On The Plastic Design Of Timber Beams With A Complex Cross-Section

Size: px

Start display at page:

Download "On The Plastic Design Of Timber Beams With A Complex Cross-Section"

Marjory Collins
6 years ago
Views:

1 On The Plastic Design Of Timber Beams With A Complex Cross-Section Maurice Brunner 1 1. INTRODUCTION For the design of steel and concrete structures, most modern international design codes allow the use of plastic stressstrain-diagrams for the analysis of the bending resistance of the cross-section in the ultimate limit state. Curiously, in Europe and most other parts of the world, timber structures are still designed for the ultimate limit state with the old, conservative concept of full elastic behavior till failure. The situation is rather funny for two main reasons. First, it is well known that under compressive loading, timber exhibits a much better ductile behavior than concrete, for example. Secondly, between the two World Wars, timber scientists developed refined calculation methods to explain test results which showed that the bending strength of small, clear timber specimens lies well above the compressive strength. It is curious that these findings have never been developed to the stage where they could be integrated into modern timber design codes. There are of course reasons why the Plastic Design Approach is not yet used for timber structures. The main problem is the unreliable tensile strength. Even small, clear timber specimens exhibit brittle behavior under tensile loading. In the case of large specimens of the size used in structures, the tensile strength can be considerably reduced by defects and by the so-called volume effect. A good description of the volume effect or weakest link concept, as it is also known is given in [1]. One of the most important factors involved is the type of stress distribution: the failure stress in the case of a bending moment is much higher for timber of the same graded quality and dimensions than the failure stress under tensile loading (Fig. 1). In the latter case a greater area is under high tension, hence the probability of early failure due to the so-called weakest link. Fig. 1: Influence of the stress distribution on the magnitude of the failure stresses In the European Design Code EC5 for timber structures, the question of plastic design has been indirectly taken into account by the introduction of an apparent bending strength f M, which has been determined directly in loading tests. The calculation of the bending resistance M R is based on the full elastic approach. The concept of the fictitious bending strength is equivalent to full plastic design for solid, rectangular timber cross-sections and also for the glulam types listed in the Code, for which extensive studies have also been carried out. Since most structural timber used in the construction industry has a solid, rectangular cross-section, the concept of the fictitious bending strength will in effect lead to full plastic design for most practical purposes. There is a small but quietly growing market for highly stressed, high-tech timber cross-sections ([2]). For such applications, the design method based on the fictitious bending strength is inadequate, because the true failure mode of timber beams as depicted in fig. 1 is not evident. Indeed, the theory of the fictitious bending strength may in fact be misleading sometimes, since the impression is created that there is no difference between the compression and the tension 1 Professor, Swiss School of Engineering for the Wood Industry, Biel, Switzerland

2 zones of a beam. It is common knowledge among engineers that the weakness of a timber beam lies in the brittle tension zone, yet when engineered, complex cross-sections are necessary, the solution is typically a symmetrical strengthening of both tension and compression zones glulam is a typical example. In comparison, the concrete engineer, aware that the brittle tensile strength of concrete is too risky, typically prescribes reinforcement for the tension zone only. In a similar way, it might make engineering and economic sense to strengthen future engineered timber beams much more in the critical tension zone than in the compression zone. Fig. 2 shows some simple examples: Grading techniques should be used to strengthen in particular the tension side of glulam beams. In the case of flanged timber beams, the boards on the tension flange should exhibit the best material properties; alternatively, the tension flange could be made thicker than the compression flange. Existing beams could be strengthened in-situ by gluing a flat steel bar on the tension side only. Fig. 2: Vision for high-tech timber cross-sections: unsymmetrical strengthening of the tension side The present widely used design concept of the fictitious bending strength does not permit the engineer to take advantage of the good plastic properties of timber under compression in order to design highly stressed, high-tech cross-sections of the future. The purpose of this paper is to present simple models to illustrate how the existing Eurocodes for timber could be supplemented not changed! - in order to let this happen. 2. OVERVIEW PLASTICITY OF MATERIALS Fig. 3: Stress-strain-diagrams for common building materials Like most metals, a steel bar under tensile loading initially exhibits a linear relationship between the stress σ and the strain ε (Fig. 3). After the yield stress f y has been attained, the stress remains constant even when the strain increases considerably. In the case of European Standard Steel Fe235, failure occurs only after an enormous strain of about 250%o

3 has been attained, compared to the yield strain of about 1.1%o. Materials which exhibit an extensive yield plateau are called ductile ; their structural behavior in the ultimate limit state is characterized by the following advantages: Failure occurs only after large deflections have been attained: there is visible warning of danger The bending resistance is increased by the so-called plastification of the cross-section. In the case of statically indeterminate structures, the failure loads are higher than in the case of purely elastic behavior because the internal forces can be redistributed by plastic joints. The Theory of Plasticity is ideal for the design of steel structures. It is however also widely used today for the design of reinforced concrete structures. In this composite building material, steel bars are placed to take up the internal tensile forces, whilst the concrete itself deals with the compressive forces. The concrete engineer uses the so-called condition of ductility to make sure that the steel is the weakest link in reinforced concrete: in the ultimate limit state the steel yields in tension before the concrete compression zone fails, hence the ductile behavior of reinforced concrete structures. In comparison to steel, concrete cubes under compression exhibit fairly brittle behavior. Failure occurs at a relatively low strain of about 3.5%o. However, in the ultimate limit state, most design codes permit the plastification of the concrete compression side, thus helping to ensure economic design. Clear timber specimens exhibit very different behavior under tensile and compressive loading. The behavior depicted in fig. 3 has been known since the 1920 s ([3]). Under tensile loading, the stress-strain-diagram runs a straight line till brittle failure. Under compressive loading, the stress-strain-diagram exhibits an initial linear stage. At a strain level of about 3%o, the buckling of the timber fibers leads to the attainment of a yield plateau. The stress remains fairly constant whilst the strain increases to about 12% at failure. Structural timber is characterized by defects such as knots and resin pockets. The tensile strength in particular is very sensitive to such defects and may be sharply reduced. The compressive strength on the other hand is usually much less affected. This phenomenon is evident in Eurocode EN338: between the timber grades C22 and C40 the compressive strength rises from 20 to 26 N/mm2, whilst the tensile strength for an axial load changes from 13 to 24 N/mm2 (Fig. 4). 3. SIMPLE STRESS-STRAIN-DIAGRAMS FOR THE PLASTIC DESIGN OF TIMBER BEAMS WITH COMPLEX CROSS-SECTIONS Fig. 4: Tensile (f t ), compressive (f c ) and bending(f M ) strengths according to Eurocode EN 338 The table of fig. 4 shows in simplified form the system of Timber Strength Classification according to the European Standard EN 338. All values refer to stresses acting in the direction of the fibers. The fictitious bending stress f M plays a central role in the system and in fact the timber grade is directly named after the value of the bending strength. As shown in fig. 1, however, the bending stress f M is not identical with the real stress conditions in the ultimate limit state. It is a fictitious value derived mathematically from tests by assuming a linear stress distribution across the cross-section: f M = [ M f / W ] / f with M f = Bending moment at failure W = Moment of resistance (Rectangle: W = b.h 2 /6) f = Safety factor As mentioned above, the concept of the fictitious bending strength f M is very practical and it is equivalent to the plastic design of rectangular beams, which form the vast majority of structural timber members in use. For many complex, hightech cross-sections which may be needed for special applications, however, a more refined approach is called for.

4 There are many calculation models for the non-elastic distribution of stresses in a timber beam. One of the simplest and best-known, developed in 1939 by Thunell ([3]), is shown in fig. 5. The tension zone is characterized by a linear, brittle model, whereas the compression zone exhibits an initial linear stress rise followed by a clear yield plateau. The E-moduli for both tension and compression are the same. Fig. 5: Trapezoid-like stress distribution over the cross-section according to Thunell ([3]). The standard stress values listed in the Eurocodes can be used to obtain working values for the Thunell model if the following simple assumptions are taken: The so-called volume effect applies to the tension zone: under bending conditions, the true failure stress f t,m is much higher than the value f t under pure tension. Research work indicates that the compression zone is much less influenced by the volume effect ([1]). For simplicity, the failure stress f c under normal compressive force is assumed to remain unchanged in the compression zone of the beam under bending. Plastification of the cross-section is possible. The value f t,m, the true failure stress on the tension side of the beam, is not listed in the Eurocodes. It can however be determined by comparing the equations for the bending resistance M R of beam with a rectangular cross-section, calculated according to Thunell ([3]), and then according to the Eurocode. Thunell: M R = f c. (b.h 2 /6). c with c = [3 + 8.m + 6.m 2 m 4 ] / (1 + m) 4 and m = f c /f t,m Eurocode: M R = f M. (b.h 2 /6) The following equation is obtained which can be solved for m, and then for f t,m (Fig. 6): (3 + 8.m + 6.m 2 m 4 ) = f M (1 + m) 4 f c

5 Fig. 6:Characteristic values (f c and f t,m ) for Thunell s stress diagram, derived from the characteristic stress values of the Eurocode 4. CALCULATION EXAMPLE A simple example will now be presented to illustrate how a Plastic Design Approach could change the outlook for hightech, timber cross-sections of the future. A timber beam, Grade C27, has a width of 100mm and a height of 300mm. Its bending resistance is: W = 100 x / 6 = 1.5 x 10 6 mm3 M R = f M x W = 27 x 1.5 x 10 6 / 10 6 = 40.5 knm Since this resistance is inadequate, it is to be strengthened by gluing a steel plate (Fe335) on the tension side. The material properties involved can be listed as follows: Timber: C27 Steel: Fe335 E T = 12 kn/mm2 E S = 210 kn/mm2 f M = 27 N/mm2 f y = 335 N/mm2 f c = 22 N/mm2 ε y = 335 / = 1.60%o ε c = 22 / = 1.83%o The calculation according to Eurocode 5 is illustrated in fig. 7. Due to the high E-modulus of the steel plate, the neutral axis is shifted below the geometrical center of the 305mm high beam. The relevant characteristic values of the composite cross-section are: Relative stiffness of steel: n = E S / E T = 210 / 12 = 17.5 Position of neutral axis: z U = 121mm (z O = 184mm) Moments of resistance: Timber compression zone: W O = 2.07 x 10 6 mm3 Timber tension zone: W U = 3.31 x 10 6 mm3 Steel tension zone: W S = 3.16 x 10 6 mm3 The calculation suggests that the timber compression zone would be responsible for failure because the fictitious bending strength f M = 27 N/mm2 would be attained there first before the tension zone. The steel yield stress of 335 N/mm2 would not have been reached when the timber compression zone fails, hence the excellent plastic qualities of steel would be useless according to this calculation approach. The bending resistance would be: M R = W O x f M = 2.07x 10 6 x 27 / 10 6 = 56 knm

6 Fig. 7: Timber beam with an attached steel plate: bending resistance M R calculated in accordance with the fictitious bending strength approach The calculation of the bending resistance of the same beam according to the Plastic Design Approach described under chapter 3 is illustrated in fig. 8. The neutral axis is shifted even further down to z U = 115mm, and the critical strains are about 4.0%o at the top compression zone of the timber, 2.3%o at the bottom tension zone of the timber, and 2.4% at the very bottom of the steel. The compressive and tensile forces corresponding to these strains are in balance. The steel plate and the timber compression zone have both reached the plastic state, but failure is actually triggered when the tension zone of the timber attains the true brittle tensile strength f t,m. The bending resistance can be calculated to be: M R = 228 x x x = 67 knm Fig. 8: Timber beam strengthened with an attached steel plate: calculation with a Thunell-like stress distribution The bending moment thus derived with the plastic design method is 20% higher that the result obtained using the fictitious bending strength f M according to the Eurocodes. The Plastic Design Method also points out that the brittle tension zone of the timber, and not the ductile compression zone, will remain the weakest link in the beam. 5. FURTHER READING [1] Steiger Rene Mech. Eigenschaften von Schweiz. Fichten-Bauholz bei Biege-, Zug-, Druck- and komplizierter M/N-Beanspruchung, IBK-Bericht Nr. 221, June 1996, Birkhäuser Verlag Basel [2] Steurer Anton Holzkonstruktionen mit Stahl- und Kunststoffverstärkung, 31. SAH-Tagung, 1999, Lignum, Zürich [3] Kollmann/Côté Principles of Wood Science and Technology, Volume 1, 1984, Springer Verlag, Berlin, Chapter 7.4

7 [4] Eurocodes EC5 Design and Construction of Timber Structures EN338 Strength Grades of Structural Timber

EXPERIMENTAL STUDY REGARDING THE BEHAVIOR OF GLUE LAMINATED BEAMS DOUBLE REINFORCED WITH RECTANGULAR METAL PIPES (RMP)

Bulletin of the Transilvania University of Braşov CIBv 2014 Vol. 7 (56) Special Issue No. 1-2014 EXPERIMENTAL STUDY REGARDING THE BEHAVIOR OF GLUE LAMINATED BEAMS DOUBLE REINFORCED WITH RECTANGULAR METAL