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Finite element analysis and test of joint at the center pole of a wooden spiral stair.

Abstract

Wood spiral stairs typically have center poles composed of several wood pieces held together by a steel tube that is screwed tight at the top. The aim of this study was to examine the influence of the prestress of the center pole on the displacements of treads to see if it was a determining factor for the behavior of the stair. A small spiral staircase was tested, and two of the treads were also tested separately. The result shows that the displacements of the treads has a small dependency on the prestress. Finite element calculations of the stair were performed with 3-D models with contact elements between the different pieces. The intention was to study if calculated results would agree with test results and find out if simulations using these models could be deployed in a product development process. The conclusion was that calculated displacements show less dependency on the prestress than what was found in the tests. But as the simulations indicated the same structural response of the stair, the model could however be used for further studies and comparisons of different designs of the stair as part of the product development.

Development of new products in the joinery industry is usually based on traditional craftsmanship and testing of prototypes. Finite element (FE) modeling can be a valuable support in the process of product development. The products can be designed using solid models in computer aided design (CAD) programs. These models can also be used for further studies with FE calculations. Stairs and furniture are examples of products that can have a complex three-dimensional geometry, which makes it preferable to use three-dimensional models.

FE simulations can enable faster, less costly, and more optimized product development, as well as examinations of product performance that would not be possible even using very detailed prototypes. In the wood industry these tools have not been greatly used, but in other industries such as the aerospace and the automobile industry they have been much used for many years. Wood is a more complicated material to model than steel, and most studies of wood product design with FE models have been published in the academic world. Gustafsson (1995) studied how the use of FE models can enhance the furniture design process. Ekstrom (1997) and Aronsson and Lindgren (2000) studied design of chairs with the help of FE. Olsson et al. (2004) examined the early stages of furniture design and the dialogue between designers and the engineers who are specialized in using FE tools. Mackerle (2005) stated that in the last four decades, the FE method has become the prevalent technique for analyzing physical phenomena in the field of structural, solid and fluid mechanics, as well as for the solution of field problems. He gives a bibliographic review of the FE method applied to the analyses of wood, and many topics are included since wood is a complex material to model. One of the references is a study by Lam et. al. (2004) of the structural performance of wood-based stair stringers with full-scale tests. They tested stair systems and also used a commercial FE package to model a stair system for further insight into its structural response. They concluded that the FE program can be used to model stair systems with configurations and material properties other than the tested ones. Blanchet et al. (2006) demonstrated the suitability of the FE method in the design process of new engineered flooring and stated that their work confirmed the potential of the FE method for product design of such products. Salokangas (2003) used a 3-D CAD model of a complex, irregular lattice structure for a wood tower in Helsinki Zoo. The complex geometry data were imported into a FE model for structural analysis using three-dimensional linear beam elements to check the ultimate and serviceability limit states for applied loads.

Comparison of results from FE analysis of an entire wood spiral staircase to full-scale laboratory tests showed that a three-dimensional solid model of the stair as one complete part was stiffer than the real stair (Pousette 2003). It was concluded that the deviations originated from deformations in the joints. The center pole consists of several wood pieces held together by a steel tube that is screwed tight at the top as shown in Figure 1. The torque thus applied creates a prestress in the center pole, which appears to be significant for the behavior of the stair. In case the prestress is not high enough, the pole parts can separate in the tension zones when the pole is bent, and the pole will be less rigid.

[FIGURE 1 OMITTED]

In this study, a small spiral staircase was tested. Two of the treads were also tested separately. The aim of this study was to further examine what happens at the center pole of the stair with the aid of tests and calculations. Finite element calculations were performed using models with contact elements between all pieces of the stair, to study if the results from these models would agree better with test results than the earlier models with the stair as one complete part.

Materials and methods

The wood spiral staircase in this study was a small staircase with only 4 treads and a railing. It had a radius of 920 mm. The treads, numbered 1 to 4 from the bottom were connected to a center pole with a diameter of 150 mm. The center pole was made of treads and spacers of wood (Fig. 1). The height of each spacer between treads was 153 mm. A steel tube in the middle of the center pole was used to screw and hold the staircase together. The tube was topped with a nut and a washer with a diameter of 80 mm close to the wood. The treads were 41 mm thick. Treads and spacers between treads were made from laminated Scots pine (Pinus sylvestris). The timber quality was grade A (Anonymous 1994), which corresponds to the European grade G4-0 (Anonymous 2002). The final glued laminated wood was not classified. The grain direction of the treads was along the front edge; grain of the spacers was in the vertical direction. The handrail was made of laminated beech (Fagus silvatica), with the grain direction along the handrail. The balusters were made of stainless steel bars with a diameter of 15 mm. The balusters connecting the treads were screwed to treads and handrail and were bearing elements in the staircase.

The staircase was mounted by the manufacturer and then tested in our laboratory in Skelleftea. The stair was placed in a steel test rig fixed to the concrete floor of the laboratory. During the tests, the temperature in the laboratory was 20 to 23[degrees]C, and the relative humidity was 30 to 35 percent.

The upper tread had a support along the outside edge under the upper newel post. The bottom of the steel tube in the center pole was fastened with four screws M8 by 60 to a 45 mm thick wood plate, which was fixed to the test rig. The lower newel post was also fastened with screws to the wood plate. The staircase was loaded with a point load or a line load placed on tread 2 or 3. The point load was placed in the center of gravity of the projected surface of the tread and the line load was placed across the tread in the same position. The load was increased up to 2.9 kN over a period of approximately 1.5 minutes. The vertical displacement was measured at several points under the tread. The horizontal displacement of the center pole was measured in two directions, x and y, where the x-direction was parallel with the front edge of tread 1. The stair was tested with different prestresses of the center pole. The torsional moment applied to the center pole was measured with a dynamometer connected to a 1 m long wrench, so that the measured load in N gave the torsional moment in Nm. At the initial prestress the moment was 67 Nm, which felt unfastened. The nut was then fastened with a wrench until it was firm, which is the usual mounting practice for this type of stair. The torsional moment was then measured to 153 Nm. The stair was also tested with a higher moment of 202 Nm and also with a very loose nut at a moment of 20 Nm.

The connection of one tread to the center pole was also tested separately with one tread at a time mounted between two spacers. Treads 2 and 3 were used in these tests. The center pole was fixed horizontally above the upper spacer by a horizontal 8 mm thick steel plate screwed to the rig. The other spacers and a steel part were mounted above the steel plate to enable screwing at the top thread of the steel tube (see Figure 2). Point loads were applied at the center of gravity of the projected surface of the tread and the displacement of tread and spacers was registered at several points. The prestress was varied in the same way as for the stair test. The treads were loaded three times for each prestress, and the torsional moment applied was controlled and adjusted if necessary after each loading.

[FIGURE 2 OMITTED]

In a special test rig, the corresponding prestressing force in the steel tube was measured with a load cell for each value of torsional moment (see Table 1).

FE calculations were made with 3D solid models in IDEAS (Anonymous 1998a), see Figure 3. The element type was 8-node brick elements (linear hexahedral elements). In the models, the treads and the spacers of the center pole were connected with contact elements. Linear elastic calculations were made, and the contact elements were linear. The program detects surface-to-surface contact from the element free faces. From each face it projects a normal, a vector that is perpendicular to the surface, and then checks to see if any of the normals intersect with another element free face. When a gap between contacting faces is zero, the contact element is closed and contact pressures are generated. The calculations are iterative until there is no change in contact status. Friction can also be defined in the contact analysis. In these calculations, the coefficient of friction between parts was assumed to be 0.2. The prestress of the center pole was simulated with a temperature load on the steel tube. The temperature was calibrated to give the reaction forces at the bottom corresponding to the measured prestressing forces for the steel tube (Table 1). The lower end of the center pole was supported in three directions and the top of the center pole was supported horizontally.

[FIGURE 3 OMITTED]

The material properties in the model were chosen to correspond to tests of other stairs from the same manufacturer where the longitudinal modulus of elasticity was estimated from three-point bending tests. The mean value for 12 rectangular treads made of pine was 10,534 MPa, which corresponds to grade K24 for structural timber in the Swedish building code, BKR (Anonymous 1998b). The SD was 758 MPa. The transverse modulus of elasticity and the shear modulus were chosen according to the relationships for structural timber in BKR. The material properties of the beech material of the handrail was chosen from tests of three rectangular treads from the same manufacturer. The mean value of modulus of elasticity from bending was 13,932 MPa, with a SD of 1359 MPa. The transverse modulus of elasticity and the shear modulus were chosen from EN 338 (Anonymous 1995) for deciduous species of strength class D50. The following values of modulus of elasticity, shear modulus and Poisson's ratio were used:

Ez = 10,500 MPa (pine), 14,000 MPa (beech)

Ex = Ey = 350 MPa (pine), 930 MPa (beech)

Gzx = Gzy = 700 MPa (pine), 880 MPa (beech)

Gxy = 70 MPa (pine), 88 MPa (beech)

[v.sub.zx] = [v.sub.zy] = 0.025,

[v.sub.xy] = 0.4

Results

The differences in measured displacements for the different values of prestress were small for the tested staircase. The point load gave larger displacements than the line load, which is reasonable. Displacements for tread 3 were smaller than for tread 2, which is logical as tread 3 was nearer to the upper support and tread 2 was in the middle of the stair.

The measured displacements of tread 2 were about 3.6 mm for line load and 4.0 mm for point load. The calculated displacement was about 3.5 mm for point load. The measured displacements of tread 3 were about 3.2 mm for line load and 3.7 mm for point load. The calculated displacement was about 2.7 mm for point load.

Horizontal displacements of the center pole increased slightly with increasing prestress, see Figure 4. The center pole was somewhat inclined in the x-direction, i.e., along the front edge of tread 1.

[FIGURE 4 OMITTED]

Results from the tests with one tread between spacers are presented in Figure 5 where the displacement is the value measured on the underside of the tread under the point load. The displacement of the treads decreased with increasing prestress of the center pole. The calculated displacements were smaller than the measured displacements and did not vary as much with the prestressing force.

[FIGURE 5 OMITTED]

In the FE calculations, the prestress compressed the center pole, and the tread was squeezed between spacers. At load 0.5 kN the tread was bent down, and on the underside the contact pressure increased at the center pole while contact pressure on the upper side decreased. At the back of the pole, where the tread was mostly squeezed together, it was kept in place.

Discussion and conclusions

It is important for the performance of this kind of stair that the tensioning of the center pole is adequate. The result of this study is that the prestress has some influence on the displacement of a stair. It is valuable to have instructions for the tensioning of the center pole, and also to be able to control and adjust the tension after a period of time. However, even if the tension is very loose, the washer and nut act as a stop at the top to prevent the treads from moving upwards, which will secure the function of this kind of stair. For the small spiral staircase with 4 treads tested in this study, displacement did not vary much with the value of the prestress of the center pole. The stair was not very flexible because it was short and supported at both top and bottom. A longer stair can be more flexible and thus more dependent on the function of the center pole.

The tests with one tread connected to the center pole showed that the prestress influenced the displacements. Also in the FE analysis the prestress had a small effect on displacements, although much smaller than in the physical tests, and the calculated displacements were also smaller than those measured physically. An additional calculation using a lower modulus of elasticity was also made, and it gave displacements closer to the test results, although the difference between low and high prestress was still less in this calculation.

The differences between test and analysis results may depend on incorrect material parameters of the wood in the calculations. The modulus of elasticity used for each of the treads and center pole parts was not verified by measurements before testing. The values were instead taken as mean values from nondestructive tests of other stairs made of the same wood material. Since wood has a large variability also in timber from one log, it can be difficult to compare test results with calculated results. For structural timber there are material properties given in building codes that should be used in calculations. For joinery products, the wood is usually not strength graded timber but classified according to appearance. In practice, this can make it even more difficult for engineers to know which material parameters to use in calculations.

For structural timber of quality K24, with modulus of elasticity of 10,500 MPa, the characteristic value for compression perpendicular to the grain is 7 MPa according to BKR. The maximum applied prestressing force of 20.6 kN gives a stress of 1.3 MPa in the center pole, which is smaller than the characteristic value. The maximum calculated contact pressure on the underside of the tread at the connection to the center pole was 3.1 MPa for load 0.5 kN, which would not give any large deformations.

Some of the differences between analysis and test of the spiral staircase may depend on the modeling of the center pole. The differences may also depend on not perfectly smooth wood surfaces in the tested stair, or deformations at the connection of the tested stair to the floor. The railing is also an important factor in the deformation of the stair. The supports and joints of the railing were not tested in this study. In the calculations, they were assumed to be perfectly connected. It is probably also necessary to model the joints of posts and balusters more exactly in order to obtain closer agreement between model and reality.

The general conclusion from this study is that the stair can be more correctly modeled with contact elements between all pieces of the stair and the center pole prestressed. Also, the approximation used when modeling the variations within the wood material remains, even if a more detailed model would be used. Although the results from tests and analyses differ in quantity, they indicate the same structural response of the stair. This means that the model can be used for further studies and comparisons of different designs of the stair as part of a product development process.

Literature cited

Anonymous. 1994. Nordiskt Tra. Sorteringsregler for sagat virke av furu och gran (Nordic Timber. Appearance Grading of Sawn Timber from Pine and Spruce). Arbor Publishing. Stockholm. ISBN 91-7322-175-9. 64 pp. (in Swedish).

--. 1995. EN 338. Structural timber- Strength classes. European standard. CEN, European Committee for Standardization. Brussels. Belgium.

--. 1998a. I-DEAS Master Series 6. Structural Dynamics Res. Corporation. SDRC. Milford. Ohio 45150. USA.

--. 1998b. BKR Boverkets konstruktionsregler (Design Regulations, BKR, of the Swedish Board of Housing, Building and Planning). Boverket Publikationsservice. Karlskrona. ISBN 91-7147-455-2 (in Swedish).

--. 2002. EN 1611-1/A1. Sawn timber--Appearance grading of softwoods--Part 1: European spruces, firs, pines, Douglas fir and larches. European standard. CEN, European Committee for Standardization. Brussels. Belgium.

Aronsson, S. and C.-M. Lindgren. 2000. Produktutveckling av mobler med hjalp av FEM (Product development of furniture with aid of FEM). Chalmers Univ. of Tech. Goteborg. Sweden (in Swedish).

Blanchet, P., A. Cloutier, G. Gendron, and R. Beauregard. 2006. Engineered wood flooring design using the finite element method. Forest Prod. J. 56(5):59-65.

Ekstrom, R. 1997. Konstruktionsdesign av mobler med FEM (Design of furniture with FEM). Lund Inst. of Tech. Lund. Sweden (in Swedish).

Gustafsson, S.-I. 1995. Furniture design by use of the Finite Element Method. Holz als Roh- und Werkstoff 53:257-260.

Lam, F., G. Lee, H. Yan, J. Gu, and A.A. Saravi. 2004. Structural performance of wood-based stair stringers. Forest Prod. J. 54(4):39-44.

Mackerle, J. 2005. Finite element analyses in wood research: A bibliography. Wood Sci. and Tech. 39:579-600.

Olsson, P., P. Eriksson, and K. Olsson. 2004. Computer-supported Furniture Design at an Early Conceptual Stage. Inter. J. of Design Computing 7, www.arch.usyd.edu.au/research/publications.shtml.

Pousette, A. 2003. Full-scale test and finite element analysis of a wooden spiral staircase. Holz als Roh- und Werkstoff 61(1):1-7.

Salokangas, L. 2003. Wooden Observation Tower, Helsinki, Finland. Structural Engineering Inter. 3/2003:160-162.

The author is a Researcher at SP Technical Research Institute of Sweden, Building Technology and Mechanics / Wood Technology, Skeria 2, Skelleftea, Sweden (anna.pousette@sp.se). This paper was received for publication in April 2005. Article No. 10038.
Table 1.--Torsional moment and force of center pole

Torsional moment (Nm) Prestressing force (kN)

20 2.4
67 7.9
100 11.2
150 16.7
202 20.6
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Author:Pousette, Anna
Publication:Forest Products Journal
Geographic Code:4EUSW
Date:May 1, 2007
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