EFFECT OF ELLIPTIC OR CIRCULAR HOLES 

ON THE STRESS DISTRIBUTION IN PLATES 

OF WOOD OR PLYWOOD CONSIDERED 

AS ORTHOTROPIC MATERIALS 

Information Revied and Reaffirmed 

March 1956 

No. 1510 
EFFECT OF ELLIPTIC OR CIRCULAR HOLES ON THE 

STRESS DISTRIBUTION IN PLATES OF WOOD OR PLYWOOD CONSIDERED AS ORTHOTROPIC MATERIALS1 

By 

Forest Products Laboratory, 2 Forest Service 

U. S. Department of Agiculture 

- - - - -

Abstract 

This is a mathematical analysis of the stress distribution existing near 

a hole in a wood or plywood plate subjected to tension, as, for example, 

near holes in the tension flanges of wood box beams. It is assumed that 

the strains are small and remain within the proportional. limit. In this 

analysis a large, rectangular, orthotropic plate with a small, elliptic 

hole at the center is subjected to a uniform tension along two opposite 

edger. By taking equal axes for the elliptic hole, the theoretical stress 

distribution in the neighborhood of a circular hole is obtained. The 

analysis shows that for a circular hole in a plain-sawn plate of Sitka 

spruce a maximum tensile stress of 5.84S is attained, S being the uniform tension applied in the direction of the grain. This maximum stress 

occurs on the edge of the circular hole at the ends of the diameter parpendicular to the direction of the uniform tension. Although this stress 

concentration may be greater than that found in isotropic materials, it 

is much more sharply localized and is relieved by plastic flow when the 

stress exceeds the proportional limit. For the reader who does not 

wish to follow the mathematical details of this report, the formula for 

computing this maximum stress for wood or plywood and an explanation 

of the constants involved is given in the appendix. The formula has not 

1 This progress report was originally preparted by the Forest Products 

Laboratory in 1944. 

2 Maintained at Madison, Wis., in cooperation with the University of 

Wisconsin. 

Report No. 1510 -1- Agriculture-Madison 
been confirmed by an extensive series of tests, but it was found to be in 

reasonably satisfactory agreement with the only test made. It is also 

demonstrated that for the Sitka spruce plate mentioned the maximum 

shear stress attained is 0.71S (fig. 5). The stress function from which 

these formulas for the orthotropic case are obtained is shown to reduce 

to the well-known stress function for the isotropic case. 

Introduction 

An orthotropic material possesses three planes of elastic symmetry at 

right angles to each other. If from such a material a flat plate is cut 

parallel to a plane of symmetry, it will have two perpendicular axes of 

symmetry in the plane of the plate. In the discussion that follows, a 

large rectangular orthotropic plate of this type with its edges parallel to 

the axes of symmetry and with a small hole at its center will be considered. A uniform tension S will be taken to act on two opposite edges of 

the plate as shown in figure 1, and the resulting stress distribution will 

be obtained. Throughout the report it is assumed that the strains are 

small and remain within the limits of "perfect elasticity." 

The solution is first derived for an elliptic hole with major and minor 

axes coinciding with the axes of symmetry. By taking the major and 

minor axes equal, the solution for a circular hole is obtained. For the 

orthotropic plate, the solution for the circular hole is but slightly more 

simple than that for the elliptic hole. This is different from the corresponding problem for the isotropic plate.3 The stress-strain distribution 

in the neighborhood of the circular hole is notably different in an orthotropic plate than in an isotropic plate as the concentration may be considerably higher and of a more localized character for the orthotropic 

than for the isotropic plate. Thus in a plate of plain-sawn Sitka spruce 

with a uniform tension acting in the grain direction, the greatest stress 

component at the edge of the hole decreases rapidly to one-half of its 

maximum value in a radial distance of less than one-tenth of the radius. 

For the isotropic plate the corresponding stress component decreases 

to one-half of its maximum value in a radial distance larger than onehalf of the radius from the edge of the hole. 

3 Timoshenko, S. Theory of Elasticity. 

Report No. 1510 -2-
The Effect of Eliptic or Circular Holes on 

the Stress Distribution in a Wood Plate 

The components of stress and strain in a plate under plane stress are 

connected by the following relations4 if the axes of x and y are taken as 

the axes of elastic symmetry of the orthotropic plate: 

(1) 

In these equations Ex and Ey are Young's moduli in the x- and y -directions, respectively. Poisson's ratio σ 

xy 

is the ratio of the contraction 

parallel to the y -axis to the extension parallel to the x -axis associated 

with a tension parallel. to the x -axis and similarly for σ yx. The quantity 

μxy is the modulus of rigidity associated with the directions of x and y. 

Since the equations of equilibrium are deduced with no reference to the 

law connecting stresses and strainns, they will be satisfied by a stress 

function F such that 

(2) 

4 March, H.W., Stress-Strain Relations in Wood and Plywood Considered as Orthotropic Materials, U. S. Forest Products Laboratory 

Report No. 1503. 

Report No. 1510 -3-
Substituting (2) in (1) and then making use of the compatibility equation 

the following differential equation, satisfied by the stress function F, is 

obtained: 

In deriving (3), the relation 4 

has been used. 

It is convenient to rewrite equation (3) 

where 

by making the substitution 

(3) 

(4) 

as 5 

(5) 

(6) 

(7) 

5 March, H, W., Flat Plates of Plywood Under Uniform or Concentrated 

Loads, U. S. Forest Products Laboratory Report No. 1312. 

Report No, 1510 -4 -
where 

(8) 

Substituting F = F(x + vη) in (5) leads to the following equation for v: 

(9) 

or 

(10) 

Now let6 

(11) 

then 

6 For wood or plywood, k as defined here is probably always greater 

than 1. 

Report No. 1510 - 5 -
Hence, the roots of equation (9) are 

Equation (5) may now be written as 

(12) 

where 

(13) 

The solution of equation (13) is facilitated by taking F of the form 

(14) 

where 

(15) 

Fo ( x, η) is any particular solution of (12), and the letter R means that 

the real part of the expression in the bracket is to be taken. That (14) 

is a solution of (12) is easily established by direct substitution. Equations (12) and (14) hold for the stress function associated with any elastic 

problem of plane stress in an orthotropic plate with x- and y- axes 

chosen as described here. 

Report No. 1510 -6 -
A suitable choice of F for the problem discussed is 

(16) 

where 

(17) 

(18) 

the boundary of the hole (fig. 1) is given by 

(19) 

and S is the uniform tension applied at the ends of the plate. 

In order that the stresses may be single-valued, the square roots in (17) 

are to be chosen so that the following relations always hold: 

Report No. 1510 - 7 -
Then 

(20) 

and 

(21) 

On the boundary of the hole x and η are given by (19), so that 

(22) 

(23) 

Similarly, on the edge of the hole 

(24) 

Consider the boundary of the hole in figure 2, and orient the positive 

direction of the normal v and of the tangent τ relative to one another in 

a way similar to the orientation of the x- and y-axis, respectively. 

Report No. 1510 -8 -
Let Xv ds and Y

v ds represent and x - and y -components of the force per 

unit thickness of the plate, acting on the element of arc ds of the boundary 

from the positive side indicated by the direction of positive v. Then 

But 

(25) 

so that 

(26) 

Report No. 1510 -9 -
Equations (26) for a boundary on which no stresses act take the form 

Hence, and are constant along a free boundry. 

Then, on the boundary of the hole, 

(27) 

Substituting (23) and (24) in (20) and (21), it follows from (27) that 

and (28) 

Taking the real parts 

(29) 

These equations must be satisfied for any value of θ, hence 

Report No. 1510 - 10 -
and 

Therefore, 

(30) 

By using equations (30), the stress components are found to be 

(31) 

(32) 

(33) 

To obtain the stress components for a circular hole, it is only necessary 

to let b = a in equations (31), (32), and (33). Then 

Report No. 1510 -11 -
(34) 

(35) 

(36) 

where 

(37) 

Expressions (31), (32), (33) are complicated and do not permit easy 

computation of the numerical values of the stresses. The most important information they contain is, of course, in the neighborhood of the 

hole. 

Stresses at the Edge of the Hole 

From equations (22), (23), and (24), it is found that over the edge of the 

elliptic hole 

Report No. 1510 -12 -
(40) 

Since, by (15) αβ = 1, the real parts are given by 

(41) 

(42) 

(43) 

where 

(44) 

It is of interest to refer these stresses at the edge of the hole to a pair of 

axes at right angles to each other having the directions of the normal v 

and the tangent τ, respectively (fig. 2). In this coordinate system, the 

shearing stress and the tension (or compression) in the v-direction must 

vanish, as this is required by the boundary conditions. It is readily 

shown that T τ of equation (44) is the tension (or compression) in the tangential direction. The component of direct stress in the tangential direction is given by 

Report No. 1510 - 13 -
From (19) and (25), this becomes 

Substituting from (41), (42), and (43), the tangential stress is found to be 

identical with Tt of equation (44). If ε ≥ 1, it follows that at the end of 

the axis perpendicular to the direction of the tension, that is for θ = 90° 

(fig. 1), Tτ attains its maximum value, while at the end of the axis 

parallel to the tension, that is θ = 0°, T τ attains its minimum value. 

Then 

(45a) 

(46) 

For an isotropic material, it follows from (6), (8), (11), and (15) that 

α = β = ε = 1. Introducing these values, (45a) and (46) become S( 2b + 1) a 

and -S, respectively, as in the isotropic case.3 

Equation (45a) gives the tension at C (fig. 1) the highest stress occurring 

in the plate. Numerical values of this stress will be computed in the two 

1 cases b 

a = 50 and b a = 50 for a plain-sawn plate of Sitka spruce whose 

elastic moduli are given in table 1. The grain direction is taken parallel 

to the line θ = 0°. 

When b 

= 50, the tension at C (fig. 1) is 243.2 times the mean tension. 

a 

When b = 

1 , the tension at C (fig. 1) is 1.097 times the mean tension.

a 50 

Report No. 1510 - 14 -
The ellipse when b 

a = 50 would appear as a straight crack perpendicular 

to the grain which might be approximated by a break in the fibers, such 

as a compression failure. A very small tension applied to the plate 

across the crack would set up a tension at the ends of very high intensity. 

If this high tension at the ends caused the crack to tear, the increase in 

length would exaggerate the stress still further, and the crack would 

continue to spread. 

When b = 

1 the ellipse would appear as a straight crack running in the a 50 

direction of the grain or of the tension. Such a crack does not produce 

great local stress, a conclusion which is almost self-evident. 

If the hole is circular ( b = 1), equations (41), (42), (43), and (44) reduce 

to a 

where 

The stress at the end of the diameter perpendicular to the direction of 

the tension for a circular hole is given by 

(45b) 

For an isotropic plate ( α = β = ε =1) this reduces to 3S, as is well 

known,3 but for a wood plate, equation (45b) may give a stress as high 

as 5.84S (fig. 8). 

Report No. 1510 -15 -
Experimental Verification of Maximum Stress 

This maximum stress was experimentally verified by compressing a 

quarter-sawn yellow-poplar board containing a circular hole. The dimensions of the board and hole and the position of the 1/8-inch metalectric 

strain gages at P on the circumference of the hole and at Q at some distance from the hole are shown in figure 3. The gage measuring the maximum strain at P was placed inside the hole as shown in figure 3, since 

the maximum strain occurs just on the edge of the hole and dies off rapidly in the radial direction. The gage at Q is placed as far from the hole as 

is feasible for the specimen used. The gage length was assumed short 

enough to give a close approximation to the actual strain at the end of the 

diameter. Table 2 gives the experimental and the theoretical values of 

the strains at pints P and Q of figure 3. The theoretical strains were 

computed in terms of S from the stresses given by formulas (35) and (45b) 

using the elastic moduli of the yellow-poplar plate. These moduli were 

obtained from coupons cut from the specimen and are given in table 1. 

The experimental value of the ratio of the strain at P to the strain at Q 

(fig. 3) is about 16 percent less than the theoretical one. This difference 

is not considered serious as the mathematical analysis was based on the 

assumption that the dimensions of the plate were large as compared to 

the dimensions of the hole. Also the impossibility of measuring strain 

at a point acted to decrease the apparent ratio. It is important to note 

that the experimental ratio is considerably higher than the corresponding 

ratio existing for a circular hole in an isotropic plate. 

Shear Stress at the Edge of the Hole 

The expression (43) for the shear component at the edge of the hole is 

highly important when considering holes in wood plates. When the uniform tension is parrallel to the grain direction, this shear acts in the 

same direction, and its distribution may indicate where failure caused 

by shear will take place if the uniform tension is sufficiently high. This 

statement has experimental verification whenb 

= 1, as this type of fail- a 

ure occurred in wood box beams with small circular holes in their tension flanges tested at the U. S. Forest Products Laboratory (see fig. 4), 

The failure occurred in the vicinity of the point of maximum shear, which 

occurs theoretically, as shown in figure 5, at about θ = 78°. The shear 

stress at the edge of the hole was evaluated numerically when a 

b 

= 1 for 

a plain-sawn plate of Sitka spruce whose elastic moduli are given in 

Report No. 1510 -16 -
table 1. The results are given in table 3 for values of θ between 0° and 

90° when the line θ = 0° is taken to be parallel to the grain of the wood. 

The shear at other points on the circle may be easily filled in from the 

given values. Figure 5 illustrates the variation of X

y 

over the edge of 

the hole for spruce and an isotropic material. 

Stress Components at Points 

Along the x- and y-Axes 

Simple expressions will now be obtained for the stress components at 

points along the x- and y-axes. 

Writing equation (31) as 

and performing similar operations on equation (32) and (33), it is found 

that 

(47) 

These equations can be reduced considerably when x = 0 or y = 0. The 

resulting equations then readily show how the stress concentration dies 

off with increasing distance from the hole. 

Report No. 1510 - 17 -
> For x = 0, y = a it follows that 

(48) 

> and for y = 0, x = a

(49) 

When b = 1 the distances from the center of the hole may be expressed as multiples a 

of the radius by writing x1 = a 

x , y1 = y 

a 

in equations (48) and (49). Hence, the 

stress components at points along the positive y-axis for a circular hole are given by 

Report No. 1510 -18-
(50) 

and at points along the positive x-axis 

(51) 

Numerical values of the stress components Xx and Yy at points along the x- and yaxis when b 

= 1 are given in tables 4 and 5 for a plain-sawn plate of Sitka spruce. a 

The variation of these components is illustrated in figures 6, 7, 8, and 9, and the 

variation of the corresponding stress components for an isotropic material is also 

shown for comparison. It is evident from these figures that considerable differ ­ ences may exist between the stress distribution developed in the orthotropic plate 

as compared to the isotropic plate. In particular, figure 8 shows that the maximum 

stress developed in the spruce plate is nearly twice the maximum stress developed

in the isotropic plate. 

Report No. 1510 -19-
Displacements in the Neighborhood 

of a Circular Hole 

The expressions for the displacements in the neighborhood of a circular 

hole ( b = 1) will now be obtained. a 

Substituting (34) and (35) in the first two equations of (1), it is found that 

(52) 

Integration of equations (52), gives 

Report No. 1510 - 20 -
(53) 

These are the actual displacements except for the addition of a linear 

function 

spectively. 

of y and a linear function of x to the expressions for u and v, re­

+ ­

This is easily shown by deriving the expression for 

exy = and comparing 

to vanish by fixing the origin and eliminating 

it with the expression These additive functions may be made 

rotation about the origin. 

Next the real and imaginary parts of and of a similar expression for z 2 and w2 will be found. TO do this, it will be useful to write 

Let 

(54) 

where 

(55) 

Report No. 1510 - 21 -
(56) 

Then 

(57) 

Similarly 

(58) 

where 

(59) 

(60) 

To keep the displacements single-valued, Y1 and Y2 must always be 

taken so that the following relations hold: 

Report No. 1510 -22 -
Taking the real parts, equations (53) become 

(61) 

(62) 

These equations for the displacements would be used to check data given 

by strain gages of finite length, since the strains computed from the gage 

readings are average strains. 

Isotropic Stress Function as the Limiting Case 

of the Orthotropic Stress Function 

It is of interest to obtain from (16) the stress function for the corresponding problem when b 

= 1 for an isotropic material. a 

Substituting from (15) and (30), (16) becomes 

Report No. 1510 - 23 -
(63) 

For an isotropic material φ becomes equal to 0, and α, β, and ε each becomes equal to unity as sh own by equations (6), (8), (11), and (15). For 

these values, parts of equation (63) take the indeterminate form 0 

0 . 

Evaluating by differentiating both numerator and denominator with respect to φ (remembering that z1 , x2, γ1, and γ2 all contain φ ), It follows 

that 

Report No. 1510 -24 -
Report No. 1510 -25 -
Let φ = 0 and ε = 1. Then (64) takes the form 

(65) 

where 

On setting b = a , parts of equation (65) assume the indeterminate form 0 . 

On evaluating (65), it is found that for the circular hole, 

(66) 

(67) 

Since the constant 2a 2 log 2 will not appear in the stresses obtained 

from (67), it may be omitted. The real. part of (67) is in polar form: 

(68) 

Report No. 1510 - 26 -

0 
which is the well-known form for the stress function in the corresponding problem for the isotropic plate.3 

The Effect of Elliptic or Circular Holes on_. 

the Stress Distribution in a Plywood Plate 

A plywood panel having the grain direction of each ply perpendicular to 

that of the adjacent plies and having a symmetrical construction may be 

considered as an orthotropic plate. The strains in such plywood may be 

obtained from the average stresses (to be defined later), or the average 

stresses from the strains,4 by exactly the same relations as exist between the strains and stresses for an orthotropic plate if the apparent 

elastic moduli of the plywood are used as the elastic moduli of the orthotropic plate. The panel is assumed to remain flat under the application 

of the loading considered. 

The x-axis will be taken parallel to the grain direction of the face plies. 

The subscript 1 will be used to denote quantities associated with plies 

having grain direction parallel to the x-axis, and the subscript 2 will be 

used to denote quantities associated with plies having grain direction 

parallel to the y-axis. 

Let h

1 denote the total thickness of all plies having grain direction parallel. to the x-axis. 

Let h2 denote the total thickness of all plies having grain direction parallel to the y-axis, 

For plies having grain direction parallel to the x-axis, the following relations hold 

(69) 

Report No. 1510 -27 -
and for plies having grain direction parallel to the y-axis corrosponding 

equations hold if the subscripts are changed. 

Since the panel remains flat, it will be assumed that (exx )

1 

= (exx) 

2 . 

(eyy) 

1 

= (eyy) 

2 , (exy)

1 

= (exy) 

2 , and it is no longer necessary to write 

the strain components with subscripts. 

Solving equations (69) for the stress components gives 

(70) 

where 

Similar relations hold in the remaining plies if the subscripts are 

changed throughout. 

Assuming the plies are all rotary cut and from the choice of axes, the 

elastic constants have the following values in the respective plies: 

Report No. 1510 -28 -
After making these changes, equations (70) become 

(71) 

where 

Report NO. 1510 -29 -
It is now important to define the mean stress components Xx and Yy by 

the following equations: 

(72) 

where h = h

1 + h2 , the overall thickness of the plywood, and the relation 

ham been used. 

Equations (72) may be rewritten as follows: 

(73) 

where 

(74) 

Report No. 1510 - 30 -
Solving equations (73) for the strain components gives 

or 

(75) 

where 7 

(76) 

It is easily established that 

7

Since σ 

xy , σxy , etc. are usually small, Ex and Ey are approximately equal to Ea and Eb , respectively. 

Report No. 1510 -31 -
In plies with grain direction parallel to the x-axis, the conditions of 

equilibrium require 

and similarly in the remaining plies 

Then 

(77) 

(78) 

and 

Report No. 1510 - 32 -
(79) 

(80) 

Adding equations (77) and (79) and dividing by h, it is found that 

(81) 

where, since in all plies the relation 

holds, the mean stress component Xy is identical with the stress component Xy. 

Similarly equation (78) and (80) yield 

(82) 

Equations (81) and (82) will be satisfied by a stress function F, such that 

(83) 

Substituting (83) in (75) and then making use of the compatibility equation 

Report No. 1510 -33 -
the following differential equation satisfied by the stress function F is 

found: 

As in the treatment of equation (3), it will be convenient to rewrite this 

equation as 

(84) 

where 

(85) 

by making the substitution 

(86) 

where 

(87) 

The differential equations (84) and (5) are of the same form mathematically, and any solution of (5) is a solution of (84) if the appropriate constants are used. It is then possible by using equations (31), (32), and 

(33) to obtain the stress distribution in the neighborhood of an elliptic 

hole situated in the middle of a plywood plate subjected to a uniform tension, provided that the axes of the ellipse are parallel and perpendicular 

to the grain of the face plies and that the tension is applied parallel or 

perpendicular to the axes of the ellipse. 

Report No. 1510 -34 -
Hence, if the K and ε calculated from the apparent clastic: moduli of plywood are used, all of the preceding discussion for orthotropic plates 

containing elliptic holes holds for plywood panels. 

Numerical values for the stresses at various points along the x- and yaxes in the case b 

= 1 for rotary-cut Sitka spruce plywood having half

a 

the material with grain direction parallel to the x-axis are given in tables 

6 and 7. Figures 10, 11, 12, and 13 illustrate how these stresses vary in 

the neighborhood of the hole. 

The foregoing analysis for plywood is based on the assumption that all 

plies are rotary cut and of the same species of wood. When these 

assumptions are not fulfilled, it is easy to make suitable modifications 

in the analysis (see appendix). 

Report No. 1510 - 35 -
Appendix 

A plain-sawn or a quarter-sawn wood plate ox a plywood plate may be 

considered to have two perpendicular axes of elastic symmetry in the 

plane of the plate. The discussion given in this appendix applies to a 

rectangular plate of this type with its edges parallel to the axes of symmetry and with a small circular hole at the center. The x- and y-axes 

are taken as the axes of symmetry. A uniform tension S directed parallel to the x-axis is supposed to act on the two opposite edges of the plate, 

and the tensile stress on the edge of the hole at the end of the diameter 

perpendicular to the uniform tension is given by 

For wood, the constants that appear in this expression are defined a3 

follows: 

ε 

α 

β

e = natural logarithmic base 

Ex = Young's modulus in the x-direction 

E y = Young's modulus in the y-direction 

μxy = modulus of rigidity In the xy-plane 

Report No. 1510 - 36-
σ xy 

= the Poisson's ratio given by the ratio of the contraction 

parallel to the y-axis to the extension parallel to the xaxis associated with a tension parallel to the x-axis. 

For plywood, equations similar to those already given for wood define 

ε , α, β, φ , and K if the remaining constants are as subsequently defined. 

The plies will be considered to be numbered consecutively from the upper 

to the lower face. The elastic moduli in the ith ply will be denoted by 

(Ex) 

i 

, (Ey) i , (μxy )

i , and (σ xy) i . The thickness of the ith ply is hi . 

That of the plate is h. 

Then approximately4 

The shear parallel or perpendicular to direction of the uniform tension 

and on the edge of the circular hole is given in the mathematical part of 

the report by equation (43) if b is set equal to a. By substituting the constants defined previously, this shear stress may be plotted as a function 

of the polar coordinate θ (fig. 5), and its maximum value read from the 

curve. 

Report No. 1510 - 37 - .4-52 
Table 1. --Elastic moduli of Sitka spruce and yellow popular 1 

Species 

Sitka spruce 

Yellow-poplar 

1 The elastic moduli of Sitka spruce are average values obtained from a 

large number of tests made at the U.S. Forest Products Laboratory. The elastic moduli of yellow-poplar are values obtained from 

the one specimen used in the experimental verification of this report. 

2 Not used. 

Table 2. --Observed and theoretical values of the strains 

at the points P and Q of figure 3 

Type Strain at P Strain at Q Ratio of strain at 

P to strain at Q 

Inch per inch 

Observed 

Theoretical 

(computed from 

equations (35) 

and (45b)) 

Report No. 1510 
Report No. 1510 
Table 4. --Values of the stress components at points 

along the x-axis exterior to a circle of 

radius a and with center at the origin 

for a plain-sawn plate of Sitka spruce -

Report No. 1510 
Table 5.-- Values of the stress components at points 

along the y- axis exterior to a circle of 

radius a and with center at the origin 

for a plain-sawn plate of Sitka spruce 

Report No. 1510 
-- 

Table 6. --Values of the stress components at points 

along the x-axis exterior to a circle of 

radius a and with center at the origin for 

a rotary-cut plywood plate of Sitka spruce 

Table 7. values of the stress components at points 

along the y-axis exterior to a circle of 

radius a and with center at the origin for 

a rotary-cut plywood plate of Sitka spruce 

Report No. 1510 
Figure 1--Elliptic hole in orthotropic plate. 

Figure 2.--Orientationof positive normal and 

tangent on the boundary of the elliptic hole. 

Z M 54246 F 
Figure 3.--Circular hole in a quarter-sawn yellow- poplar board subjected to a uniform compression. 

Z M 54249 F 
Figure 4.--The tension flange of a loaded wood box beam showing shear failure caused by vertical holes. The beam is 

subjected to a lower load in A than in B. The small cracks 

occurred in the stress-coat, which was applied to the beam 

before testing. The small round objects paralleling the 

edge of the flange are Whittemore strain gage points. 

Z M 54250 F 
Figure 6.--Variation of the normal stress component χx at points along the x-axis exterior to a circular 

hole of radius a and with center at the origin for a plain-sawn plate of Sitka spruce and for an isotropic plate. 

Z M 54261 F 
Figure 7.--Variation of the normal stress component Yy at points along the x-axis exterior to a circular 

hole of radius a and with center at the origin for a plain-sawn plate of Sitka spruce and for an isotropic 

plate. 

ZM 54252 F 
Figure 8.--Variationof the normal stress component Xx at points along the y-axis exterior 

to a circular hole of radius a and with center at the origin for a plain-sawn plate of 

Sitka spruce and for an isotropic plate. 

Z M 54253 F 
Figure 9.--Variation of the normal stress component Yy at points along the y- axis exterior to a circular 

hole of radius a and with center at the origin for a plain-sawn plate of Sitka spruce and for an isotropic 

plate. 

Z M 54254 F 
Figure 10.--Variation of the normal stress component Figure 11.--Variation of the normal stress component

Xx at points along the x-axis exterior to a cir- Yy at points along the x-axis exterior to a circular hole of radius a with center at the cular hole of radius a and with center at the 

origin for a rotary-cut plywood plate of Sitka originfor a rotary-cut plywood plate of Sitka 

spruce spruce. 

Z M 54256 F 
Figure 12.--Variation of the normal stress component Xx at points along the y-axis exterior to a 

circular hole of radius a and with center at the origin for a rotary-cut plywood plate of Sitka 

spruce. 

Figure 13.--Variation of the normal stress component Yy at points along the y-axis exterior to a 

circular hole of radius a and with center at the origin for a rotary-cut plywood plate of Sitka 

spruce. 

Z M 54256 F