Deflection and Stresses of Tapered Beam

U.S. FOREST SERVICE RESEARCH PAPER FPL 34 SEPTEMBER 1965

U. S. DEPARTMENT OF AGRICULTURE • FOREST SERVICE

FOREST PRODUCTS LABORATORY • 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 theoretical 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 interaction 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 2 4

10 13 14 21 30 31 33 36

37

39

41 42 42

44

47 47 48

50 50 51

52 52

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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 in a member under any external condition.

Phase II consisted of substantiating the mathematical analysis of Phase I by experimentally evaluating two specific cases, the double-tapered beam and the haunched beam, under a concentrated center load. The beams in this 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 theoretical 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:

Mf =

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 determine 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:

h y -2f =

12M

bh 3 (reference axis chosen as in fig. 1) (2)x

2Timoshenko, S. Theory of Elasticity. McGraw-Hill Book Co., New York, 1934.

FPL 34 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:

6Mf x = +

bh 2 (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 d f

dx x from equation (3) and equating it to zero, yielding:

h dM

- 2M dh = 0dx dx

Therefore the section has a depth h given by:

at such a section is:

(4)

The value of f x max

(5)

where dM and dh represent the shear force and slope of taper at the section,dx dx respectively.

3

Shear Stress

The approximate mathematical treatment of the shear stress distribution at any section in a tapered beam was formulated by Norris 3 and its derivation is presented in Appendix I. It is shown in general that:

(6)

where fxy is the shear stress at any pointy (measured positive from the non

tapered face) in a section of breadth b, depth h, subjected to a moment M. The

slope of the taper and the shear force at the section are represented by dh and dM dx

respectively. dhA 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:

(1) At y 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

is linear; that is, f xy is a function of only y h. This can be seen by equating:

Therefore, the section at which this linear distribution occurs has a depth h given by:

(7)

B., Engineer, U.S.3 ForestNorris, Products Laboratory, Madison, Wis.

FPL 34 4

The shear stress at this section is given by the formula:

(8)

which has its largest value at y h

= 1, and is given by:

(9)

(3) There is also a section across which the shear stress f is a function xy

of 2

alone. This occurs when

or

(10)

at a section with depth h given by:

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 h = 1 into the left-hand side, yielding the

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,

when h y = 1/2. Substituting this value in (13) yields:

M dh h dx

= 0 (16)

dhwhich is only true for M = 0 since dx

¹ 0. Therefore the maximum shear stress

will occur at the neutral axis at all sections where the moment is zero and have a value

which can be determined by substitution of M = 0 and h y = 1/2 in equation (6).

For 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 maximum value of f

xy occurring somewhere along the taper. The section at

d f which this maximum occurs can be found by taking xy from equation (17)

dxand equating it to zero. This results in:

(18)

FPL 34 6

and f has a maximum value: xy

(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;

dhand consequently: dx

= tanq

d2h = 0 } (21)

dx2

Substitution of equations (20) 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 h = 1/2 and has the value:

(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 = Ci α, where Ci is determined at various beam cross sections.

FPL 34 8

Proceeding along the span from the the equation for the locus of points of maximum shear stress can be determined by substituting expressions (20) and (21) in equation (13), yielding:

Considering a section at which the point of maximum shear stress lies within h o 7the beam such as where h = 8’ the point of maximum shear stress occurs at

y 6= h 10 and f xy at this point has a maximum value of:

fxy = 63 α80

The value of shear stress at the taper of this section is:

f xy

= 7 16

α

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:

h = 4 h3 o

and the shear s t r e s s has a value at the taper or point of intersection of:

f xy

= 3 4 α

The depth of section at which the shear distribution is linear can be found by substituting expressions (20) and (21) in equation (7), yielding:

3h = h2 o

The maximum shear stress at the taper has a value

fxy = 8 9

α

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

distribution is a function of ( y h )

2 alone will coincide with the section at which the

9

absolute maximum shear stress in the beam occurs. This section will have a depth:

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 > 2ho 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

can be evaluated from equation (23) for h y = 1 yielding:

f = 3 α 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 y

= 0:h

1 d2

Mf y

= b dx

2 (29)

and at y

= 1:h

(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

df y

tapered edge (h y

= 1), take the dx y from equation (30) and equate to zero, yielding:

(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

shown in figure 1. The stresses at the taper, that is, when y

= 1, can be transh 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 assumption 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)

where f 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.

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 constructed 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 x, ε

y, and ε

xy are strains; f x, f y , f xy are stresses; E is the elastic

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 distributions are presented in figures 3, 4, and 5. The observed strain readings from the experimental evaluation are also shown in their appropriate position to facilitate a direct comparison. It is interesting to note that while 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 pounds. 15 ZM 128 971.

Figure 4.--Vertical strain distribution for aluminum tapered beam under a midspan load of 1,000 pounds. ZM128953

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 Appendix 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. ZM 128 972

decreases over the region 0 < x <

maximums within the physical

In this region, f and therefore x fxy and f

y , by previous analogy, have no structure

h oof the beam, but increase as the ratio h increases. This can be most readily

seen perhaps by rewriting the expressions for the normal, vertical, and shear

17

h ostresses in terms of h , realizing that for the haunched beam of figure 6:

h = h - x tanθo dh dx = -tan θ

M = -VX (41)

dM = -Vdx

Substitution of expressions (41) into the general expressions (2), (6), and (33) derived in Phase I results in:

(42)

Evaluating equations (42) at h y

= 1 yields:

(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 increases quite rapidly as the ratio

h o increases. For example, at

h o

= the 3Vvalue for f

xy at

h y = 1 is 8 .

2bh or eight times the maximum shear stress ooccurring at the reaction.

Figure 7.--Typical shear andvertical stress distributions in a haunched beam under a concentrated midspan load of 2V. ZM 128 966

To substantiate this theoretical stress distribution in a haunched beam, a procedure was followed identical to that described for the symmetrical double-tapered beam. A haunched beam, with dimensions as given in figure 6, was constructed from the same sandwich material used previously. Rosette type,

h oSR-4 electrical strain gages were placed at a section where h = 2. Expressions

(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 250 pounds. ZM 128 967

FPL 34 20

Figure 10.--Shear strain distribution in aluminum haunched beam under a midspan load of 250 pounds. ZM 128 970

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 members.

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 x , E y

, G xy, µ xy, and µ yx are elastic properties as previously defined

and F xt, F xc, Fyt, F yc, and F xy are strength properties; the subscripts x and

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 distribution of three sections--where h1 = 1.25ho; h2 = 1.5ho; and h3 = 1.75ho. The

section h1 is located in the region 0 < x < and, as a result, should have the

maximum shear stress and corresponding strain occurring within the beam. Section h2 was selected for observation since, from the analysis, a linear shear

strain distribution was expected here, with the maximum occurring at the taper.

4Recommended ASTM Standards used in determining the wood properties of specimens evaluated. 5U.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 h3, represents the typical distribution

between sections located at depths 1.5h o < h < 2h o, The critical section ( h = 2ho )c was not evaluated for this particular beam since it coincided with the section containing the concentrated load. The final dimensions of beam No. 1 are presented in figure 11. 2v

Figure 11. --Dimensions of experimental beams of Sitka spruce used in strain and strength evaluation. ZM 128 955

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 Lwould 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 of8beam 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 a8

concentrated midspan load of 2V = 7,000pounds. ZM 128 961

FPL 34 24

LFigure 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

LFigure 15.--Comparison of theoretical and observed shear strain distributions at section 4

of beam No, 1 under a concentrated midspan load of 2V = 7,000 pounds.

ZM 128 968

25

ZM 128 954

Figure 16.--Comparison of theoretical and observed bending and vertical strains at section 3L 8 of

beam No. 1 under a concentrated midspan load of 2V = pounds,

Figure 17.--Comparison of theoretical and observed shear strain distributions at section 3L 8 of

ZM 128 965beam No. 1 under a concentrated midspan load of 2V = 7,000 pounds.

FPL 34 26

Figure 18.--Comparison of theoretical and observed bending (ε x) strain distribution at section L 4 of beam No. 2 under a concentrated midspan load of 2V = 3,500 pounds. ZM 128 959

Figure 19.--Comparison of theoretical and observed vertical (ε y) strain distribution at section L of beam No, 2 under a concentrated midspan load of 2V = 3,500 pounds. ZM 128 9584

27

Figure 20.--Comparison of theoretical and observed shear strain (ε xy) distribution at section

4 L of beam No, 2 under a concentrated midspan load of 2V = 3,500 pounds. ZM 128 957

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 compared 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 relationships, 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 x , f y , and f xy

are the bending, vertical, and shear stresses, respectively,

existing at some point and F x , F y , and F xy are their respective failing stresses,

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

and tan 2 θ, respectively, the interaction relationship (44) can also be written:

or

where

(45)

(46)

(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 additional 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

8Norris, C.B. Strength of orthotropic materials subjected to combined stresses. Forest Products Lab. Rpt. No. 1816. 24 pp., illus,

29

ZM 128 983

Figure 21.--Comparison of theoretical and observed values of o as defined in equation

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.

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 of a simply supported, single- or double-tapered beamunder uniformly distributed

load. If from design requirements, the span L, the slope of taper 1 L ( h c - h o ) for

2single-tapered beams or L (h c - h o) for double-tapered beams, the width b, the

deflection limitation ∆ B , and the loading conditions are known, a value for γ can

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

beam, the process can be reversed by computing a γ where g = h 1

o (h c - h o) and then

reading an ordinate value and solving for ∆ B.

Tapered beams, of course, deflect due to shear as well bending. It has been 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 concentrated 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 parabolixy 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 pointsy 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 Symmetrically-Double-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 shear stress at the reaction is the neutral axis and is given by:

If in this region: (1) the largest h is such that h > 2h o, then the maximum stresses occur at

the section h = 2h and are given byo

} (51)

or (2) if the largest h is such that 4 3

h o < h < 2h o

, then the maximum

stresses occur at the largest h and are given by (50); or (3) if the largest

h is such that h o < h < 4 3 h o, then at the largest value of h, the bending

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, can be used:the following equations

(52)

and has a value:

(53)

GPO 821–625–5FPL 34 34

The, bending stress at this point within the beam is given by:

(54)

Case II.--Single or Symmetrically Double-Tapered 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, and vertical stresses all occur at the taper of a section with depth h given by:

(55)

and have values as given by equations (50).

Case III.--Haunched Beam Under Concentrated Center Load

Figure 25.--Haunchedbeam under concentrated center load, ZM 128 979

In the region 0 < x < (L 2

- a), the bending stress f x

and consequently f xy and

f y have no absolute maximum values within the physical structure of the beam

but increase as the ratio h o h

increases. As in the previous cases considered and

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

Assuming the bending stress relationship MC

is valid, the stress on AC (fig. 26)Iis given by:

where f is the bending stress at any point y (measured from the nontaperedx 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 FBF

yields:

∆h n

By series expansion and neglecting values of ( h ) where n > 1, FBF becomes:

Expanding further and neglecting products of differentials (∆M)(∆h), FBFbecomes:

and

since:

resulting in finally:

FPL 34 38

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 to be in equilibrium must have the following forces (fig. 28.) acting upon it:

Figure 28.--Forces acting upon the element AEFB when in equilibrium ZM 128 976

39

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 (3-3) and (3-2) in equation (3-1) and evaluating, results in:

(3-4)

x 1The midspan deflection, DB, is obtained by substituting L =

2 in equation

(3-4), yielding

(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)

where f xy is as defined by expression (3-6) and G is the shear modulus.

The internal shear strain energy over the whole beam can be obtained by integrating equation (3-7) throughout the volume of the beam as:

(3-8)

2Integrating and realizing that tanq = L (h c - h o) , it can be found that:

(3-9)

The external energy can be represented by:

where DS is the midspan deflection.

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 loading. o

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) and (4-2) in equation (4-1) and evaluating, yields:

(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

zposition L . From figure 31, a single-tapered beam with dimensions such that

z γ = 1.0, and under a concentrated load P located at L = 0.30, will have a maximum 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

the slope of the elastic curve, dy dx, as determined by either equations (4-4) or

x z(4-6), has a value of zero. This condition will occur at a value of L = L and

can be determined by the solution ofthe following equation for a given value of γ :

(4-7)

zBy taking the limit, for example, as γ 0, the value of L 1 2 will be deter-=

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 984128

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


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)


(5-8)

LEquating equation (5-8) at x = 2 , to obtain midspan deflection ∆B

, gives:

(5-9)


The shear stress in a haunched beam over the region 0 < x < (L

- a), as2given 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 c, yielding: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)

3P2 awhere the quantity 10bh G represents the internal shear-strain energy over the

c uniform portion 2a of the beam. Performing the integration yields:

(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 ∆ S is the midspan deflection under the concentrated load P.

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.


Assuming

(6-1)

where for 0 < x < L : 2

(6-2)

(6-3)

and


FPL 34 50 GPO 821-625-3

(6-4)

x 1Evaluating equation (6-4) at L = 2 to obtain the midspan deflection D

B, results

in:

(6-5)

Equation (6-5) is represented in graph form in figure 22.


The expression for the total elastic shear-strain energy is given by:

(6-6)

The displacement ∆ S , at the point of a dummy load q can be determined by

Castigliano’s Theorem as:

(6-7)

where G is the shear modulus and dv the differential volume. The expression for the shear stress f xy, in a beam of varying rectangular

section is given by equation (6) as:

(6-8)

To obtain the 1

shear deflection at L x

= 2, the dummy load q is placed at midspan,

Lyielding the following equations for moment and shear for 0 < x < 2 :

(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 oload.

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.


Assuming

(7-1)

(7-2)

(7-3)

where

FPL 34 52


(7-4)

The value of L x at which the maximum deflection ∆B occurs can be obtained

d yBby taking the d ( x

L) and equating it to zero, and solving the resulting equation for

L . The maximum deflection D B as related

figure 22.

to γ is presented in graph form in

53 2.-55

x

Deflection and Stresses of Tapered Beam

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