STRAIN-DEFLECTION RELATIONSHIPS OF FREELY •• VIBRATING WOOD BEAMS by Ray Carl Minor Thesis submitted to the Graduate Faculty of the Virginia Polytechnic Institute in candidacy for the degree of MASTER OF SCIENCE in Agricultural Engineering Approved: CMirinan, H. T. HJh&l Prof. E. T. Swink Dr?J. P. H. Mason Dr. G. If"(j (J \Jf\ Prof. D. D. Hamann September 1966 Blacksburg, Virginia
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APPEND IX ••••••••••••••••••••••••••••••••••••••••••••••••••••••• • • • • •• 64
LIST OF FIGURES
Figure
1. Single-span beam with elastic supports •.•••.••••••••••••••• 11
2. Single-span beam with end moments proportional to rotation ................................................... 18
3. Natural Frequency vs. Rigidity of end supports ••••••••••••• 21
4. Fixed-end beam test apparatus with steel test beam and instrumentation for measuring displacement, acceleration, and strain at midspan of beam •••••••••••••••• 24
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Figure
5. Arrangement used for calibrating LVDT with dial gage .................................................. 25
6. Arrangement of vibration sensing devices including LVDT, accelerometer, and strain gage .•••••.•..••.........•.......•..••••••..•.••••••••••••. 27
7, Testing apparatus· for fixed-end tests .••••••.•••••••••••••• 29
8. End clamp for fixed-end tests ..•••.••.•.••••••••••••••••••• 30
9. Testing apparatus for pinned-end tests .•••••••••••••.•••••• 31
ANOVA of ed/Yd obtained from the fixed-end tests eslYs
on all three wood beams ..........................•...•.•••. 52
10. ANOVA of edfYd obtained from the pinned-end tests eslYs
11.
on al 1 three wood beams . .......................•.........•. 53
ANOVA of ed/yd obtained from the semi-rigid end esfYs
tests on all three wood beams •••••••••••••••••••••••••••••• 53
A
B ·c
c
D
E
e
£ g
I
k
L
M
n
p
p(x) t
w
x
y
Yd Yo Ys y' y"
-6-
LIST OF SYMBOLS
Arbitrary constant Arbitrary constant Arbitrary constant Distance from neutral axis to extreme fiber Arbitrary constant Modulus of elasticity Flexural strain Flexural strain of vibrating beam Flexural strain of beam with static load applied at midspan Primary natural frequency Acceleration of gravity Moment of inertia of beam Spring constant Span of beam Moment of beam
Concentrated load applied to beam Load applied to beam per unit length Time Weight of beam per unit length Longitudinal distance from end of beam Deflection of beam Deflection of vibrating beam Maximum deflection of vibrating beam at midspan Deflection of beam with static load applied at midspan Slope of beam, ~
dx M/EI, d2y
dx2 Circular primary natural frequency
•
-7-
I N T R 0 D U C T I 0 N
Residential floors are usually designed on the basis of allowable
stress in the floor joists or allowable deflection, whichever may be
the limiting factor. Allowable deflection is usually the limiting
factor in wooden floor design.
Floors designed on the basis of allowable stress and allowable
deflection may, however, have undesirable vibrational behavior.
Increased usage of lighter types of construction in floors for economy
purposes has occasionally resulted in floors with insufficient stiff,ness
or dampening to prevent noticeable vibration from being induced by
human impact (6).*
This concern about residential floor vibrations has resulted in
a need for methods of analyzing and controlling these vibrations. In
order to experimentally analyze floor vibrations, methods must be
devised for measuring various vibrational properties such as frequency
amplitude, and duration of vibration. Human reaction to residential
floors is, in part, dependent on each of these properties (6).
Accelerometers, photo-electric cells, electromagnets, and a
recording stylus mechanically attached to the test specimen are some of
the sensing devices that have been used to measure vibration of wooden
structural elements (3). Some researchers in the area of wooden floor
vibrations have used bonded SR-4 strain gages on the floor joists to
*Numbers in parenthesis refer to appended references.
-8-
sense these vibrations (1, 2, 4). The use of bonded SR-4 strain gages
is desirable because they are readily available, relatively inexpensive,
and easy to use.
These experiments using strain gages as vibration sensing devices
resulted in a need to be able to determine the vibration amplitude
(or deflection) from the vibration data obtained using strain gages.
Empirical relationships found between midspan flexural strain and mid-
span deflection of freely vibrating floor joists were different from
those relationships found when a static load was applied at the center
of the joists (2, 4).
This investigation deals primarily with the theoretical and
experimental analysis of the relationship between flexural strain and
lateral deflection of selected, freely vibrating wood beams with
various end conditions (lateral deflection is the deflection in the
plane of the depth of the beam). One steel beam was also used as a
check on the vibration theory and testing equipment.
The end conditions investigated were pinned, fixed, and semi-rigid.
Semi-rigid end conditions were included because many end conditions
encountered in actual practice are neither pinned nor fixed, but some-
where in between.
-9-
R E V I E W 0 F L I T E R A T U R E
Several researchers have studied the vibrational behavior of
wood (3, 5, 7). Most of these studies were directed toward the use of
vibration testing for commercial grading of structural lumber. Two
leaders in this area are Jayne (5) and Pellerin (7). Both of these men
have done considerable work in detennining the modulus of elasticity
and other strength properties of structural lumber through vibration
techniques. They demonstrated that the modulus of elasticity of
structural lumber can accurately be determined by vibration testing.
Youngquist (11) and Walker and Dale (10) investigated the perfor-
mance of bonded wire strain gages on wood. Their investigations dealt
primarily with static strain measurement. Youngquist found that the
reliability of strain data obtained with bonded wire gages mounted on
wood is somewhat leas than the reliability of these gages mounted on
steel. He also found that gages of 13/16 in. length (Type A-1) gave
more accurate results on wood than 1/4 in. long gages (Type A-7).
Tests were performed at the NAHB Research Institute Laboratory (1)
to determine the effects of cross bridging on the vibrational character-
istics of a typical wood joist floor. Vibration was induced by suddenly
releasing a weight attached at the midspan to the underside of one of
the joists. Vibration was sensed by a SR-4 strain gage located at
midspan bonded to the underside of one of the joists. In these tests,
no attempt was made to find the relationship between deflection and
strain; however, in discussing their results, they assumed that strain
is directly proportional to midspan deflection.
-10-
Hurst (4) investigated the effects of various components and
structural elements on residential floor vibration, This was accomplished
by performing floor vibration tests at 14 stages of construction of a
typical full-scale house. Vibration at midspan of the floor joists was
sensed simultaneously by SR-4 strain gages bonded to the underside of
the joists and a linear variable differential transformer (LVDT) with
the moveable core attached to the bottom of the joist. For static
loading, the LVDT and the strain gage yielded essentially the same
results. However, for dynamic loading (vibration) the strain gage
averaged 34% less than that indicated by the LVDT.
Thompson (9) developed and solved the differential equation which
describes the free vibration of beams. He began with the well-known
differential equation of the relationship between deflection, y, and
applied load per unit length, p(x), for a beam of constant flexural
rigidity, EI,
EI d4y • p(x). dx4
((1))
For a beam vibrating under its own weight, the effective load per
unit length changes along the length of the beam and is equal to the
inertia load due to its own mass and acceleration. By assuming that a
beam vibrating under its own weight has simple harmonic motion, its
acceleration is w2y where u is the circular frequency.
Then
((2))
where w is the weight of the beam per unit length and g is the
acceleration of gravity.
-ll-
Combining equations 1 and 2 gives
EI d4t • ~ w2y. dx g
Substituting n4 • w ~2 Elg
the fourth order differential equation
is obtained.
The general solution to equation 5 is
Yd • A cosh nx + B sinh nx + C cos nx + D sin nx,
((3))
((4))
((5))
((6))
where yd • deflection of freely vibrating beam. The constants A, B,
C, and D are arbitrary constants which must be determined in every
particular case from the conditions at the two ends of the beam.
Rogers (8) set up the determinant associated with the single-span
beam shown in Figure 1. The supports shown in Figure 1 are all elastic,
which means that the amount of shear force (or bending moment) present
is proportional to the amount of deflection (or rotation). The spring
constants are k1 and k2 for the translational springs and k3 and k4 for
the rotational springs.
L
Figure 1. Single-span beam with elastic supports.
-12-
The determinant for this general case is given below.
-k3/EI
-n3 cos nL - k2/EI sin nL
-n sin nL + ict./EI cos nL
·k3/EI
n3 cosh nL - k2/EI sinh nL
n sinh nL + k4/EI cosh nL
k1/EI
-n
n3 sin nL - k2/EI cos nL
-n cos nL - k4/EI sin nL
n • 0 ((7))
n3 sinh nL - k2/EI cosh nL
n cosh nL + k4/EI sinh nL
-13-
0 B J E C T I V E S
The first objective of this investigation was to theoretically
and experimentally determine the relationship between midspan flexural
strain and midspan deflection of freely vibrating wood beams having the
following end conditions:
(1) Fixed
(2) Pinned
(3) Semi-rigid.
The second objective was to determine how accurately existing
vibration theory for ideal elastic materials could be used to predict
the strain-deflection relationship of freely vibrating wood beams.
-14-
T H E 0 R E T I C A L C 0 N S I D E R A T I 0 N S
In this section, theoretical relationships between deflection and
. strain of freely vibrating simple beams and fixed-end beams will be
developed. Theoretical relationships between the rigidity and the
natural frequency of beams with semi-rigid end connections will also be
developed.
The Simple Beam
For a simple beam, the boundary conditions to be applied to
equation 6 are
at x • O; y • O,
y" • o,
at x • L; y • 0,
y" - 0.
which yields
yd • D sin nx. ((8))
Rogers (8) showed that n •11/L for a simple beam vibrating at its
primary frequency. Equation 8 then becomes
yd • D sin '1'~ • ((9))
By setting the deflection at the midspan of the beam equal to
Yo sin .,t, equation 9 becomes
Yd • (y0 sin Wt) sin ~ • ((10))
The equation relating flexural strain, e, to deflection is
-15-
where c • distance from neutral axis to the extreme fiber.
Differentiating equation 10 twice with respect to x, and substitut-
ing into equation 11 yields
ed • c(1'f /L)2 (y0 sin Ut) sin ff'x, L
(( 12))
where ed • flexural strain of a freely vibrating beam.
The relationship between midspan deflection and strain found by
dividing equation 12 by equation 10 and substituting x • L/2 is
(( 13))
This relationship will now be compared with the relationship between
deflection and strain at the midspan of a simple beam with a concentrated
load at the center. This relationship, as derived from static equations,
is
((14))
The relationship between deflection and strain of a vibrating simple
beam can now be compared with that relationship of a simple beam with a
concentrated load at the center by dividing equation 13 by equation 14:
- rr2 - o. 824· TI
The Fixed-end Beam
((15))
The boundary conditions for a beam with both ends fixed are
at x • O; y • O,
y' • O,
at x • L; y • O,
y' • o.
-16-
Rogers (8) used these four end conditions to develop the frequency
equation
cos nL • __ l __ cosh nL
( ( 16))
for a fixed-end beam. The lowest non-zero value of nL satisfying this
equation was found to be 4.730, which is the value of nL for a fixed-
end beam vibrating at its primary frequency.
Applying the first and third boundary conditions to equation 6
results in C • -A, and D • -B.
Equation 6 applied to a beam with both ends fixed and vibrating at
its primary frequency has now become
Yd• A(cosh 4.730x - cos 4.730x) + B(sinh 4.730x L L L
_ sin 4 • 7 30x) • L
( ( 17))
Applying the second boundary condition to equation 17 results in
B • -0.9825A. Substituting this value for B in equation 17 gives
yd• A(cosh 4 .13ox - cos 4.730x - 0.9825 sinh 4.730x L L L
+ 0.9825 sin 4. 730x). L
((18))
The deflection at midspan of the vibrating beam can be expressed as
Yd • Yo sin Wt. ((19))
Substituting x • L/2 and equation 19 into equation 18 gives
y0 sin Wt • A(cosh 2.365 - cos 2.365 - 0.9825 sinh 2.365
+ 0.9825 sin 2.365)·
Evaluating equation 20 yields
A • 0.6297 y 0 sin Wt.
(( 20))
((21))
-17-
The deflection equation can now be written as
Yd • (y0 sin Wt) (0.6297 co sh 4.730x - 0 6297 cos 4.730x L . L
-0.6187 sinh 4.730x + 0.6187 sin 4.730x). ( ( 22)) L L
Differentiating equation 22 twice with respect to x, and substituting
into equation 11 gives the equation for flexural strain,
ed • c(4.730/L)2 (y0 sin Wt) (0.6297 cosh 4.730x + 0.6297 cos 4.730x L L
-0.61B7 sinh 4 ·~30x - 0.6187 sin 4.~30x). ((23))
The relationship between midspan deflection and strain found by
dividing equation 23 by equation 22 and evaluating at x • i is
(( 24))
The relationship between midspan deflection and strain for a fixed-
end beam with a static load applied at the center is found from static
equations to be
(( 25))
The relationship between deflection and strain of a vibrating
fixed-end beam can now be compared with that relationship of the same
beam with a concentrated load applied at the center by dividing
equation 24 by equation 25:
ed/Yd • 17.131/24.00 • 0.714. e/ys
Beam With Semi-Rigid End Connections
((26))
The end conditions being considered now are shown in Figure 2.
-18-
~ -ky' +ky' ( j---__ L __ --'·I~
,,,.())7 J~J
Figure 2. Single-span beam with end moments proportional to rotation.
Determinant 7 for general end conditions can be applied to the
above beam by letting k1 • k2 • oO, and kJ • k4 • k. Dividing the
first row of the determinant by k1, the third row by k2, and substi-
tuting the above values into the determinant gives the following:
0 l/EI
-k/El -n l - sin EI nL 1
EI cos nL
sin k cos nL -n nL +- -n cos nL - k sin nL EI EI
0 l/EI
-k/EI n - o • (( 27))
.L sinh nL 1 cosh nL EI EI n sinh nL + .!_ cosh nL n cosh nL + ~ sinh nL
EI EI
Expanding the determinant gives
[~1] 2 [1 - (cos nL) (cosh nL~ + 2kn Gs in nL) (cosh nL) EI
The theoretical relationship between the rigidity of the end
connections and the rotation of the ends, y', of a semi-rigid beam with
a static load at the center was determined from static equations to be:
PL2 y I • ------...,,
16EI [l + ~LE~ • (( 29))
-20-
This equation was rearranged and plotted in Figure 3 to compare the
effect of ~nd rigidity on end rotation with its effect on frequency.
. u ~ CD
........ CD ~ ....
•' u >-u ..
~ ~
>-u c: ~ :I t1' ~ ... p.,
0.07 3.56 .--------------------------------------
3.20
2.80
Note: Frequency curve approaches 2.40 3.56 and end rotation curve
approaches zero as kL/EI approaches oo(fixed-end).
2.00
1.60
End Rotation
1.20 0 10 20 30 40 50 60 70 80 90
(Pinned-end) End Support Rigidity (kL/EI)
0.06
0.05
I 0.04
j 0.03
0.02
0.01
0 100
Figure 3. Natural frequency and static end rotation vs. end support rigidity.
CD i:: aJ
-..-4 "d aJ ~
r;;i Hr~ ~ i:i.. -CJ
i:: 0
-..-4 ... as I ... N 0 .... ~ I
"d c: ~
-22-
T H E E X P E R I M E N T
Scope
Free vibration tests were performed on three Douglas Fir wood beams
and one steel beam. The steel beam was included primarily as a check to
see how closely theoretical and experimental strain-deflection relation-
ships agreed. The three Douglas Fir wood beams were tested with pinned-
end (simple beam), fixed-end, and semi-rigid end conditions. Twenty
repetitive tests were conducted on each beam with each end condition.
The steel beam was tested only under fixed-end conditions, which,
according to theory (equation 26), gives the largest difference between
the static and dynamic strain-deflection ratios.
Vibration was induced by suddenly releasing a load from the center
of the beam. Deflection was measured with a linear variable differential
transformer (LVDT) and strain by a bonded SR-4 strain gage. Signals
from both sensing devices were amplified and recorded using a two-channel
high speed oscillograph. The relationship between deflection and strain
was then determined from the oscillograph recordings.
Apparatus
The test beams. It can be seen from equations 15 and 26 that
according to theory the ratio ed/yd is independent of any dimensions es/Ys
of the beam itself and is dependent only upon the end conditions.
Having this in mind, the size of the beams was based primarily upon the
ease with which the beams could be subjected to the experimental tests.
It was desirable to size the beam such that its depth/length ratio
was relatively low so that measurable strain and deflection could be
-23-
obtained with a small load. Low depth/length ratios also result in
lower natural frequency which is generally easier to measure and record
than higher frequencies.
The size of wood beams selected was 1-5/8 in. x 3-1/2 in. x 10 ft.
span, using the 1-5/8 in. dimension as the depth.
Three clear Douglas Fir beams, 1-5/8 in. x 3-1/2 in. x 12 ft.,
were selected and purchased·at a local lumber company. These beams had
been dressed and kiln dried.
The steel beam selected was 1/2 in. x 2 in. x 8 ft. span, using
the 1/2 in. dimension as the depth.
The wood beams will herein be referred to as Beams No. 1, 2, and 3,
and the steel beam, No. 4.
Instrumentation. Instrumentation was used for measuring and
recording deflection, strain, and acceleration at the midspan of a
vibrating beam. All the instrumentation is shown in Figure 4 connected
to the steel beam. Acceleration was measured only on the steel beam.
The deflection was measured with a Sanborn Model 585DT1000 linear
variable differential transformer (LVDT) having a linear range of two
inches. The LVDT was forced to follow the movement of the beam by
being attached to a shaft, one end of which contacted the beam, and the
other end being spring loaded. The spring loaded LVDT exerted a force
of nine to eleven ounces on the beam, which was considered to have
negligible effect on the free vibration of the beam. An Ames Dial
gage was used to calibrate the LVDT as shown in Figure 5.
-24-
Figure 4. Fixed-end beam test apparatus with steel test berun and instrumentation for recording displacement, acceleration, and strain at midspan of beam.
-25-
'Figure 5. Arrangement used for calibrating LVDT with dial gage.
-26-
The flexural strain was measured with an Al-S6, SR-4 strain gage.
One strain gage was bonded to each beam at least 24 hours prior to the
tests. The gage was located on the underside of the beam at midspan
and oriented so as to measure longitudinal or flexural strain. The
gage was connected to the amplifying and recording equipment using a
four arm bridge circuit. The bridge arms consisted of the one active
gage mounted on the beam, one compensating gage mounted on another
piece of the same material, and two other unbonded gages to complete
the bridge circuit. All strain gages used throughout the experiment
were from the same lot having a gage factor of 2.02 and a resistance
of 120.8 ohms.
The amplifying and recording equipment for the strain gage and
LVDT consisted of a two-channel Baldwin-Lima-Hamilton (BLH) Model BSA-250A
Meterite oscillograph with two BLH Model PR-401 carrier amplifiers. The
oscillograph recorded in ink with a choice of six chart speeds ranging
from 0.5 to 200 mm/sec. The chart speeds used during the vibration
tests were 5 and 20 mm/sec.
The acceleration at midspan of the vibrating steel beam was sensed
with an Endevco Model 2219 accelerometer. The signal from the accel-
erometer was amplified and recorded with a Kistler Model 568 charge
amplifier, Model 567 power amplifier, and a Honeywell Model 1406 Visi-
corder oscillograph. The amplifying and recording equipment was
calibrated with a Kistler Model 541A hand calibrator. The accelerom-
eter followed the motion of the beam by being attached to the top of
the LVDT shaft as seen in Figure 6.
-27-
·Figure 6. Arrangement of vibration sensing devices including LVDT, accelerometer, and strain gage.
-28-
Fixed-end supports. An overall view of the test apparatus with
the fixed-end supports is shown in Figure 7. The supporting frame con-
sisted of a 6 x 6@ 25 lb./ft. WF beam, 12 ft. long, bolted to a
concrete floor and two 12 x 5@ 31.8 lb./ft. I beams, 1 ft.-8 in. long,
One end clamp which held the test beam is shown in Figure 8.
Twelve inches of the beam was held by each end clamp leaving a span of
10 ft. for the wood beams and 8 ft. for the steel beam. The clamps
consisted of half-inch steel plates bolted together with four half-inch
steel bolts. In order to apply equal pressure to the clamped part of
the wood beam, the bolts were tightened equally to 40 ft.-lbs. torque.
' Pinned-end supports. The supporting structure used for the
fixed-end supports was also used for the pinned-end supports except
the end clamps were replaced by pinned connections as shown in Figure 9.
One end of the test beam was free to rotate but restrained from
any other motion using a steel shaft and bearings as shown in Figure 10.
The other end of the beam was free to rotate and free to translate
horizontally using the arrangement of shaft and bearings shown in
Figure ll.
Semi-rigid end supports. Investigation of methods of achieving
well-defined semi-rigid end conditions led to the decision to use
torsion bars.
The torsion bars were designed so that the rigidity of the end
would be midway between fixed and pinned ends. It was thought that this
midway point might be achieved by designing the torsion bar to provide
-29-
Figure 7. Testing apparatus for fixed-end tests.
-30-
jigure 8. End clamp for fixed-end tests.
-31-
..Figure 9. Testing apparatus for pinned-end tests.
~ Figure 11. Pinned-end connection which also permits horizontal translation. The shaft sup-porting the beam has four ball bearings on it. The two inner bearings are used to support the shaft and the two outer bearings are used to prevent the shaft from rising off the support during vibration.
-~-
a tesisting moment equal to half the end moment of the fixed-end beam
when the bar is rotated half the rotation of the end of the simple beam.
The end rotation, y', of a pinned end beam with a concentrated
load, P, applied at the center is
y' • PL2/16EI ((30))
The end moment of a fixed-end beam with a concentrated load at the
~enter is
M • PL/8 ((31))
The spring constant for the torsion bar can now be found as follows:
k • M/y' • PL/16 • 2EI/L PL2/32EI
((32))
Since the modulus of elasticity was not the same for all three wood
beams, a different torsion bar would be needed for each beam. However,
only one size of torsion bars was selected, its size being based on the
average flexural rigidity, EI, found from static deflection tests on
all three beams. The average value of EI was found to be 2.45 x 106 lb.-
in. 2. Using this value and a length of 120 in. gives k • 4.08 x 104 in.-
lb./rad.
The torsional spring constant, k, can be calculated for a circular
torsion bar by the following formula: Gd4
k - ~--.- ((33)) 10.3L
where
G •modulus of rigidity (12 x 106 psi for steel),
d • diameter of bar,
L • length of bar.
-35-
Based on this equation, a steel torsion bar 0.627 in. in diameter and
4.5 in. long was used to give the desired spring constant.
The torsion bar was made by machining down a 4\ in. section of a
3/4 in. square bar to 0.627 in. One end of the torsion bar was rigidly
supported and the other end fastened to the beam by using the clamping
arrangement shown in Figure 12. With this arrangement, one end of the
torsion bar rotated with the beam while the other end was completely
fixed.
Torsion bars were used on both ends of the beam as can be seen in
Figure 13.
Loading device. Vibration of the wood beams was achieved by
suddenly releasing a load attached to the center of the beam. The load
was attached to the beam by a string and released by cutting the string
with scissors as shown in Figure 14. A 25 lb. load was used on the
fixed-end beams and a 16 lb. load on the pinned and semi-rigid end
beams.
It was found that when the above method of exciting vibration was
used on the steel beam, the second natural frequency of the beam was
also excited which caused a lower amplitude, higher frequency vibration
to be super-imposed on the primary vibration. The strain gage picked
up this secondary vibration, but the LVDT did not. This behavior can
be seen in Figure 15. It was found, however, that the secondary vibra-
tion was not picked up by the strain gage if vibration of the beam was
excited by pulling the beam down by hand and suddenly releasing it.
Therefore, for the steel beam tests, vibration was excited by the latter
method.
-36-
figure 12. Semi-rigid end connection using torsion bar.
-37-
· Figure 13. Testing apparatus for semi-rigid end t~sts.
-38-
Figure 14. Vibration excitation by cutting ~
string to release load.
-39-
r- -
H ,, l 1 :U :\
·~ \ r.t . ,1 11. t ~
., ~ I -~ !I
t
" i 1i:1 ' .. . .. \ r.;:i -
t• VJ r,~ ~ ~~ ! I"' ., ... ~ ~ -1
:i:,c.IT-.I u
~ " EI ' v~ · ~n · ;t
rrn rrr :1 l! <'.""" ... .,
~--
Strain (Strai Got•) "lI,J ~t er
·Figure 15. Oscillogram of vibrating steel beam when excited by method shown in Figure 14. Chart speed-200 mm/sec.
·• I '
-40-
Test Procedure
Calibration. It was very important that the instrumentation be
calibrated in such a manner that the relationship between deflection
and strain of the vibrating beam could be easily determined from the
vibration records. The method used to accomplish this goal will now
be described.
Prior to testing, the strain gage and LVDT circuits were connected
to the amplifiers and balanced according to the manufacturer's instruc-
tion manual. The strain gage was calibrated according to the amplifier
instruction manual so that the chart deflection could be converted to
strain. The factor for converting chart deflection, in mm., to strain,
in inches/inch, was 8 x lo-6.
The load was applied to the beam using the loading device described
earlier. With the load applied, the chart deflections of the strain
gage and the LVDT were equalized using the amplifier controls. The
results of this procedure can be seen in Figure 16 in the portion of
the chart where the static load was applied.
After having adjusted the instrumentation as just described, the
LVDT was calibrated using a dial gage as shown in Figure 5, page 25,
so that the chart displacement could be converted to inches of deflec-
tion of the beam. After calibration, the dial gage was removed as it
was found that it caused unwanted dampening.
For the special tests on the steel beam in which the accelerometer
was used, calibration of the accelerometer instrumentation was
necessary in addition to all the above procedures. The accelerometer
equipment was calibrated according to the manufacturer's instruction
-41-
manual using a hand calibrator. The hand calibrator was connected to
the amplifiers in place of the accelerometer and provided a signal
equal to a known amount of acceleration. This signal caused a chart
deflection which could then be converted to acceleration.
Conducting test. After completing the calibration procedure,
vibration tests were perfonned by applying the load, turning on the
chart drive motor, and cutting the string to release the load, the
results of which can be seen in Figure 16. Twenty replicated tests
were conducted on each beam with each end condition.
Having conducted the tests as described, the ratio of edlYd could eslYs
be determined directly from the oscillogram by dividing the strain gage
vibration amplitude, as recorded on the oscillogram, by that of the
LVDT. (The strain-deflection ratio of the beam with the static load
applied equals es/Ys, and the strain-deflection ratio of the vibrating
It was noted that the half cycle during which the strain gage was
in compression was consistently of lower magnitude than the other half
cycle where the strain gage was in tension. This difference can be
seen in Figure 16. This behavior of the strain gage occurred on the
steel beam as well as all the wood beams.
Static tests were conducted on each beam to detennine the effect
of the strain gage giving a smaller signal when in compression. These
tests were performed by hand loading the beam at the center so that
both strain gage and LVDT gave equal chart deflections of 15 divisions.
The beam was hand loaded in the opposite direction (by pulling up) so
Comparisons between the theoretical and experimental strain-
deflection relationships for the fixed-end and pinned-end conditions
are summarized in Table 8. The average experimental value and the
standard deviation for each beam and end condition given in the table
are based on the results of 20 tests.
End and
Table 8. Comparison of theoretical and experimental strain-deflection relationships.
Condition Corrected Theoretical % Difference of Beam No. Average
edf Yd Experimental From
edfYd Theoretical edf Yd es/es es/Ys es7Ys
Pl o. 794 0.824 -3. 6 P2 0.848 0.824 +2.9 P3 0.849 0.824 +3.0 Fl 0.687 o. 714 -3. 8 F2 o. 707 o. 714 -1.0 F3 o. 710 o. 714 -0.6 F4 o. 719 o. 714 +o. 7
Standard Deviation
0.017 0.040 0.020 0.013 0.012 0.022 0.010
The wood beams gave results which varied as much as 3.8% from
theory, yet the results from the steel beam differed from theory by
only 0.7%. As indicated by the standard deviations, the results from
the wood beams were also spread out more than those from the steel
beam. A logical explanation for the greater deviation of the wood
beams from theory is that wood deviates more from the basic assumptions
made in developing the theory. For example, two assumptions in the
theory are that modulus of elasticity and density of the beam are
-52-
uniform along its length. However, both of these properties may vary
considerably along the length of a wood beam.
The average of the three corrected strain-deflection relationships
found from the wood beams under pinned-end conditions is 0.830. Com-
paring this average to the theoretical value of 0.824 shows a difference
of less than one per cent. Similarly, the average of the three wood
beams for fixed-end tests is 0.701, which, compared to a theoretical
value of 0.714, shows a difference of less than two per cent. The
average edlYd of all three wood beams under pinned-end conditions is esf Ys
considerably closer to theory than any of the beams considered
separately.
Statistical analysis of variance (ANOVA) for each end condition
shows a significant difference in the strain-deflection relationships
between the wood beams. The analysis of variance tables are shown in
Tables 9, 10, and 11. The uncorrected data given in the Appendix were
used in the ANOVA. Based on the statistical analysis, it can be said
that the difference in the test results for each end condition is
more significant between beams than within beams.
Table 9. ANOVA of edlYd obtained from the fixed-end esfYs
tests on all three wood beams.
Source of Variation df SS MS F Critical F*
Beams Error Total
2 57 59
0.00260 0.01560 0.01820
Mean - 0.6831 Standard Deviation - 0.0176 Coefficient of Variation - 2.57%
*All levels of significance used are 5%.
0.00130 0.00027
4.81 3.16
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Table 10. ANOVA of ed/Yd obtained from the pinned-end tests es/Ye
on all three wood beams.
Source of Variation
Beams Error Total
Mean - 0.8109
df
2 57 59
SS
0.031559 0.043760 0.075319
Standard Deviation - 0.0357 Coefficient of Variation - 4.40%
MS
0.015779 0.000767
F Critical F
20.57 3.16
Table 11. ANOVA of ed/Yd obtained from the semi-rigid end esfYs
tests on all three wood beams.
Source of Variation
Beams Error Total
Mean - 0.7758
df
2 57 59
SS
0.007566 0.037593 0.045159
Standard Deviation - 0.0264 Coefficient of Variation - 3.55%
MS
0.003783 0.000659
F Critical F
5. 74 3.16
The average of the three corrected strain-deflection relationships
found from the three beams under semi-rigid end conditions is 0.802.
Comparing this value with 0.830 for the pinned-end and 0.701 for the
fixed-end conditions, it is seen that the vibrating strain-deflection
relationship for the semi-rigid end conditions was considerably closer
to the pinned-end conditions than to the fixed-end conditions. These
results indicate that beams having end conditions with only a small
amount of resistance to rotation (kL/EI less than two) can be considered
pinned-end beams without introducing an error greater than five per cent
in their vibrating strain-deflection relationship.
-54-
Frequency
As shown in Table 6, it was found that the natural frequency for
the semi-rigid end conditions was considerably closer to the pinned-end
conditions than to the fixed-end conditions.
As described previously, the rigidity of the semi-rigid ends was
designed from static equations (equations 30, 31, and 32) to be midway
between pinned-end conditions and fixed-end conditions. This midway
point was found to result in a rigidity, kL/EI, of 2.00.
Referring to the frequency-rigidity curve shown in Figure 3, page 21,
a rigidity kL/EI of 2.00 should have a frequen~y considerably closer to
pinned-end conditions than to fixed-end conditions. From Figure 3,
the frequency equation for kL/EI of 2.00 is f • 2.02~!~f Using
average values of the three beams in this equation, the average fre-
quency for the semi-rigid tests is calculated to theoretically be
12.8 cycles/sec. which when compared with 12.2 cycles/sec. (the average
frequency obtained from the tests) is about five per cent higher.
Modulus of Elasticity
Comparison of the modulus of elasticity found for each beam by the
six methods, the results of which are stmnnarized in Table 5, shows some
large discrepancies.
The modulus of elasticity found from the fixed-end static deflec-
tion test and that found from the pinned-end static deflection tests
for the same beam were expected to agree very closely. However, as can
be seen in Table 5, fixed-end static deflection tests consistently gave
a lower modulus· than the corresponding pinned-end test. This same
-55-
relationship is also found when comparing the modulus found by fixed-end
and pinned-end vibration tests (rows 3 and 4 of Table 5).
After making these observations and studying equations 34, 35, 36,
and 37, the only explanation for the discrepancies described above is
that the effective span of the fixed-end beams was longer than that of
the pinned-end beams. The center of the pins used in the pinned-end
supports shown in Figure 9, and the edge of the clamps used for the
fixed-end supports shown in Figure 7, were both within one-sixteenth of
an inch of ten feet apart. Some bending of the beam inside the clamp
could have increased the effective span of the fixed-end beams.
This effect can be illustrated by using equations 34 and 35 and
the static deflection data obtained from beam No. l. The modulus
determined from pinned-end tests for beam No. 1 was 2.22 x 106• Using
this value in equation 34 along with other data obtained from the
fixed-end tests and solving for L gives a value 124.5 in. This means
that if the effective span of beam No. l during the fixed-end tests
extended 2.25 in. into each clamp, the same modulus of elasticity would
have been found from the fixed-end tests as was found from the pinned-
end tests.
Secondary Vibration
. A logical explanation will now be given as to why the secondary
vibration induced by cutting a string to suddenly release the load
from the steel beam was picked up by the strain gage and accelerometer,
but not by the LVDT.
-56-
According to Rogers (8), the only way a beam can be excited to
vibrate only at its primary frequency is to give it an initial deflec-
tion the exact shape of its deflection when vibrating. Otherwise, it
will vibrate at its secondary as well as primary frequency, the second-
ary motion being superimposed on the primary motion. The equation for
the deflection of a beam vibrating at all of its natural frequencies is
represented by an infinite series. The rate of convergence of the
series is more rapid for the deflection than for its derivatives. This
means that the effects of the secondary vibration will be more pro-
nounced on any of the derivatives of deflection than on the deflection
itself.
The strain gage in effect measures moment, which is the second
derivative of the deflection with respect to distance along the beam.
The accelerometer measures acceleration, which is the second derivative
of the deflection with respect to time. Hence, the strain gage and
the accelerometer both measure a second derivative of the displacement,
Thus, the above information obtained from Rogers (8) may well explain
why the strain gage and the accelerometer picked up the secondary
vibration even though the LVDT did not.
-57-
SUMMARY AND C 0 N C L U S I 0 N S
Theoretical relationships between midspan flexural strain and
midspan lateral deflection of freely vibrating beams with pinned-end
and fixed-end conditions were derived from existing vibration theory.
These relationships were theoretically independent of any of the beam
properties and dependent only upon the end conditions.
Results of tests determining the strain-deflection relationship
of a freely vibrating steel beam with both ends fixed agreed closely
with the theoretical relationship. Free vibration tests on three wood
beams with pinned-ends as well as fixed-ends also gave results which
were in agreement with theoretical strain-deflection relationships.
However, the strain-deflection results from the wood beam tests
showed a significant variation between beams.
The theoretical relationship between the end rigidity and natural
frequency of beams with semi-rigid end connections was derived. Vibra-
tion tests performed on wood beams with semi-rigid end connections
produced frequency-rigidity results which differed from theory by five
per cent.
The semi-rigid ends were achieved by using a torsion bar on each
end designed so that the beam would have a static behavior midway
between pinned-end conditions and fixed-end conditions. However, it
was found both theoretically and experimentally that these torsion bars
resulted in a dynamic behavior (strain-deflection ratio and frequency)
much closer to pinned-end conditions than to fixed-end conditions.
-58-
Calculation of modulus of elasticity of the wood beams based on
the results of fixed-end tests consistently gave lower values than
similar calculations based on pinned-end tests. These results indicate
that some bending of the beam may have taken place inside the clamps
causing the effective span of the fixed-end beams to be greater than
the distance between the end clamps. (In the calculations, the distance
between the end clamps was used as the span.)
It was learned from this research that the strain gages and accel-
erometer used were more sensitive to secondary vibration than the
liqear variable differential transformer was. This behavior appears to
be due to the secondary vibration having a more pronounced effect on
the strain and acceleration than on the deflection of freely vibrating
beams.
In conclusion, the empirical relationships between deflection and
strain of freely vibrating wood beams agree well with the strain-
deflection relationships developed from vibration theory. The vibrating
strain-deflection relationships found under laboratory conditions can
be predicted within five per cent using existing vibration theory for
ideal elastic materials.
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A C K N 0 W L E D G E M E N T S
The author wishes to express his sincere appreciation to his
·major professor, Professor Homer T. Hurst, for his guidance, assistance,
and encouragement, and for suggesting this investigation. Deepest
thanks also go to the members of the Graduate Committee for their sug-
gestions and advice and to the Agricultural Engineering Department for
supplying the facilities and space needed to perform the investigation.
The author also wishes to acknowledge Professor Frank J. Maher
of the Engineering Mechanics Department for assisting in part of the
theoretical analysis.
Special thanks is extended to the National Science Foundation for
their financial support through the N.S.F. Graduate Traineeship which
the author received.
-60-
L I T E R A T U R E C I T E D
1. Bridging of Residential Floors, Research Institute Laboratory Report No. 6, National Association of Home Builders, 1961.
2. Burkholder, James R., '~etermining Vibration Amplitude with Bonded SR-4 Strain Gages," Unpublished Senior Project, Agricultural Engineering Department, Virginia Polytechnic Institute, 1964.
3. Galligan, William L., Proceedings of the Second Symposium on Nondestructive Testing of Wood, Washington State University, Pullman, Washington, 1965.
4. Hurst, Homer T., The Wood Frame House as a Structural Unit, National Forest Products Association Technical Report No. 5, 1965.
5. Jayne, Ben A., "Vibrational Properties of Wood," Forest Products Journal, Volune 9, No. 11, November, 1959.
6. Lenzen, Kenneth H., "Vibration of Steel Joist-Concrete Slab Floor Systems - Final Report," Studies in Engineering Mechanics, Report No. 16, University of Kansas, 1962.
7. Pellerin, Roy F., "A Vibrational Approach to Nondestructive Test-ing of Structural Elements," Forest Products Journal, Volume 15, No. 3, March, 1965.
8. Rogers, Grover L., Dynamics of Framed Structures, John Wiley and Sons, Inc., New York, 1959.
9. Thompson, William T., Vibration Theory and Applications, Prentice• Hall, Inc., Englewood Cliffs, N. J., 1965.
10. Walker, J. N. and Dale, A. C., "Interpretation and Measurement of Strain on Wood," Paper No. 62-418, American Society of Agri• cultural Engineers, 1962.
11. Youngquist, W. G., "Performance of Bonded Wire Strain Gages on Wood," U.S.D.A. Forest Products Laboratory Report No. 2087, Madison, Wisconsin, August, 1957.
-61-
L I T E R A T U R E R E V I E W E D
1. Keightley, Willard O., "Vibration Tests of Structures," Ph.D. Thesis, California Institute of Technology, July, 1963.
2. Lestingi, Joseph F., "Effect of Semi-Rigid Connections on the Natural Frequency of Beams and Simple Frames," Unpublished Master of Science Thesis, Virginia Polytechnic Institute, 1959.
3. Rathbone, Thomas C., "Human Sensitivity to Product Vibration," Product Engineering, August, 1963.
4. Timoshenko, S. P., Vibration Problems in Engineering, D. Van Nostrand Co., 1955.
-64-
APPENDIX
Strain and Deflection Results
Obtained From Vibration Oscillograms
The notation used to denote each test is as follows:
The letter at the beginning ind~cates the end conditions,
"P" denoting pinned-end, "F" denoting fixed-end, and "S"
denoting semi-rigid end. The number following the letter
denotes the beam on which the test was performed. The num-
ber following the hyphen denotes the particular test number
for the end conditions and beam indicated by the number and