U.S. FOREST SERVICE RESEARCH
PAPER FPL 34 SEPTEMBER 1965 U. S. DEPARTMENT OF AGRICULTURE
FOREST
PRODUCTS
LABORATORY
•
•
FOREST SERVICE
MADISON,
WIS.
DEFLECTION AND STRESSES OF TAPERED WOOD BEAMS SUMMARY Approximate mathematical relationships based on elementary Bernoulli-Euler
theory of bending are developed for the general cases of shear and vertical
stresses existing in flexural members with varying cross sections. The theore­
tical analysis was then substantiated by experimental evaluation on specific
beams having uniformly varying cross sections.
The investigation was expanded to determine the applicability of an inter­
action
formula in predicting the ultimate strength of tapered, timber bending
members. This study also showed good correlation with results received from
experimental
evaluation, considering the variability of the material involved.
Good correlations were also observed between the theoretical and observed
deflection
relationships
studied.
i CONTENTS
Page
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase 1.--Approximate Mathematical Relationships for Stresses in Tapered Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bending Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflection of Tapered Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase II.--Evaluation of Tapered Beams of Isotropic Material . . . . . . . . . . . Phase III.--Evaluation of Wood Tapered Beams. . . . . . . . . . . . . . . . . . . . . . . . . Phase IV.--Design Criteria for Tapered Beams . . . . . . . . . . . . . . . . . . . . . . . . Design Determined by Deflection Limitations. . . . . . . . . . . . . . . . . . . . . Design Determined by Stress Limitations . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix I.--Shear Stresses in a Beam of Rectangular Cross-Section Having a Variable Depth Along Its Length . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix II.--VerticalStresses in a Beam of Rectangular Cross-Section Having a Variable Depth Along Its Length . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 111.--Deflectionsof a Simply Supported. Double-TaperedBeam Under Concentrated Midspan Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflection Due to Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflection Due to Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix IV.--Deflections of a Simply Supported, Single-TaperedBeam
Under a Concentrated Load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix V.--Deflectionsof a Simply Supported, Haunched Beam Under Concentrated Midspan Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflection Due to Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflection Due to Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix VI.--Deflectionsof a Simply Supported. Double-TaperedBeam
Under Uniformly Distributed Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflection Due to Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflection Due to Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix VII.--Deflections of a Simply Supported. Single-TaperedBeam Under Uniformly Distributed Load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflection Due to Bending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii 1
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42
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50
50
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52
DEFLECTION AND STRESSES OF
TAPERED WOOD BEAMS
By
A. C. MAKI, Engineer
and
E. W KUENZI, Engineer
Forest Products Laboratory 1 , Forest Service
U.S. Department of Agriculture
INTRODUCTION
During recent years there has been some demand from the timber industry
and associated firms for adequate design criteria on flexural members with
varying cross section, It was believed that in tapered bending members there
were possibly stress concentrations existing at the tapered face, but the extent
of these stresses was unknown. One of the purposes of this study, therefore,
was to investigate the behavior of tapered beams of uniformly varying cross
section to determine to a certain degree the extent of these stress combinations
and their effect upon the ultimate strength of the beam.
To achieve this purpose, the study was conducted in four phases. Phase I
was concerned primarily with the development of an approximate mathematical
analysis relating the principal stresses (normal, vertical, and shear) existing
condition.
in a member under any external
Phase II consisted of substantiating the mathematical analysis of Phase I
by experimentally evaluating two specific cases, the double-tapered beam and
under a concentrated center load. The beams in this
the haunched beam,
phase were aluminum sandwich loaded on edge. It was believed that a comparison
of the theoretical analysis with experimental data from a beam of an isotropic
material would better substantiate the theory or reflect any error due to the
approximate analysis,
1 Maintained at Madison, Wis., in cooperation with the University of Wisconsin.
Phase III closely followed the format of Phase II, with the exception that only
one case was studied--that of symmetrical, double-tapered beams of Sitka
spruce, orthotropic in nature, This phase reflects the applicability of the theore­
tical analysis to timber beams in which material variability is encountered.
Deflection relationships were also compared, as well as the results of using
an interaction formula in determining the ultimate strength of the bending
member.
In Phase IV, specific cases are evaluated for stresses and deflections and
are presented in graph form, adaptable for design criteria purposes.
PHASE 1.--APPROXIMATE MATHEMATICAL RELATIONSHIPS
FOR STRESSES IN TAPERED BEAMS
Bending Stress
It has been standard engineering practice to analyze beams of variable cross
section, where the variations from uniformity are not too great, on the basis of
the Bernoulli-Euler theory of bending; that is, on the assumption that plane
sections before bending remain plane sections after bending. The result of this
theory is the familiar relationship:
f=
M
S
(1)
where f is the bending stress, M is the bending moment, and S is the section
modulus.
In applying this relationship in the analysis of nonuniform beams, problems
arise in determining the limits of taper to which formula (1) applies. If it is
realized that the tapered beam problems are analogous to wedge problems as
treated by Timoshenko,2 theory of elasticity relationships is available to deter­
mine these limits. For example, in an analysis of a wedge with an extreme
slope of 1:4, an error of only 1-1/2 percent would be encountered by using the
Bernoulli-Euler relationship.
For the range of slopes encountered in usual tapered beam problems, therefore,
it was assumed that the bending stress is given by the following formula:
12M y f =
x
2Timoshenko, S.
FPL 34
bh
3
h
2
(reference axis chosen as in fig. 1)
Theory of Elasticity. McGraw-Hill Book Co., New York, 1934.
2
(2)
where fx is the bending stress, M is the bending moment, b is the beam width,
h is the beam depth, and y is the distance from the nontapered surface to the
point where stress is fx ,
The neutral surface would then coincide with a plane containing the locus of
‘points of the centroids of the cross section and normal to the sides of the beam.
The orientation of the cross section was chosen to be normal to the nontapered
face of the beam. This permits an approximate analysis for determining shear
stress in a tapered beam, which is particularly suitable for timber beams, as
the reference axis then coincides with the natural wood axis for beams as
normally used.
From equation (2) it can be seen that the maximum value of bending stress
at any section, h, will occur at the extreme fibers or when y = 0 and y = h, and
is given by:
f =+
x
6M
2
bh
(3)
If both M and h are functions of x, as in tapered members, it is reasonable
to assume there may be a section in the beam at which an absolute maximum
value of fx will occur at its extreme fibers. This section can be found by taking
df
x
from equation (3) and equating it to zero, yielding:
dx
h
dM
dh
= 0
- 2M
dx
dx
Therefore the section has a depth h given by:
(4)
The value of f
x
at such a section is:
max
(5)
dh
dM
and
represent the shear force and slope of taper at the section,
dx
dx
respectively.
where
3
Shear Stress
The approximate mathematical treatment of the shear stress distribution at
3
any section in a tapered beam was formulated by Norris and its derivation is
presented in Appendix I. It is shown in general that:
(6)
where f
is the shear stress at any pointy (measured positive from the non­
xy
tapered face) in a section of breadth b, depth h, subjected to a moment M. The
dh
slope of the taper and the shear force at the section are represented by and
dx
dM
respectively.
dh
A sign convention was imposed on M and such that a positive moment
creates tensile normal stresses in the fibers near the tapered surface and a
positive slope is one in which h increases as x increases. A positive shear
stress, therefore, will be one such that a cut section will have a shear force
acting in the same direction as the measured y.
From equation (6) it can be seen that a component of shear stress can be
induced in a bending member by a change in section as well as by a change of
moment, Further inspection of equation (6) reveals interesting facts about the
shear stress distribution at various sections; for example:
y
(1) At h = 0, f xy = 0 for all sections and loadings. This relationship satisfies
the boundary condition for shear stress along the nontapered surface.
(2) There is a section where the shear stress distribution across the section
y
is linear; that is, f
is a function of only . This can be seen by equating:
xy
h
Therefore, the section at which this linear distribution occurs has a depth h
given by:
(7)
3 Norris,
FPL 34
B., Engineer, U.S. Forest Products Laboratory, Madison, Wis.
4
The shear stress at this section is given by the formula:
(8)
which has its largest value at
y
= 1, and is given by:
h
(9)
(3) There is also a section across which the shear stress f is a function
xy
2
of
alone. This occurs when
or at a section with depth h given by:
(10)
The shear stress at this section is given by the formula:
(11)
which has its largest value at
h
= 1, and is given by:
(12)
(4) The
shear
stress can also have a maximum within the section. This
maximum can be found by taking
from equation (6) and equating it to zero.
This gives the relationship:
(13)
which is the equation of the locus of points of maximum shear stress, realizing
that y <
The depth of section at which this equation intersects the taper can
5
be found by substituting the value of = 1 into the left-hand side, yielding the
h
relationship:
(14)
The maximum shear stress at this section is given by:
(15)
It is also of interest to determine from equation (13) the depth of section at
which the point of maximum shear stress occurs at the neutral axis, that is,
y
when = 1/2. Substituting this value in (13) yields:
h
M dh
=0
(16)
h dx
dh
¹ 0. Therefore the maximum shear stress
dx
will occur at the neutral axis at all sections where the moment is zero and have
a value
which is only true for M = 0 since
which can be determined by substitution of M = 0 and
For
h
y
= 1/2 in equation (6).
h
= 1, equation (6) reduces to:
(17)
If both M and h are functions of x, it is reasonable to assume there may be a
occurring somewhere along the taper. The section at
maximum value of f
xy
df
xy
which this maximum occurs can be found by taking
from equation (17)
dx
and equating it to zero. This results in:
(18)
FPL 34
6
and f
xy
has a maximum value:
(19)
The significance of these shear stress distributions can best be visualized
by referring to figure 1 where the expressions are evaluated for a particular
beam and represented graphically. For purposes of illustration, the particular
beam chosen was one in which the width was constant but the depth varied
uniformly, and the beam was subjected to loading such that a reaction
was
induced at one end. Such loading is representative of cantilever beams under
end load or simply supported beams under concentrated loads. For clarity and
purposes of illustration, the following computations are developed:
1.--Shear stress distributions for a tapered bean
If x is the distance from the reaction, the moment at any section is:
(20)
and
therefore:
7
If the beam depth at the reaction is noted as ho and the slope of taper by tanq;
then the depth at any section is:
h =ho + tanq;
dh
= tanq
dx
and consequently:
2
d h
=0
2
dx
Substitution of equations
(20)
}
(21)
and (21) into equations (2) and (6) results in
(22)
and
(23)
or
(23a)
At the reaction,
x = 0
and h = ho;
therefore, the shear stress is given by
(24)
which is maximum for y = 1/2 and has the value:
h
(25)
which is the familiar shear stress formula for straight beams. For convenience
in presenting the information derived here in graphic form, the abbreviation
can be made that:
(26)
and fxy = C α, where C
i
i
FPL 34
is determined at various beam cross sections.
8
Proceeding along the span from the
the equation for the locus of
points of maximum shear stress can be determined by substituting expres­
sions (20) and (21) in equation (13), yielding:
Considering a section at which the point of maximum shear stress lies within
h
o 7
the beam such as where
=
the point of maximum shear stress occurs at
h 8’
y = 6
and f at this point has a maximum value of:
xy
h 10
fxy =
63
α
80
The value of shear stress at the taper of this section is:
7
f
=
α
xy 16
Evaluating equation (14) for the particular case to determine the depth of
section at which the locus of points of maximum shear stress intersects the
taper, gives:
4
h= h
3 o
and the shear s t r e s s has a value at the taper or point of intersection of:
3
f =
α
xy 4
The depth of section at which the shear distribution is linear can be found by
substituting expressions (20) and (21) in equation (7), yielding:
3
h
2 o
The maximum shear stress at the taper has a value
h =
fxy =
8
α
9
By examination of equations (10) and (18), it can be seen that for the particular
case given by equations (20) and (21), the section at which the shear stress
y 2
distribution is a function of
alone will coincide with the section at which the
h
()
9
absolute
depth:
maximum shear stress
in the beam occurs.
This section will have a
h = 2ho
and the absolute maximum value of shear stress in the beam occurring at the
taper is:
f
=α
xy
It may also be of interest to examinethe shear stress distribution at a section
beyond the one at which the absolute maximum shear stress occurs. Such a
section can be chosen where
h > 2h o
For such sections, it is seen that there is a point of zero shear stress within
the section. This occurs at a value:
(27)
At h = 4h , for example, this becomes:
o
y
4
=
h 10
The maximum value of shear stress at the section still occurs at the taper and
y
= 1 yielding:
can be evaluated from equation (23) for
h
f = 3α
xy 4
Vertical
Stress
A method similar to that used by Norris in the determination of shear stress,
that is, considering the equilibrium of the beam element, can be used to determine
an approximate relationship for the vertical stress existing in the beam. This
analysis is presented in Appendix II. It was found that, in general, the vertical
stress is given by:
(28)
FPL 34
10 and at
and at
y
= 0:
h
y
= 1:
h
2
1 d M
f =
2
y b
dx
(29)
(30)
and this can be shown to be the maximum vertical stress at any section. To
find the section at which the absolute maximum f in the beam occurs at its
y
df
y
y
from equation (30) and equate to zero, yielding:
tapered edge ( = 1), take the
dx
h
(31)
The maximum vertical stress in the beam at this section then has a value:
(32)
If the expressions (28) to (32) are evaluated for the particular beam treated
in the discussion of shear stress, the relationship for fy becomes
(33)
The depth of section h at which the absolute maximum value of f occurs is
y
h = 2h
o
and the stress has a value:
In summarizing the preceding discussions, it was found that in general the
11
values of the bending, shear, and vertical stresses existing at the taper are given
by:
(34)
For a particular beam with uniformly varying depth, these relationships can be
written:
(34A)
(where tanθ represents the slope of taper).
It should be pointed out that the stresses under discussion here refer to the
components of these stresses with respect to the x-y coordinate system as
y
shown in figure 1. The stresses at the taper, that is, when = 1, can be trans­
h
formed to the x' - y' coordinate system by means of the transformation equations
(for the case of uniformly varying depth of cross section):
(35)
where it can be found that the shear and vertical stresses are zero, as expected
from boundary conditions.
Finally, for the particular beam as given by equations (20) and (21), it can be
found that the absolute maximum values of the bending, shear, andvertical
FPL 34
12
stresses in the beamoccur at the taper of a section with depth h given by h = 2ho ,
and have values:
(36)
Deflection of Tapered Beams
The deflection formulas for tapered beams were based on the original assump­
tion that plane sections before bending remain plane sections after bending. On
the basis of this assumption, the rate of change of slope of the elastic curve is
given by the familiar equation:
(37)
The equation for the elastic curve can then be obtained by integrating equation
(37) twice, realizing that both M and I are functions of x. The constants of
integration are determined from the beam's boundary conditions.
Timber beams, however, deflect significantly due to shear as well as bending.
The shear deflection relationships can perhaps be most conveniently determined
by the use of Castigliano’s theorem where the total elastic-strain energy due to
shear is given by:
(38)
and the shear deflection yq at the point of a dummy load q is:
(39)
is the shear stress, G the shear modulus, and dv the differential
xy
volume (dv = bdydx).
The deflection formulas for some of the most commonbeamproblems
encountered are derived and presented in Appendixes III, IV, V, VI, and VII.
where f
13
PHASE II.-EVALUATIONOF TAPERED BEAMS OF
ISOTROPIC MATERIAL
With the establishment of a theoretical analysis for tapered members, an
experimental evaluation was made to determine the applicability of such an
analysis, To achieve this purpose, a symmetrical double-tapered beam was con­
structed from a sandwich construction for which the elastic constants were
known, The sandwich panel was comprised of 0.064-inch, 2024T3 aluminum
facings bonded to a resin treated, cotton honeycomb core, The sandwich was
loaded on edge and it was assumed that the core's only function was to provide
stability to the facings. The experimental beam, with dimensions as presented
in figure 2, was center loaded and strain data were obtained from three SR-4
rosette type, electrical strain gages, which were placed at convenient heights
at a section of depth h = 2h ,
o
ZM 128 963
Figure 2.--Symmetrical aluminum, double-tapered beam--dimensions and gage locations.
FPL 34
A 0.0001-inch-deflection dial was stand mounted and placed so that the
maximum deflection at midspan was obtained.
The theoretical stress distributions at the section h = 2h were computed from
o
expressions (2), (6), and (33) and converted into a theoretical strain distribution
by the established relationships:
where ε , ε , and ε
are strains; f , f , f are stresses; E is the elastic
x y xy
x
y
xy
modulus; G is the shear modulus; and µ is Poisson's ratio. The property values
for the aluminum facings were assumed to be:
E = 10,400,000 pounds per square inch, G = 4,000,000 pounds per
square inch, and µ = 0.3.
These strain
observed strain
their appropriate
note that while
distributions are presented in figures 3, 4, and 5. The
readings from the experimental evaluation are also shown in
position to facilitate a direct comparison. It is interesting to
a compressive vertical stress fy acts at y = h, the resulting
strain is positive (in tension) due to Poisson's effects.
Figure 3.--Bending strain distribution for aluminum tapered beam under a midspan load of 1,000
ZM 128 971.
pounds.
15
Figure 4.--Vertical strain distribution for aluminum tapered beam under a midspan load of
ZM128953 1,000 pounds.
Figure 5--Shear strain distribution for aluminum tapered beam under a midspan load of 1,000
pounds.
16 The theoretical deflection relationship for this beam, as presented in Appen­
dix III, was also evaluated. The midspan deflection was calculated to be
inch, and the experimental deflection was measured as 0.110 inch at a midspan
load of 1,000 pounds. The shear deflection in this case amounted to approximately
7 percent, whereas usually for an isotropic material it would be neglected.
The verification of expressions (2), (6), and (33) was given a further test by
considering the stress distribution at a section in a haunched beam, as shown
in figure 6. In this type of simple beam the moment increases as the section
Figure 6.--Aluminum haunched beam dimensions and gage locations.
decreases over the region 0 < x <
ZM 128 972
In this region, f and therefore
x
fxy and f , by previous analogy, have no maximums within the physical structure
y
h
o
increases. This can be most readily
of the beam, but increase as the ratio
h
seen perhaps by rewriting the expressions for the normal, vertical, and shear
17
h
stresses
in
terms
of
o,
realizing that for the haunched beam of figure 6:
h
h = h - x tanθ
o
dh
= -tan θ
dx
(41)
M = -VX
dM
= -V
dx
Substitution of expressions (41) into the general expressions (2), (6), and (33)
derived in Phase I results in:
(42)
Evaluating equations (42) at
y
= 1 yields:
h
(43)
Typical distributions of these stresses at any section are presented in
figure 7, from which it can be seen that the shear stress existing at the taper
FPL 34
18
h
h
o
o
increases quite rapidly as the ratio
increases. For example, at
h = the
y
3V h
value for f
at = 1 is 8 .
or eight times the maximum shear stress
h
xy
2bh
o
occurring at the reaction.
Figure 7.--Typical shear andvertical stress distributions in a haunched beam under a concentrated
ZM 128 966
midspan load of 2V.
To substantiate this theoretical stress distribution in a haunched beam, a
procedure was followed identical to that described for the symmetrical doubletapered beam. A haunched beam, with dimensions as given in figure 6, was
constructed from the same sandwich material used previously. Rosette type,
h
o
SR-4 electrical strain gages were placed at a section where
= 2. Expressions
h
(42) were evaluated for this section for a midspan load of 2V = 250 pounds and
theoretical strain distributions were calculated by using equations (40), with
the elastic constants of the aluminum facing remaining the same as previously
used.
Figures 8, 9, and 10 present a comparison of the observed strain distributions,
as obtained by experimental evaluation, and the theoretical distributions.
19 Figure 8.--Bendiong strain distribution in aluminum haunched beam under a midspan load of 250 pounds.
ZM 128 969 Figure 9.--Vertical strain distribtuin for aluminum haunched beam under a midspan load of ZM 128 967 250 pounds.
FPL 34
20 Figure 10.--Shear strain distribution in aluminum haunched beam under a midspan load of 250
ZM 128 970
pounds.
With the good results achieved in evaluating these two different type beam
problems, it is believed that expressions (2), (6), and (33) do provide a close
approximation of the stress situation existing in tapered bending members, or
at least in those beams isotropic in nature and consisting of uniformly varying
cross section under concentrated loads.
The deflection expression for the haunched beam is derived in Appendix V.
The comparison between the observed and theoretical maximum beam deflection
at midspan was favorable. The theoretical value was calculated to be 0.0830 inch,
while an experimental value of 0.087 inch was observed at a midspan load of
250 pounds.
PHASE III.-EVALUATION OF WOOD TAPERED BEAMS The function of phase II was satisfied in that it substantiated the mathematical
treatment as applied to beams constructed from an isotropic material. The
purpose of phase III was to determine its applicability to anisotropic wood mem­
bers.
For evaluation, three beams were constructed from planks of Sitka spruce,
carefully chosen to be straight grained and free from defects, to obtain the
most reliable results possible. The elastic constants and properties of the beams
were taken to be:
21 where E , E , G , µ , and µ
are elastic properties as previously defined
xy
xy
x
yx
y
F , and F
and F , F , F
are strength properties; the subscripts x and
yt, yc
xt
xc
xy
y indicate parallel and perpendicular to grain, respectively, and the subscripts
t and c represent tension and compression properties, respectively. The strength
properties are an average of minor specimens4 taken from the excess material
used in the construction of the beams. The elastic properties agree closely
with those published in the Wood Handbook5 and presented in Forest Products
Laboratory reports.6, 7
The symmetrical double-tapered beam, which is one most commonly used,
was chosen as the beam to evaluate.
Since it was learned from previous discussion that the shear stress distribution
at a section, as given by equation (6), changes significantly as the distance from
the reaction increases (refer to fig. 1), it was thought advisable to evaluate
various sections.
Beam No. 1, therefore, was designed primarily to observe the strain distribu­
tion of three sections--where h1 = 1.25ho; h2 = 1.5ho; and h3 = 1.75ho. The
section h is located in the region 0 < x <
1
and, as a result, should have the
maximum shear stress and corresponding strain occurring within the beam.
Section h was selected for observation since, from the analysis, a linear shear
2
strain distribution was expected here, with the maximum occurring at the taper.
4Recommended ASTM Standards used in determining the wood properties of specimens evaluated.
5 U.S. Forest Products Laboratory. Wood handbook, U.S. Dept. Agr., Agr. Handb. No. 72, 528 pp.,
illus. 1955.
6Drow, J. T., and McBurney, R.S. The elastic properties of wood. Young’s moduli and Poisson’s
ratios of Sitka spruce and their relations to moisture content. Forest Products Lab. Rpt.
NO. 1528-A.13 pp., illus. 1946.
7Doyle, D.V., McBurney, R.S., and Drow, J.T. The elastic properties of wood. The moduli of
rigidity of Sitka spruce and their relations to moisture content. Forest Products Lab. Rpt.
No. 1528-B.7 pp., illus. 1946.
FPL 34
22
The. shear strain distribution at section h ,
3
between sections located at depths 1.5h < h <
o
was not evaluated for this particular beam
containing the concentrated load. The final
presented in figure 11.
2v
represents the typical distribution
2h , The critical section ( h = 2h )
o
o
c
since it coincided with the section
dimensions of beam No. 1 are
Figure 11. --Dimensions of experimental beams of Sitka spruce used in strain and strength evalua­
ZM 128 955
tion.
Beam No. 2 was designed primarily to investigate the critical section at
h = 2ho , where the absolute maximum values of the bending, shear, and vertical
stresses in the beam all occur at the taper. It was felt that the best results
L
would be obtained if this section coincided with the section at 4 (midway
between concentrated load and reaction). The beam had final dimensions as
presented in figure 11.
23
The data were again in the form of strain readings from SR-4 rosette type,
electrical strain gages placed at the various sections. Deflection readings at
midspan were obtained by observing the movement between a graduated scale
mounted at midspan and a thin wire stretched between points above the reaction.
The comparisons between theoretical and observed strain distributions at
the various sections are presented in figures 12 through 20.
Figure 12.--Comparison of theoretical and observed bending and vertical strains at sectionL of
8
beam No. 1 under a concentrated midspan load of 2V = 7,000 pounds,
ZM 128 956
Figure 13.--Comparisonof theoretical and
observed shear strain distributions at
section L of beam No. 1 under a
8
concentrated midspan load of
ZM 128 961
2V = 7,000pounds.
FPL 34
24 L
Figure 14.--Comparison of theoretical and observed bending and vertical strains at section 4 of beam No, 1 under a concentrated midspan load of 2V = 7,000 pounds.
ZM 128 960 L
Figure 15.--Comparison of theoretical and observed shear strain distributions at section of
4
beam No, 1 under a concentrated midspan load of 2V = 7,000 pounds.
ZM 128 968 25 3L
Figure 16.--Comparison of theoretical and observed bending and vertical strains at section
of
8
beam No. 1 under a concentrated midspan load of 2V =
pounds,
ZM 128 954
3L
Figure 17.--Comparison of theoretical and observed shear strain distributions at section
of
8
beam No. 1 under a concentrated midspan load of 2V = 7,000 pounds.
ZM 128 965
FPL 34
26
Figure 18.--Comparison of theoretical and observed bending (ε ) strain distribution at section x
L
of beam No. 2 under a concentrated midspan load of 2V = 3,500 pounds.
ZM 128 959 4
Figure 19.--Comparison of theoretical and observed vertical (ε ) strain distribution at section
y
L of beam No, 2 under a concentrated midspan load of 2V = 3,500 pounds.
ZM 128 958
4
27 Figure 20.--Comparison of theoretical and observed shear strain (ε ) distribution at section
xy
L of beam No, 2 under a concentrated midspan load of 2V = 3,500 pounds.
ZM 128 957
4
While perfect agreement between theoretical and experimental strains was
not achieved, it is believed that the tendency toward agreement is present and
should be the prime consideration. It is possible that the theoretical curves
could be manipulated to obtain closer correlation by choosing slightly different
values for the elastic constants.
The equation for the deflection at midspan for beams Nos. 1 and 2 can be
determined from the derivation presented in Appendix III. For beam No. 1,
a calculated value for maximum deflection of 0.383 inch was obtained as com­
pared to 0.386 inch obtained from experimental evaluation. Similarly, beam
No. 2 had a calculated maximum deflection of 0.383 inch and anobserved
experimental value of 0.376 inch. It should be noted that in both beams the
deflection due to shear amounted to approximately 27 percent of the bending
deflection and therefore cannot be ignored.
As a general conclusion, therefore, it is believed that expressions (2), (6),
and (33) very Closely approximate the stress situation existing in tapered timber
members with uniformly varying cross section and that the deflection relation­
ships, as derived in appendix III, represent the equation of the elastic curve.
FPL 34
28
Since it has been established that it is possible for bending, shear, and
vertical stresses to be combined at one point in the beam, the question arises
as to the possible effect of these combined stresses on the ultimate strength
of the member. In his work on orthotropic materials subjected to combined
stresses in a two-dimensional stress system, Norris8 proposed the use of an
interaction equation to determine the strength of the material. For the case of
a timber beam, such as treated in this report, subjected to bending, vertical,
and shear stresses at the taper, this equation would take the form:
(44) where f , f , and f are the bending, vertical, and shear stresses, respectively,
x y
xy
existing at some point and F , F , and F are their respective failing stresses,
xy
x
y
that is, the stress at which the member would fail were this stress existing
alone.
Since it was determined from previous discussions that the shear and vertical
stresses at the taper are related to the bending stress at the taper by the tanθ
2
and tan θ, respectively, the interaction relationship (44) can also be written:
(45) or
(46) where
(47) From these relationships, curves can easily be drawn for different species,
such as those shown in figure 21, for the Sitka spruce of which the tapered
beams were made.
Figure 21 contains the information in graph form to provide a comparison of
the actual observed beam strength with the strength predicted by the interaction
formula. Beam No. 3 in the figure was included in the study to provide an addi­
tional specimen for the evaluation of the effect of stress combinations when
occurring with the tapered surface on the tension side of the beam and had
8
Norris, C.B. Strength of orthotropic materials subjected to combined stresses. Forest Products
Lab. Rpt. No. 1816. 24 pp., illus,
29
ZM 128 983
as defined in equation
o
dimensions as presented in figure 11. On the basis of this limited comparison
it is believed that the interaction formula does provide an approximate method
for determining the ultimate strength of tapered beams.
Figure 21.--Comparison of theoretical and observed values of
PHASE IV.--DESIGN CRITERIA FOR TAPERED BEAMS
With the establishment that the mathematical treatments as presented in
Appendixes I through VII approximate the stress and deflection behavior occurring
in tapered beams, it is now desirable to combine these results in a manner
which might serve as a basis for design criteria.
Since either the stress or deflection limitations may be the critical factor
considered for design purposes, both of these factors will be investigated.
FPL 34
30 Design Determined by Deflection Limitations
For design purposes it is often desirable to have mathematical formulations,
which are time-consuming to evaluate, presented in graph form; therefore, the
bending relationships derived in Appendixes III, IV, VI, VII are handled in this
manner.
Figure 22, for example, can be used directly in determining the end depth h
o
or
double-tapered
beamunder
uniformly
distributed
of a simply supported, single1
h - h
for
load. If from design requirements, the span L, the slope of taper
o
L c
2 h
- h for double-tapered beams, the width b, the
single-tapered beams or
o
L c
deflection limitation ∆ , and the loading conditions are known, a value for γ can
B
be obtained from the graph, and a value for h calculated.
o
(
(
)
)
Figure 22.--Graphfor determining tapered
beam size based on deflection under
uniformly distributed load.
ZM 128 982
31 If it is desired to determine the maximum deflection due to bending in such a
1
h - h and then
beam, the process can be reversed by computing a γ where g =
o
h c
o
reading an ordinate value and solving for ∆ .
B
bending. It has been
Tapered beams, of course, deflect due to shear as well
found that the midspan shear deflectionof a double-tapered beam under uniformly
distributed load can be computed with small error by the formula:
(
)
(48)
where G is shear modulus and the other terms are as previously
It should be noted that the shear deflection value as given by (48) is conserva­
tive and represents the shear deflection in a uniform beam of depth h . Thus,
o
after finding a beam size based on bending deflection, the shear deflection
should be calculated and, if necessary, the beam size increased so that the total
deflection
+
does not exceed allowable values.
Figure 23.--Graph for
determining tapered
beam size based on
deflection under
concentrated
midspan load.
ZM 128 978
A similar process can be used in the design of tapered beams under concen­
trated load by the use of figure 23, drawn for the specific case of a concentrated
load at midspan. The shear deflection for tapered beams under concentrated
midspan load can be determined with small error by the formula:
(49)
which again is conservative, and represents the value for a uniform beam of
depth h under identical loading.
o
Design Determined by Stress Limitations
As previously pointed out, the neutral axis of the beam lies midway between
its tapered and straight sides. The bending stress f varies linearly from the
x
neutral axis and can be evaluated at any section by substitution of appropriate
values in equation (2). The shear stress f in the beam is distributed paraboli­
xy
cally at the reaction, having its maximum value at the neutral axis. Away from
the reaction point, the shear stress distribution changes because of the taper
and can become maximum at the tapered edge; also, because of the taper, there
are small vertical stresses f in the beam at the tapered edge. Thus, at points
y
where longitudinal compression or tension stresses occur, there may be shear
and vertical stresses. The design must then consider possible interaction of
these
stresses.
To utilize the relationships developed in phase I, consider their application
to the following:
Case I.--Straight Single or SymmetricallyDouble-Tapered Beams Under Concentrated
Load
Figure
--Straight single or symmetrically double-tapered beams under concentrated load.
ZM 128 977
33 (a) In the region, 0<x<z; the stresses at the tapered edge are given by:
(50)
the neutral axis and is given by:
The shear stress at the reaction is
If in this region:
(1) the largest h is such that h > 2h , then the maximum stresses occur at
o
the section h = 2h and are given by
o
}
(51)
or (2) if the largest h is such that 4 h < h < 2h , then the maximum
3 o
o
stresses occur at the largest h and are given by (50); or (3) if the largest
4
h is such that h < h < h , then at the largest value of h, the bending
o
3 o
stress is a maximum at the tapered edge, while the maximum value of
shear stress lies within the beam The values of the stresses at the
taper remain as given by equations (50).
If it is desired to check the location and value of the maximum shear stress,
the following equations can be used:
(52)
and has a value:
(53)
FPL 34
34
GPO 821–625–5
The, bending stress at this point within the beam is given by:
(54)
Case II.--Single or Symmetrically DoubleTapered Beams Under Uniformly Distributed
Load
At the reaction the shear stress is a maximum at the neutral axis and is
given by:
The absolute maximum bending, shear,
the taper of a section with depth h given by:
and vertical stresses all occur at
(55)
and have values as given by equations (50).
Case III.--Haunched Beam Under
Concentrated Center Load
Figure 25.--Haunchedbeam under concentrated center load,
( )
L
- a , the bending stress f and consequently f and
x
xy
2
have no absolute maximum values within the physical structure of the beam
In the region 0 < x <
fy
ZM 128 979
h
but increase as the ratio
o
increases. As in the previous cases considered and
h
35
prior discussion, the relationships between the stresses at the tapered edges
remain as given by equations (50).
The stress distributions throughout a section differ, however, from those
previously considered. The shear stress distribution is parabolic at the reaction,
but as x increases, the distribution takes the form as shown in figure 7.
The vertical stress in this region has a similar distribution, with the maximum
again occurring at the taper.
For design purposes, therefore, it is necessary to check the result of the
interaction of the combined stress condition at the point of least section to
determine whether that imposed limitation will yield an allowable stress- less
than that occurring at midspan, assuming uniform beam theory applies in the
region 2a.
The deflection relationship for haunched beams under concentrated center
loading in which shear deflection is considered is presented in Appendix V.
This analysis utilizes the assumption that uniform beam theory applies in the
region 2a.
CONCLUSIONS On the basis of the strain comparisons presented in figures 12 through 20,
the stresses in tapered beams of constant width and uniformly varying depth
can be closely approximated by the use of formulas (2), (6), and (33), and these
stresses, combining at one point, will affect the strength of the beam as given
by formula (44). Deflections can be calculated by means of the appropriate
formulas derived.
FPL 34
36 APPENDIX I.--SHEAR STRESSES IN A BEAM OF RECTANGULAR CROSS-SECTION HAVING A VARIABLE DEPTH ALONG ITS LENGTH Figure 26.--Typicalforce components in a tapered beam.
128 952
MC
Assuming the bending stress relationship
is valid, the stress on AC (fig. 26)
I
is given by:
where f is the bending stress at any point y (measured from the nontapered
x
face) in a section of width b, depth h, and subjected to a moment M. A positive
moment is one that produces tensile stress normal to the section at the tapered
face.
The force on AE is therefore:
The force on
is then:
37 The horizontal shear stress on plane EF is:
A positive shear stress then is one such that a cut section will have a shear
force acting in the same direction as positive y.
Expanding the expression for F yields:
BF
By series expansion and neglecting values of (
n
∆h
becomes:
) where n > 1, F
h
BF
Expanding further and neglecting products of differentials (∆M)(∆h),
becomes:
and
since:
resulting in finally:
FPL 34
38
F
BF
APPENDIX II.--VERTICAL STRESSES IN A BEAM OF RECTANGULAR
CROSS-SECTION HAVING A VARIABLE DEPTH ALONG ITS LENGTH Figure 27.--Typical force components in a tapered beam.
ZM 128 975
From previous discussion, it was determined that from an analysis of an
elemental section, such as shown in figure 27, a shear stress was derived of the
form
The element AEFB
(fig. 28.) acting upon it:
to be in equilibrium must have the following forces
Figure 28.--Forces acting upon the element AEFB when in equilibrium
39 ZM 128 976
therefore:
where:
and
expanding in general:
neglecting terms containing
Substituting:
FPL 34
40 Neglecting
terms containing
products of
differentials, i.e., ∆M · ∆h, yields:
At the limit, assume
Therefore:
APPENDIX III.--DEFLECTIONS OF A SIMPLY SUPPORTED,
DOUBLE- TAPERED BEAM UNDER CONCENTRATED MIDSPAN LOAD ZM 128 981
Figure
--Simplysupported, double-taperedbeam under concentrated midspan load.
41
Deflection Due to Bending
Assuming
(3-1)
(3-2)
(3-3)
where
Substituting expressions
results in:
(3-3)
and (3-2)
in equation (3-1)
and evaluating,
(3-4)
The midspan deflection, DB, is obtained by substituting
(3-4), yielding
x 1
= in equation
L 2
(3-5)
Equation (3-5) is represented in graph form in figure 23.
Deflection Due to Shear
The shear stress in a tapered beam of uniformly varying cross section is
given by equation (6) as:
FPL 34
42
GPO 821–625–4
(3-6) The internal shear strain energy in a differential volume of the beam is given
by:
(3-7)
is as defined by expression (3-6) and G is the shear modulus.
xy
The internal shear strain energy over the whole beam can be obtained by
integrating equation (3-7) throughout the volume of the beam as:
where f
(3-8)
Integrating and realizing that tanq =
(
)
2
h - h , it can be found that:
o
L c
(3-9)
The external energy can be represented by:
where D is the midspan deflection.
S
Since:
it can be found that:
(3-10)
Letting : equation (3-10) can be written: 43
(3-11) Evaluating equation (3-11) for
values of γ, it can be found that the
shear deflection can be conservatively given with small amount of error by:
(3-12)
which represents the value of equation (3-11) at the limit as γ approached zero,
or the value of shear deflection for a uniform beam of depth h under similar
o
loading.
APPENDIX IV.--DEFLECTIONS OF A SIMPLY SUPPORTED, SINGLE-TAPERED BEAM UNDER A CONCENTRATED LOAD ZM 128 974
Figure 30.--Simply supported, single-tapered beam under concentrated load.
Assuming
(4-1)
where:
(4-2)
FPL 34
44 For 0 < x < z:
(4-3)
expressions (4-2)
and (4-3) in equation (4-1) and evaluating,
yields:
(4-4)
For z < x < L:
(4-5)
Substituting expressions (4-5)
yields:
and (4-2) in equation (4-1) and evaluating,
(4-6)
It should be realized that for this beam, the point of maximum deflection
changes as the load position changes. The carpet plot (fig. 31) presents values
of the maximum deflection coefficient, K', as a function of γ and the load
z
position
. From figure 31, a single-tapered beam with dimensions such that
L
z
γ = 1.0, and under a concentrated load P located at
= 0.30, will have a maxi­
L
mum deflection y
equal to:
max
Numerical calculations show that the absolute maximum deflection in a given
beam will occur when the concentrated load P is located at a section for which
45
dy
, as determined by either equations (4-4) or
dx
x
z
(4-6), has a value of zero. This condition will occur at a value of
=
and
L
L
can be determined by the solution ofthe following equation for a given value of γ :
the slope of the elastic curve,
(4-7) z
1
By taking the limit, for example, as γ
0, the value of
= will be deter2
L
mined as expected from elementary beam theory.
The solution to equation (4-7) is represented by the dashed line for various
values of γ in the carpet plot of figure 31.
ZM
128
984
Figure 31.--Carpetplot of coefficient K' for determining maximum bending deflection, ymax., of
a single tapered beam under concentrated load,
FPL 34
46
APPENDIX V.--DEFLECTIONS OF A SIMPLY SUPPORTED,
HAUNCHED BEAM UNDER CONCENTRATED MIDSPAN LOAD
Figure 32.--Simplysupported, haunched beam under concentrated midspan load,
ZM 128 980
Deflection Due to Bending
Assuming
(5-1)
For
(5-2)
(5-3)
where
Substituting expressions (5-2) and (5-3) in equation (5-1) and evaluating,
yields for the equation of the elastic curve:
(5-4)
47
For
(5-5)
where
(5-6)
(5-7)
Substituting expressions (5-6) and (5-7) in equation (5-5) and evaluating,
yields for the equation of the elastic curve:
(5-8)
Equating equation (5-8) at x =
L
, to obtain midspan deflection ∆ , gives:
2
B
(5-9)
Deflection Due to Shear
L
The shear stress in a haunched beam over the region 0 < x < ( - a), as
2
given by equation (42) is:
(5-10)
realizing that
The shear stress in the uniform section (region 2a) of the haunched beam can
be determined from equation (5-10) by letting h = h = h , yielding:
c
o
FPL 34
48
(5-11) where bh is the cross-sectional area,
c
The internal shear-strain energy in a differential volume of the beam is given
by:
(5-12)
The internal shear-strain energy over the whole beam can be obtained by
integrating equation (5-12) throughout the volume of the beam as:
(5-13)
2
3P a
represents the internal shear-strain energy over the
10bh G
c
uniform portion 2a of the beam. Performing the integration yields:
where the quantity
(5-14)
where K is the same as previously defined, and G is the shear modulus.
The external energy can be expressed by:
(5-15)
where ∆
is the midspan deflection under the concentrated load P.
S
Equating the internal and external energy expressions, yields:
49 APPENDIX VI.--DEFLECTIONS OF A SIMPLY SUPPORTED,
DOUBLE-TAPERED BEAM UNDER UNIFORMLY
DISTRIBUTED LOAD
ZM 128 951
Figure 33.--Simplysupported, double-taperedbeam under uniformly distributed load.
Deflection Due to Bending
Assuming
(6-1)
where for 0 < x < L :
2
(6-2)
(6-3)
and
Substituting expressions (6-2) and (6-3) in equation (6-1) and evaluating,
yields for the equation of the elastic curve:
FPL 34
50
GPO 821-625-3
(6-4) x 1
= to obtain the midspan deflection D , results
Evaluating equation (6-4) at
L
2
B
in:
(6-5)
Equation (6-5) is represented in graph form in figure 22.
Deflection Due to Shear
The expression for the total elastic shear-strain energy is given by:
(6-6)
The displacement ∆ , at the point of a dummy load q can be determined by
S
Castigliano’s Theorem as:
(6-7)
where G is the shear modulus and dv the differential volume.
The expression for the shear stress f , in a beam of varying rectangular
xy
section is given by equation (6) as:
(6-8)
x 1
= , the dummy load q is placed at midspan,
2
L
L
yielding the following equations for moment and shear for 0 < x < :
2
To obtain the shear deflection at
(6-9)
(6-10)
Substituting expressions (6-9) and (6-10) in equation (6-8), and performing
51
the operations as defined by equation (6-7) yields:
(6-11)
Evaluating equation (6-11) for small values of γ, it can be found that the shear
deflection can be conservatively given with small amount of error by:
(6-12)
which represents the value of equation (6-11) at the limit as γ approaches zero,
or the value of shear deflection for a uniform beam of depth h under similar
o
load.
APPENDIX VII.--DEFLECTIONS OF A SIMPLY SUPPORTED, SINGLE-TAPERED UNDER UNIFORMLY DISTRIBUTED LOAD ZM 128 973
Figure 34.--Simplysupported, single-taperedbeam under uniformly distributed load.
Deflection Due to Bending
Assuming
(7-1)
where
(7-2)
(7-3)
FPL 34 52 Substituting expressions (7-2) and (7-3) in equation (7-1) and evaluating,
yields for the equation of the elastic curve:
(7-4)
The value of
x
at which the maximum deflection ∆ occurs can be obtained
B
L
d y
B
and equating it to zero, and solving the resulting equation for
x
d
L
x
as related to γ is presented in graph form in
. The maximum deflection D
B
L
figure 22.
by taking the
( )
53
2.-55