Holz Roh Werkst (2007) 65: 223–229 DOI 10.1007/s00107-006-0159-z
ORIGINALARBEITEN · ORIGINALS
Strength of wood versus rate of testing A theoretical approach Lauge Fuglsang Nielsen
Published online: 18 November 2006 © Springer-Verlag 2006
Abstract Strength of wood is normally measured in ramp load experiments. Experience shows that strength increases with increasing rate of testing. This feature is considered theoretically in this paper. It is shown that the influence of testing rate is a phenomenon, which depends on the quality of the considered wood. Low quality wood shows lesser influence of testing rate. This observation agrees with the well-known statement made by Borg Madsen that weak wood subjected to a constant load, has a longer lifetime than strong wood. In general, the influence of testing rate on strength increases with increasing moisture content. This phenomenon applies irrespective of the considered wood quality such that the above-mentioned order of magnitude observations between low and high quality wood are kept.
h¨oherer Festigkeit. Im Allgemeinen nimmt der Einfluss der Belastungsgeschwindigkeit auf die Festigkeit mit h¨oherer Holzfeuchte zu, wobei der Einfluss der Holzqualit¨at unver¨andert erhalten bleibt. List of symbols The notations most frequently used in this note are listed below. Sometimes the sub/superscripts indicated are not used – only, however, when the proper meaning is obvious from the text associated. Sub/superscripts R D
Die Festigkeit von Holz in Abh¨angigkeit von der Belastungsgeschwindigkeit – ein theoretischer Ansatz Zusammenfassung Die Festigkeit von Holz wird in der Regel in Versuchen mit konstanter Belastungsgeschwindigkeit bestimmt. Aus Erfahrung ist bekannt, dass die Festigkeit mit zunehmender Belastungsgeschwindigkeit zunimmt. Dieses Ph¨anomen wird hier theoretisch untersucht. Es wird gezeigt, dass der Einfluss der Belastungsgeschwindigkeit von der Qualit¨at des Holzes abh¨angt. Bei Holz geringer Qualit¨at ist der Einfluss geringer. Dies stimmt mit der bekannten Feststellung von Borg Madsen u¨ berein, dass bei konstanter Langzeitbeanspruchung Holz geringerer Festigkeit eine gr¨oßere Belastungsdauer bis zum Bruch aufweist als Holz L. F. Nielsen (u) Department of Civil Engineering, Technical University of Denmark, 2800 Lyngby, Denmark e-mail:
[email protected]
Ramp load (increasing proportional with time) Dead load (constant)
General σL σCR FL = σCR /σ L σ SL = σ/σCR E t
Theoretical strength Short time strength (high testing rate) Strength level (Materials quality) Load (stress) Load (stress) level Young’s modulus Time in general
Creep (in damaged area) c(t) = (1 + (t/τ)b)/E Creep function b Creep power τ Relaxation time q = [(1 + b)(2 + b)/2]1/b Time shift parameter Defects δ δCR o
Damage opening Critical damage opening Damage size at t = 0
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= (t) CR κ = /o tS tCAT
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Damage size (half crack length) Critical damage size Damage ratio Time to start of damage propagation Time to catastrophic failure
1 Introduction The DVM-theory (Damaged Viscoelastic Material) was developed by Nielsen (1991, 1992, 2000) to predict the strength of wood subjected to static and variable loads. Because of the non-dimensional formulation of the theory it applies for a number of loading modes such as tension and bending, see for example Nielsen (1992). The quality of DVM-predictions has often been shown to be quite good. Examples are shown in Figs. 1 and 2 reproduced from Nielsen (2000, 2005a) with experimental data reproduced from the unique works of Hoffmeyer (1990, 2003) and Bach (1975) on ‘duration of load’ and fatigue, respectively. (The material parameters, strength level FL, creep power b, and relaxation time τ, are explained in the subsequent text and in the list of symbols). The DVM-theory is based on the mechanics of crack expansion in wood as a viscoelastic material. The basic (most ‘dangerous’) crack model is the one of Dugdale, illustrated in Fig. 3 and further explained in Nielsen (1992). In the DVM-theory a crack expands as outlined/explained in Fig. 4: Two stages characterize strength degradation in damaged materials: 1) The initial cracks start propagating at time t = t S ; 2) Propagating cracks cause catastrophic fail-
Fig. 2 Fatigue of clear spruce subjected to square wave compressive loading with stress levels SL = 0 − SL MAX parallel to grain. Strength level FL = 0.4, creep power b = 0.25, relaxation time τ = 1 day. Experimental data from Bach (1975) Abb. 2 Dauerfestigkeit von Fichtenholz bei rechteckf¨ormig pulsierender Druckbeanspruchung in Faserrichtung mit Spannungsniveau SL = 0 − SL max. Festigkeitsniveau FL = 0,4, Kriechexponent b = 0,25, Relaxationszeit τ = 1 Tag. Versuchsdaten von Bach (1975)
Fig. 3 Dugdale crack loaded perpendicular to crack plane Abb. 3 Senkrecht zur Rissebene belasteter Dugdale Riss
Fig. 1 Lifetime of spruce lumber in dead load bending. Strength level FL = 0.25, creep power b = 0.2, relaxation time τ = 25 days. Experimental data: Hoffmeyer (1990, 2003) Abb. 1 Belastungsdauer von Fichtenholz bis zum Bruch bei konstanter Biegebeanspruchung. Festigkeitsniveau FL = 0,25, Kriechexponent b = 0,2, Relaxationszeit τ = 25 Tage. Versuchsdaten aus Hoffmeyer (1990, 2003)
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ure at time t = tCAT where the rate of propagation becomes infinitely high. The quality (or strength level FL = σCR /σ L ) of the considered wood is quantified as short time strength1 of wood, σCR , relative to the theoretical strength, σ L , of (uncracked) wood material. This quantity can be predicted by the Dugdale load capacity graph presented in Fig. 5, see Nielsen (1991). We notice that a wood quality of FL = 0.25 1 Short time means very short time used for testing, see subsequent Section 2.4.
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Fig. 4 Stages of crack propagation. In the initial stage, t < t S , the crack has a constant length. Due to creep the thickness becomes thicker and thicker until the crack front opening becomes critical, δCR . Then, with a constant crack opening, the crack starts propagating until its length becomes critical, lCR , and the rate of propagation becomes infinite at t = tCAT Abb. 4 Stadien des Risswachstums. Im Anfangsstadium, t < ts , ist die Rissl¨ange konstant. Aufgrund von Kriechen wird der Riss im¨ mer breiter bis die Offnung an der Rissspitze kritisch wird, δCR . Mit dann folgender konstanter Riss¨offnung wird der Riss gr¨oßer bis er eine kritische L¨ange lCR erreicht hat und der Rissfortschritt bei t = tcat unendlich wird
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Fig. 6 Ramp load is load increasing proportional with time. Dead load is a constant load Abb. 6 Konstante Belastungsgeschwindigkeit bedeutet eine mit der Zeit proportional ansteigende Belastung. Bei der Dauerlast handelt es sich um eine konstante Belastung
ation time τ. b 1 t creep function c(t) = 1+ ⇒ (power − law creep) E τ b t C(t) = E ∗ c(t) = 1 + normalized creep function τ (1) Normally, see Nielsen (1991, 2000, 2005a), the creep power b is 1/5–1/4 . The relaxation time τ is 1–25 days depending on moisture content, loading mode (bending, compression, tension, perpendicular or parallel to grain). In general, τ decreases with increasing moisture content. 1.1 Scope of paper, notations, and material parameters
Fig. 5 Wood quality, FL, estimated from damage size, l, relative to the damage nucleus (inherent defect) d = 0.3 mm, see Nielsen (1991) Abb. 5 Holzqualit¨at, FL bestimmt aus der Schadensgr¨oße l bezogen auf die Ausgangssch¨adigung d = 0,3 mm (materialspezifische Vorsch¨adigung), siehe Nielsen (1991)
estimated for the analysis of Hoffmeyer’s data in Fig. 1 corresponds to high quality structural wood, while FL = 0.4 estimated for the analysis of Bach’s data in Fig. 2 corresponds to high quality clear wood. Viscoelasticity in damaged wood areas is characterized by the so-called Power-Law creep function described in Eq. 1 and further considered in Nielsen (2005a). This function is quantified by the creep power b and the relax-
The DVM model just outlined is used in this paper to develop strength results for wood subjected to ramp load, see Fig. 6, which simulates very well strength determination in practice. The DVM-expressions needed for this analysis are summarized in subsequent sections 2.1 and 2.2 with notations explained in the list of symbols. Among the more significant notations are: Stress level, SL = σ/σCR , with stress (σ) relative to short time strength (σCR ). The strength level FL = σCR /σ L has already been defined. More specific notations defining the load histories considered in this paper are explained in Fig. 6. The influence on strength of test rate, creep, and wood quality will be demonstrated in this paper. The solutions will be compared with dead load solutions previously developed by the author. If not otherwise indicated, a creep power of b = 0.2 and a relaxation time of τ = 25 days are used in this pa-
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per together with a strength level of FL = 0.25. These are the same quantities as were estimated for the analysis of Hoffmeyer’s lifetime data in Fig. 1.
2 Analysis 2.1 Time, t S , to start of damage propagation The following results on time to start (t S ) of damage propagation are reproduced from Nielsen (2005b). It is noticed that t S is independent of FL for both ramp and dead load tests. ⎧ 1/b ⎪ 1 ⎪ − 1 (dead load) ⎨ SL 2 tS 1/b (2) = D 1/b ⎪ τ ⎪ 1 ⎩ (2+b) ! −1 (ramp load) 2b! SL 2 R,S
Fig. 7 Lifetime of wood subjected to ramp load as described in Fig. 8 Abb. 7 Belastungsdauer von Holz bis zum Bruch bei konstanter Belastungsgeschwindigkeit, entsprechend Abb. 8
Remarks: We notice that both t S are proportional with τ – and also that dead load t S and ramp load t S are proportional to each other. For the so-called Maxwell materials (b = 1) ramp load t S is three times longer than the dead load t S . 2.2 Time, tCAT , to catastrophic failure As previously indicated a crack stops resting at t = t S . Then the crack starts moving until its rate of expansion becomes infinitely high at t = tCAT . The period of time under expansion (tCAT – t S ) can be calculated by Eq. 3 reproduced from Nielsen (1991, 2000).
1/b dt 8qτ 1/ κSL 2 − 1 = dκ (πFL)2 κSL 2 l 1 lifetime expires when damage ratio κ = = l0 SL 2 (3)
Fig. 8 Ramp load history as indicated in Fig. 7 Abb. 8 Verlauf der Abb. 7 zugrunde liegenden Belastung bei konstanter Belastungsgeschwindigkeit
Numerically:
1 8qτ 1 κSL 2 − 1 b ∆t = ∆κ κSL 2 (πFL)2 with ∆κ =
1 − 1 1000 (example) SL 2
Notice: For ramp load the above calculation is made from t = t S with SL = k ∗ t where k = SL R,S /t S = SL R /tCAT Now, total lifetime is determined as t S determined by Eq. 2 plus (tCAT − t S ) determined by Eq. 3. Examples of predicted ramp load lifetimes are shown in Fig. 7. An example of assumed ramp loads is shown in Fig. 8. For comparison the results of a dead load lifetime analysis (also with Eqs. 2 and 3) are shown in Fig. 9.
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Remark: We notice that Eq. 3 predicts (tCAT − t S ) to be proportional with τ. This means that total lifetime also becomes proportional with τ. It is noticed that wood quality (FL) does influence elapsed lifetime when damages are expanding. We re-call from the previous section that such influence does not apply for the time to start of crack propagation, t S . 2.3 Ramp strength versus strength level (wood quality) In practice it is of interest to know the influence of testing time on measured strength. Estimates on this feature can be made by the analysis performed in this paper. Some results are summarized in Figs. 10 and 11 with a relaxation time of τ = 25 days and τ = 1 day, respectively. According
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Fig. 9 Lifetime of wood subjected to dead load Abb. 9 Belastungsdauer von Holz bis zum Bruch bei konstanter Belastung
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Fig. 11 Strength obtained by ramp load tests on wet wood (τ = 1 day). Influence of wood quality and time used in experiment Abb. 11 Festigkeit von feuchtem Holz bei Versuchen mit konstanter Belastungsgeschwindigkeit (τ = 1 Tag). Einfluss der Holzqualit¨at und der Versuchsdauer
notice that σCR refers to the considered wood quality). The influence, just mentioned, of testing rate and wood quality on measured strength becomes stronger with increasing moisture content.
3 Ramp strength versus strength distribution
Fig. 10 Strength obtained by ramp load tests on dry wood (τ = 25 days). Influence of wood quality and time used in experiment Abb. 10 Festigkeit von trockenem Holz bei Versuchen mit konstanter Belastungsgeschwindigkeit (τ = 25 Tage). Einfluss der Holzqualit¨at und der Versuchsdauer
to Sect. 1 the latter τ represents a larger moisture content than the former τ. 2.4 Intermediate conclusion It is obvious from Figs. 10 and 11 that ‘short time strength’ (σCR ) for the considered wood can only be obtained theoretically using extremely fast tests. In practice, however, short time strength can be determined reasonably well by modifying the strength obtained in 5 minutes tests often used in practice. It is observed that strength results obtained in ramp load tests on structural wood are closer to σCR than similar results obtained from ramp load tests on clear wood (please
The theoretical solutions presented above with various strength levels (FL) can be related to strength and strength distributions as expressed by Eq. 4 with only one reference strength level. In order to reflect the most genuine (true, creep independent) strength properties the distribution function (σCR ) must be based on fast experiments (short time strengths). π
s σCR (ϕ) = σ + log E tan ϕ strength distribution 2 ⎧π ⎪ ⎨ϕ accumulated distribution with σ strength (at ϕ = 0.5) ⎪ ⎩ s standard deviation σC R (ϕ) FL(ϕ) = FL(0.5) strength level (quality) (4) σ¯ with reference strength level FL(0.5) at ϕ = 0.5. 3.1 Example For the purpose of demonstration we choose the strength distribution presented in Fig. 12. This distribution is an approximate description of the distribution applying for the wood population Q1 tested in the work of Hoffmeyer (1990) previously referred to. The influence of wood quality and rate of loading on ramp strength for a whole
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Fig. 12 Strength distribution according to Eq. 4. Dashed lines indicate how mean strength and standard deviation can be estimated from experimental data Abb. 12 Festigkeitsverteilung entsprechend Gl. 4. Die gestrichelten Linien geben an, wie die mittlere Festigkeit und Standardabweichung aus den Versuchdaten gesch¨atzt werden kann
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by Madsen (1992)) from his ramp load bending experiments on Douglas-Fir lumber boards: ‘ . . . the stronger boards show an increasing strength as the rate of stressing increases, but this effect becomes less pronounced for weaker boards’. Spencer (1979) and Madsen (1992) noticed as curiosum that very weak boards showed a decreasing strength with increasing rate of testing. In the present author’s opinion such behavior is hard to believe. The statement might very well be the result of difficulties showing up when experiments are performed on very low quality wood – and from reading the sensitive data from such tests. The effect of decreasing wood quality to slow down the effect of testing rate on measured strength is worthwhile noticing. Obviously, the theoretical reason for this phenomenon is the quality influence on the rate of crack propagation expressed by Eq. 3. The rate of crack propagation increases with increasing FL. Basically this observation is consistent with the well-known statement made by Madsen (1992) that weak wood subjected to constant loads has a longer lifetime than strong wood. In general, the influence of testing rate on strength increases with increasing moisture content. This phenomenon applies irrespective of the considered wood quality such that the above-mentioned order of magnitude observations between low and high quality wood are kept. Acknowledgement The author appreciates very much the support he was given by the “Danish Research Agency, project no. 2020-000017” to write this paper.
References Fig. 13 Ramp strength as related to testing time. A wood population is considered with the strength distribution described in Fig. 12 Abb. 13 Festigkeit in Abh¨angigkeit von der Versuchsdauer bei einer Holzgrundgesamtheit mit der in Abb. 12 beschriebenen Festigkeitsverteilung
wood population can now be calculated as explained in Sect. 2. The results are shown in Fig. 13. As previously indicated the creep parameters have been maintained at (b, τ) = (0.20, 25 days). The reference strength level is FL(0.5) = 0.25.
4 Conclusion and final remarks The observations made in this paper are consistent with the overall conclusions made by Spencer (1979) (commented
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Bach L (1975) Frequency-dependent fracture under pulsating loading. Build Mat Lab, Tech. Univ. Denmark, Tech. report 68, 1979, (presented at For Prod Res Soc Annual Meeting 1975, Portland, Oregon, USA) Hoffmeyer P (1990) Failure of wood as influenced by moisture and duration of load. Doctoral Dissertation. State University of New York, College of Environmental Science and Forestry Hoffmeyer P (2003) Strength under Long-Term Loading. In: Thelandersson S, Larsen HJ (eds) Timber Engineering. John Wiley & Sons, Ltd, New York Madsen B (1992) Structural Behavior of Timber, Timber Engineering Ltd., North Vancouver, BC, Canada Nielsen LF (1991) Lifetime, Residual Strength, and Quality of Wood and other viscoelastic materials. Holz Roh- Werkst 49: 451–455 Nielsen LF (1992) DVM-analysis of wood (Damaged Viscoelastic Material). In: Madsen B (1992) Structural Behavior of Timber, Chapter 6. Timber Engineering Ltd., North Vancouver, BC, Canada Nielsen LF (2000) Lifetime and residual strength of wood subjected to static and variable load, Part I: Introduction and analysis,
Holz Roh Werkst (2007) 65: 223–229 Part II: Applications and design. Holz Roh- Werkst 58:81–90, 141–152 Nielsen LF (2005a) On the influence of moisture and load variations on the strength behavior of wood’, presented at the international conference on Probabilistic models in timber engineering – tests, models, applications held in Arcachon, France, 8.–9. September 2005, Proceedings, Association ARBORA, France
229 Nielsen LF (2005b) Strength of wood versus rate of testing – a theoretical approach. Technical report R122, Department of Civil Engineering, Technical University of Denmark Spencer A (1979) Rate of loading effect in bending for Douglas-Fir Lumber. First International Conference on Wood Fracture, Banff, Alberta, August 1978, Proceedings, Forintek Canada Corp, Vancouver, Canada
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