William Macfarland Patton.

A treatise on civil engineering online

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a body by the change or strain in that particular dimension con-

j /^ i« . i. * x-i»^ intensitv of stress

sidered. Coefficient of stiffness = r-^ .

strain

The coefficient of elasticity is the same as the coefficient of



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THB ELASTLCITT AND RESISTAXOE OF HATEBIALS. 275

stiffness when the intensity of the stress is such that the stress
and strain vary in such a manner that the ratio is sensibly con-
stant for all values of the stress^ the elasticity of the body being
sensibly perfect. It yaries with the kind of material and kind
of stress.

This coefficient is sometimes given as the force which, applied
to a bar whose cross-section is unity, would produce an elongation
equal to the original length of the bar, its elasticity being perfect
up to this limit. This is purely a theoretical force.

The coefficient of elasticity simply expressing the ratio of stress
to strain, assumes the body to be perfectly elastic, and the ratio to
be constant. But no body is perfectly elastic, nor possesses a per-
fectly constant coefficient of elasticity. Yet, within certain not
well-defined limits, the assumptions are sufficiently near the truth
to give results of great value.

This limit is called the elastic limit or limit of elasticity, which
may be defined as that intensity of stress within which the ratio of
stress to stnun is constant.

DUCTILITY. — PERHAKBm? SBT.

236. Ductility is that property by which a solid material is en-
abled to change its form beyond the limit of elasticity before
fraature occurs. It is measured by the permanent set or stretch,
in the case of a tensile stress, which the test-piece possesses after
fracture, and also by the decrease in area found in the piece at the
place of fracture. Under whatever strain the determination is
made, the permanent set is the strain which remains in the piece
when the external forces are removed or cease to act. In many
cases this so-called permanent set decreases immediately after the
removal of the stress, and under small strains it may disappear en-
tirely, although it is commonly claimed that a permanent set is
produced under any degree of stress whatever.

In many grades of wrought iron and steel the limit of elastic-
ity can be quite accurately determined. In other materials there
seems to be no simple relation between stress and strain for any
condition of stress. For such materials there is no definite elastic
limit or coefficient of elasticity. Between these are found all
grades of material.



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276 €K>V£BKMBNT TESTS OF TIKBBB.

AET. XXV.
ELASTICITY AND KBSISTANCE AS DETERMINED BY EXPERIMENT.

TIMBEB.

237. The resnlts of experiments made to determine the elas-
ticity and resistance of timber yary so greatly^ owing to the sizes
of the speoimens nsed, their condition as to seasoning, and manner
of making the experiments, that but little reliance can be placed
upon them.

In timber especially is it important that the experiments
should be made upon pieces as nearly in those conditions which
are necessarily used in practice, such as conditions of growth and
seasoning, with the usual small defects, and in sizes suitable for
columns, posts, and beams. The large majority of tests have been
made upon specimens of from i of an inch to 2 inches round or
square sections. The speoimens were usually free from defects,
and commonly well seasoned.

Becently the United States Government has established a bureau
known as the Forestry Division of the Department of Agriculture,
under the direction of Mr. B. E. Pemow, Chief of Division. One
object of this bureau is to determine "the interrelation between
physical condition, anatomical structure, and mechanical proper-
ties" of American timbers. Although up to date experiments have
only been made on what is generally known as the Long-leaf Pine
of the Southern States, and these only very limited in extent, the
results thus far obtained have great value, and will be instructive
as showing the proper methods of investigation and the character
of information required. All that will be said on this subject ip
taken from the reports of Mr. B. E. Fernow, entitled " Timber
Physics/' which he has kindlv sent the writer.

238. The following: questions indicate the scope of the investi-
gations being carried on :

What are the essential working properties of our various woods,
and by what circumstances are they influenced? How does age,
rapidity of growth, time of felling, and after treatment change



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DETEBMIKATIOK OF SLA8TICITT AJfD BBiySTAKCE.



877




Fio. 89/
Precautions were taken to



qaality in different timbers? How far is weight a criterion of
strength ? What macroscopic or microscopic aids can be devised
for determining qnality from physical examination ? What differ-
ence is there in wood of different parts of the tree ? How far do
climatic and soil conditions influence quality? In what respect
does tapping for turpentine affect quality of pine timber ?

The te^-specimens were taken from the logs as shown in sections
Figs. 89 and 90. The small sticks were nominally 4 inches square,
dressed down to 3^ inches square. The large sticks varied from
from 6 X 12 to 8 X 16 inches in cross-section* The logs varied from
12 to 18 feet in length. The ''green
tests'' were usually made within two
months after sawing. The " dry tests *'
were made at various subsequent timea
One end of each small stick was tested
green, and the other end tested after
seasoning. The seasoning was hastened
in some cases by means of a drying-box.
Temperature of inflowing air, 100'' F.
IH^ent checking of the wood.

The testing apparatus consisted of a 1,000,000-pound column-
testing machine; one 100,000-pound beam-testing machine; one
100,000-pound TJniversal testing-machine of Eiehle's "Harvard*'
pattern ; various steam-engines, planes, lathes, etc.

289. The Cross-breaking Tests.— The large beam-testing ma-
chme is shown in Fig. 91. The base of this machine consists of
two long-leaf pine sticks {Pinus palustris), 6 X 18 inches and 24
feet long, with steel plate three fourths of an inch by 18 inches by
20 feet long, all bolted up as one beam. The poWer is applied by
hydraulic pressure upon a plunger below, to the cross-head of
which are attached the two side-screws, on which the upper cross-
head is moved by sleeve-nuts and spur-gearing. The beam to be
tested rests on pivots at the ends, placed on top of the base-beam,
Jind the upper cross-head is moved down by means of the gearing
nntn the central pivot attached to it comes in contact with the
beam, or rather with the distribution-blocks placed on the beam at
this point. The test then be^ns, the power originating in a double
pluTig:er-pump, operated by hand or by steam power, in another part
of the room.



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278 C^SS-BBEAKIXG TESTS OP TIMBER.






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DETERMIKATION OF ELASTICITY AND RESISTANCE.



279



In the tests of all beams, both large and small, the load is put
on at the same uniform rate, so as to eliminate the time effect,
which is very great in timber tests.

The load on the small beams is increased at snch a rate as to
produce an increase in the deflection of i inch per minute without
any pause until rupture occurs. This causes rupture in from ten
to fifteen minutes' time. The time required for the large beam
tests is about the same, the deflection rate being greater when the
total deflection is to be greater, which is the case with 4x3 inch
sticb 12 feet long.

Small Beams, — The small beams, which are nominally 4 inches
square and 60 inches long between supports, are tested on the small




BiiALLBCAM TESTING MACHI

OAPACITY-«000 LBS. ON LENGTH OF FEETi

HICfOMETEN KCAO^ lO 0.091 INCH.

SaUe of Inch—.



Ti



« 9 It

Fig. 92.



machine shown in Fig. 92. The load is put on by the hand-wheel
Mid power-screw, and the weighing-beam kept in balance by putting
on overweights and moving the poise. One man moves the power-
wrew, which has one-fourth-inch pitch, so as to make one revolution
erery two minutes, and this uniform motion continues until rup-
ture occurs. Another keeps the scales balanced and calls off the



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280 MOISTURE^ TENSION^ 8HBAEIN0 TESTS.



even hundreds of pounds. A third keeps the micrometer*8crew
in contact with the head of the power-screw, reads it for certain
even hundred-pound loads called off, and records the time of each
6Qch reading to the nearest minute, the load, and the corresponding
reading of the micrometer-screw.

240. Moisture Tests, — After rupture the sticks are bored about
20 inches from each end a^id at about one third the width from
either side, in order to get samples for the moisture te6ts. These
are weighed, and then dried at a temperature of 212** F. until they
reach a constant weight. If the original weight is twice the drj
weight, there were equal quantities of water and woody fibre, and
the moisture, if computed on the basis of the original weight, would
be 50 per cent of the original; but if computed on the basis of the
dry weight, it would be 100 per cent. This latter is the method
used.

241. The specific gravity is determined by measuring very
carefully one of the end-pieces, usually 4x4x8 inches, and with
its volume and its weight the weight of a cubic foot is calculated.
This weight divided by the weight of a cubic foot of distilled
water gives the specific gravity.

242. The Tension Test. — A piece 16 inches long and 2^ x li
inches cross-section is cut from one end of the broken beam. Its
thickness at the centre is jeduced to about 2^x1 inches by cutting
out circular segments. This is then tested similar to the test of a
bar of iron in the Universal machine. (See Figs. 93 and 94.)

Care is taken to cut the specimen parallel to the grain of the
wood, so that failure will occur in a condition of pure tension.

243. Compression Across the Grain, — Specimens 4 inches square
and 6 inches long are tested in compression across the grain. An
arbitrary limit of distortion, namely, 3 per cent of the height, has
been chosen as a reasonable maximum in practice. The load, then,
on the specimen is called the compressive strength across the grain.
This load is indicated by the ringing of an electric bell. The test
is then continued until the distortion has reached 15 per cent of
the height; both results are recorded. (See Pig. 95.)

244. Tfie Shearing Tests.— Since timber fails by shearing or
splitting oftener than in any other way, the shearing test is of great
importance. The specimen is taken 2 inches square and 8 inches
long; rectangular holes are mortised 1 inch from each end, and at



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DRTSRMIKATIOK OF SLASTIOITT AKD BE8I8TAN0E. 281



right angles to each other (see Fig. 96). The specimen is then
palled in the Universal machine (see Fig. 98) by means of suitable
gtirnips and keys^ as shown in Fig. 96. The ends are kept from
spreading or splitting by putting on small clamps with just enough
initial stress in them to hold them in place. After one end shears
out two anziliary hoops or stiJTups are used to connect the key^
which is sheared out to a pin put through the hole at the centre of
the specimen as shown. The other end is then sheared out^ and



Pio. 98.

two results are obtained at right angles to each other. The details
of connections^ gripe, stirrups, and keys are shown in Fig. 96.

245. Endwise Compression Tests. — Most of the tests for com-
pressive strength were made on sticks 4 inches square and 8 inches
long. The ends were cut true and square and at rig^t angles to
the sides. They are tested in the Universal machine (see Fig. 93).
The compression is continued until the stick has been visibly
cmshed and has passed its maximum load. The crushing usually
manifests itself over a plane section by crushing down or bending
oTer all the fibres at this section, which may be either a right or
oUiqne section. The section of failure, however, is seldom at the
Tery end. The slightest source of weakness may determine its
position, as for example a very small knot — for knots are a source
of weakness in compression as well as in tension.



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282



COMPRESSIYE TB8T8, FULL SIZED COLUMIfS.



Some tests were made on columns 40 inches long by 4 inches
square on the large beam machine. These usually ftdled as in
case of the short blocks, and not by bending sidewise. (See Fig.
97.)

Compressive Testa of Full-sized Columns. — To make these
teats requires a machine of at least 1,000,000 pounds capacity



TENSION TEST




iD;^ c::l



c.:p:.:



^



^i



enfUSHINO ENOWIM




ORUtHINO ACROSS GRAIN



capable of crushing to failure columns from 12 to 14 inches
square and at least 30 feet long. It will receive a column of 36 feet
in length or less. The sides or tension members of this machine are
made of four long-leaf yellow-pine sticks {Pinus palustris), from
Georgia, each 8 X 12 inches by 45 feet long. The power is applied
by the same hydraulic pump which operates both the large beam
machine and the 100,000-pound Uniyersal machine. The loads are



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DETERMINATION OF ELASTICITY AND RESISTANCE. 28S

▼eighed on this latter machine the same as for the beam tests.
The plunger in the colnmn machine has jnst ten times the area of
that in the weighing-machine^ and hence the loads in the column
t^ts are just ten times those indicated on the weighing beam^
vhereas in the beam machine they were the same. The tail-block
is of cast^ iron resting in a spherical socket^ which is carried on a
car, and can be held by struts resting in slots in the timber.
This is to make the distance between the face-plates any even
number of feet from two to thirty-six. The spherical socket in
the tail-block will produce an accurate adjustment of the end-bear-
ings at the beginning of the test> but after the load is on it is -
thought that this joint will remain rigid. This socket is not
intended to serve as a round end-bearing for the column. No
tests ha?e been made on this machine up to the present time.

246. Significance of Results, — From the cross-breaking tests
are obtained the cross-breaking modulus of rupture, the modulus
of elasticity, or measure of , stiffness, and the elastic resilience, or
measure of toughness.

The loads and their corresponding deflections are plotted as
rectangular co-ordinates, and the modulus of elasticity and the
elastic resilience are obtained from a study of this strain diagram.

The following is an example of the record made for every beam
test This is a record of a test made on a 4 by 8 inch stick of long-
leaf pine, 12 feet long, which was placed on supports 140 inches
apart.



(16

Length, 140.0 inches.
Height, 8.04 inches.
Breadth, 402 hichea.



Cross-bbbakino Tbst.
Strength of extreme tibre, where

oTrrr

/ = ^TTj = 10,910 pounds per square inch. . (105)

Modulus of elasticity = 2,070,000 pounds per sq. inch.

Total resilience = 85,440 inch-pounds.

Resilience per cub. in. = 7.88 inch-pounds.

Total elastic resilience = 8650 inch-pounds.

Elastic resilience, per cubic inch = 1.91 inch-pounds.



The obeerved data are given in the columns headed ^' Time/'
** Load," and " Scale Reading.'* These results are recorded on this
sheet in ink as they are observed. The result in the " Deflection "



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284



T0U0HKBS6, DEFLECTION^ RESILIENCE.



Table XV.
(Number of annual rings per inch = 14.)



Au^trr.


LOAd.


Defleetioo.


Scale
Reftding.


Bmnarki.


h. m.










1 58


1,000


0.17


11.02


1 T


59


2.000


0.84


11.19






B^fea


50


8.000


0.60


11.86






i^^^




2 00


4,000


0.66


11.51






Wf^m




00


6.000


0.82


11.67






|{v^M




01


6.000


0.96


11.81






^^^




02


7.000


1.18


11.98




^^t^^




02


8.000


1.27


12.12


Butt end. Top end.


08


9,000


1.46


12.81




08


10.000


1.66


12.60


Figs. 98.


04


11,000


1.98


12.78




06


12,000


2.27


18.12




07


18.000


2.86


18.70




00


18,600


8.86


14.70


Maximum load.







colamn is computed from the scale-reading. It is placed next to
the column of " Loads '* for convenience in plotting the strain dia-
gram, which is done on the ruled squares at the bottom of each
sheet. These plotted results fall in all cases on a true curve,
similar to the one shown in Fig. 99. The total area of this curve,
D Ey properly evaluated by the scales used, represents the total
number of foot-pounds or inch-pounds of work done upon the
stick before rupture occurred. This is called the Total Cross-
breaking Resilience of the stick, and when divided by the volume
of the stick in cubic inches it gives approximately the total cross-
breaking resilience of the stick in inch-pounds per cubic inch of
timber.

A better criterion of toughness, or resistance to shock, is some
definite portion of this strain-diagram area, as OPK, Fig. 99, for
example. This amount of resilience or spring can be used over and
over again, and is a true measure of the toughness of the timber as
a working quality. To locate the point P, the following arbitrary
rule has been followed :

Draw a tangent to the curve at the origin, as OA, Lay oS AC
= iBA, and draw OC. Draw nin parallel to OC and tangent to



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DETERMIKATION OF ELASTICITY AND RESISTANCE. ^85

the curve. Take the point of tangency as the point P, and draw
PK. The area OPK is then called the Relative Elastic Eesili-



mce,*

Th«re is no ^elastic limit ^^ in timber as there is in rolled
metals. In this respect it is like cast iron. The point P is the
point where the rate of deflection is 50 per cent more than it is at
first, and usually falls on that part of the curve where it begins to
change rapidly into a horizontal direction or where the deflection
b^ms to increase rapidly. The areas of these curves are measured



Fio. 99.

▼ith a planimeter and reduced to inch-pounds. Thus, if X inch
vertically represents 5000 pounds, and 1 inch horizontally repre-
sents 1 inch deflection, then 1 square inch represents 5000 X 1 =
5000 inch-pounds. If the area OPK is 1.73 square inches, then
the corresponding resilience is 8650 inch-pounds. This means
that a weight of 100 pounds, falling 86.5 inches, or 1000 pounds
Wling 8.65 inches, would have strained the beam up to the point
P or it would have deflected it 1.66 inches, and the beam would
have been then resisting with a force of 10,000 pounds, since P falls

* This term has been coined to define this particular portion of tho resili-
ence which will be used for comparing the relative elasticity or toughness of
different timbers.



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^86 BLA8TIC LIMIT, DEFLECTION, RESILIENCE.

on the 10,000-pound line. If this result — 8650 inch-pounds — be
divided by the number of cubic inches in the stick between end-
bearings, the result is the true Relative Resilience in Cross break'
ing in inch-pounds per cubic inch. This result is independent of
the dimensions of the test specimen, and is therefore a true meas-
ure of the quality of timber which is usually known as toughness.
It depends, as toughness in the usual understanding does, on both
the strength and the deflection; in fact, it is very nearly the half-
product of the strength developed and the deflection produced at
this particular point P, It is probably the nearest quantitative
measure of the toughness that can be arrived at.

247. The strength of the extreme fibre is computed by the ordi-
nary formula

/=S' <"«>

where / = stress on extreme fibre in pounds per square inch,
W = load at centre in pounds,
I = length of beam in inches,
b = breadth of beam in inches,
h = height of beam in inches.

At the time of final rupture this formula by no means repre-
sents the actual facts. It assumes that the neutral plane remains
At the centre of the beam till rupture occurs, which is far from
correct. In green timber, where the crushing strength is greatly
reduced by the presence of the sap, the crushing resistance is only
about one third as much as the resistance to tension, so that the
stick invariably begins to fail on the compression side. • This causes
the neutral plane or plane of no stress to be lowered, and at the time
of final rupture this plane may be from one fourth to one sixth the
depth from the bottom side of the beam. The value of / computed
by this formula from a cross-breaking test, therefore, will always be
intermediate between the crushing strength and the strength in
tension. Thus the crushing strength of a given stick was found
to be 5820 pounds per square inch, while the tensile strength was
15,780 pounds; the cross-breaking strength was found by this test
to be 10,900 pounds.



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DETEKMIKATION OF ELASTICITY AND EESISTANCE. 287

The modulus of elasticity is computed from the formula

^ ~ 48/)/ ~ WW ~ D ' 4W • ' ' ^^^^'

▼here E — modulus of elasticity, W, I, b, and A as in eq. (106),
D = deflection of beam, and

/= moment of inertia of the cross-section = -^A' for rec-
tangular sections.

To find this modulus, a tangent line is drawn to the strain dia-
gram at its origin, as OA, and the co-ordinates of any point on this
line need as the PTand D from which to compute B,

The modulus is thus seen to vary directly as the load and in-
versely as the deflection, hence it is a true measure of the stiffness
of the material. It is the most constant and reliable property of
all kinds of engineering materials,* and is a necessary means of
computing all deflections or distortions under loads.

In using the modulus of elasticity of timber for computing
deflections, it must be remembered that in this case the time effect
is very great (it is nearly zero in metals) and that this factor can
only be used to compute the deflection for temporary loads. The
deflection of floor or roof timbers, for instance, under constant
loads is a very different matter, as it increases with time.

Relation between Strength and Stiffness, — In Fig. 100 is shown
the relation found by Professor Bauschinger f between the modulus
of elasticity (stiffness) and the cross-breaking strength, from tests
on pine, larch, and fir timber. Although the results show a wide
range, there is evidently a general relation between these two quan-
tities, as indicated by the straight line drawn through the plotted
points. The algebraic expression of the law shown by this line,
rendered into pounds per square inch, is, in round numbers.

Cross-breaking strength = 0.0045 modulus of elasticity + 460. (108)



* The wide range of values of the modulns of elasticity of the various
netals, found in published records of tests, must be explained by erroneous
Betbods of testing.

f See PI. n, vol. 16, of Professor Bauschinger's Reports of Tests made at
6tiifeniment Testing Laboratory at Munich.



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288



STRENGTH, 8TIFFKE8S, WEIGHT.



If it should be found that there is snch a law for all kinds of
timber, then there may be derived an eqnation of this form, but
with different constants, for each species.

CroBB-breaklng Strength = 0,0016; Modulus of Elasticity (or Stifltaess) + 4B0.




OwM htiwHnif ttr—gth in atmocpheres.

FkG. lOO.^Relatioo between Cross-breaking Strength and Modulus of Elaa»

ticity or Stiffness.

Oompretiive Strength = 18,800; Sp. G. ~ 900.




Sffeolflo gnifitj (ra4iioed to U pex o«iit noistarei

Pig. 101.— Relation between Ck)mpre8sive Strength and Sp. G. or Weight.

Relation between Strength and Weight. — In Fig. 101 is shown
the relation between the crushing strength and the specific grarity,
when both are reduced to the standard percentage of moisture,
which was taken at 15 per cent.

These results are also taken from Professor Bau8chinger*8 pub*



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DETERMIirATIOK OF ELASTICITY AND RESISTANCE.



2S9



lished records of tests on pine, larch, and iSr timbers, and they con-
clusiyely show that the greater the weight the greater the strength
of the timber. The law here is a well-defined one, so far as these
timbers are concerned. When rendered into English units (pounds
per sq. in.), the equation of this line is

Crushing strength = 13,800 specific gravity — 900, . (109)

when the timber contains but 15 per cent of moisture. This equa-
tion would also vary in its constants for each species of timber.

Relation Between the Compressive Strength and the Percentage



Online LibraryWilliam Macfarland PattonA treatise on civil engineering → online text (page 26 of 145)