L) 56_5
U. S. Department of Agriculture, Forest Servic e
FOREST PRODUCTS LABORATOR Y
In cooperation with the University of Wisconsi n
MADISON, WISCONSI N
WOOD-BEAM DESIGN METHOD PROMISES ECONOMIE S
By J . A . NEWLI N
Principal Enginee r
G. E . HEC K
Engineer
and
H. W . MARC H
Special Consulting Mathematician
Published i n
ENGINEERING NEWS-RECORD
May 11, 1933
By
J . A. NE?LIN, Principal Enginee r
G. E . HECK, Engil .ee r
and
H. W . MARCH, Special Consulting Mathematicia n
A new design method for calculating the horiz-ontal smear im
wooden beams, developed by the Forest Products Laboratory th--reu .mathematical analysis and tests, assumes that, because of the s+hear
distortion in the vicinity of the ' base of checks or fissures0uza :t ar e
present in all beams, the upper and ldwer halves of the beam act t o
some extent as independent beams . The result is to relieve the .mean
shearing stress in the neutral plane, and since this reduced sheatin g
stress is not comerok in present design methods, they are often direr conservative and therefore uneconomical to use . In certain cases, suc h
as floor beams of highway bridges and railway ties, usual design method s
predict stresses that are two or three times the ultimate sheaxing :stres s
of the material . still these members are carrying their loacl .s withou t
failure . In the discussion presented here an attempt is male to emp3 :4im
i
the elastic behavior of a checked beam under load and this to 3 :Aai4
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pr*d:io
=
the discrepancy existing between the facts of experience and t
.
The
background
fo
t
tions of the usual methods of calculating shear
this explanation is furnished by an approximate mathematical andlys s o f
the problem combined with the results of a series of about 200 tests, i m
which the loads causing shearing failure were observed on built-u p
artificially checked beams varying from 3/4 x 1-1/2 in . to $ x 16 in. in
cross section and with varying amounts of checking . On the basis of the
theory and the results of tests, practical directions are given for th e
more realistic calculation pf loads that will cause shearing failure*
In the tests, built-up artificially checked beams of cares Iy
matched material were used . A typical section is shown in Figure 1 . The checks are placed in the middle of the lateral faces, as this is th e
position in which they are most likely to occur and to cause failur• .e by
shear . A characteristic failure, by horizontal shear, of a naturall y
checked structural timber loaded at two points is shown in Fire 2 .
IA all other tests referred to in this paper the beams were loaded at
one point only .
Two-Beam Actio n
An explanation of the behavior of a checked beam is feuxid iYn. .
the fact that, owing to the shear distortion in the material in t
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Figure 1 .---(left)--Cross-section of a built-up artificiall y
checked beam such as was used in the test, The four parts a were ,
glued to the central portion or web b . The joints c were' paraffine d
to prevent sticking of any glue that might protrude from adjacen t
joints and to minimize friction .
Figure 2 .---(right)---Characteristic failure of a timbe r
beam, failure occurring in horizontal shear in the neighborhood o f
the neutral axis .
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vicinity of the base of the checks, the upper and lower halves of th e
-beam act to some extent as independent beads . An approximate mathemati=
cal analysis indicates a vertical distribution of shearing stress as
w
shown in Figure 3 . The . shearing stress _plotted . is the mean sl a
The
exstress at failure averaged over the full width of the beam .
istence of the "two beam action" is evident from the curves . The
43 action
mean shearing stress in the neutral plane is relieved .
of the upper and lower halves of ihd beam . as partly i epO .ent beams .
The significance of this action is emphasized by the f, t:11er- result
of the mathematical analysis that the reaction R at the supp-ett . nearer
the load may be expressed as the slam of two terms, of which only th e
first is associated with shearing stress in the neutral plane . Thus
it is found that
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(1) R = B + A/a2
where
(e, B - 2/3 Jbh,
1 1
J being the moan of the shearing , stress in the neutral plane over t-V
1 ~■
full width of the beano ; and b and h being the width and depth . 7e, I 6
r
of the beam. The portion B of the reaction is precisely the reatt
that would be associated with the mean shearing stress J in the neit'ka,l
plane in the usual theory of beams and may be referred .+to as the. gars t P - '.
beam portion" of the reaction .
x the second term of equation 1, a is the distance ftem Ake
?i,,
%s the nearer support, and A is a quantity determined c .4 e.y by
the dimensions ' of the beam, the longitudinal Youn gs s modulus ,' -aid. the
mean longitudinal displaceient oh the lower face of the upper half of
the beam at points imt ediatelf o+rer the supports The portAan of th e
reaction expressed: by this term is attributed to the inAepend!enn as ±b n
of the upper and lover halves of the beam and is not assoc sated wit h
shear in the neutral planes It is the "two-beam porttom" of the reaction .
The values of the two'beem and single -,beam portions, Of the-reaction at
,a .
the ihsta it of failure were determined by testing a :large h-u er of
1eawsi ar4l$er combining the results of tests of
is ,of carefully matdhe d
beams loaded to failure) the concentrated. loads be '.hg ap aliec ai different
points . On entering the results of each pair of tests it egtt ;on 1
twd equations were obi-aimed. that douL be solvdd. &'t r B axle.. s
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TestResults Suggest Design Procedure
1
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In this procedure it was assumed that tie values o$ A and 66
the mean shearing stress J in the neutral plane at faiiuee we-e ,indepe n
of the position of tha load,. The justification of bota ass ns i s
found in the fact tkgl 'the results. of the tests were representei . appDxiiec . Th e
matteln by eft-ion Z' "th A and B as constants fox the. ash©le
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tests showed that as a concentrated load approaches a support, th e
two-beam portion et' the reaction increases rapidly, while the single beam portion associated with shear in the neutral plane, remains prac tically constant and becomes a correspondingly smaller fraction of th e
reaction . As a consequence, the point of application of the minimu m
concentrated load to produce failure by shear is not just inside th e
support, as is commonly assumed because of the usual simple beam theory ,
but is at some distance from the support . The results of the tests o f
three,series of beams are given in the accompanying table and are repre sented graphically in Figure )4e The single-beam and two beam portion s
of t64 reaction shown in the table were computed by combining in pair s
the mean reaction corresponding to a = 7 with that corresponding t o
each of the other points of loading . If these reactions are combine d
in pairs in other ways, results varying somewhat from those shown ma y
be obtained .
The results of the numerous tests may be summarized in th e
following statements : For checked beams with a span-depth ratio of 9
to 1 the point of application of the minimum concentrated load causin g
failure by shear is at a distance from the support approximately thre e
times the height of the beam . The distance of the critical point flor a
the support is somewhat greater for longer spans and somewhat less fo r
shorter spans ; but in any case, for loads applied at three times th e
height of the beam from the support, the two-beam portion of ta e
reaction at the nearer support is approximately one-sixth .
Design Recommendations
1. If there are moving loads, place them so that the heavies t
concentrated moving load is at a distance from the support of thre e
times the height of the beam . After this has been done the followin g
recommendations apply both to static loads and to the moving loads .
2. Neglect all concentrated loads within a distance from th e
support equal to the height of the beam .
3. Consider all concentrated loads that are from one to thre e
times the height of-the beam from the support as being at three time s
the height of the beam from the support and compute the resultin g
reaction . Neglect one-sixth of the reaction .
This one-sixth is th e
two beam portion of the reaction and is not associated with shear i n
the neutral plane .
(Note : For very small span-depth ratios -- less than G _to I -place the loads designated in recommendations 1 and 3 above at. the midffle +
of the span .)
4.
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Consider all other concentrated loads in the usual manner .
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Results of tests of built-up, artificially checked spruce beam s
(Various lengths ; breadth, 2-1/2 in . ; height 4-1j2 in . ; depth of checks, I in . )
Distance
Load at failure
(a) froth } ..
:
load to i 2 g-in. : 42-in. : 63--in . :
nearer : span : span : span
:
support
Inches
7 .0
:Pounds, :Pounds : Pounds :
: 5,770
Mean reaction
at
nearer
support
Pounds
: Single- : Two- : Two: beam : beam
: beam
: action : action : action
:
x a2
: Pounds
: Pounds
Pounds
1,872
: 91,70 0
4,226
10 .5
5,460 : 4,240 : 3,560
:
•
:
3,18 6
2,354
832
: 91 :700
14.0
5,680 : 4,430 . 3,780 •
2,91 1
2,473
43 g
85, 80 0
:
2,64 9
: 2,349
300
: 91,900
4,800 : 3,630 •
2,41 0
2,183
:
221
100,100
• 2,140
:
170
102,000
17 .5
5,050 : 4,660
: 4,800 : 3,460
21 .0
24.5
; . .
:
: 3,780
:
2,31 0
28 .0
: ;
: . .
; 4,050
•
2,250
31 .5
:
:
: 3 :970
K O05
1)985
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• 1,869
:
116
115,100
S
5. Neglect 10 percent of the uniform load . This 10 percent
is taken care of by two-beam action .
6. Assume lateral distribution of the various loads to adjacent parallel beams or stringers on the basis of their assumed placement .
7. Calculate the shearing stress in the neutral plane by th e
usual formula, using the full width of the beam and all of the reaction ,
obtained as directed, that is not included in the two-beam portion .
S . Use 90 percent of the safe shearing stresses previousl y
recommended by the Forest Products Laboratory, since these recommenda tions did not take into account the effect of two-beam action ,
The .use of these recommendations will result in a considerabl e
saving of material . The highway bridge, on account of the effects o f
lateral distribution of the loads to adjacent stringers, is probably th e
structure that will show the greatest difference in the results of th e
application of the new and old methods . An application of the tw o
methods of calculation to a highway bridge follows :
Consider a single concentrated load P moving along a floo r
beam or joist . Assume the span to be 16 ft ., the height of the joist ,
11, 16 in ., and its width, by 5 in., and all dimensions full n fina l
size . Assume also that when the load is in the center of the span th e
floor carries one-fourth of it to each of the two adjacent beams, on e
on either side, In such a case it has been calculated that he sid e
beams will carry a little more than 40 percent of the load when th e
load is at three times the height of the beam from the support . The
shearing stress, J, in the neutral plane is given b y
J=x
2
6 x bh
(4)
where R = reaction due to the load P placed at three times the heigh t
of the beam from the support .
But, according to the assumptions as to lateral distribution ,
R=
10
. x1~}xP=
OP
(5)
Assuming 100 lb . per sq. in. as the tabular value for the
safe shearing stress and reducing it by 10 percent as explained earlier ,
it follows from (4) and (5) that
90 = 3 x.2x
x_ P
2
20 5 x 1 6
(6 )
Hence, P = 12,800 lb ., the safe load in shear .
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tabular value
noting that at the en d off' tl? beh m
the safe load P is found frog
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This method of computatio n '.
one-half that permitted b the new_ m11d,.
r
11
90 2 x "
Then P
= 7,680
i4
. P
5 x 16
lb .
This load is more than 40 percent greater than that
by the old method of calculation s
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