Mastering the Monumental Stair

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Studio Museum in Harlem, New York, which was completed this year and will open in 2025, has more than doubled its space for exhibitions and programming with a new iconic building that will soon become a cultural anchor for the community. The building was designed by architects Adjaye Associates and Cooper Robertson. Simpson Gumpertz & Heger Associates, Inc. P.C. (SGH) was the Engineer of Record, and they collaborated with consulting engineer Guy Nordenson and Associates (GNA). GNA started the structural design for the building during the initial phases of the project, and they completed the construction documents for the monumental stair. Rowan Williams Davies & Irwin Inc. (RWDI) served as the vibration consultant for the stair, and Plan B Engineering (Plan B) was the specialty structural construction engineer for the contractor, Sciame Construction. The six-story building has a steel superstructure and is supported by a concrete mat foundation. The facade is clad with precast panels that evoke the masonry architecture of Harlem (Fig. 1).

The new building features a four-story monumental steel-framed stair clad with terrazzo panels. As an architectural focal point of the museum, the stair allows visitors to view artwork along an 85 foot-tall vertical art gallery. (Paintings and sculptures are supported on the walls enclosing the stair). At the center of the stair, balustrade trusses, concealed within the architectural finishes, form a helical column (a central spine) that extends from the top of the stair to the mat foundation (Fig. 2). Large platforms with additional balustrade trusses cantilever off the central spine of the stair to create gathering spaces for visitors as they view works of art or watch events in the lecture hall below (Fig. 3). However, these visitors would both cause and feel vibrations from the stair. To mitigate the vibrations, a series of Tuned Mass Dampers (TMDs) are incorporated inside the balustrades between truss members.

Cladding the stair with terrazzo panels provides a contextual material and color that harmonizes with the distinctive exterior precast panels of the building (Fig. 4). To support the terrazzo, the architects designed light-gauge framing members that span from the treads or landings to the top of the balustrades. This back-up wall is sheathed with plywood, to which the terrazzo panels are adhered. The architects showed narrow, grouted joints between precast panels. As the joint sizes between terrazzo cladding panels are greatly affected by truss deflections and vibration characteristics, in-situ load testing of the steel stair was required to validate the expected results from structural modeling and to determine final joint sizes and jointing materials.

To determine the behavior of the complex stair, GNA modeled the entire stair, including the central spine, mat foundation, truss balustrades, in-plane bracing, and concrete-on-metal deck slabs at the landings using finite element analysis (FEA) software, SAP2000. SGH performed an independent review. The model represents steel members as frame elements and concrete landings as thin shell elements. Because of the large, cantilevered platforms and the proposed terrazzo architectural finish, it was critical to understand and manage vertical deflections under dead and live loads (Fig. 5).

The model incorporated adjacent base-building slab bays at the stair support points to accurately capture the floor stiffness and mass that can be activated by pedestrian footfall. The engineers assumed viscous modal damping at one percent of critical damping (typical for steel monumental stairs), applied walking forces as a percentage of body weight at corresponding modal frequencies, and determined acceleration responses. These procedures were iterated several times with different member sizes until an optimal design was found that met recommended peak acceleration limits, as further described here.

The central spine and balustrade trusses have significant strength and stiffness compared to typical long-span trusses. However, the four-story height and the long-cantilevered landings create vibration characteristics similar to a slender monumental stair. Because the American Institute of Steel Construction (AISC) Design Guideline 11: Vibrations of Steel-Framed Structural System Due to Human Activity does not provide tolerance criteria for stairs, SGH and GNA used the analysis procedure described by Davis and Murray (2009). In their paper, they performed a literature review, an experimental study, and a finite element analysis of a slender stair structure, considering ascent and descent harmonic forces that are often much larger than those on a flat surface. Their method consists of calculating the response acceleration that would be generated by a single individual or a group of individuals walking on the stair. The calculated accelerations are then compared to tolerance limits for human comfort of people standing on the stair.

Walking footfall rates that produce vibrations of floor structures typically vary between 1.0 Hz and 2.5 Hz (or footfalls per second). However, people commonly ascend and descend stairs at frequencies that are higher than the frequencies of walking on a flat surface. The forcing frequencies can be as high as 4.0 Hz to 4.5 Hz, as in the case of a person “trotting” down a stair. In their procedure, Davis and Murray considered the maximum frequency for one person at 4.0 Hz. Therefore, the second, third, and fourth harmonics of the walking frequencies are 8.0 Hz, 12.0 Hz and 16.0 Hz. The stair can be excited by the forces in all these harmonics, though to lesser degrees as their contributing mass decreases.

The walker’s location (excitation) and the affected occupant’s location (response) must be identified using engineering judgement. Typically, the worst accelerations are obtained when both the excitation point and the response point are taken at the maximum mode shape. The number of steps required to develop resonant response is highly variable, and AISC Design Guideline 11 recommends considering 7 to 8 consecutive steps (or jumps). Taking this into account, the walker’s position should be as close as possible to the maximum mode shape. The affected occupant must be stationary, so the guide recommends that the affected occupant location should be at an intermediate landing or at mid-span as close as possible to the maximum mode shape (i.e. maximum likely amplitude).

The analysis indicated that the maximum mode shapes typically corresponded to the far corners of each landing. The affected occupant location was chosen at those points for further study. Five landings were initially identified as sensitive locations based on the modal shapes from the model, and for every landing, four walker locations were analyzed:

The initial results showed that individuals walking up and down the steps would not pose vibration concerns for the stringers. Nevertheless, the long-cantilevered landings are susceptible to peak accelerations that could cause discomfort for people standing on them, when others rapidly descend stairs, especially because of the low inherent damping in the steel structure.

To minimize vibrations, a series of custom tuned mass dampers (TMD) with viscous dampening devices were designed to be integrated into the stair at five locations with risk of high accelerations. RWDI, the vibration consultant, performed an assessment of the dynamic response due to pedestrian traffic and detailed the custom-shaped dampers so that they could be seamlessly incorporated into the design and concealed in the balustrade. The effective dynamic mass of each tuned mass damper is around 450 pounds, and the total mass (including bearing, springs and viscous damping devices, etc.) is about 600 pounds. RWDI designed a total of five tuned mass dampers with frequencies tuned to target stair natural vibration modes 2, 3, 4, 7, and 12 (between 5-8Hz), which are most susceptible to human-induced vibrations.

Plan B assisted the design team in specifying the terrazzo panel joint size, with particular attention to the calibration of joint widths with the local deflections of the complex stair structure under design dead and live loads.

The initial review involved a geometric study of the relationship between the steel truss nodes and the terrazzo joint spacing (5 feet on center truss nodes versus 14 inches on center joint spacing +/-). The stair framing included several cantilevered conditions that were identified as critical regions for local deflection analysis. Vertical and lateral nodal deflections of these trusses were tabulated from FEA models provided by SGH. The relative nodal deflections between adjacent nodes were calculated and further processed to determine the relative movement of one terrazzo panel to the next. This enabled an understanding of the maximum local movement that panel joints could accommodate without closing the gap and risking damage to the panels. The output from the modeling also included pattern loading to ensure that no isolated live-load conditions governed the joint movement.

The theoretical deflection shown in the FEA model from one cantilevered end of the stair framing to the opposite end was predominantly vertical. A review of the relative horizontal movement between truss node points of the same chord indicated a value that was a nominal displacement relative to the vertical—indicating a low amount of chord lateral movement. A further comparison between top and bottom chords indicated little differential in lateral movement, suggesting a low amount of relative geometric rotation. This relative movement between truss panel nodes should be divided by the number of terrazzo panels between nodes (60 versus 14 inches) to determine the relative respective terrazzo panel movement.

While there was a concern that axial strain in the truss chords could magnify relative movements between nodes, the FEA analysis proved that this was a non-factor. In fact, all of the vertical movement was a product of the truss flexural stiffness and the rotational stiffness (fixity) of the end supports.

Theoretical FEA results pointed to the suitability of a 1/16-inch gap; however, Plan B recommended a scaled load test on the already erected stairs. Survey points were established to measure vertical deflections of the steel truss framing. Digital strain gauges were also placed across mockup “terrazzo” (plywood) panels. This allowed for both calibration of the FEA model based on the global stiffness and a direct measurement of movement between panels, in-situ. The load test was set to a scaled load of 60% of the full live and superimposed dead loads to optimize the size and scope of the surcharge (understanding that the results of the loading were linearly related to the actual loads). The contractor filled water barrels on site to alleviate the logistics of mobilizing solid weights and located the barrels on the stair framing to simulate the loading associated with the largest computed vertical deflection, based on the FEA analysis. The load test results showed that the system was stiffer than modeled with tested deflections at 75% of the scaled theoretical movement (likely because of local rigidity at truss nodes). Most panel joints closed only a fraction of the joint design width.

The load-test instrumentation used was sensitive enough to capture changes in size due to thermal expansion, correlating with temperature changes, as well as loading/unloading of water barrels. The data also showed daily variations in displacement that matched the frequency of temperature changes throughout the day. Plan B recommended sizes of terrazzo and plywood joints that accounted for the omission of finished grouting.

The Studio Museum in Harlem’s newly unveiled architectural marvel not only marks a significant expansion in space for exhibitions and community engagement but also stands as a testament to visionary design and collaborative engineering prowess. At its core, the building features a robust steel superstructure supported by a concrete mat foundation, providing a stable base for the building. One of the most striking elements is the four-story monumental stair, crafted from structural steel members and adorned with terrazzo panels. With careful attention to detail and innovative solutions, the Studio Museum’s new building will not only serve its community as a cultural hub but will also showcase exemplary architectural and engineering design. Its impact is expected to resonate beyond Harlem, enriching lives, and inspiring future generations. ■

About the Authors

Alexander Stephani, PE, is a Structural Engineer with Simpson Gumpertz & Heger (SGH), specializing in new design, repair and rehabilitation, and flood resiliency design.

Andrew Angelilli is a licensed Professional Engineer at Plan B Engineering, a structural engineering firm specializing in construction means & methods.

Xiaoxiao Wu is a licensed Professional Engineer and Associate Partner at Guy Nordenson and Associates, a New York-based structural engineering practice.

References

Davis, Brad and Murray, Thomas. “Slender Monumental Stair Vibration Serviceability” Journal of Architectural Engineering, American Society of Civil Engineers, 2009.