Stability of glulam arches Lars WOLLEBÆK Graduate student NTNU Trondheim, Norway

Kolbein BELL Professor NTNU Trondheim, Norway

[email protected]

[email protected]

Lars Wollebæk, born 1973, received his civil engineering degree from the Norwegian University of Science and Technology (NTNU) in 1999, and is currently pursuing a PhD at NTNU.

Kolbein Bell, born 1939, received his civil engineering degree from the Norwegian Institute of Technology (NTH) in 1962, and his dr.ing degree from the same university in 1968. He has been professor of structural mechanics at NTH/NTNU since 1981.

Summary In order to indicate the nature and complexity of buckling of large glulam arches, the problem is studied both as a curved plate buckling problem and as an arch problem modelled by a series if straight 3D beam elements implemented into a 3D nonlinear finite element code using a co-rotated formulation. Results from two cases are presented, one of which is a 3D model of a network arch bridge analysed by a simpler 2D model in another paper by the same authors and presented at this conference. Keywords: Buckling and stability, glulam arch, network arch, timber bridge.

1.

Introduction

Buckling, and in particular lateral torsional buckling, of timber arches is a complex problem for which the various timber codes offer little assistance to the design engineer. An ongoing PhD study is aimed at investigating both the nature and the importance of the problem as well as proposing adequate methods of analysis and design. In this paper we concentrate on two case studies. The first is a typical “deep” arch (h/b = 6), analysed both as an orthotropic curved plate, and as a 3D “frame” problem, in which the arch is modelled by a series of straight 3D beam elements. In addition to the comparison between the two models, the arch geometry and the out-of-plane bracing of the arch are studied. The second case is a timber bridge in which two network arches form the main support structure. This problem, where each arch is made of four individual arches, two of which are made of 3 parts and two of four parts (to accommodate production and transportation), and mechanically joined together in such a way that the joints are staggered along the arch, are dealt with in another paper by the same authors at this conference [1]. The main purpose of this second case is to study the out-of-plane stability of the arches.

2.

Theoretical background

The plate buckling analyses are carried out by a fairly simple finite element plate program, FEMplate [2]. The plate bending part offers 6 different triangular thin plate (Kirchhoff) elements, whereas 7 elements (3 triangular and 4 quadrilateral) are available for the membrane (or in-plane) analysis. The plate buckling analyses reported here use a higher order (quintic) plate bending triangle with 18 degrees of freedom for the bending part, and a standard 12 degree of freedom linear strain triangle for the in-plane stress analysis. Orthotropic material properties, ranging from highly orthotropic to isotropic, are used. The frame type analyses use a 3D nonlinear frame program (FEMframe) being developed in connection with an ongoing PhD study [3]. A key feature of the program is that all external loading is lumped into statically equivalent concentrated nodal forces, and that many short and straight elements with constant properties are used to model curved and non-uniform members. In addition

to several different beam elements, implemented within the Element Independent Co-Rotational framework, including Euler-Bernoulli, Timoshenko and Mindlin-Reissner type beam elements and bi-linear bar elements, the program offers Mindlin type elements with finite rotations formulated in the total Lagrangian approach by Simo et. al.[5] and Cardona and Gerardin[6], respectively. Both element and load eccentricities are available and handled in a consistent manner. Different linear and nonlinear spring type elements are under development and will be included in the near future. Presently the program can perform geometrically nonlinear analyses, linearized buckling analyses and natural frequency and eigenmode analyses. Ultimate structural stability is presently determined by use of a simple bisection-algorithm. Although implemented, the generalized arc-length routine is as yet not tested and verified.

3.

Case 1 - a simple glulam arch

We consider a typical glulam arch with a span of 40 m and a fairly deep cross section with b×h = 200×1200 mm. The first analyses apply to a 2-hinge arch with no other out-of-plane support than that provided by the end supports. At these points the arch is free to rotate about an axis normal to the plane of the arch, but it cannot rotate about any axes in its own plane. The only loading is the self weight of the beam. Material properties are assumed to be: E0 = 14500 N/mm2, E90 = 960 N/mm2, G = 830 N/mm2 and γ = 4,4 kN/m3. A series of plate buckling analyses, with 1344 triangular elements, are carried out for different radius of curvature or h/L ratios, ranging from 0 (straight beam) to 0,5 (half a circle). Results are shown in Figure 1, along with a typical buckling mode. If we consider the 1,6 Buckling coefficient

1,4 1,2 1 0,8 0,6

mode 1

0,4 0,2 0 0,0000

0,1000

0,2000

0,3000

0,4000

0,5000

h/L Beam Elements

FEMplate

Figure 1 Buckling of an unsupported 2-hinge, deep glulam arch subjected to its own weight “optimal” h/L ratio (= 0,2), the buckling load is about 6 times the weight of the beam, for the assumed material properties. What is the effect of orthotropic material properties on the buckling load? If we set E90 = E0 (isotropy) the buckling load coefficient becomes 6,038, and if we set E90 = E0/14500 = 1N/mm2, we find a buckling load coefficient of 6,006 From this it seems fair to state that orthotropic material properties have no significant influence on the buckling load. Next we study the effect of a transverse spring support at the top of an arch with a radius of curvature R = 30 m (h/L = 0,191). How stiff must a spring supporting the top (compression) fibres of the midsection of the arch be in order to force the arch into the second buckling mode, and what is the corresponding buckling load? The answer is: k = 25,35 kN/m, and the buckling load coefficient is increased to 13,77 If we move the spring to the centerline of the mid-section we find that the arch buckle for a load coefficient of 12,98, for the same spring stiffness, and in order to force the arch into the second buckling mode we need to increase the spring stiffness to 28,35 (an increase of 12%). We repeat the analysis with the spring at the bottom (tensile) fibres of the mid-section, and find a buckling load coefficient of 11,97. The smallest spring stiffness that will force the arch into its second buckling mode is now 33,45, which is 132% of that required when applied to the top fibres. Figure 2 shows how the buckling load varies with the stiffness of a transverse spring attached at the mid-height (neutral axis) of the mid-section of the arch. The second buckling mode is also shown. A similar story is told in Figure 3, where 3 equal springs are placed at the mid-height of sections in the “quarterpoints” of the arch (measured along the arch). Note the difference in buckling load, for similar buckling modes obtained for the cases of infinitely stiff springs and no spring stiffness at all, and note also that very stiff springs are required to produce the buckling mode in the middle of Figure 3.

1 4 ,5 0 1 3 ,5 0

mode 2

Buckling coefficient

1 2 ,5 0 1 1 ,5 0 1 0 ,5 0 9 ,5 0 8 ,5 0 7 ,5 0 6 ,5 0 5 ,5 0 4 ,5 0 0

10

20

30

40

50

S p rin g s tiffn e s s [k N /m ]

Figure 2 Buckling of a 1-spring supported 2-hinge, glulam arch subjected to its own weight

44,5

mode 4

Buckling coefficient

39,5 34,5 29,5

mode 3 mode 2 (kspring = 0)

24,5 19,5 14,5 9,5

kspring = ∞

kspring = 0

mode 1

4,5 0

200

400

Spring stiffness [kN/m ]

mode 1

mode 4

buckl.coeff = 38,06

buckl.coeff = 44,05

Figure 3 Buckling of a 3-spring supported 2-hinge glulam arch subjected to its own weight The arch above (L = 40 m and R = 30 m) is also analysed with a 3D beam model in which the arch is approximated by 300 straight beam elements of the Timoshenko type (classical beam theory with shear deformations included in an approximate manner). The results are, as indicated by the graphs in Figure 1, quite similar to those obtained by plate buckling analysis. Some of the difference can be explained by the difference in torsional stiffness defined by the two models (0,333hb3 and 0,298hb3, respectively). With the beam model, the arch is also analysed with a hinge at the top (i.e. a 3-hinge arch). Assuming the hinge only permits a moment-free rotation about an axis normal to the plane of the arch, while the other 5 kinematic degrees of freedom are continuous, the buckling coefficient for an otherwise unsupported arch is 5,94. This compares with 5,91 for the same 2-hinge arch. The loading is still self load only. The lowest buckling coefficient for in-plane buckling is 207,6 and 252,7, for 2-hinge and 3-hinge, respectively.

mode 1 buckl.coeff = 83,6

mode 2 (in-plane)

buckl.coeff = 115,4

Figure 4 Buckling of an unsupported 2-hinge, massive glulam arch subjected to its own weight Finally we study an arch with a square (800×800 mm) cross section, which is more typical for a bridge arch. We adopt the same geometry and boundary conditions as above, that is L = 40 m and R = 30 m, and all, but one (rotation about an axis normal to the plane of the arch) of the six degrees of freedom

are fixed at the base points. The loading is still only the self weight of the arch, and we consider both a 2-hinge and a 3-hinge design. The arch is analysed by a 3D beam model with 320 elements. Figure 4 shows the lowest out-of-plane and in-plane buckling modes for the 2-hinge design. The buckling load coefficients are 83,6 and 115,4, respectively. For the 3-hinge design the corresponding numbers are, 84,4 and 95,1. From this it is seen that if we can achieve supports that behave like those in the model, the torsional stiffness of the arch is quite effective in resisting lateral buckling. The results presented in this section are indicative (in a qualitative sense) of arch behaviour. In order to draw conclusions for practical design, more results, for different geometries, loading and boundary conditions need to be considered. This work is in progress and will be reported in [3].

4.

Case 2 - a network arch bridge

Figure 5 shows a network arch bridge which is investigated, by 2D analyses, in another paper at this conference [1]. It is shown in [1] that if the overall stability of the system can be demonstrated, the design may be a viable one. The bridge, which is a hypothetical one, adopts the deck solution used at

B A

C

D hanger Ø50 mm 55

o

16 × 5 = 80 m

880×800 mm E R = 54 m 11,7 m

Figure 5 Network arch bridge the newly built timber bridge at Tynset [4]. A 7 m wide stress laminated timber deck with asphalt constitute the two road traffic lanes. The bridge also has a 3 m wide pedestrian/bicycle lane. The deck rests on steel cross beams which are here spaced at an equal distance of 5 m (slightly shorter than at Tynset). Each cross beam is supported by two inclined steel hangers at each end. The two parallel arches are modified 2-hinge arches, each made up of four 220×800 mm glulam arches mechanically joined at five points, A, B, C, D and E, in such a way that the two inner arches are joined at A, C and E, while the two outer arches are joined at B and D. The longest part of an arch is thus about 30 m, and its total height is under 3 m. Production and transportation should therefore not present any problems. Between the individual arches are placed 8 mm steel plates, on to which the hangers are fastened, and everything is bolted together, with bolts and shear plates (see [1]), at the intersection between each hanger and the arch. Figure 6 shows a 3D model of the entire bridge. It contains about 3500 beam elements and 60 bar elements, and a total of about 20000 degrees of freedom.

Figure 6 3D model of network arch bridge

With reference to Figure 5 each arch in Figure 6 is modelled as four individual arches next to each other. All 4 “sub-arches” are forced to have the same displacements (including rotations) at every point where a hanger is fastened to the arch. Between these points, however, they are completely unconnected. At points A, B, C, D and E the two joined sub-arches are connected by “hinges” that cannot transmit moments about any axes. The hanger force is assumed to be equally distributed between the 4 sub-arches. The hangers can only take tension. Hence, hangers with compressive forces are removed from the model. Each cross beam is modelled with its appropriate stiffness, and the deck is modelled by four longitudinal (timber) beams, rigidly connected to the cross beams. Two of the longitudinal beams, each with a cross section of 1000×250 mm, are placed in connection with the fastening of the hangers to the cross beams, and the other two beams, with cross section 2000×250 mm, are placed such as to be in the correct position for traffic loading in each lane, assuming the loads to be as far over to the most loaded arch as possible. Each of the 4 longitudinal beams is rigidly connected to all cross beams. The deck model is clearly an approximate one, but it is believed to represent the deck stiffness with adequate accuracy, particularly with respect to the load distribution. At the arch supports, the only degree of freedom that is not fully constrained is the rotation about an axis normal to the arch plane, which is unconstrained. This is perhaps optimistic, particularly for the two in-plane rotations. Two load conditions are considered: the dead or self load of the entire system, and the self load in combination with traffic load, both distributed and concentrated, at the middle of the bridge (see Figure 3 in [1]). The same traffic load is placed in both lanes. A linearized buckling analysis is carried out for both load combinations. For self load only, a buckling coefficient of 19,24 is found for the buckling mode (No. 14) shown in Figure 7a. This corresponds quite well with the buckling factor of 18,66 found for the plane model in [1]. For combined dead load and traffic load we find a buckling factor (corresponding to the total loading) of 4,28 for the buckling mode shown in Figure 7b. This is significantly higher than the 2,49 predicted by the 2D model with spring reinforced hinges at the joints ( k θ = 50000 kNm/rad). Since the 3D model exhibits a mode of

buckl.coeff = 19,24 mode 14

(a)

buckl.coeff = 4,29 mode 1

(b)

Figure 7 Buckling modes for a) dead load only, and b) combined dead load and traffic load buckling with both in-plane and out-of-plane components, it seems reasonable to expect a lower buckling load than that predicted by the 2D model, which is forced to in-plane buckling only. We will return to this discrepancy shortly, but first we mention that if we perform a linearized buckling analysis in which the geometric stiffness due to self load is included in the material stiffness and the geometric stiffness matrix in the eigenproblem only accounts for the stiffness resulting from the axial forces caused by traffic alone, we find that we need to multiply the traffic load by 8,15 in order for the system to buckle. We modify the model in Figure 6 by replacing the mechanically joined arches by two identical, fictitious glulam arches made of a massive 880×800 mm cross section, that is two continuous and perfect 2-hinge arches. With these arches the lowest buckling coefficient for the bridge when subjected to self load only is 20,16, and when subjected to combined self load and traffic the coefficient is 4,92. These numbers compare to 7,83 and 4,28, respectively, for the case of joined arches. Figure 8 compares the lowest buckling modes for the two different arch designs, when the bridge is subjected to self load only. It seems clear that much of the difference in buckling load is due to the inability of the mechanically joined arches to transmit shear between the sub-arches except at

mode 1

mode 1

buckl.coeff

buckl.coeff

= 7,83

(a)

= 20,16

(b)

Figure 8 Buckling modes for a) joined arches and b) massive arches discrete points. The relatively moderate difference between the two models, for self load plus traffic (4,28 compared to 4,92), seems to indicate that the 2D model in [1] may have a conservative estimate of the stiffness ( k θ )assumed for the rotational springs that reinforce the joint hinges. On the other hand it may be optimistic to assume that the four “sub-arches”, in Figure 6, are completely and rigidly joined at each point of hanger fastening. Some movement in these joints is inevitable, the question is how much this will influence the results. This problem can and will be looked into more closely, by relaxing some of the constraints in the present model for joined arches. However, the analyses presented in this section seem to indicate that the suggested network arch design for an 80 m span bridge has sufficient stability properties. However, other problems need to be looked into more carefully before this bridge design can be given a clean bill of health.

5.

Concluding remarks

Some aspects of the stability of glulam arches have been demonstrated. A more comprehensive and systematic series of analyses is necessary in order to provide information and guidelines for practical design work. The aim is to include (code based) capacity controls coupled with full fledged nonlinear static analyses that account properly for the stiffness of the entire (3D) structural system, including all bracing and geometric imperfection, and thus eliminate the use of some rather cumbersome k-factors. The bridge case presented above indicates the potential of a flexible and visual 3D analysis and design tool. However, 3D modelling of real structures is not always a straightforward process, as will be demonstrated in [3], and it will probably take some time before the powerful, state-of-the-art computational engines, are developed into robust, easy to use and understand and, to some extent, foolproof tools, for the ordinary practising engineer.

6.

References

[1] Bell, K. and Wollebæk, L. “Large, mechanically joined glulam arches”, paper (6 pages) presented at WCTE 2004 in Lahti, and contained in these proceedings. [2] FEMplate - a finite element program for linear analysis of elastic plates http://www.femtech.no [3] Wollebæk, L., Computer aided analysis and design of 3D timber structures, PhD dissertation in progress at the Norwegian University of Science and Technology, Trondheim. [4] Bell, K. and Karlsrud, E., “Large Glulam Arch Bridges - A Feasibility Study”, Proceedings of the IABSE conf. on Innovative Wooden Structures and Bridges (pages 193-198), Lahti, Finland 2001. IABSE report, Volume 85. [5] Simo, J.C. and Vu-Quoc, L., “A three-dimensional finite-strain rod model. Part II: Computational aspects”, Comp. Met. Appl. Mech. Engn., 58, (79-116), 1986. [6] Cardona, A. and Gerardin, M., “A beam finite element non-linear theory with finite rotations”, Int. J. Num. Met. Engn., 26, (2403-2438), 1988.