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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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T A B L E 0 F C 0 N T E N T S

Subject

INTRODUCTION • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

REVIEW OF LITERATURE. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 9

OBJECTIVES ••••••••••••••••• , •••••••••••••••••••••••••••••••• , •••••••• 13

THEORETICAL CONSIDERATIONS ••••••••••••••••••••••••••••••••••••••••••• 14

The Simple Beam ••••••••••••••••••••••••••••••••••••••••••••••••• 14

The Fixed-end Beam •••••••••••••••••••••••••••••••••••••••••••••• 15

Beam With Semi-Rigid End Conditions •••••••••••••.••••••••••••••• 17

THE EXPERIMENT ••••••••••••••••••••••••••••••••••••••••••••••••••••••• 2 2

Scope •••••••••••••••••••.••••••••••••••••••••••••••••••••••••••• 22

Apparatus •••••••••.••.•••••.•••••••••••••••••.•••••••••••••••••• 22

The Test Beams •••••••. • ••••••••••••••••••••••••••••••••••••• 22

Instrwnentation •..••.••.••••••••••••••.•••.••••.••••••••••• 23

Fixed-End Supports .•.•••••••••••.••••••.•••••.••••••••••••• 28

Pinned-End Supports ••.•••••.••••••••••••••••.•••••••••••••• 28

Semi-Rigid End Supports ••••••••••••••••••••.••.•••••••••••• 28

Loading Device ••••••••••••••••••••••••••••••.••.••••••••••• 35

Test Procedure •••••••••.•••••••••••••••••••••••••••••••••••••••• 40

Calibration •••••••••••••••••••••••••••••••••••••••••••••••• 40

Conducting Test •••••••••••••••••••••••••••••••••••••••••••• 41

TEST RESULTS ••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 46

The Strain-Deflection Relationship •••••••••••••••••••••••••••••• 46

I

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Subject

Modulus of Elasticity. ••• 4 7

Frequency . ..•....................•..•......•...........•........ 48

Secondary Vibration ............................................. 49

Acceleration ••••••••• ••••.••••••••••••••.••••••••••••••••••••• 4 9

ANALYSIS OF RESULTS ••••••• . ...................................•.... 51

The Strain-Deflection Relationship •• ........................ • ••• 51

Modulus of Elasticity • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • 54

Frequency . ...................................................... 54

Secondary Vibration ........................................••... 55

SUMM:ARY AND CONCLUSIONS • ••••••••••••••••••••••••••••••••••••••••••• , • 5 7

ACI<:t~'C)l•ILEDGEl-rlENTS • ..................................................... 59

LITERATURE CITED . •••••.••.••••.....•..••••..•..•.•••••••••••••••••••• 60

LITERATURE REVIEWED • ••.•.•...........••.••••.•.••••••.•.•...•..•••••• 61

VITA • •••••••• • ••••••••••••••••••••••••••••••••••••••••••••• • ••• • • • ••• 62

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

10. Non-translational pinned-end connection ••••••••••••.••••••• 32

11. Pinned-end connection which also permits horizontal movement ..............................•...•..... 33

12. Semi-rigid end connection using torsion bar •••••••••••••••• 36

13. Testing apparatus for semi-rigid end tests •.••••••••••••••• 37

14. Vibration excitation by cutting string to release load ............................................... 38

15. Oscillogram of vibrating steel beam when excited by method shown in Figure 14 .••.••••••.•••••••••••• 39

16. Typical oscillogram of a wood beam test •••••••••••••••••••• 42

17. Points A, B, and C at which deflection was measured to determine if end clamps moved

Table

during vibration ........................................... 45

LIST OF TABLES

1. Effect of rigidity, kL/EI, on frequency •••••••••••••••••••• 19

2. Correction factors due to lower strain indication by the strain gage when in c 0tnp res a ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . • . . . 4 3

Table

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3. Summary of strain-deflection relationships found frcnn tests . ..•............••.....•.....•.••...•.••.. . 46

4. Data for calculating modulus of elasticity, E •••••••••••••• 48

5. Modulus of elasticity calculated from test data ....................................................... 49

6. Natural frequencies of the wood beams obtained from the tests ........................................•..•. 49

7. Comparison between vibration amplitude measured by the LVDT and calculated from the acceleration ••••••••••• 50

8. Comparison of theoretical and experimental strain-

9.

deflection relationships ..••••••••••••••••••••••••••••••••• 51

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"

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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

•

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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.

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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.

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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.

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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.

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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

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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.

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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

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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.

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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))

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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.

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~ -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

- (sinh nL) (cos nL~ + 2n2 (sin nL) (sinh nL) - 0. ((28))

The ends of a simple beam are completely free to rotate (k • O) and

the ends of a fixed-end beam cannot rotate at all (k • 00). Semi-rigid

ends will fall between these two extremes. From Rogers (8), nL • 3.142

for simple beams and nL • 4.730 for fixed-end beams.

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The value of nL for semi-rigid ends will be between 3.142 and 4.730,

the exact value depending upon the rigidity of the ends. In order to

find the relationship between nL and the rigidity (kL/EI) of the end

connections, equation 28 was theoretically evaluated for eight values

of nL spaced between 3.142 and 4.730. The frequency for each value of

kL/EI can then be determined using equation 4 and multiplying W by

2 rt to get the frequency, f; in cycles per second.

These results are expressed in Table 1, and are represented by a

curve in Figure 3.

Table 1. Effect of rigidity, kL/EI, on frequency.

nL

3.142 3.200 3.400 3.600 3.800 4.000 4.200 4.400 4.600 4. 730

1.570 1. 629 1.839 2.063 2.298 2.547 2.807 3.081 3.367 3.560

kL/EI

0 0.199 1.009 2.155 3.863 6.545

11. 35 7 22.100 66. 241

g()

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))

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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

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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.

-32-

. Figure 10. Non-translational pinned-end connection.

-33-

~ 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

-42-

.. _,_,_.. .......... '"" :nr- ·1 i . ::

~ ~ - ·,.. .. ;t :~ ! '

... "' ·~· ~,t :;-.:- J __ ~

:1:.J A :t '1 ~"'i ~

• >r: 1 ..

ti::+ I ... 1.., ta~IT 1

~· ~

: ~ l :o. +

~•I' ... ~ • ;'I!

Jl=t :..;

:~"' 1"1 1 ~ ~· ·]:. ~ 1.ii!EE ' .... :r u..~rt l ·+ ,.~It !fii ~ J_ - ~ - l·l p

No Load Static Free Vibration _l!11•c.._ · •.a.Mi i~ ! ~~-, ,_· u ri:n ~I i1f1 ~ --

21 rP I ~ .".j

: 1""~ .. ' .L ; cJ :'1:

t: r±:i s. 4

"~•1 ,;

•+ ~ ; ~ 'i .... bt :!:'

•+ :j:

I i :c LL !lft .~

CT ' .! .. •: iii - ti:;;: -· 1 ti tt ··:o- -r' ; ;.; ~;l ... i ' . -. ~ +t ··~- t ~ . ~ h "'"""" . h-- ... ~.

..t f.•" £ 1H' !:J:l lL.·.·· +:P :;: 1:-:+l --

. Figure 16. Typical oscillogram of a wood beam test. Chart speed-20 mm/sec.

-43-

that the LVDT gave a chart deflection of 15 divisions on the opposite

side of the zero line and the strain gage deflection was noted. A

correction factor was determined for each beam and each end condition

by dividing the total divisions deflection of the LVDT chart in the

tests just described (30 divisions) by the total divisions deflection

of the strain gage chart. These correction factors {herein to be

referred to as compression correction factors) are shown in Table 2.

Table 2. Correction factors due to lower strain indication by the strain gage .when in compression.

Beam No. End conditions 1 2 3 4

Fixed 1.020 1.027 1.034 1.034 Pinned 1.020 1.034 1.020 Semi-rigid 1.034 1.034 1.034

It was learned from preliminary tests that the value obtained for

es/Ys from the tests could be changed considerably by interchanging

the channels on the recorder to which the LVDT and strain gage were

connected. Investigations to determine the source of this difference

led to the conclusion that the damping of the pen motors which drive the

recording pens was out of adjustment. The damping was then adjusted

for both pen motors by adjusting the damping control according to the

manufacturer's instruction manual.

It was found, however, that slight differences could still be

obtained by interchanging the channels. In order to minimize the effect

of this difference, ten tests were conducted on each beam and end

condition, then the channels were interchanged without changing anything

-44-

else and ten more tests were conducted. By basing conclusions on the

average of these 20 tests, the effect of imperfect damping of the pen

motors should be eliminated.

The differences caused by imperfect dampening of the pen motors

can be seen in the test results tabulated in the Appendix.

To determine how rigidly the end supports were during vibration,

the LVDT was placed on the clamps at points A, B, and C, shown in

Figure 17. When the steel beam was vibrating at its maximwn amplitude

(immediately after releasing the load) a vibration of 0.0002 in. total

amplitude was measured at point A, and 0.00015 in. at point C, both of

which were considered negligible. No measurable vibration occurred at

point B. SimilAr tests conducted with a wood beam produced no measur-

able vibration at any of the three points.

-45-

, Figure 17. Points A, B, and C, at which deflec-·· tion was measured to determine if end clamps moved during vibration of beam.

-46-

T E S T R E S U L T S

The Strain-Deflection Relationship

The value of ed/Yd was determined for each test from the vibration eslYs

oscillogram similar to that shown in Figure 16, page 42. Evaluation of

several tests showed that the ratio of the strain gage reading to the

LVDT reading on the oscillogram remained constant during each test from

the initial vibration until the vibration damped out. Since high a.mpli-

tudes can be read from the oscillogram more accurately, the maximum

vibration amplitude of the strain gage and LVDT was determined from the

oscillogram for each test.

Using the oscillogram shown in Figure 16, page 42, as an example,

it can be seen that the maximum strain gage vibration amplitude was 21.5

divisions and the maximum LVDT amplitude was 27.0 divisions. These values

were used to determine ed/Yd as follows: ed/Yd • 21.5/27.0 • 0.796. esfYs esfYs

The results of all tests and the corresponding values

included in the Appendix and are smnmarized below in Table 3.

Table 3. Summary of strain-deflection relationships found from tests.

A B AxB End Conditions Average Compression Cor- Corrected and Beam No.* edf Yd rection Factor edf Yd

esfYs (From Table 2) esfYs

Pl o. 779 1.020 0.794 P2 0.821 1.034 0.848 P3 0.833 1.020 o. 849 Sl 0.765 1.034 o. 791 S2 o. 771 1.034 0.797 S3 0.791 1.034 0.817 Fl 0.674 1.020 0.687 F2 0.689 1.027 0.707 F3 0.687 1.034 0. 710 F4 0.698 1.034 o. 719

*Prefix "P" denotes pinned-end, "S II denotes semi-rigid end, and "F" denotes fixed-end conditions.

-47-

Modulus of Elasticity

Test results provided six methods for calculating modulus of

elasticity, E, for each wood beam. The methods consisted of three for

the pinned-end conditions and three for the fixed-end conditions.

These methods are: (1) static deflection, (2) natural frequency,

and (3) stress-strain telationships.

Modulus of elasticity can be calculated from static deflection

tests using the equation

E • p13 1921Ys

((34))

for fixed-end beams, and

E • PL3 48Iys ((35))

for pinned-end beams.

Modulus of elasticity can be calculated from the vibration tests

using equations from Rogers (8).

E • f2w14 12.67Ig

for fixed-end beams, and

((36))

E • . £2w14 ((37)) 2.465Ig

for pinned-end beams. In equations 36 and 37,

f •primary natural frequency in cycles/sec.,

w •weight of beam per unit length in lb./in.,

L • length of beam in inches,

I• moment of inertia of beam in (inches)4,

g •acceleration of gravity, 386 in./sec.2

-48-

Modulus of elasticity can be found from the static strain measure-

ments using the equation

E • R1£_ 8esl

for the fixed-end beams, and

for the pinned-end beams.

((38))

((39))

Data from the tests to be used in these six equations are given in

Table 4.

Table 4. Data from tests for calculating modulus of elasticity, E.

Test Static Load Static Defl. Static Strain Frequency w No. P (lbs.) Ys (in.) es (in.fin.) (cps) (lb.fin.)

Pl 16 0.208 134.4 x 106 10.5 0.115 P2 16 0.229 148.0 II 10.2 0.113 P3 16 0.240 140.0 II 9.9 0.111 Fl 25 0.091 112.8 II 22.5 0.115 F2 25 0.093 120.0 II 22.1 0.113 F3 25 0.104 112.0 II 21. 7 0.111 F4 25 0.174 120.0 " 11. 3 0.284

Moment of inertia, I, was calculated to be 1.25 in.4, and the

distance from the neutral axis, c, was 0.8125 in. for the wood beams.

For the steel beam (F4), I• 0.0208 in.4 and c • 0.25 in.

Results from the different methods described above for calcula-

ting modulus of elasticity are shown in Table 5.

Frequency

The effect of the different end conditions on natural frequency

is summarized in Table 6.

-49-

Table 5. Modulus of elasticity calculated from test data. (E x 106 psi).

Test Beam No. 1 2 3 4

Static Deflection Fixed-end 1.98 1.94 1. 73 31. 78 Pinned-end 2.22 2.01 1.92

Vibration Fixed-end 1.97 1.87 1. 77 30.23 Pinned-end 2.21 2.05 1.90

Static Strain Fixed-end 2.16 2.03 2.17 29.98 Pinned-end 2.32 2.11 2.23

Table 6. Natural frequencies of wood beams obtained from tests. (cycles/ sec.) •

End Conditions

Fixed Semi-rigid Pinned

Secondary Vibration

1 2 22.5 22.l 12.5 12.2 10.5 10.2

Beam No. 3 Average

21. 7 22.l 12.0 12.2 9.9 10.2

An interesting behavior was noted when the strain gage, LVDT, and

accelerometer were used simultaneously to record vibration of the steel

beam. Secondary vibration induced by cutting a string to suddenly

release the load was picked up and recorded by the strain gage and the

accelerometer, but not the LVDT. This secondary vibration as picked

up by the strain gage can be seen in Figure 15, page 39.

Acceleration

The accelerometer was used primarily as another means of determin-

ing deflection to check the LVDT. For simple harmonic motion, displace-

ment can be obtained from acceleration simply by dividing acceleration

-50-

by circular frequency squared (W2), However, due to the effect of the

secondary vibration on the acceleration, pure simple harmonic motion

did not exist. Five displacements were calculated from acceleration

tests for which no secondary vibration was apparent on the oscillogram.

The results of these tests were compared with the corresponding deflec-

tion measured by the LVDT. These results and comparison are shown in

Table 7.

Table 7. Comparison between vibration amplitude measured by the LVDT and calculated from the acceleration,

.Acceleration Amplitude Measured &; sa: 1 ~a: at i QD Per cent Test No. with LVDT (in.) w2 Difference

1 0.0792 0.0841 6.2 2 0.0616 0.0642 4.2 3 0.0880 0.0887 0.8 4 0.0550 0.0581 5.6 5 0.1144 0.1193 4.3

-51-

ANALYSIS 0 F R E S U L T S

The Strain-Deflection Relationships

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

-53-

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.

-59-

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

letter preceding the hyphen.

Test No.

Fl- 1

- 2 - 3 - 4 - 5

- 6 - 7 - 8 - 9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20

-65-

/

Table 1. Beam No. 1, Fixed-end conditions

A B Strain Vibration Deflection Vibration

Amplitude (divisions)* Amplitude (divisions)*

18.0 25.5 17.8 25.5 18.1 26.3 16.7 24.6 16.0 23.7 17. 2 25.5 17.0 25.0 17.0 25.2 17.3 25.7 17 .4 25.5 17.2 26.0 17.8 27.0 18.0 27.3 18.2 27.3 17.2 25.8 18.3 27.6 18.0 27.2 17.5 26. 7 . 17.1 26.0 18.0 26.5

Average of Tests 1-10: 0.6847 Average of Tests 11-20: 0.6631 Overall average: 0.6739

A/B ed/Yd es/es

0.706 0.698 0.688 0.679 0.675 0.675 0.680 0.675 0.673 0.698 0.662 0.659 o.659 0.667 0.667 0.663 0.662 0.655 0.658 0.679

*Figure 16, page 42, shows how these values were obtained from the oscillograms.

· Test No.

F2- 1

- 2 - 3 - 4 - 5

- 6 - 7 - 8 - 9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20

-66-

;,_j



Amplitude (divisions) Amplitude (divisions)

19.0 27.2 18.7 26.0 18.2 26.2 18.0 26.2 17.7 26.0 18.5 27.0 19.0 26.0 18.0 26.2 18.2 26.3 18.5 27.3 16.7 24. 3 16.7 24.4 17.5 25.5 17.2 25.0 17.2 25.2 17.0 25.0 17.0 25.0 17 .4 25.3 16.3 24.4 17. 0 25.0

Average of Tests 1-10: 0.6954 0.6820 0.6887

Average of Tests 11-20: Overa11 average:

A/B edfYd eslYs

0.699 o. 719 0.695 0.687 0.681 0.685 0.731 0.687 0.692 0.678 0.687 0.684 0.682 0.688 0.683 0.680 0.680 0.688 0.668 0.680

Test No.

F3- 1 - 2 - 3 - 4 - 5

- 6 - 7 - 8 - 9 -10 -11

-12 -13 -14 -15 -16 -17 -18 -19

-20

-67-z :.,;




19.0 27.0 18.5 26.5 18.5 26.3 18.0 25.5 18.0 25.0 18.5 26.5 19.0 26.5 19.0 27.0 19.0 26.5 19.0 27.5 18.3 26.5 17.5 26.5 17.5 26~2

16.5 25.0 16.5 25.2 17.0 26.4 18.0 26.3 17.0 25.2 18.0 26.5 17.0 25.7


A/B ed/Yd esfYs

0.704 0.700 0.703 0.706 o. 720 0.700 o. 717 o. 704 o. 717 0.691 0.690 0.660 0.668 0.660 0.655 0.644 0.684 0.675 0.679 0.661

Test No.

F4- l

- 2 - 3 - 4

- 5 - 6 - 7 - 8 - 9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20

-6,~-!J)r-.V \




20.5 29.5 21.5 30.7 24.0 34. 7 19.5 28.0 20.0 29.0 19.0 27.0 19.7 28.0 20.2 29.0 20.2 28.7 20.5 28.3 21.8 31.5 21.5 31.0 18. 5 26.5 19.5 28.5 21.5 30.0 20.5 29.7 21.5 31. 2 22.0 32.6 23.5 34. 0 20.0 29.5


A/B edf Yd e/ys

0.695 0.700 0.692 0.696 0.690 o. 704 0.704 0.697 0.704 0.724 0.692 0.694 0.698 0.684 o. 717 0.690 0.689 0.675 0.691 0.678

Test No.

Pl- 1

- 2 - 3 - 4 - 5

- 6 - 7 - 8 - 9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19

-20

-69-

I ;;;-.

Table 5. Beam No. 1, Pinned-end conditions



22.0 28.0 23.8 29. 8 22.0 28.0 20.5 26.0 23.0 29.5 19.5 24.5 22.0 28.0 21.0 26.5 23.7 29.3 21.0 26.0 23.0 30.0 23.0 29.5 22.2 29.5 22.2 29.4 23.0 29. 7 22.9 29.5 22.0 29 .o 23.3 30.1 21.9 29.0 22.6 29.7

Average of Tests 1-10: o. 7930 0.7654 o. 7792

Average of Tests 11-20: Overall average:

A/B ea/Yd es/ys

0.786 o. 798 0.786 0.788 0.780 o. 797 0.786 0.793 0.808 0.808 0.767 0.780 0.752 0.755 o. 775 o. 776 0.759 o. 775 0.755 0.760

Test No.

P2- 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8

- 9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20

-70-~ 1..-




26.8 25.6 26.3 26.6 24. 3 26.0 26.0 26.0 27.0 27.0 25.7 25.4 26.8 27.0 26.3 25.6 26.4 26.3 26.5 26.2

Average of Tests 1-10: Average of Tests ll-20: Overall average:

32.3 30.8 30.8 30.4 28.9 30.0 30.2 30.2 31.0 30.6 32.0 32.6 33.7 34.0 34.0 33.5 33.8 33.4 34.1 33.0

o. 8571 0.7847 0.8209

A/B ed/Yd eslYs

0.829 0.831 0.854 0.874 0.841 0.867 0.861 0.861 0.871 0.882 0.803 o. 778 0.795 0.794 0.773 o. 764 0.782 0.787 o. 777 0.794

Test No.

P3- 1

- 2 - 3 - 4 - 5

6 7

- 8 - 9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20

-71-;,> >




25.6 30.5 24.3 28.7 26.5 30.5 26.6 31.3 27.4 32.0 27.2 32.0 27.0 31.8 26.0 30.8 26.0 30.7 26.6 31.3 24. 7 30.4 25.2 31.4 25.1 31.0 26.5 32.0 25.6 30.7 25.5 31.2 26.5 32.5 25.0 31.0 26.6 32.0 25.0 31. 3

Average of Tests 1·10: o. 8496 0.8157 0.8326


A/B ed/Yd eshs

0.838 0.846 0.869 0.850 0.855 0.850 0.848 o. 844 0. 846 0.850 0.812 0.803 0.810 0.828 0.834 0.818 0.816 0.807 0.831 0.798

Test No.

Sl- .1

- 2 - 3 - 4

- s - 6 - 7 - 8 - 9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20

-72-,

Table 8. Beam No. 1, Semi-rigid end conditions



21.9 22.0 22.0 22.0 21.9 22.0 22.0 21. 7 22.0 21. 7 22.0 21.6 22.0 21.5 22.5 22.0 22.0 22.0 21.4

21.8

Average of Tests 1-10: Average of Tests 11-20: Overall average:

27.8 28.0 27.8 28.0 27.8 28.0 28.0 27.6 28.0 27.8 29.8 29.8 29.8 29.2 29.8 29.8 29.3 29.3 28.3 29.0

0.7864 0.7440 0.7652

A/B ed/Yd es7Ys

0.788 0.786 o. 792 0.786 0.788 0.786 0.786 0.785 0.786 0.781 0.738 o. 725 0.738 0.736 0.755 0.738 0.751 o. 751 0.756 0.752

Test No.

52- 1

- 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20

-73-"';

.;




22.0 28.0 22.0 27.7 22.0 27.8 21.8 27.3 22.0 28.5 21.8 27.8 21. 8 27.3 22.2 28.0 21.8 27.5 22.0 27.8 21.5 28.7 21.3 28.1 21.3 28.l 21.2 28.0 21. 3 28.3 21.4 28.5 21. 7 29.0 21.0 28.0 21.0 28.0

21.2 28.1


A/B ed/Yd es7Ys

0.786 o. 794 0.791 0.798 o. 772 0.784 0.798 0.792 0.792 0.792 o. 748 0.758 0.752 o. 757 o. 753 0.751 0.749 0.750 0.750 0.754

Test No.

S2- 1

- 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20

-74-




20.8 24. 5 20.3 24.5 20.0 24. 8 20.6 25.2 20.7 25.0 20.3 25.0 21.2 25.5 20.5 25.3 20.7 25.5 20.3 25.3 19.8 26.0 19.0 25.8 19.7 24.6 19.5 25.6 19.5 25.5 19.5 25.8 19.4 25.6 20.2 26.1 19.4 25.8 20.0 26.0

Average of Tests 1-10: 0.8193 0.7635 0.7914


A/B edfYd es7Ys

o. 847 0.828 0.806 0.817 0.828 0.812 0.831 0.810 0.812 0.802 0.762 0.736 0.801 0.762 0.765 0.756 0.758 o. 774 0.752 0.769

STRAIN-DEFLECTION RELATIONSHIPS OF FREELY VIBRATING WOOD BEAMS

by

Ray C. Minor

ABSTRACT

Several researchers engaged in family housing have recently become

concerned about the vibrational behavior of residential floors. This

concern resulted in a need for methods of sensing floor vibrations.

Some investigators have sensed floor vibrations with electric resistance

strain gages bonded to the underside of the floor joists. These experi-

ments using strain gages as vibration sensing devices resulted in a need

to be able to determine the vibration amplitude (or deflection) from

strain vibration data.

The objectives of this project were to theoretically and experiment-

ally determine the relationship between midspan flexural strain and

midspan deflection of freely vibrating wood beams with various end

conditions,

Theoretical strain-deflection relationships of freely vibrating

wood beams with pinned-end and fixed-end conditions were derived from

vibration theory. Free vibration tests on three wood beams with pinned-

ends and fixed-ends gave results which were in agreement with theory.

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 agreed with theory within

five per cent.

-~

The semi-rigid end connections 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.

It was established that the strain-deflection relationship of

freely vibrating wood beams can be predicted from vibration theory if

the rigidity of the end connections is known.

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