effects of shear coupling on shear properties of wood

8/13/2019 Effects of Shear Coupling on Shear Properties of Wood

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EFFECTS OF SHEAR COUPLING ON SHEAR PROPERTIES OF WOOD

Jen Y. LiuResearch General Engineer

U.S. Department of Agriculture, Forest ServiceForest Products Laboratory

One Gifford Pinchot DriveMadison, WI 53705-2398

(Received October 1999)

ABSTRACT

Under pure shear loading, an off-axis element of orthotropic material such as pure wood undergoesboth shear and normal deformations. The ratio of the shear strain to a normal strain is defined as theshear coupling coefficient associated with the direction of the normal strain. The effects of shear

coupling on shear properties of wood as predicted by the orthotropic elasticity theory were validatedusing our recently developed shear test fixture. The validation also serves to demonstrate that theshear test fixture possesses the capability to introduce a state of pure shear to the critical section of aspecimen as required and that orthotropic elasticity theory should be used to describe the mechanical

properties of pure wood.

Keywords: Orthotropic elasticity, shear coupling, shear modulus, shear strength, shear test.

INTRODUCTION

For a two-dimensional square element oforthotropic material such as clear wood with

its grain oriented obliquely with respect to theedges, a pure shear loading applied to the edg-es will induce two normal strains perpendic-ular to the edges. The ratio of the shear strainto a normal strain is defined as the shear cou-

pling coefficient associated with the axis of thenormal strain (Tsai and Hahn 1980). Likewise,a normal loading applied to two opposite edg-es will induce a shear strain to distort the el-ement. The ratio of the normal strain in the

loading direction to the shear strain is definedas the normal coupling coefficient (Tsai andHahn 1980). Some authors have made no dif-ferentiation between the two and have calledthe latter phenomenon shear coupling (due tonormal loading) (Daniel and Ishai 1994; Pin-dera and Herakovich 1986), or just couplingeffects (Pierron et al. 1998).

The theory of orthotropic elasticity has beenfully developed (Jones 1975; Tsai and Hahn

1The Forest Products Laboratory is maintained in co-

operation with the University of Wisconsin. This articlewas written and prepared by U.S. Government employeeson official time, and it is therefore in the public domainand not subject to copyright.

Wood and Fiber Science, 32(4), 2000, pp. 458-465

1980; Daniel and Ishai 1994), and equationsfor the coefficients of normal coupling andshear coupling are well documented in it. Inthe experimental aspect, observation of the ef-

fects of normal coupling requires only a sim-ple tension test with a specimen of a long stripof wood. When the grain is aligned with theloading direction, the specimen will elongatein the same direction and contract in the trans-verse direction due to Poissons ratio. Whenthe grain is oblique, the specimen will distortinto a curved shape while being stretched dueto the normal coupling coefficient (Pinderaand Herakovich 1986). To observe the effectsof shear coupling, we have used a recently de-veloped shear test fixture (Liu et al. 1999),which is capable of determining the shearstrength and shear modulus of clear andstraight-grained wood specimens.

The current shear strength values of clearwood (Forest Products Laboratory 1999) were

based on the shear block test (ASTM 1978),that actually does not give the shear strength

because of high stress concentrations (Youngs1957). The test cannot determine the shearmodulus, nor can it be used to study the effectof grain slope, a parameter of special impor-tance in the mechanics of orthotropic materi-


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Liu EFFECTS OF SHEAR COUPLING ON WOOD 459

are axial, transverse, and shearstrains, respectively,are components of compliancematrix, and

are axial, transverse, and shearstresses, respectively.

Unlike the matrix for isotropic materials, thecompliance matrix for orthotropic materials isfully populated. A normal stress will induce anormal strain in the same direction, anothernormal strain in the transverse direction, anda shear strain; a shear stress will induce ashear strain and two normal strains.

als. The current shear modulus values of clear Components of the compliance matrix can

wood were obtained using the method bybe expressed in terms of the engineering con-

March et al. (1942). The method is called thestants (Daniel and Ishai 1994):

plate twist test (ASTM 1976). For solid wood,it is difficult to make plate specimens of prac-tical size with uniform properties. Also, the

plate twist test does not consider the effect ofgrain slope. These problems can all be solvedusing the shear test fixture recently developed

by Liu et al. (1999).In this paper, we present the results of a

small test program on shear coupling and com-pare them against analytical predictions. Theagreement of the comparison has demonstrat-ed not only that the effects of shear couplingon shear properties of wood are as predicted

by the orthotropic elasticity theory but, moresignificantly, that our shear test fixture is prop-erly designed to make this comparison suc-cessful.

STRESS-STRAIN RELATIONS

Let the 1-2 coordinate system represent theprincipal material axes and the x-y coordinatesystem the geometrical or loading axes withangle from the x to the 1 axis as shown inFig. 1. The stress-strain relations with respectto the x-y axes are expressed in the form

(1)

where

(2)

where

are Youngs moduli along the

x and y directions, respectively,is shear modulus in the x-y

plane,are Poissons ratios (first sub-script denotes loading directionand second subscript, strain di-rection), and

are normal and shear couplingcoefficients; normal = first twocoefficients and shear = last

two (first subscript denotesloading or loading directionand second subscript, strain orstrain direction).

From symmetry consideration of the com-pliance matrix, we obtain

(3)

Using the compliance transformation rela-tions, we obtain the following equations forthe transformed engineering constants (Danieland Ishai 1994):


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460 WOOD AND FIBER SCIENCE, OCTOBER 2000, V. 32(4)

(4)

where the basic engineering constants E1, E2 ,G12, and v12 are referred to the principal axesin Fig. 1. (Note: v1 2 /E1= v2 1 /E2.)

SHEAR TEST FIXTURE

The newly developed shear test fixture (Liuet al. 1999) uses the Iosipescu specimen of theV-notched beam specimen. The purpose of the

design was to introduce a pure shear state tothe critical section of the specimen during test-ing. Other shear test fixtures (e.g., Walrath andAdams 1983; Conant and Odom 1995) obvi-ously have the same purpose. The difference

lies in the extent to which a design can achieveits goal.

The test method for shear properties ofcomposite materials by Walrath and Adams(1983), later adopted by the American Societyfor Testing and Materials (ASTM 1993) as astandard, was found to cause specimen twist-ing, among other things, resulting in unevenstress distributions across the thickness of thespecimen. In their effort to correct the short-comings of the standard, Conant and Odom(1995) designed a series of prototype fixtures.They succeeded in eliminating the specimen

twist problem, but they restricted the fixturehalves to move only in the vertical directionduring testing. Such restriction introduces nor-mal stresses to the critical section of the spec-imen, adversely affecting the accuracy of theshear strength estimate, which should be a pri-mary goal of any shear test fixture.

Figure 2 presents the shear test fixture byLiu et al. (1999). The right upper and left low-er parts are the conventional antisymmetrical

halves. The right lower and left upper partsare the two controlling blocks. Each block isconnected to both halves of the fixture by two

pairs of shafts that move on ball bushings. Theleft upper block is connected to the left lower

part by two vertical shafts (Fig. 2A) and to theright upper part by two horizontal shafts (Fig.2B; only the front shaft is partially visible).The right lower block is similarly connectedto both halves of the fixture. Figure 3 shows

the tips of the two horizontal shafts that con-nect the right lower block to the left lower

Part.Two heavy plastic doors, which can slide in

the plane of Fig. 2, are used to mount anddismount the specimen. When closed, thedoors also help locate the specimen in the hor-izontal direction of Fig. 3. Location of thespecimen in this direction is also controlledthrough two knobs that are connected to steel

components (Figs. 3 and 4). The specimen iscentered in the fixture by two spring-con-trolled pegs that rest in the valleys of the V-notches when the specimen is properly locatedin the horizontal direction (Fig. 2). The load-


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FIG. 2. Shear test fixture with specimen in place (Liuet al. 1999).

ing devices rest on the top of the right upper

part and at the bottom of the left lower partand are controlled by screws (Fig. 2). Thesescrews must be tightened before the test.

The deflection of the spring between the fix-ture and the lower frame of the test machineis calibrated to support half of the total fixtureweight. At the start of a test, the initial loadon the specimen can be adjusted to be closeto zero. During testing, the lower frame of thetest machine moves downward, applying a

tensile load to the fixture, which is transmittedas shear loading on the critical section of thespecimen.

At the end of the test, the lower frame ofthe test machine returns to its initial position.

FIG. 3. Right side view of shear test fixture.

The screws controlling the loading devicesand the knobs controlling the alignment of thespecimen are released, the plastic doors are

opened, and the specimen is removed from thefixture.The test fixture operates like the Arcan

shear test (Arcan et al. 1978) but uses thespecimen geometry suggested by Iosipescu(1967) for efficient gripping and short turn-around time. The fixture has corrected the ten-dency to twist and misalign experienced by theArcan test and several versions of the Iosipes-cu test when specimens are not of homoge-

neous materials. There was no restriction toprevent relative movement of the two fixturehalves in the horizontal direction in the de-signs by Arcan and Iosipescu. Our shear testfixture has introduced none.


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462 WOOD AND FIBER SCIENCE, OCTOBER 2000, V. 32(4)

FIG. 4. Rear view of shear test fixture.

FIG. 5. Elasticity and shear moduli of Sitka spruceversus grain slope (E1= 11,800 MPa, E2 = 2,216 MPa,

G12 = 910 MPa, v12 = 0.37).

SPECIMENS AND TEST PROCEDURES

Specimen dimensions were as follows: totallength, 95 mm; width, 38 mm; thickness, 12.7

mm; depth of 90 notch, 9.5 mm; and widthof critical section between notches, 19 mm.The surface of the specimen was oriented inthe longitudinal-radial plane; the slope of thegrain was oriented at a specified angle to thecritical section. The specimen thickness wasin the tangential direction.

A board of Sitka spruce (Picea sitchensis)of unknown history but of the desired grainand growth ring orientations was selected

from storage at the Forest Products Labora-tory. The board was cut into five samples byslope of grain (0, 30, 45, 60, and 90). Eachsample consisted of 20 specimens. Before test-ing, the specimens were stored in a condition-ing room at 23C and 50% relative humidityfor several weeks until stabilized.

All specimens were tested for shearstrength. Tensile loading was applied with anInstron test machine, which pulled the left half

of the fixture downward while the right halfremained stationary. Crosshead speed was1.27 mm/mm. Load and crosshead displace-ment data were recorded electronically.

RESULTS AND DISCUSSION

We intended to use the shear test fixture tomake a comprehensive study of shear failureand grain slope relations. However, many

specimens did not fail at the critical section asexpected, making it impossible to conduct ameaningful statistical analysis. Nonetheless,all failures can be satisfactorily explained andthe study has yielded conclusive results con-cerning shear coupling, which have also ful-filled our study objectives.

In a previous study (Liu and Ross 1998)with the same material processed as in the

present study, the values of the basic engi-

neering constants in Eq. (4) were found to beas follows: E1 = 11,800 MPa, E2 = 2,216MPa, G12 = 910 MPa, and v12 = 0.37. Withthese values as input, we obtained the numer-ical results plotted in Figs. 5 and 6 from Eq.


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FIG. 6. Poissons ratio and shear coupling coefficientof Sitka spruce versus grain slope.

(4). From these figures, results for the othertransformed engineering constants can readily

be derived. These figures and equations facil-

itate the analysis of the test results. (5)Our test results can be demonstrated using

the two specimens in Fig. 7. If the lower spec-imen is flipped over from right to left or from

bottom to top, the two specimens are geomet-rically identical. As shown in the figure, the

upper specimen has a grain slope of 30, andthe lower specimen has a grain slope of 30,with respect to the critical section between the

notches, which is on the x-axis. When testedin the shear test fixture, the upper specimenfailed at the critical section, but the lowerspecimen did not. In the latter case, as the ap-

plied load is increased, the grain of the spec-imen was squeezed, and the specimen bulgedout of plane at the critical section. The relativeapproach of the fixture halves in the horizontaldirection was readily visible. The specimen fi-nally failed at the upper right and lower left

edges away from the critical section as a resultof tensile stresses, as shown in Fig. 7. Thesame phenomenon was observed in all speci-mens with 0 or 90.

This phenomenon can be explained from

FIG. 7. Specimens with failure locations (top: = 30,

bottom: = 30).

the stress-strain relations in Eq. (2), which

gives

for pure shear loading. When the left half ofthe fixture is lowered with respect to the righthalf, the shear at the critical section of thespecimen s is negative (Tsai and Hahn 1980).With = 30, s x/Gxy = 2.67E 4MPa

- 1,

as calculated from Eqs. (4). Therefore, xispositive in Eq. (5). Likewise, we can show for = 30, y is also positive with s y/Gxy =

5.04E 5MPa

-1

. A positive strain corre-sponds to a positive stress. Therefore, the ten-sile stresses x and y due to the shear cou-

pling coefficients sx and syare acting togeth-er with the applied shear s at the critical sec-tion as shown in Fig. 8a to cause the specimento fail.

When = 30, the signs for sx and s ychange from negative to positive, but theirmagnitudes remain the same. As a result of s

or xy, x and y are both negative, as shownin Fig. 8b. The compressive stresses x and yare acting together with the applied shear s atthe critical section to delay the failure of thespecimen. In this case, as the applied load in-


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464 WOOD AND FIBER SCIENCE, OCTOBER 2000. V. 32(4)

FIG. 8. Response of material elements with grain slope under pure shear loading: (a) + , (b) .

creased, failure occurred elsewhere but not atthe critical section.

These results show that when the grainslope changes from positive to negative, the

induced normal stresses change from tensile tocompressive due to a change in sign of theshear coupling coefficients. This results in in-creased strength of the material to resist theapplied shear loading, which is in line with theTsai-Wu strength criterion (Daniel and Ishai1994). Although we were not able to show themagnitudes of the compressive stresses thatcause failure together with the applied shearloading as described above, the trend of

change in material shear resistance was evi-dent.

These results also demonstrate that for anyshear test fixture, the two halves of the fixturemust be allowed to move horizontally whileone is pulling away from the other in the ver-tical direction so that pure shear can occur atthe critical section of the specimen. Evenwhen = 0 where both sxand sy, are zero,the relative horizontal movement of the two

parts cannot be avoided during testing unlessthe material of the specimen is extremely brit-tle and its shear strength is very low.

While the shear test fixture can only deter-mine the shear strength at = 0, it can de-

termine the shear modulus Gxyat any value of. Referring to Fig. 1, we obtain (Jones 1975)

(6)

where ' is the angle from the x to the X' axis.By setting ' = 45, it follows

(7)

Equations (6) are derived from the laws ofstress and strain transformation, which are in-dependent of and any other material prop-erties. Therefore, we can obtain s using the45 strain rosette or the shear gage by Ifju(1994) based on Eq. (7). With s known forany value of in Fig. 1, the corresponding Gxycan be obtained. The data for Gxy in Fig. 5were based on Eqs. (4) and validated experi-mentally using the shear gage (Liu and Ross1998).

CONCLUSIONS1. The shear coupling effects demonstrated in

the present study clearly show that any im-provements over the Arcan and the Iosi-pescu shear test fixtures must observe the


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2.

3.

requirements that the fixture halves be an-tisymmetric and the relative movement inthe horizontal direction be unrestricted dur-ing testing.For orthotropic materials such as wood, theshear test fixture can determine the shearstrength with respect to a principal materialaxis in a principal material plane. In anyother directions in the same plane, shearcoupling coefficients will introduce normalstresses to the critical section of the speci-men together with the applied shear. How-ever, in such situations, the shear modulus

can still be determined using the shear testfixture.Shear coupling is a direct consequence ofgrain slope, as demonstrated in the study

presented here. In any mechanical designof structural wood members, the effect ofgrain slope on stress development should

be taken into consideration.

REFERENCES

effects of shear coupling on shear properties of wood

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