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Tenth U.S. National Conference on Earthquake Engineering 1210NCE Frontiers of Earthquake Engineering

IMPROVING THE SEISMIC PERFORMANCE OF DIAGRID STEEL STRUCTURES USING FRICTION MASS DAMPER

Garlan Ramadhan1 and André R. Barbosa2

ABSTRACT

The steel diagrid structural system is a recent load bearing and lateral resisting structural system for tall building structures that is relatively unexplored in the western United States. One possible reason for the little use of diagrid systems in earthquake prone regions is the lack of guidelines and application examples illustrating the design and analysis of these structures. In this work, a prototype building with 72 stories is used as an example for which the design and analysis of the diagrid system is performed. Friction mass dampers are provided at the top of the building to mitigate the possible large displacement and base shear demands that these structures may undergo under seismic events. Using a nonlinear finite element model of the mass damper, which is connected to a linear model of the building structure, the effectiveness of the friction mass damper system is studied. The mass damper system consists of an additional concrete tank containing sand or water. The tank is placed in between the building reinforce concrete structural core and the exterior steel diagrid system. This mass damper is connected to the structure using friction pendulum system (FPS) isolators, which are chosen due to their ability to undergo large deformations. Parametric studies are carried out to optimize the mass damper design. Optimization of the seismic performance is assessed in terms of minimization of displacements, base shear, and floor accelerations. The end result provides designers with a first example for development of guidelines for designing and analyzing diagrid structures using FPS mass dampers.

1 Graduate Student, School of Civil and Construction Engineering, Oregon State University, Corvallis, OR 97331 2Assistant Professor, School of Civil and Construction Engineering, Oregon State University, Corvallis, OR 97331

Ramadhan G, Barbosa AR. Improving the Seismic Performance of Diagrid Steel Structures using Friction Mass Damper. Proceedings of the 10th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.

Tenth U.S. National Conference on Earthquake Engineering 1210NCE Frontiers of Earthquake Engineering

IMPROVING THE SEISMIC PERFORMANCE OF DIAGRID STEEL STRUCTURES USING FRICTION MASS DAMPER

Garlan Ramadhan1 and André R. Barbosa2

ABSTRACT

Introduction

Tall buildings appeared in United States of America in the late nineteenth century [1]. Today, many tall buildings have been built around the world. Several structural systems have been developed to realize mankind’s dream in pursuing new heights, and the steel diagrid structural system is but one of them. The diagrid structural system consists of a structural system in which diagonal steel members are placed on the exterior face of buildings while a reinforced concrete core or a structural steel gravity system in placed in the interior. Some prime examples of this kind of structures are the Hearst Tower, in New York City, the China Central Television (CCTV) Headquarters in Beijing, China, and “The Gherkin” in London, UK.

This structural system is known for its redundancy, continuous, and uninterrupted load paths. Diagrid structures are considered very efficient [2], but these efficiencies make the

Ramadhan, G., Barbosa, AR. Improving the Seismic Performance of Diagrid Steel Structures using Friction Mass Damper. Proceedings of the 10th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014. structures lighter and more flexible than conventional tall building systems, thus can suffer large horizontal displacements, under lateral loadings. The diagrid structural system is relatively new and unexplored in the western United States, and hence engineers lack the guidelines and examples that can be used to promote the design of tall buildings using such a structural system.

In this paper an introduction to the design and analysis of diagrid structural systems is presented for a prototype 72-story building. For the seismic design, the focus is placed on optimizing the design by mitigating large displacements and shear forces that may appear in these structures through the use of a tuned mass damper placed at the top of the building. The mass damper is connected to the structure with FPS isolators, which are chosen due to their ability to undergo large deformations. This paper also presents parametric studies that are carried out in order to optimize the mass damper design in improving the seismic performance of the building. In all, this paper provides a first example of a design of diagrid structures in regions prone to seismic loading, using tuned mass dampers.

Diagrid Steel Structures and Options for Improving Seismic Performance

Diagrid is an abbreviation of diagonal grid. The structural system gets its name from the diagonal columns that form triangular trusses connected by horizontal rings (steel beams) that provide support for the floors and column buckling restrains. This system has been shown by other authors to be ideal for use in skyscrapers [2]. The diagonal members carry gravity loads as well as lateral loads, and thus steel is typically used in diagrid structures. The main difference between conventional steel exterior-braced frame trusses (X, K, V, and Chevron type braces) is that in the diagrid structural system almost all vertical columns are typically eliminated.

To improve the seismic performance of the diagrid system, several alternatives have been proposed to date, which included the use of base isolation and viscous dampers. In the first alternative, Arup [3] proposed a diagrid structure combined with a base isolation system as a method for reducing the potential for damage induced by earthquake shaking. In this solution, a 20-story office building was completed in 2006 in Sony City, Japan. The isolation solution was effective since the period of the base isolated building was shifted and the seismic lateral forces applied to the structure were substantially reduced. In a second alternative, Lago et al. [4] proposed a vertically distributed isolation system. In this solution, the diagrid exterior structure was isolated from the main seismic mass of the building interior along the height of the structure. The distributed isolation was achieved by attaching viscoelastic dampers between floor diaphragms and horizontal rings of the diagrid structure. Lago et al. claimed that this system has the potential to significantly reduce the damage to the architectural façades.

The two systems introduced in the previous paragraph are unique and have several advantages, but are not suited for very tall building structures. A base isolation system is only effective for relatively stiff structures since the period of the base isolated structures is typically set in the 2.0 sec to 3.0 sec range [5]. Tall building structures are typically very flexible and often have fundamental periods close to and above 5.0 sec, and therefore base isolation is not effective for very tall buildings. Following similar discussions, Lago et al. [4] also stated that the vertically distributed isolation system would not be effective for tall building structures. Based on purely numerical results, the authors showed that for a 20 story building, the dampers had already experienced a strokes on the order of 0.8 m. Any form of extrapolation to the prototype 72-story building, would translate roughly to the need for dampers with approximately 4 m in length, which is beyond the scope of the proposed solutions.

Herein, a third alternative for mitigating seismic demands is proposed for use in tall buildings consisting of a diagrid structural system, which makes use of tuned mass dampers (TMD). Even though the particular system being proposed is new, the concept has been applied in many skyscrapers built around the world, such as: Taipei 101 in Taipei, Taiwan; One Wall Centre Hotel in Vancouver, Canada; and Shanghai World Financial Center in Shanghai, China. The TMD systems installed in these three buildings are all unique. Taipei 101 featured the heaviest TMD in the world with 660 metric-tons; the One Wall Centre has tuned liquid (water) damping system; and the Shanghai World Financial Center holds a double TMD system. All TMDs have been installed at the top of the buildings and have been shown to successfully mitigate the effects of the lateral loading. The TMD concept in this paper is somewhat similar to the used in the One Wall Centre Hotel [6]. In this newly proposed system, the mass of the TMD is provided by a concrete container with sand or water inside it and the container is supported by friction isolators, which are connected to the building structure. The volume of sand or water can be adjusted base on the optimal friction/mass, which is iteratively obtained from modeling. Further explanation of the concept and its modeling details are provided in the next section.

Methodology Design

The prototype building is designed to be a 72 story office building (Risk Category III [7]) with uniform story height of 4m. The building has 36m×36m floor plan, which are supported by diagonal columns that cross every four floors. With this configuration, the diagonal columns form isosceles triangle with an angle of 69o. This is the optimal configuration for slender diagrid structures greater than 60 stories according to empirical studies carried out by Moon [8]. All beams, except the horizontal rings in floor diaphragms, are designed to carry gravity load only and thus are pin-connected. The horizontal rings connect the diagonal elements, providing both lateral strength and stiffness. Slabs and reinforced concrete cores consist of 4,000 psi (27.6 MPa) and 5,000 psi concrete (34.5 MPa), respectively. Structural steel used in the design is A992Fy50 (Fy = 345 MPa).

The design of dead and live loads are 2.64 kN/m2 and 2.40 kN/m2 respectively. The design for wind load is computed for basic wind speed of 110 mph and exposure 1.0 (ASCE 7- 10). Snow load are assigned as ordinary flat roof snow load based on ground snow load of 0.958 kN/m2. The design seismic load is computed based on ASCE 7-10 response spectrum method with seismic coefficients Sds = 0.911g and Sd1 = 0.529g, and with TL = 6 sec (site in Seattle, Washington).

A two staged design and design verification was performed: (1) In a first stage, the building was designed not considering the effects of the TMDs, as these were not included in the design. The dimensions of the building were estimated using engineering judgment and through an iterative process in which the response spectrum method was used only for prototype building design (without the TMD). For this case, after three design iterations, dimensions shown in Figure 1 were obtained. Results showed that the diagrid steel structure is the main lateral resisting system and it absorbs approximately 80% of seismic load; (2) in a second stage, the FPS TMD system was incorporated to the basic model obtained from the design performed in the first stage. A concrete container was placed at story 68 and extended to story 72 (see Figure 2).

(b) (a) Figure 1. Prototype building drawing: (a) plan view, (b) truncated elevation view; all dimensions are in mm.

(a) (b)

Figure 2. The TMD: (a) plan view at story 69 to 71, (b) cross section 1-1; dimensions in mm.

The placement of the TMD in top area is optimal, since the first mode of the building has contribution to the displacement response while also having an important contribution to the base shear forces. Parametric analyses are performed to optimize the location, mass, and friction of the TMD. According to the numerical work performed herein, for this structural system, the mass damper should be extended for 4 floors which allows for the transfer of the load from the mass damper to the stiffer stories, that is, stories at which the diagonal steel members cross. Design verifications are done by performing nonlinear time-history response analyses of the model subjected to a suite of seven, three-component, ground motion accelerograms. The container also allows for additional required mass that can be provided including sand or water. It is suggested that water be placed in the container since it allows for fire protection also. Systems for water treatment/circulation would have to be designed, but were beyond the scope of this paper. Protective shock absorbers (rubber bearings or an equivalent system) with thickness of 1m are placed between the outer horizontal rings of the diagrid exterior and the TMD and also between the TMD and the inner reinforced-concrete core. The absorbers are placed in stories 69, 70, and 71. The gap is set to allow for a TMD movement of ±500 mm before hitting the absorbers. Absorbers are also placed above the container to prevent impact due to overturning. Lastly, additional truss beams (over two floors) are applied below TMD for support and to provide for the gravity load transfer to the inner reinforced-concrete core.

Modeling

All simulations are done using SAP2000 [9]. Diagonal columns and beams are modeled as linear elastic frame elements with dimensions defined in Figure 1a. The inner reinforced concrete core is modeled using shell elements, with nominal thickness of 450mm, and assuming the Young’s modulus to be 35% of the design value modulus of elasticity of concrete Ec. Floor diaphragms consist of a grid of steel beams with concrete slab (see Figure 1a), and are modeled as rigid diaphragms. For modeling purposes, the concrete container (with its content) is represented by a single wall, which is placed at mid-distance between the outer diagrid and the inner core. The single layer is modeled by shell elements with thickness and mass defined as to represent the container and its contents. It should be noted that sloshing effects of the sand/water surface are neglected as these will not have a considerable effect in seismic loads and performance of the TMD, due to the relatively small distance between walls. The reference mass ratio of the TMD to the main structure is approximately 4.67%. In the reference model, shown in Figure 3, the properties used to define the FPS links provided in SAP2000 are also listed. Friction coefficients in this model are velocity-dependent according to [10] and [11].

Above the container, uniaxial linear springs are provided at the connection between the container and slabs, with stiffness values of 1 kN/mm at global Z axis. To model the behavior of absorbers and gaps surrounding the container, two gap links are placed in parallel to each other. The first gap link corresponds to a gap of 500 mm followed by a linear elastic stiffness branch. The second gap link has a gap of 1500 mm, followed by a large stiffness to model the full compression/crushing of the rubber. The stiffness of first gap links is 1 kN/mm with 500 mm opening (707.1 mm for diagonal/corner links). The stiffness of second gap links is 2449 kN/mm (1725 kN/mm for diagonal links) with 1500 mm opening (2121.3 mm for diagonal links).

Parametric Study

There are 3 variables addressed in the design of the TMD: friction coefficients of the isolators, location, and mass of the TMD. The parameter levels considered were: (A) 3 levels of friction coefficient values for parametric studies are: (i) fslow = 0.01, ffast = 0.02; (ii) fslow = 0.04, ffast = 0.06; (iii) fslow = 0.08, ffast = 0.12. (B) 2 levels for height distribution of the TMD: (i) 4 stories (reference); (ii) 2 stories, (C) Mass of the TMD (3 levels): (i) reference; (ii) increase by 20%; (iii) decrease by 20%.

Seven (7) earthquake motions were used in the analysis of each parameter set: NGA 161, NGA 173, NGA 183, NGA 187, NGA 1045, NGA 1044, and NGA 1605. The 7 earthquake motions are chosen from the 2011 PEER Ground Motion Database [12] using the PEER ground motion selection web-tool, which were selected since they were the ones that produced the smallest root mean square error between the geometric mean of the 5% linear response spectra of two horizontal ground motion components and the ASCE 7-10 target spectrum (see Figure 4). Each of the earthquake motions has three components, which were assigned as follows: fault normal (FN) to the X-direction, fault parallel (FP) to the Y-direction, and vertical to the Z- direction.

Friction Isolator Properties (reference model): Linear analysis properties*: o ke U1 = 3500 kN/mm o ke U2 ,U3 = 0.2 kN/mm o ke R1 = 1130000 kN/mm Nonlinear analysis properties*: o k U1 = 3500 kN/mm o k U2 ,U3 = 5.5 kN/mm o Friction coefficient fslow U2 ,U3 = 0.04 o Friction coefficient ffast U2 ,U3 = 0.06 o rate U2 ,U3 = 0.0429 sec/mm o Radius of sliding surface U2 ,U3 = 6 m * The positive local axis 1 is parallel to the positive global Z axis, the positive local 2 axis is parallel to the positive global X axis and the positive local 3 axis is (a) (b)

Figure 3. Properties of the reference building model in SAP2000: (a) 3D views of the model, and properties of the FPS isolators assigned to the mode, (b) mode shapes. Modal periods and participation ratios for system (c) without TMD and (d) with TMD.

Results

Design Verification

There are two design aspects that are verified for the reference structure containing the TMD, which are (i) demand over capacity (D/C) ratios of the steel diagrid exterior members, and (ii) peak displacements of the TMD unit. For the member design check, the results from the 7 crustal earthquake motions are averaged and incorporated in the design combinations following ASCE 7-10. The design check was performed based on AISC 360-05 [13]. A peak D/C ratio of 0.90 was obtained over all diagrid members. The TMD has limited movement and the absorbers were verified to safely limit the TMD movement.

Comparison with Reference Model

The improvements in terms of seismic response from the model without the TMD unit to the one with the TMD unit can first be examined from the modal parameters and mass participation factors shown in Figure 3. By placing the TMD at the top of the building, the mass participation ratios of the first modes in the X- and Y-direction are decreased by 29.3% and 28.5%, respectively. Minimal seismic performance improvements are expected for any records that excite the building more at the higher modes, while significant improvements are expected for the first mode dominated response.

NGA 161 NGA 173 NGA 183

1 NGA 187 NGA 1045 NGA 1044 NGA 1605 ASCE 7-10 Mode Mode 1 Mode 3 Mode 2 2 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 1 Sa (g) Sa (g) 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 5 10 0 5 10 T (sec) T (sec)

(b) (a) Figure 4. Response spectrum of each tested ground motions: (a) fault normal, (b) fault parallel; straight line legend: (i) without TMD [–]; (ii) reference model (with TMD) [- - -].

Comparative studies are performed for the following observed seismic responses: base shear, floor displacement, inter-story drift, and floor absolute accelerations. Figure 5 shows that the utilization of TMD system provides improvements in the peak base shear response to all ground motions, averaging (over both directions) 17.6% reduction. Significant improvements can be seen for a few earthquakes, with reductions of 30%. These improvements are related to the fact that three earthquake motion records excite mainly the first mode, which is the mode that the TMD is tuned to. On the other hand, poor improvements can be seen for the base shear of NGA 1045 FN, NGA 1044 FN, and NGA 1605 FP, since these earthquakes provide large contributions to higher mode building response. Figure 6 shows the geometric mean (over two directions) of peak inter-story drift, peak floor displacements, and peak absolute floor acceleration responses. The reference structure with TMD has 19.8% smaller displacements (on average over all floors) and 17.5% smaller inter-story drift compared to the structure without TMD unit. However, this TMD system is not very effective in reducing floor accelerations as it only provides 8.1% of reduction (on average over all floors). This is because floor accelerations are usually controlled by higher modes that are not affected by this TMD system design.

-10% 0% 10% 20% 30% 40% -10% 0% 10% 20% 30% 40%

NGA 161 NGA 161

NGA 173 NGA 173

NGA 183 NGA 183

NGA 187 NGA 187

NGA 1045 NGA 1045 Ground Motions Ground NGA 1044 NGA 1044

NGA 1605 NGA 1605

Average Average

Reference Low friction Reference Low friction High Friction Decrease Mass High Friction Decrease Mass Increase Mass Reduce Height Increase Mass Reduce Height

(a) (b)

Figure 5. Base shear improvements from basic diagrid building: (a) fault normal component the ground motions, and (b) fault parallel component of the ground motions

70 70 70 60 60 60 50 50 50

40 40 40 Floor Floor 30 30 Floor 30 20 20 20 10 10 10 0 0 0 0 0.005 0 500 1000 0 0.5 IDR Displacement (mm) Acceleration (g)

(a) (b) (c)

Figure 6. Geometric means of envelopes of responses for: (a) inter-story drift (IDR), (b) floor displacement, (c) absolute floor acceleration. Legend: (i) without TMD [–]; (ii) reference model [- - -]. Friction Coefficient Variation The parameters used in the reference structure with the TMD unit are fslow = 0.04 and ffast = 0.06. To study the effect of changing the friction coefficients of the FPS isolators two new levels of friction coefficients are introduced: low friction (fslow = 0.01, ffast = 0.02); and high friction (fslow = 0.08, ffast = 0.12). By observing the base shears (Figure 5), it can be seen that the structure benefits when increased friction is specified. This conclusion had also been made elsewhere [14]. Improvements can be seen at almost all tested ground motions. Significant reduction in base shear is shown in Figure 5a, in which improvements of approximately 25% are achieved. The friction coefficient of ffast = 0.12 can be produced by the friction of two lubricated hard steel materials [15]. In contrast, lower friction corresponds to the TMD producing smaller friction forces than those required to counteract the seismic forces. Although results are not shown here due to space limitations, it is worth noting that the envelopes of inter-story drift, floor displacement, and peak absolute floor acceleration parameters are not sensitive to friction coefficient changes. By reducing the friction, the observed floor responses only increase by approximately 3%. Also, increasing friction introduces negligible changes in floor responses.

Height Distribution and Mass Variation As stated in the methodology section, ideal configuration for the TMD system needs the mass damper to be extended for four floors for optimal load transfer. Nonetheless, concentration of the same mass over half the height, results in a decrease in base shear forces of approximately 13.7%. Even though this slight improvement is noted, it is worth saying that the 4.67% of tall building weight would have to be condensed in 2 stories only. No significant changes are observed in displacement and acceleration floor responses compared to the reference model that has the TMD ( 3% differences).

Finally,≤ by maintaining the height and friction parameters fixed to the values of the reference model, the mass was varied by two levels. By reducing the mass by 20%, the overall base shears improvements for both directions are approximately 16% compared from the results of 17.6% improvement obtained for the reference model (with a TMD unit). Additionally, increasing the mass by 20% corresponds to improvements of only 19% versus 17.6% of average improvements obtained for the reference model. However, the changes in mass are also not affecting the displacement and acceleration floor response much. The observed floor responses only fluctuated by ±3.3% around the results of the reference structure with the TMD unit. In all, these observations confirm that changes in mass do not affect greatly the results when friction pendulum system isolators are used.

Conclusions

This paper presents an analysis of two 72-story diagrid steel structures, with or without friction tuned mass dampers (TMD). The TMD system consists of a reinforced concrete container with sand or water inside it, which is connected to the main structure by FPS isolators. The TMD unit is placed at the top of the building and provides for significant mitigation of displacement and shear seismic demands when compared to the demands experienced by the building without TMD. Maximum peak base shear reductions of 30% are achieved with the proposed TMD design; overall (average for 7 ground motions, 2 directions, and over all floors) reductions of 17.6% in base shear, 20% in the roof displacement, and 17.5% of inter-story drift ratios are observed. However, the single TMD can only be tuned to one mode, typically the first mode, and therefore multiple TMD systems may be ideal. Furthermore, it is important to highlight that the building location and soil characteristics play a big role in tuning the FPS TMD design.

A parametric study is performed to characterize the response sensitivity and to optimize the settings for the friction TMD system. Friction, mass, and height of the TMD were parameters analyzed. First, with respect to the friction, doubling the friction coefficients of friction isolator provide the best results and further decreased the overall base shears to about 22.0%. Second, increasing the mass tends to slightly reduce the base shears, while reducing the mass reduces the effectiveness of the TMD, but overall change in mass by 20% did not impact greatly results, which suggests that if cost is considered, smaller masses may be warranted. Nonetheless these results highlight the fact that when friction pendulum systems are used, the mass plays a relatively small part in the tuning of the friction mass damper. Lastly, reducing the height of the mass damper reduced the forces, but also the constructability of the TMD unit and future studies should focus on this topic.

This works provides for a reference for design and analysis of diagrid structures and presents an innovative approach towards mitigating the seismic performance of tall buildings using the diagrid structural system. Friction and constructability of the TMD unit seem to be points of future research. For example, triple FPS isolator [14] should be investigated. Other considerations, such as use of multiple TMDs should also be further explored. To improve dissemination of this structural system the seismic performance factors (R, Ω0, Cd) also need to be developed for this system.

References

[1] M. M. Ali and K. S. Moon, “Structural developments in tall buildings: current trends and future prospects.” Architectural Science Review, vol. 50, no. 3, pp. 205–223, 2007. [2] McCain, I. "DiaGrid: Structural efficiency & increasing popularity" 2009, URL: www.dsg.fgg.uni- lj.si/dubaj2009/.../Diagrid%20tehnologija.pdf, Accessed Jan. 2013. [3] Arup, “Sony City, Tokyo: A Diagrid Combined with Base Isolation,” The Arup Journal, V2, pp. 49–51, 2009. [4] A. Lago, T. J. Sullivan, and G. M. Calvi, “A novel seismic design strategy for structures with complex geometry,” Journal of Earthquake Engineering, vol. 14, no. S1, pp. 69–105, 2010. [5] M. C. Constantinou and I. G. Tadjbakhsh, “Optimum characteristics of isolated structures,” ASCE Journal of Structural Engineering, vol. 111, no. 12, pp. 2733–2750, 1985. [6] Glotman Simpson, “Project Showcase - One Wall Centre,” Glotman Simpson Consulting Engineers, 2001. [Online]. http://www.glotmansimpson.com/Commercial/One-Wall-Centre. [Accessed: 03-Nov-2013]. [7] American Society of Civil Engineers, Minimum design loads for buildings and other structures. Virginia: ASCE, 2010. [8] K. S. Moon, “Optimal grid geometry of diagrid structures for tall buildings,” Architectural Science Review, vol. 51, no. 3, pp. 239–251, 2008. [9] Computer and Structures, SAP2000 v15.1.0. Berkeley: Computer and Structures, 2011. [10] Y.-K. Wen, “Method for random vibration of hysteretic systems,” Journal of the Engineering Mechanics Division, vol. 102, no. 2, pp. 249–263, 1976. [11] Y. J. Park, Y. K. Wen, and A. Ang, “Random vibration of hysteretic systems under bi-directional ground motions,” Earthquake engineering & structural dynamics, vol. 14, no. 4, pp. 543–557, 1986. [12] PEER, “2011 PEER Ground Motion Database – Beta Version,” PEER Ground Motion Database, 2011. [Online]. http://peer.berkeley.edu/peer_ground_motion_database/spectras/new. [Accessed: 20-Aug-2013]. [13] ANSI, “AISC 360-05-Specification for Structural Steel Buildings,” Chicago, AISC, 2005. [14] T. A. Morgan and S. A. Mahin, “The optimization of multi-stage friction pendulum isolators for loss mitigation considering a range of seismic hazard,” in Proceedings of the 14th World Conf. on Earth. Eng., 2008. [15] R. Ramsdale, “Reference Tables - Coefficient of Friction,” Engineer’s Handbook. [Online]. Available: http://www.engineershandbook.com/Tables/frictioncoefficients.htm. [Accessed: 20-Oct-2013].