STRESS DISTRIBUTION AND FRACTURE CRITERION OF TAPERED WOOD BEAM

Transcription

1 STRESS DISTRIBUTION AND FRACTURE CRITERION OF TAPERED WOOD BEAM Hitoshi Kuwamura ABSTRACT: The stress distribution of a tapered beam was derived b the method of stress function in order to eliminate the errors in the conventional solution of elementar mechanics. Bending tests were conducted for ten specimens of tapered wood beams with two species of wood and different taper gradients ranging from 0. to. It was found that splitting fracture occurs when the combination of stresses in the taper edge satisfies the strength criterion of Norris or Tsai-Hill. KEYWORDS: Tapered wood beam, Splitting, Fracture criterion, Stress function INTRODUCTION 3 The load carring capacit of a tapered wood beam whose ultimate state is governed b splitting at the taper edge in tension still remains unpredictable due to the presence of two problems. One is that the exact stress distribution along the taper edge is unknown. It was figured out b the author [] that the conventional solution given b Maki and Kuenzi [] and later b Sawada and Maruama [3] is associated with an unacceptable level of errors because their elementar solution does not satisf the strain compatibilit as well as the boundar condition. The other is that the fracture criterion of splitting is unknown. There proposed several fracture criterions in the fields of wood and composite materials, such as Hankinson s law [], Norris s spherical formula [5] and ellipsoidal formula [6], and Tsai-Hill s formula [7,8]. This paper introduces the exact stress solution of tapered beams b the stress function method and demonstrates which formula is the best criterion for the splitting of tapered wood beams. STRESSES IN TAPERED BEAMS. DEFECTS IN CONVENTIONAL SOLUTION The distribution of stresses in a tapered beam was solved first b Maki and Kuenzi [] and later b Sawada and Maruama [3]. While their mathematical approaches are slightl different, both solutions are the same, and their formulae are adopted in American and Japanese manuals of wood engineering [9, 0]. The reason wh their Hitoshi Kuwamura, Professor of Department of Architecture, School of Engineering, The Universit of Toko, 7-3- Hongo, Bunko-ku, Toko , Japan. kuwamura@arch.t.u-toko.ac.jp O Figure : Tapered beam upper edge b h o r r A σ o r () h a Q r taper edge B solutions coincide is that both came from elementar mechanics assuming Bernoulli-Euler s hpothesis. Recentl, however, a doubt was placed on that elementar solution []. The elementar solution tells us that the maximum stress along the taper edge appears at the height as twice as the depth h o at the support. However, bending tests of tapered beams often provide an embarrassing result that splitting failure takes place at a height less than h o, or closer to the support. In fact, all of the ten specimens of tapered beams, which were tested b the Universit of Toko, were split at heights less than h o as demonstrated in this paper. Another question ma arise about the elementar solution: when the entire depth of a tapered beam is less than h o, where is the maximum stress observed along the taper? The elementar mechanics cannot answer to these questions. The elementar mechanics on the basis of Bernoulli- Euler s hpothesis provides the following radial normal stress along the taper edge of Figure. It is noted that the circumferential normal stress as well as the shear stress in the polar coordinate sstem is 0 along the taper edge, because the taper edge is a free surface.

2 6Q r r σ o r( ) = r r r o () sin cos r where σ r() = radial stress along the taper edge (MPa), Q = shear force per unit width of the beam (N/mm), = taper angle (rad), r = radial distance from the origin O to a point on the taper (mm), r o = radial distance from the origin O to the support A with the height h o, and r = radial distance from the origin O to the taper bottom B with the height h. It is easil found that the radial stress has a peak at r=r o or h=h o, after that it graduall diminishes to 0 asmptoticall towards an infinite radial distance. Obviousl, the above elementar solution has two problems. One is that it violates the compatibilit condition of strains. Suppose the points a and b, the intersections of a vertical line to the taper edge and the upper horizontal edge, respectivel, the displacements of a and b caused b bending be horizontal vectors. Thus, the point a is going to leave from the solid bod, which is not allowed. The other problem involved in the elementar solution is that it does not satisf the boundar condition at the bottom end B of the taper. There should be no stress at B due to the equilibrium of stresses, while the elementar solution gives a non-zero stress at B. The current manuals and specifications on timber engineering are aware of the above-mentioned defects involved in the elementar solution, and thus the impose some limitations in the use of the solution such as the taper slope be less than 0.. However, practitioners ma want to know how to calculate the stresses when the taper slope is steeper than the limit or when the entire beam depth is less than double the depth at the support.. EXACT SOLUTIONS.. Brief note on stress function In this paper, the exact solution for the stress distribution in a tapered beam is derived b means of Air s stress function. First, the bases of stress function analsis are summarized herein for readers. In the r-θ polar coordinate sstem in Figure, when the stress function φ satisfies both compatibilit and boundar conditions, the stresses are represented b the following equilibrium equations: φ = r r r θ φ σ r () = r φ σ θ (3) φ φ φ τ r θ = = () r θ r r θ r r θ where σ r and σ θ = normal stresses in the radial and tangential directions, respectivel, and τ rθ = shear stress. The signs of these stresses are defined such that the directions shown in Figure are positive. The compatibilit conditions are represented b the following biharmonic equation: φ = 0 where = r r r r θ The general solution of Equation (5), or a group of biharmonic functions was derived b Michell []. Thus, if we find a biharmonic function satisfing the boundar conditions of our tapered beam in the Michell s functions, Equations (), (3), and () give the exact stresses in the tapered beam. O x r o (5) The stresses in the x- coordinate sstem are given b the following equations through the well-known process of transformation: σ x = σ r cos θ σθ sin θ τ rθ sinθ cosθ (6) r A' A Q Figure : Tapered beam with infinite taper length σ = σ r sin θ σθ cos θ τ rθ sinθ cosθ (7) ( σ σ ) sinθ cosθ τ ( cos θ sin θ ) τ x = r θ rθ (8) Since σ θ =τ rθ =0 in the taper edge, the stresses σ x, σ, and τ x in the taper edge can be easil calculated onl from σ r b using the above equations... Solution for infinite taper For a tapered beam with an endless taper, that is, r is infinite, the stress function is represented b the following biharmonic function in the Michell s general functions []: rθ φ = a o θ a θ θ τ rθ sin( θ ) ( b cosθ b sin ) Q (9) From the boundar conditions, i.e., the taper edge is free, the moment resultant at the support A is zero, the shear resultant is Q everwhere and the axial resultant is zero everwhere, the coefficients a o, a, b, and b are determined, and then Equations (), (3), and () give the stress distribution in a beam having an infinitel long taper. The stress formula for this infinite taper will be included in the formula for the finite taper in the following article. σ θ σ r τ x σ σ r () σ x

3 ..3 Solution for finite taper In practice the taper is finite as shown in Figure, in which the end of the taper, i.e., the point B is free from an stress as previousl explained in the criticism on the elementar solution. Thus, an adequate term φ R must be added to the stress function of Equation (9). This additional stress function φ R must satisf two requirements. One is that the radial stress at B produced b φ R is equal to the negative of the radial stress produced b the stress function of Equation (9) in order to vanish the stress at B. The other is that the stresses produced b φ R do not ield an stress resultant, in other words, the stresses b φ R is self-equilibrating like the residual stresses in a bod as shown in Figure 3. Otherwise, the loading sstem will be alternated from the original set. Such stress function can be found in the Michell s functions as follows []: taper angles such as tan =0. and 0., because the values of λ r for small taper angles are ver large and then the effect of the boundar condition of zero-stress at the taper end B is limited to a small area around B. But the values of λ r for large taper angles are small, and consequentl the released stress extends to a larger O r o r A' A σ rb B B' where λ φr = r f ( θ ) (0) Figure 3: Self-equilibrating stress in finite taper ( λ ) θ c cos ( λ ) θ f ( θ ) = c cos () Table : Eigenvalues where c, c, and λ are complex coefficients. It can be verified that the resultant forces and moment produced b φ R are all zero, irrespective of the values of c, c, and λ, and then the stresses are self-equilibrating. The stresses produced b φ R must satisf the boundar condition that the upper horizontal edge and the taper edge are free, from which the following eigen-equation is derived in order to make c and c non-zero: λ sin sin λ = 0 () The coefficient λ satisfing the above equation is complex and then is written b: λ = λ r iλ i (3) There are infinite combinations of λ r and λ i, but the firstorder eigenvalue, i.e., the combination of minimum positive numbers is adopted herein, because the are most influential. The are given in Table. From the requirement that the radial stress at B given b φ R is equal to -σ rb where σ rb is the radial stress at B of the infinite taper, the coefficients c and c are determined, and then the radial stress along the taper given b φ R is represented b λr r r σ rr( ) = σ rb cos λi ln r r In the above equation, the periodical term, cos( ), can be neglected, because it has little contribution due to the ver rapid attenuation of the first term. Then the released stress can be simplified as follows: rr( ) = r r λ rb r σ σ () It is noticed from Table and Equation () that the attenuation of the released stress is ver rapid for small tan λr λi (rad) (deg) (real part) (imag. part) length from B. Finall, the radial stress along the taper edge for the finite taper is the sum of the stress given b the stress function of Equation (9) and the released stress of Equation (), and is summarized as follows []: σ r( ) = asin sin r r λr r r ( b cos b ) c Q (5)

4 where the coefficients are given b = r a o, sin b =, sin b ( tan ) = (6) sin sin and c = asin r r ( b cos b sin ) (7) It is noted that the stress for the infinite taper is given b omitting the third term, i.e., the term of released stress, in the bracket of Equation (5)... Comments on the stress solution Equation (5) is the exact solution for the tapered beam satisfing the strain compatibilit as well as the boundar conditions. The onl deficienc is that anisotrop of wood is not regarded, in other words, the solution is valid for isotropic beams. At this moment, the stress distribution of an anisotropic tapered beam is not theoreticall given. Finite element analsis b the author demonstrated a slight difference in the stress distribution between isotropic and orthotropic beams. For example, the maximum radial stress in the taper of an orthotropic beam whose x and axes are the L and R directions, respectivel, of wood is smaller than that of an isotropic beam b the order of 0%. Therefore, Equation (5) can be emploed with a minor error to wood beams of orthotropic elasticit. Another comment about the stresses in the vicinit of support A might be necessar. The stress function method gives non-zero stresses in the vertical section at the support, although their resultants in terms of horizontal force and bending moment are perfectl zero. This is because the section at the support is not assumed free, but is assumed continuous up to the origin O of the coordinate sstem. The geometrical detailing of the beam support is case b case in practice, and then the stresses at the support section are also case b case. However, as far as the stress resultants at the support are reserved, in this case horizontal force and bending moment are both zero and vertical shear force is Q, the stresses at a point with an adequate distance from the support are not influenced b the support geometr according to the Saint-Venant s principle. In addition, stress concentration ma appear in more or less degree depending on the configuration of the support. Thus, the stress solution derived herein should be emploed to the region with an adequate distance, sa more than h o, from the support. It might be practicall important to know the limit of the elementar solution provide b Maki and Kuenzi. From numerical calculations for various tapered beams, it was shown that the elementar solution has an unacceptable level of errors in the estimation of the stress along the taper when tan > 0. and/or h /h o < 3 []. But the elementar solution is alwas conservative, in other words, it gives a larger value of stresses along the taper than the exact solution. A numerical example is shown in Figure, in which the maximum radial stress and its location are compared between the elementar solution given b Equation () and the stress-function solution b Equation (5) for the cases h o =00mm, h =00, 300, and 00mm, and tan =0. to. It is observed that the error involved in the elementar solution is enhanced with the increase of the taper slope and the decrease of h /h o. exact to elementar of max radial stress σr() beam hight at max stress, h/ho h/ho= h/ho= (exact) h/ho= taper slope, tan h/ho=3 (exact) taper slope, tan Figure : Numerical comparison of elementar and exact solutions (top: maximum radial stress, bottom: location of maximum stress) 3 AVAILABLE FRACTURE CRITERIA 3. FRACTURE CRITERIA The real nature of fracture of composite materials represented b woods and various fiber-reinforced plastics are still unknown at this moment. Man formulae for the fracture criteria are proposed, but the are hpothetical. Thus when the formula is applied in practice, the applicabilit should be verified in advance for the issue of interest. Here, tpical stress criteria for the strength of orthotropic materials are examined for the splitting problem of tapered wood beams. Examined are Hankinson s formula that is well known in wood engineering, Norris s spherical formula that is adopted in American Wood Handbook, Norris s ellipsoidal formula 0.8 elementar solution h/ho= h/ho= (exact)

5 that was modified later b himself, and Tsai-Hill s formula that is popular in composite materials. 3. HANKINSON S FORMULA The strength of wood in the direction with an angle to the grain is formulated b Hankinson as follows []: Fk ( H ) Fx F =, n =.5,.0 n n F sin F cos x (8) where F k(h) = tensile strength in the direction with an angle to the x-axis after Hankinson, F x = tensile strength in the x-direction, and F = tensile strength in - direction. The x and axes are correspondent to the longitudinal (L) and radial (R) directions, respectivel, of wood and k =tan herein. It is noted that the effect of shear strength is not involved in the Hankinson s. The Hankinson s formula is commonl emploed in wood connections [9,0]. However, this formula was originall developed for compressive strength, where the power number was found, that is n=. Later research demonstrated that n= to.5 for compression, and n=.5 to.0 for tension [3]. Then, the power number of.5 and.0 is assumed here for the taper subjected to tensile stress. 3.3 NORRIS S FORMULAE Norris and McKinnon proposed the following strength criteria in the form of spherical equation for woods under bi-directional stresses [5]: σ σ τ x x = (9) F x F Fx where F x and F are the same as before, F x = shear strength in the x- plane, σ x and σ = normal stresses in x and axes, respectivel, and τ x = shear stress in x- plane. This formula is adopted in conjunction with the design of tapered beams in American wood manual [9], and was also emploed in the design of wood aircraft structures long time ago []. Substituting Equations (6), (7), and (8) with noticing σ θ =τ rθ =0 and replacing σ r with F k, we obtain, Fk ( N) cos Fx sin F sin cos Fx = (0) where F k(n) = tensile strength in the direction with an angle to the x-axis after Norris s Sphere. Later, Norris modified the spherical formula to the following ellipsoidal formula based on the idea of ield criterion of Henke and von Mises: σ σ τ x σ xσ x = () F x F Fx FxF This equation is transformed into the following equation for the strength of an inclined direction from the same procedure above mentioned: cos = Fk ( N ) Fx sin F sin F F x xf cos () where F k(n) = tensile strength in the direction with an angle to the x-axis after Norris s ellipsoid. 3. TSAI-HILL S FORMULA One of the most popular strength formula in the field of composite materials is the following proposed b Hill [7] and Tsai [8]: σ σ τ x σ xσ x = (3) Fx F Fx Fx The same mathematical procedure gives cos = Fk ( T H ) Fx sin F sin Fx Fx cos () where F k(t-h) = tensile strength in the direction with an angle to the x-axis after Tsai and Hill. EXPERIMENT. STRENGTH PROPERTIES OF WOODS Five timber plates of Japanese cedar and two of S-P-F were emploed to the tapered beams. Their strengths in the natural axis and related properties are summarized in Table. Each of the strengths given in the table is an average of three samples. It is noted that the subscripts x and denote the longitudinal and radial directions, Table : Properties of emploed wood Species Timber Fx F Fx Densit Moisture Band width of (MPa) (MPa) (MPa) (kg/m 3 ) content (%) annual ring (mm) J. cedar C J. cedar D J. cedar D J. cedar F3' J. cedar N S-P-F SPFA S-P-F SPFB

6 respectivel of the original log. Thus, x is replaced with L and is R in wood engineering.. TEST SETUP OF TAPERED BEAMS Bending tests of ten beams with various taper angles from 0. to in terms of tan were carried out. The test set-up is three-point bending or four-point offset bending as shown in Figure 5. The longitudinal axis and the vertical depth of the beam are aligned to the L and R directions of the wood. The test conditions of tapered beam specimens are summarized in Table 3 in the order of tan. It is noted that the shear force is / and /3 of P/ Figure 5: Set-up of bending test L P L/ L/ () three-point bending P double taper loading beam P/ P/3 P/3 L/3 L/3 L/3 P/3 L/3 L () four-point offset bending the applied load P, respectivel, and the Q in the Equations () and (5) for calculating the radial stress is further divided b the width of the beam..3 TEST RESULTS OF TAPERED BEAMS All of the beams failed in splitting fracture due to the sudden propagation of a crack initiated at the taper edge as shown in Figure 6. The maximum loads P m governed b splitting and the crack-start position h f of the fracture normalized b support depth h o are enumerated in the columns P m and h f /h o of Table. More details of the experiment are described in Japanese journal paper [5].. TAPER STRESS OF THE SPECIMENS It is noticed from Table that the value of h f /h o is less than.0, which suggests that the maximum stress along the taper does not appear at h f /h o =.0 as supposed from the Equation () of elementar mechanics. This experimental observation can be explained b the exact solution obtained from the stress function method. The Equation (5) gives the curves of the stress distribution along the taper of the 0 specimens at their maximum loads, which are shown in Figure 7. The right end of each curve is correspondent to the bottom end of the taper where the stress is zero due to the boundar condition. It is noticed that the peak of each curve is located at the position h/h o <.0. Especiall, the curves of specimen Nos., 7, 9, and 0 have peaks apparentl far from h/h o =.0, because in these specimens tan is much larger than 0. and/or h /h o is much less than 3. 5 PREDICTABILITY The maximum load carried b a tapered wood beam ma be predicted b equating the maximum radial stress along the taper with the tensile strength of the wood in Table 3: Conditions of tapered beam specimens Beam ID No. L Measured h o h Timber tan width Taper (mm) (mm) (mm) (mm) Test setup F3' single 3-point bending D double 3-point bending 3 N double 3-point bending D single -point offset bending 5 SPFB 0.5, single 3-point bending 6 F3' single 3-point bending 7 SPFA single 3-point bending 8 D single -point offset bending 9 C double 3-point bending 0 SPFA single 3-point bending

7 h o = h o h o = h o No. tan=0., スギ No.6 tan = 0., スギ h o = 5.88 h o h o = h o No. t a n = 0., スギ No.7 tan = 0.5, S P F h o = h o h o = 5.80 h o No.3 t a n = 0., スギ No.8 tan = 0.5, スギ h o = 5 h o = 0.6 h o. 5 h o No. t a n = 0., スギ No.9 tan = 0.9, スギ 0 0 h o = h o h o = 35.9 h o No.5 t a n = 0. 5, S P F No.0 tan =, S P F Figure 6: Splitting failure of tapered wood beams the direction of the same angle to the grain. The former, i.e., σ r() ma be calculated b Equation (5), and the latter, i.e., F k ma be b Equations (8), (0), (), or (). This operation gives the maximum or failure loads P m(h), P m(h), P m(n), P m(n), and P m(t-h), where (H) and (H) indicate Hankinson with n =.5 and.0, respectivel, (N) and (N) Norris spherical and ellipsoidal, respectivel, and (T-H) Tsai-Hill. These predictions are normalized b the experimental P m as shown in Table and Figure 8. Regarding the inevitable heterogeneit of wood, the demonstrate fair good agreements with the

8 radial stress, σ r (MPa) position along taper, h/ho 3.0 Figure 7: Distribution of the radial stress along the taper of the specimens on the basis of the stress-function analsis Table : Comparison of splitting failure loads between experiment and predictions b Hankinson, Norris, and Tsai-Hill formulas Experiment Hankinson Norris Tsai-Hill Beam ID No. Pm Pm(H) Pm(H) Pm(H) Pm(H) Pm(N) Pm(N) Pm(N) Pm(N) Pm(T-H) Timber tan hf/ho Pm(T-H) (kn) n=.5 n=.0 /Pm /Pm sphere ellipsoid /Pm /Pm /Pm F3' D N D SPFB F3' SPFA D C SPFA average test results except Hankinson with n =.0. Hankinson s formula seems out of date, because the effect of shear stress is not apparentl included in the equation and the value of the index n ma fluctuate in accordance with the degree of shearing effect. It ma be beneficial to know wh Norris s spherical formula, ellipsoidal formula, and Tsai-Hill s formula produce a ver small deference between their predicted loads. The reason is that the term /F x F in Norris s Ellipsoidal formula () and the term /F x in the Tsai- Hill formula () is much smaller than /F x and is negligible because the longitudinal strength F x is much larger than the shear strength F x as observed in Table. Thus, in such a case, i.e., F x >> F x, Norris s spherical formula (0) is good enough. 6 CONCLUSIONS This paper shows that the load carring capacit of tapered wood beams susceptible to split at the taper edge can be predicted when the strengths of the wood in its natural axes and the geometr of the beam are given. The prediction is based on the fact that the splitting failure takes place when the maximum of the radial tensile stress along the taper reaches the strength of the wood in the direction of the same angle to the grain as

9 the taper. The stress along the taper can be precisel calculated from the formula that was derived b stress function method. The strength in an angle to the grain at the fracture can be calculated from the stress criteria of Norris or Tsai-Hill. The maximum error in the predictions was found to be about 30% from the experiment on ten specimens of tapered wood beams whose taper slopes range from 0. to. Pm(H)/Pm.0.5 n=.0 Hankinson ACKNOWLEDGEMENT The author wishes to acknowledge the help of Dr. T. Ito, former assistant professor of the Universit of Toko, in the process of the experiment of tapered wood beams and also the help of Mr. Y. Ehara, former graduate student of the Universit of Toko, in the numerical simulation of tapered beams n= tan 0.8 REFERENCES [] Kuwamura H.: Stress distribution in tapered beam (Stud on steel-framed timber structures Part 3). Journal of Struct. Constr. Eng., AIJ, 7(635): 83-90, 009. [] Maki A. C., Kuenzi E. W.: Deflection and stresses of tapered wood beams. Research Paper, US Forest Service, 3: -53, 965. [3] Sawada M., Maruama T.: Studies on tapered wood beams Part. Flexural rigidit and strength of tapered laminated wood beams. Research Bulletins of the College Experiment Forests, Hokkaido Universit, Vol. 7, No.: 395-7, 970. [] Hankinson R. L.: Investigation of crushing strength of spruce at varing angles of grain. Air Service Information Circular, 59: 3-5, 9. [5] Norris C. B., McKinnon P. F.: Compression, tension and shear tests on ellow-poplar plwood panels of sizes that do not buckle with tests made at various angles to the face grain. Forest Products Laborator, Report No. 38: -6, 956. [6] Norris C. B.: Strength of orthotropic materials subjected to combined stresses. Forest Products Laborator, Report No. 86: -0, 96. [7] Hill R.: A theor of the ielding and plastic flow of anisotropic metals. Proceeding of the Roal Societ of London, ser. A, 93: 8-97, 98. [8] Tsai S. W.: Strength characteristics of composite materials. NASA Contractor Report, 96. [9] American Institute of Timber Construction: Timber Construction Manual (Fifth Edition). John Wile & Sons, Inc., Hoboken, Jew Jerse, 005. [0] Architectural Institute of Japan (AIJ): Standard for Structural Design of Timber Structures (Fourth Edition). Toko, 006. [] Kuwamura H.: Taper paradox of wood beams. Summaries of Technical Papers of Annual Meeting, Architectural Institute of Japan, C-: 85-86, 008. [] Michell J.H.: On the direct determination of stress in an elastic solid with application to the theor of plates. Proceedings of the London Mathematical Societ, Ser., Vol.3: 00-, 899. [3] Forest Products Laborator: Wood Handbook - Wood as an Engineering Material. General Pm(N)/Pm Pm(T-H)/Pm sphere ellipsoid tan 0.6 tan Norris Tsai-Hill Figure 8: Predictabilit of failure load b Hankinson, Norris, and Tsai-Hill Technical Report, FPL-GRT-3, U.S. Department of Agriculture, WI, 999. [] Arm-Nav-Civil Committee on Aircraft Design Criteria: Design of Wood Aircraft Structures. ANC Bulletin No. 8 (ANC-8), 9. [5] Kuwamura H.: Splitting criteria of wood induced from failure load of tapered beam (Stud on steelframed timber structures Part ). Journal of Struct. Constr. Eng., AIJ, 7(6): 95-30, 009.

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