the structure after its installation and before the addition of the non-structural components such as cladding, plywood, cover, etc., only the structural elements of the stair will be described here. The staircase consists of stringers forming a truss with HSS12X3X5/16 for the top and bottom chords and HSS3X3X5/16 verticals at 1.22 m (4 ft) on center. The total depth (end to end) of the chords in the landing area is 1.10 m (43.5 in). Treads and risers are made of 6.4 mm (0.25 in) thick steel plates. Steel plates, 6.4 mm(0.25 in) thick, placed between the top and bottom chords of the stringers in the first two web panels from each end of the staircase. The plates are welded to the chords and verticals of the truss. In addition, HSS5X3X5/16 are used as diagonal members in the remaining truss panels. The bottom chords of the stair stringers are supported at their ends by the concrete beams through rigid connections. Figure 1 is a photo showing a partial view of the stair during the vibration tests, which occurred about two weeks after its installation.

Fig. 1 Partial View of the Staircase During the Vibration Tests.

Descriptions of the Dynamic Tests A modal test using an electrodynamic shaker (APS 113) and a number of accelerometers (PCB393C and PCB393B04) were conducted to estimate the dynamic properties of the structure. In addition, a number of walking tests were performed to evaluate the vibration acceptability of the staircase. The shaker was placed on a force plate, which was located at the corner of the stair landing (see Figure 2).

Uniaxial accelerometers were oriented along three perpendicular directions. For vertical direction measurements, 393C accelerometers were placed on their bases, however, for lateral directions they were clamped to angle pieces as shown in Figure 2. The accelerometers were placed along the exterior and interior stringers and at the middle of the steps (all along the truss verticals). Using a roving accelerometer approach, they were relocated from one position to another during the modal testing.

A burst-chirp excitation of 6-19.5 Hz with 30 seconds on and 15 seconds off was selected. A measurement frequency resolution of 0.025 Hz (40 sec duration) was used.

In addition to the modal tests, a number of controlled vibration tests with the help of three human subjects were conducted. An individual and the entire group ascended and descended the stair at various step frequencies. Using a metronome, the movements of the subjects on the stair were synchronized. Two different step frequencies based on the apparent first mode resonance frequency were selected: 146 spm (steps per minute) and 195 spm, representing the fourth and third sub-harmonics of the first mode resonance frequency of 9.8 Hz, respectively. In addition, an average walking step frequency of 120 spm, representing the most common walking speed was used. The subjects were also asked to move the fastest possible in addition to random walks (three persons only), and ascending while skipping a step (one person only). For each test, the subject(s) ascended and descended the stair. At the end of each ascent/descent, the individual(s) stopped motionless while the measurement was completed.

Fig. 2 Placement of the Shaker and Accelerometers During the Modal Tests.

Analytical Modeling of the Staircase SAP2000 [20] structural analysis program was used to create a computer model for dynamic analysis. To develop a computer model, which is consistent with the standard structural engineering design office practice, two noded frame elements were used to model the linear elements such as the stringers’ top and bottom chords, diagonals, and verticals. The stair steps (treads and risers) were approximated by orthotropic shell elements. In addition, the landing area and steel infill plates within the stringers were modeled using isotropic shell elements. Figure 3 shows the analytical model.

Fig. 3 Analytical Model of the Stair Using SAP2000.

Comparison of the Estimated Analytical and Measured Dynamic Properties Using the ME’scope modal analysis software [21], the modal properties of the structure were estimated. Since the building was not completely enclosed during the tests and measurements, there were small variations in the resonance frequencies of the frequency response functions (FRF) due to temperature variations. For this reason, a local polynomial curve-fitting technique was adopted to estimate the natural frequencies and damping ratios. The estimated natural frequencies for the first two modes of vibration were 9.8 Hz and 18.3 Hz, respectively. It is clear that the second mode natural frequency cannot get excited by people walking or running on the stair [19]. Therefore, this study focuses on the evaluation of the first mode dynamic parameters and response of the structure. The estimated first mode damping ratio, ξ1, from the results of the modal analysis was 0.42%.

The structural analysis using the computer software resulted in the first two modes’ natural frequencies of 11.55 Hz, and 21.05 Hz, respectively. They are about 15% more than their corresponding measured values. Even though this discrepancy can be

considered acceptable, in particular, since a simplified model of the structure was used, parametric studies were conducted to estimate the effects of some of the modeling assumptions.

As indicated before, the treads and risers were modeled using the flat shell elements with orthotropic properties. It is clear that the bending stiffness of the step in the transverse (along the width of the stair) is much larger than the bending stiffness of a flat steel plate, due to the folded shape of the deck. Setareh, and Jin [22] conducted a parametric study to check the effects of changing the stiffness of steps in two orthogonal (longitudinal and transverse) directions through the use of bending stiffness property modifiers in the SAP2000 structural analysis software. From this study, it was found that, within a practical range, first mode natural frequency of structure is not sensitive to variations of step stiffness. Therefore, it was concluded that the orthotropic properties of the shell elements do not contribute to the discrepancies between the measured and computed natural frequencies. Thus, the shell bending stiffness in the longitudinal direction was assumed to be equivalent to the stiffness of the 6.4mm (0.25in) steel plate. The bending stiffness of the shell was increased in the transverse direction to incorporate the increase in the moment of inertia due to the folded shape of the steps.

Further studies showed that the folded geometry of the stair can have a greater effect on the natural frequencies of the structure [22]. However, this requires a much more elaborate modeling scheme and incorporation of solid structural elements, which are not usually considered by practicing engineers, as they are more costly.

Comparison of the Measured and Analytical Mode Shapes The computed and measured mode shapes were compared by using the Modal Assurance Criterion (MAC). The MAC matrix   for the first two modes is  0.62 0.07  . It is clear that the MAC value for the second mode is not acceptable and the first  0.006 0.092  mode is also low as the ideal value for MAC is 1.0.

Since there was a concern about the quality and accuracy of the measured modal amplitudes for the lateral degrees of freedom, three methods were used to identify the degrees of freedom that may have resulted in the low MAC value (Note that only the first mode response is of interest in this study since higher modes cannot get excited by people walking or even running on the stair). The first method was to use cumulative (or running) MAC (CMAC) value of various measured degrees of freedom in the structure, defined as: 2

n

 i

CMAC(i, j) 

mj i

 aj (1)

i1 n

n

i1

i1

 i  mj i  *mj  ( i  aj )2

The second and third methods used the ECOMAC and COMAC for each degree of freedom as defined by [23]: L

 ECOMAC(i) 

COMAC(i)  1

i

 mj  i  aj (2)

j1

2L

  L  i  mj i  aj    j1 L

 i

j1

mj i



* mj

2

(3)

L

(  i

aj

)

2

l1

where i  mj and i  aj are the measured and analytical modal amplitudes for mode j of the ith degree of freedom, respectively, n is the total measured degrees of freedom, * represents complex-conjugate, and L is the total number of modes considered.

Since the considered modes at 9.8 Hz and 18.3 Hz are well separated and only the first mode is of interest, the CMAC, ECOMAC, and COMAC for the first mode are computed here. Figure 4 shows the CMAC, ECOMAC and COMAC for all 129 measured degrees of freedom (d.o.f.) during the modal tests. To better understand the effects of inaccuracies in the lateral measurement, the d.o.f.s are sorted such that the first group (43 d.o.f.s) represents the z-measurements (vertical) followed by x and y measured d.o.f.s, respectively. From the results shown, it is apparent that the CMAC can best identify the d.o.f.s that may result in large decreases in the MAC value. Even though for the most part, the ECOMAC and CMAC show a similar pattern, for a few d.o.f.s, the ECOMAC indicate reductions in the MAC value, which are not substantiated by the CMAC value. Figure

4 also shows that the COMAC cannot accurately predict the d.o.f.s that may have low correlations between the analysis and measurement and result in a drop in the MAC value. This is consistent with the conclusion reached by [23] and [24].

Fig. 4 Comparison of CMAC, ECOMAC and COMAC for the First Mode of the Structure.

Figure 5 shows the CMAC variations after the identified d.o.f.s that resulted in large drops in the MAC value have been   removed from the measurements. The resulting MAC matrix is  0.84 0.04  . The MAC value for the first mode seems to  0.002 0.32  be acceptable for civil structures. As indicated before, the second mode of vibration is not of interest to this study.

Fig. 5 Computed First Mode CMAC Value of the Structure After the Removal of the Low Quality Measurements.

It is interesting to note that the second mode MAC value improves greatly if all the measured lateral d.o.f.s are removed. The   resulting MAC is  0.99 0.33  . However, this result in spatial-aliasing as the off-diagonal elements are large, which is  0.10 0.78    substantiated by the analytical Auto-MAC of  1.00 0.33  . Figure 6 shows the measured and analytical first mode shapes  0.33 1.00  of the structure.

(a) Measured

(b) Analytical

Fig. 6 Measured and Analytical First Mode Shapes of the Structure.

Comparison of the Analytical and Measured Responses To check the level of accuracy of the analytical model to predict the structural response of the stair, the SAP2000 model was subjected to the same dynamic excitation during the field measurements and the responses were compared. To achieve this, a steady-state analysis of the structure was conducted by applying a harmonic excitation with a unit amplitude at the corner of the landing. The excitation frequency was varied in order to compute the frequency response functions (FRF) of any desired point on the stair structure. The measured first mode damping ratio was used to compute the responses of the analytical model.

Figure 7 compares the largest measured and computed FRFs in the landing area. As can be noted, the responses are different by about 23%, which can be considered within the acceptable range. It has to be noted that further refinement of the analytical model, in particular inclusion of the folded geometry of the stair, may result in improvements in the estimation of the dynamic properties of the structure and accuracy in prediction of its response subjected to dynamic loads (such as people’s footfalls).

Evaluating the Stair Vibrations The measured vibrations with human subjects ascending or descending the stair were used to evaluate the design of the stair for everyday use. The vibration limits as recommended by [4] were adopted. It has to be noted that the construction of the stair was not complete at the time of the tests as the non-structural elements were not yet installed.

Fig. 7 Comparison of the Measured and Analytical FRFs.

From the results of the measurements, it was found that when one person ascended or descended the stair rapidly at 146 spm, or 195 spm, the maximum stair vibration could exceed the limit. However, when one or three people ascended or descended the stair at the normal walking speed of 120 spm, only one location on the stair reached a level more than the limit. The vibrations due to random walks were all within the acceptable range. The largest vibrations were found when three people descended the stairs at 195 spm or the fastest they could. Vibrations up to three times the acceptable limits were recorded.

However, such speeds of motion represent running down a stair, which has a low probability of occurrence especially involving a group of people at a museum (except in the case of an emergency). From the analysis of the vibrations measured in the field, it was concluded that at its condition during the tests, when a person ascended or descended the stair at a slow to normal rangeof-speed, the movements were within the acceptable range.

Conclusions This paper presented a study of vibrations of a monumental stair due to human movements. It considered various field testing and computer modeling of the structure. Using the standard structural engineering practice for modeling such structures, a computer model of the structure was created. Comparison of the results of the dynamic tests and computer analysis showed that the computer model could reasonably predict the dynamic behavior of the structure.

A parametric study of the variations in the modeling assumptions showed that improvements in the structural response can be achieved by refining the computer model. To identify the degrees of freedom with low quality of collected data, a CMAC value was defined and computed. Comparing the value of CMAC with ECOMAC and COMAC for the first mode showed its superiority in identifying the problem degrees of freedom over the ECOMAC and COMAC. The measured and analytical FRFs for the location with the largest stair response were compared and found to be close within a practically acceptable range.

From the analysis of the collected responses from a number of tests when human subjects ascended and descended the stair, it was concluded that the stair vibrations (at the time of the tests) were within the acceptable range as long as people walked at or below the normal range of speed.

Acknowledgements This study was supported by the National Science Foundation under Grant No. CMMI-1335004. This support is gratefully acknowledged. Any opinions, findings, and conclusions expressed in this paper are those of the writers and do not necessarily reflect the views of the National Science Foundation. The authors would like to acknowledge the assistance of Mr. Paul Dannels and Ms. Elisabeth Wang from Structural Design, Inc., Ann Arbor, Michigan, during the various stages of this study. Ms. Emma Jin contributed to part of the presented study.

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