Static and dynamic properties of a reinforced concrete building model

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Static and dynamic properties of a reinforced concrete building model

? 2,

3

3

1

,

,

STATIC AND D Y N A M I C P R O P E R T I E S

OF

A REINFORCED C O N C R E T E BUILDING M O D E L

by

G . H a r t l e y ,

J.

H. Rainer and

H.

S . Ward

The structure investigated w a s one of four r e i d o r c e d c o n c r e t e

models, b u i l t by the Canada Emergency Measures Organization

(EMO)

during the summer of 1969, and w a s made available t o the Division of

Building R e s e a r c h for further studies. The model is a seven-storey,

flat slab structure, a y m e t s i c a l about both axes.

The concrete used in the m o d e l was 4 0 0 0 p s i micro-concrete, r e i d a r c e d with steel w i r e . Details of the construction of the model

a x e described in Reference 1. The over-alldimensions of the s t r u c - tur e ar s shown in Figure 1.

THEORETICAL CONSIDERATIONS

F o r the purpose of a lateral dynamic analysis, the model can b e re- garded a s a system of seven rigid masses (flaox slabs), jnterconnec ted by springs and d a m p e r s (columns), Each mass is c a p a b l e of displacing in the transvers e or longitudinal directions or of executing a rotational motion (Figure 1). W i t h the above simplification, the structure has 2 1

degrees of freedom.

The differential equations of motion far the s y s tern are, in

matrix form:

w h e r e [m] is the mass matrix for the structure, whose elements cor- respond t o the masses of the slabs f o r the translational d e g r e e s of f r e e -

dom, and polar moments of inertia f o r the t o r s i o n a l degrees of freedom;

[ c ] is t h e matrix of damping coefficients; [ k

1

is the matrix of s tiffnes s

coefficients; F i s the c o l matrix ~ of generalized forces applied to each degree of f r e e d o m in the ~ y s t e m ; and x is the generalized d i s p l a c e -

ments corresponding to the degrees of freedom.

NOTE: This Note is based on an Internal Report of the Division of Building Research. A s the I n t e r n a l Report was issued in 1972, no

SI

u n i t s

are contained in this N o t e . The following conversions a r e applicable: 1 in. = 25.4 mrn

1 psi = 6 . 9 k P a

For an undamped svstem undergoing free vibration, _{Eq. (1)}

reduces to

F o r natural vibrations, the solution of Eq. ( 2 ) is of the f o r m

x 2 =

A2

sin (wt

+

m

2 )

w h e r e

A

,

A represent the amplitudes of vibration,

1 2

W 2 are the natural frequencies (radlsec], and

cPl*lcP~ are phase angles,

Substitutimg Equations ( 3 ) into (2) gives

2

A =

a

c r n ~ ~ k ~ - l A.

-

1

w h e r e [ f ] =

[k]

iscalledtheflexibilitymatrix. T h e f l e x i b i l i t y f i , , j i s

defined as the displacement corresponding to the location and direction designated by 1, resulting f r o m a unit load corresponding to the location and direction designated by j.

In

the present study, the flexibility coefficients for the building

model a r e determined by static testing. These are then u s e d to compute

the eigenvalues, w . , , and thecsrresponding eigenvectors,A f s o m E q , ( 4 )

11

using the Stodola method. 1'

EXPERIMENTAL PROCEDURE FOR STATIC TESTING

Preparation of the M o d e l

Eleven triaxial strain gauges w e r e a p p l i e d t o slab IV of the n ~ o d c l as shown in F i g u r e 2, and 16 uniaxial strain gauges w e r c a p p l i e d tu both

ends and an the longitudinal and transverse faces of c o l u m n s A3 and I33 above and below slab

IV,

Thin metal plates w e r e £as tened wkth epoxy to

a U

locations in which deflections w e r e t o be measured. A t a r g e t w a s marked on each plate at the centre of the slab thickness a d approximately at t h e c e n t r e -

line of the adjacent c o l w n n line. Two "DexionIt frames were b u i l t f o r mounting the deflection gauges.

Five small mirrors ( F i g u r e 3 ) were mounted in l o c a t i o n s V, W,

X, Y

and

Z

shown in F i g u r e 1. The mirrors w e r e used t o measure the

r o t a t i o n s at these points. A 35 =mm slide was pr epased with h o r i z o n t a l and vertical g r i d lines. The h a g e was projected onto the rotation

mirrors and r e f l e c t e d from t w o larger mirrors, located at opposite ends of t h e laboratory, onto a screen, The length of the optical a r m was appsaxbnately 150 ft. B y measuring the displacement an the screen of a s h g l e g r i d line, the rotation at the column could be estimated. F i g u r e 4 shows the location of the s l i d e projector and one of the l a r g e m i r r o r s ,

For each test it was necessary t o reposition the model. The base

slab was clamped f i r m l y t s t h e f l o o r of the l a b o r a t o r y in each case. F o r

the translational tests, l o a d s w e r e applied to the centre of each slab by

means of hydraulic jacks reacting against loading frame columns which w e r e bolted to the floor of the l a b o r a t o r y ( F i g u r e 3 ) . The positions of t h e jacks and "Dexion" frames f o r the rotational static t e s t a r e i l l u s t r a t e d

in Figure 5.

T e s t No, 1

-

Transverse Loading

The hydraulic jack was applied at the c e n t r e of each s l a b and the two "Dexion" f r a m e s each containing seven dial gauges w e r e p o s i t i o n e d at the extremeties uf t h e model as shown in F i g u r e 4. A load was ap- plied to eachfloor slab in turn in the transverse direceon, and f o r each

load application the deflections w e r e recorded at the 14 dial gauge l o -

cations, The rotations af c o l ~ s and strains w e r e measured only when

slabs

a

IV

and Vll. w e r e loaded. The m i r r o r s w e r e moumted s o t h a t the rotations jn a vertical plane parallel to the transverse direction c o u l d b e d e t e c t e d , Slabs

I,

Ua

a

/V, and VII w e r e loaded on the f i r s t d a y of t e s t i n g ;

slabs

VI

and V on the following day. The sequence of operations is l i s t c d i n detail in Table

I.

T e s t No. 2

-

Longitudinal Loadimg

The positions of the m o d e l and the testing e q u i ~ r n c n t arc similar t o those in Test

No.

I . Loads w e r e applied in the longitudinal d i r e c t i o n and corresponding deflections w e r e recorded again at 14 dial gauge le- cations. Only column rotations at

V,

W, and

Z

w e r e considered. T h e

three rotation mirrors were mounted s o that t h e rotations about a hari- zontal axis parallel to t h e longitudinal direction w e r e measured. The t e s t was performed in_{a single day. The procedure is summarized} in Table 2.

T e s t

No.

3

-

Torsional loading

The positions of t h e model and t e s t i n g equipment are shown in

Figure 5. Equal and opposite forces to form a couple with a moment

arm af 60 in., w e r e applied in the transverse direction to each f l o o r slab in turn. The distance between the two r o w s of dial gauges w a s 8 3 . 9 in.

The

mirrors were mounted s o that rotations about vertical a x e s were measured at locations X and Y. The experiment was per

-

formed in one day; supplernentaxy testing was perfaxmed two days

later. Tables 3 and 3a summarize the testing procedure, RESULTS

Laad-Deflection Properties

Tables 4a, b and c s h o w the mean slab displacements per u n i t

of applied force.

In

the transverse and longitudinal directions (Tables

4a and b), these values are computed by averaging the t w o deflections on a slab and dividing the average by the magnitude of the load. Only t h e results far the highest load level are given. F o r the torsion c a s e (Table 4121, the slab rotation has been computed by adding the deflec- tions for the two dial gauges on a slab, and dividing that sum by the

distance between the dial gauge lines, shown in F i g u r e 5, The quantities

shown in Table 4 c are the rotations divided by the couple that produced

the rotation, Only the torsional results f r o m the largest applied forces

(75 lb) are presented in this paper, Column Rotations

The:

coluslln rotations about a horizontal axis p a r a l l e l to the

transverse direction at locations

V,

W, X , Y , and Z, shown

in

F i g u r e I, a r e presented in Table 4d, whereas the colurnn r o t a t i o n s about a hori- zontal axis parallel to the longitudinal direction are given in Table 4e. Table 4f shows the rotations about a vertical axis of columns at lo-

cations

X

and

Y

of F i g u r e 4 when the model is subjected to torsional loads. Strain G a u g e R e a d i n g s

The strains observed on s l a b

IV

and adjoining columns were d i f - ficult t o interpret because of their low level. A s it was not desirable to increase the lateral loads any further for fear of damaging the model prematurely, no useful information has yet been gained.

It

is expec tea, however, that under higher load levels the recorded strains will provide

some insight i n t o the model's structural behavioux at a later date. Some aspects of strain gauge observations a r e discussed, however, in the next

DlSCUSSION

OF

RESULTS

A s y m m e t r y in Deflection Readings

For the i d e a l linear s t r u c t u r e , the flexibility matrix is s y m - metrical because of Maxwell's reciprocal relationship. The three flexibility matrices d e t e r m i n e d f o r the model a r e obviously not per - f e c t l y eyrnmetrical (Tables 4a, b, and c). The lack of symmetry in t h e s e results is examined in Tables 5a,b,and c. Each e n t r y is com-

puted by subtracting an element in the f l e x i b i l i t y matrix above the main diagonal from its diagonally apposite counterpart below and dividing the

result by the mean of the two values, The result occupies the same lo- cation as the coefficient below the main diagonal in the flexibility matrix.

There are several possible sources of experimental error which

may contribute to the asymmetry of the matrices of deflection readings.

The smallest deflections measured w e r e of the order of 1 J l O O O in. T h i s is the smallest division on the dial gauge and i t is Believed that

sticky dial gauge fieedlqs could account for close to 1

/

1000 of an inch error, B y far the l a r g e s t deviations from symmetry occur in the f i r s t r o w and column of each flexibility matrix a n d these correspond to t h e smallest dial gauge readings. A s the model is loaded, the dial gauge pin has a tendency to s h i f t off the target point of the bearing plate, the

s u r f a c e roughness of which c o u l d account for s o m e portion of the error. Furthermore, the Loads could be measured only to within a f e w pounds.

The

load levels werenot largeenough to reduce this source of error to

negligible limits.

The most s i g n s i c a t source of deviation from s y m m e t r y , however,

can b e explained as f trllows. In each of the three static t e s t s , Loads w e r e

applied to t h e model in sequence, beginning with the bottom f l o o r slab. Because the m o d e l had not previously been tested, this procedure had a

tendency ta produce a systematic progression of cracking. (It should

b e noted that this cracking is at a small and acceptable level and is produced by loads that are w e l l below those t h a t the structure was d c - signed to carry. ) W h e n slab

1

was loaded, new cracks formed in the columns between slab

I:

and the base slab. Deflections at slabs

II

t h r o u g h

V n

have j u s t been measured for a m o d e l in which cracks appear only in the bottom colurrms.

As

each successive slab is loaded the d e f l e c t i o n s measured below the loaded slab correspond to the cracked state; t h o s e

above t o the luncracked state, The symrnetr y of the flexibility matrix,

however, arises from an assumption of linearity of the structure, The

Although there a r e same exceptions, there is a tendency for t h e

flexibility below the diagonal coefficients in Tables 4% b and c to b e larger than their supposedly symmetric a1 equivalents above t h e diagonal

that

w e r e measured

in

an earlier s t a g e of the t e s t , Thus f o r the ffexi- bility matrix of the uncracked model, the upper diagonal matrix should be u a e h and for the cracked mode4 the lower diagonal matrix would b e appropriate, This pre-supposes a svccessive loading pattern f r o m the bottom to the top.

C

olurnn

Rotations

The

moat striking feature of the column

r crtation

readings (Table 4d) is that when the madel was unloaded, the mirror readings did n o t r e t u r n even closely t o their o r i g i n a l position. The results of the slab deflection measurements, however, show

that

fie structure did r e k r n

almost completely t o its original codigmration after unloading, There-

fore, yielding in the structure could not have produced these residual

column rotations. Either jarring movements in the structure or the

column rotations themselves may

have

caused slight shifting at the mir-

ror clamping points. M o s t likely, however, local column cracking prohibited the rotations from returning t o their unloaded position, U n -

loaded conditions were not recorded for mirror readings in Tables 4 e and f,

A comparison of the mirror rotations W and Z for loading in the

transverse direction, given in Table 4d, shows that W is consistently

larger than 2.

As

W and Z are fie rotations of the columns at the upper

and lower sides of slab

IV,

t h i s difference h d i c a t e s that substantial

relative deformations must have occurred, Two possible explanations fox

this are cracking near the colurnn-slab interface or elastic deformations within the column- slab joint. Because, upon unloading, the mirror

readings did not return to zero the occurrence of cracking is likely to

be the predominant reason for the difference in the W and Z readings

in Table 4 d ,

The angle readings of mirrors

W

and Z in the longitudinal direction

(Table 4e) indicate identical readings for slab IIl loaded and for the 100-lb

l o a d o n s l a b W , T h e 2 0 0 - 1 b l o a d o n s l a b I V s h o w s ~ a t Z i s l a r g e r ~ a n W, which is opposite to t h e general trend of other readings and should t h u s b e viewed with some suspicion, W i t h slab

V U

loaded, Z is consis-

tently larger than W, t h u s follovging the s a m e trend as

in

the transverse direction.

The mirror readings under torsional loads,

X

and

Y

in Table 4f, give the rotation at the top and bottom, respectively,

of

the center column

B3

between slabs

I T

and

V

(see Figure 1). F o r SlabIII and

IV

loaded, the

top of the colurrln is seen

t o

rotate consistently through a larger angle than the bottom, although the diffexencea are v e x y small. This follows t h e same trend as the slab rotations under torsional leading

in

Table 4c,

which shows a elight increase in the rotations of the slabs that are

located immediately above the loaded slab. F o r lab

V U

loaded, Table 4f indicates that the top of the colurnn, location

Y,

shows substantially

g r e a t e r rotations than the bottorrq location

X,

This is again in agreement with the trend exhibited by the corresponding slab rotations r e c o r d e d in Table 4c.

A quantitative comparison between slab rotations in Table 4 c

and column rotation

in

Table 4f for slab

VU:

loaded gives the following:

_/from Table 4c)

Rotation Slab

V

= 6.1178 x 10

-8

Difference = 1.2178 x 1 0

-8

rad/in. lb

-2

= 0 . 3 1 4 x 1 0 degrees f o r 7 5 - l b tarsional loads.

(from Table 4f)

Y

=

1 . 9 7 3 x 10 -2 -2

Difference = 0 . 3 0 7 x 10 degrees for 75-lb torsional loads.

T h e s e figures a r e seen t o be in good agreement. F o x the 50-lb torsional

load t h e corresponding f i g u r e s ar.. 0 . 2 2 x 1 degrees from the slab

rotations versus 0.123 x degrees from Table 4f f o r the 50-lb loads. The quantitative agreement between these l a t t e r figures is not as s a t i s -

factory.

The results f r o m the optical measurements of column rotations

have demonstrated that i t is possible to obtain useful information about

localized behaviour of constituent elements af a structural model.

In

the present work the results have been l a r g e l y qualitative and preliminary.

However, with s o m e improvements in experimental techniques, optical measurements should enable one to obtain reliable quantitative data on the progression of member deformations, including local cracking or

yielding.

Strain G a u g e Readings

All s t r a i n gauges on the model w e r e checked t h o r o u g h l y b e f o r e the static testing to ensure that they were properly applied. None w a s

faulty. It appears, however, that the error in the measuring equipment was

g r e a t e r than the actual strains, which were very low. L o w levels of l o a d - ing w e r e maintained in order to prevent any serious localized cracking of the model. T h e s e comments apply especially t o the s l a b s in which values of only a few micro inches per inch were obtained,

An error of 3r _{5 pin. Jin. could be expected from t h e strain indi-} cator. All the at-rain gauges were balanced to the same z e r o point through

switching units.

It

was found before testing, that some of the contacts in the potentiometers of the switching units w e r e not clean, This c a u s e d a

drift in t h e strain indicator needle. Some of the channels,

in

which this

was quite noticeable, were not u s e d in the testing.

k

t

others this effect w a s imperceptible, and would only have been noticed after an hour o r so

of testing. This is perhaps one r e a s o n why strain readings were n o t close

to zero when the model was unloaded. However, both the strain gauge

and mirror results might better b e taken as evidence of small local crack-

ing

in

the concrete.

DYNAMIC

TESTING O F

EM0

MODEL

The fundamental frequencies of t h e E M 0 b u i l d i n g m o d e l in the three directions of vibration (longitudinal, transverse, and torsional) w e r e obtained b y two methods:

1. The f l e x i b i l i t y coefficients w e r e m e a s u r e d by static t e s t i n g

as described previousky. These coefficients were then u s e d as data

f o r the computation of natural f r e q u e n c i e s and mode shapes, using thc

program of Reference 2 . Each of t h e t h r e e directions were considcrcrl

to be uncoupled and the natural frequencies and correspunding mode

shapes computed separately.

2. Simple impact t e s t s were perf orrned and frequencies obtained

from the recorded vibration traces.

The computer program from Reference 2 computes the n a t u r a l frequencies and the corresponding mode shapes for three modes of vi- bration using the Stodola-Vianello method.

The

data required by the

program are: the masses of the slabs; t h e p o l a r moments of inertia; and Ule f l e x i b i l i t y coefficients obtained in the static t e s t s . Symmetrical rnartric e s w e r e used, utilizing only the coefficients below t h e main

diagonal in Tables 4a, b, and c. The results of the analysis are given

in Table 5 .

P r oc edur e f o r Dynamic Testing

Two MB velocity transducers w e r e placed on Slab VII, with

both transducers aligned to record vibrations i n the transverse d i r e c t i o n only, The signals from t h e transducers were recorded on a C e n t u r y G P O 4 6 0 Recorder. T h e modal was excited in t h e transverse direction by striking the middle of the t o p slab horizontally with a hard rubber h a m e r .

The transducers were rotated by 9 0 deg. to measure vibrations

in the longitudinal direction that were induced by striking the middle of the top slab.

T o vibrate the m o d e l in the torsional direction, a 1 0 - f t board w a s

placed across the top slab and clamped

with

C-clamps as shown in F i g u r e 6. By hitting the extending arm of the board, t o r s i o n a l as well as longitudinal oscillations w e r e excited. The transducers w e r e placed as s h o w n in F i g u r e

6

and w e r e aligned with the transverse direction, so t h e y would not respond

to longitudinal vibrations.

In one

t e s t , an operational amplifier was u s e d to subtract the s i g n a l s from the transducers so that the transverse m o t i o n of the m o d e l was essentially cancelled and the torsional motion doubled.

It was

found that there w a s n a d i f f e r e n c e in the records, and that t o r s i o n a l vibrations could be measured without the use of t h e operational amplifier.

The vibration r e c o r d s obtained in the dynamic testing a r e given in

Figures 7, 8 and 9 f o r the transverse, longitudinal, and t o r s i o n a l motions, respectively.

The

fundamental frequencies d e r i v e d f r o m t h e s e r e c o r d s are given in Table 5.

Comparison of Computed and Measured Frequencies

The comparison between t h e measured and computed frequencies of the fundamental modes in _{each of the t h r e e principal directions i s pro-} vided

in

Table 5. It may be observed that the t w o s e t s of figures a r e in c l o s e agreement. T h e most important component of the computational data, the flexibility matrix, w a s determined e x p e r k e n t a l l y by the static t e s t s ; thus the g r e a t e s t potential unc ertaknty has effectively been elirni- nated f r o m the c omputations,

The close agreement between computed and measured frequencies, however, confirms the validity of using vibrational data f o r assessing

Damping in the Model

The

degxee of damping in the model was estimated by comparing successive peak amplitudes of the vibration records. Considering two

maxima, x and x

1 n + l separated by n c y c l e s of oscillation, t h e logarithmic decrement,

'

A,may be computed as:

The fraction of critical damping,

5

,

can be found from the relation:

These r e s u l t s are presented in Table 5 .

CONCLUSIONS

The model t e s t s have demonstrated close agreement between ob-

served dynamic behaviour and dynamic properties computed f r o m measured

flexibility matr ic es,

The

static flexibility matrices were not found to be completely

symmetrical. This w a s ascribed m a i n l y to progressive cracking of the

m o d e l as the various f l o o r s were loaded.

The optical rotation measurements demonstrated that it is p o s s i b l e

to obtain qualitative as well as quantitative information on localized c om- ponent rotations atvarious load levels by this technique. F o r greater

reliability, however, improvements in the experimental techniques are

necessary.

A s the asymmetry of both flexibility matrices and r o t a t i o n readings has indicated, the loading sequence for a concrete model t e s t may r e q u i r e careful consideration, The results obtained may depend on t h e particular loading sequence adopted.

Results from the strain gauges produced limited information d u e

to the small load levels applied.

ACKNOWLEDGEMENT

The model tested was one of four models b u i l t by the Canada

ErnergencyMeasures Organization ( E . M . O . ) in 1969. It was made available b y E.M.O. to the Division of Building R e s e a r c h for further testing. The work wag carried out by G. Hartley while employed as a summer student at the Division of Building Research under the supervision of 3. H. Rainer

.

Ref. 1

-

Model c o n s t r u c t i o n

REFERENCES

1 . Zelrnan, M. I. , e t al. Practical Problems and C o s t s of Fabricating Multi-Storey Models, Models far Concrete Structures, Publication S P - 2 4 , A m e r i c a n C o n c r e t e I n s t i t u t e , D e t r o i t , Michigan,

197Q,

p. 159-185.

2. W a r d , 33. S . Calculation of the M o d e s and Frequencies of V i b r a t i o n f o r Multi-Degree-of -Freedam Systems. National Research Council

of Canada, Division of B u i l d i n g Research, DBR Computer P r o g r a m

TABLE

1. TEST SEQUENCE FOR TRANSVERSE DIRECTION

TABLE

2 -

TEST

SEQUENCE FOR LONGITUDINAL DIRECTION

Slab Loaded I I I 111 IV

VL

I V

VI

Sequence of Operations

-

initial d i a l gauge r e a d i n g s - load increased to 100 lb

-

d i a l gauge readings

-

load i n c r e a s e d to 200 l b I I I I I I I V

v

V I VI I I , - i n i t i a l d i a l gauge r e a d i n g s - Isad to 100 lb

-

d i a l gauge readings

-

load to 200 Pb - d i a l gauge readings - unload

-

-- i n i t i a l d i a l gauge r e a d i n g s - load to 100 1 b

-

d i a l gauge readings, column r o t a t i o n s , strain gauges - load to 200 1b

- dial gauge readings, c o l u m rotations, s t r a i n gauges - unload

-

{

as f o r I and IT above - initial d i a l gauge readings - load to 100 Ib

- dial gauge r e a d i n g s , column r o t a t i o n s , strain gauges

- laad to 150 l b

-

dial gauge r e a d i n g s , column rotations, strain gauges

- u n l o a d - unload

-

i n i t i a l d i a l gauge readings

- load increased to 100

Xb

- d i a l gauge r e a d i n g s , column r o t a t i o n s , strain - l o a d increased to 200 l b

- dial gauge readings, column rotations, s t r a i n

-

unload

-

i n i t i a l d i a l gauge readings

-

load increased to 100 I b - d i a l gauge readings

-

load increased to 200 1b - d i a l gauge readings I

-

unload

TABLE 3 . TEST SEQUENCE FOR

TORSIONAL

LOADING Sl ab

Loaded Sequence of Operations

- i n i t i a l d i a l gauge readings - load (on both jacks) to 50 I b - dial gauge readings

I

-

load t o 7 5 l b - d i a l gauge readings - load to 100 1b - i n i t i a l d i a l gauge readings - load to 50

lb

-

dial gauge r e a d i n g s - load t o 75 l b

- dial gauge readings

-

initial d i a l gauge r e a d i n g s > - l o a d t o 50 I b

d 111 - dial gauge r e a d i n g s , column rotations, strain gauges

o IV - load t o 75 I b

- dial gauge readings, column rotations, s t r a i n gauges

- unload

+

V) * c i , u LL V

-

initial d i a l gauge r e a d i n g s - l o a d t o 31 l b - d i a l gauge r e a d i n g s - l o a d t o 50 I b - d i a l gauge readings - unload VI VI I - i n i t i a l d i a l gauge readings

-

load t o 50 lb

-

dial gauge readings - load t o 75 l b

- dial gauge r e a d i n g s

-

unload

- initial dial gauge r e a d i n g s - l o a d t o 50 1 b

-

d i a l gauge r e a d i n g s , column r o t a t i o n s , strain gauges

- load t o 75 1 b

-

d i a l gauge r e a d i n g s , column rotations, strain gauges - unload

TABLE 3a. SUPPLEMENTARY TORSIONAL LOADING TESTS

- i n i t i a l dial gauge r e a d i n g s

-

load t o 50 l b

- d i a l gauge readings

-

load t o 100 Xb

-

initial dial gauge readings - l o a d s to 50 Ib - d i a l gauge readings

-

l o a d s to _{100 l b} Z 0 u Lrl vl 11 I V

- initial dial gauge readings

-

load to _{50 l b}

- d i a l gauge readings

-

load to 75 l b

- dial gauge r e a d i n g s - unload

TABLE 4a

MEAN SLAB DEFLECTIONS PER UNIT WAD IN THE TRANSVERSE

DIRECTION

Deflection of Slab No. ( i n . x

S l a b Loaded

1 I I I L L

IV

v

1; I VIT

TABLE 4b

MEAN

SLAB

DEFLECTIONS

PER UNIT LOAD I N THE LONGITUDINAL DIRECTION

D e f l e c t i o n 05 Slab No. (in, x Slab

Loaded

TABLE

4c

SLAB ROTATIONS PER UNIT TORQUE, OBTAINED FROM THE 7 5- LB

.

LOADING CASE

Slab R o t a t i o n o f Slab No. ( r a d x 10'~) Loaded

I I T TI1 I V V VI _{V I I}

V I I 0.9267 2.5156 3 . 6 8 2 2 4.9000 6.1178 7.4689 8.5022

TABLE 4d

COLUMN ROTATIONS UNDER LOADING IN TRANSVERSE DIRECTION

S

X

ab Rotation a t Mirror (degrees x

Loaded Load, l b .

_v

_{1v X}

_Y

_Z

I11 200 4 . 2 9 1 3.281 1.735 1 -487 1.893 un l o aded 4.039 3 . 0 2 9 1.487 1 . 2 4 0 1.514 1 0 0 0.884 1.010 1 . 3 6 4

-

2.524 IV 200 2.019 2.524 2.107 2 . 2 2 3 3 . 9 1 3 unloaded 2 .a19 1.514 1.735 2.355 2 . 5 2 4 100 0-63 1 1.010 0.867 0.867 1.010 VL I 200 1 . 8 9 3 2 . 2 7 2 2.603 2.355 2 -524 unloaded 0.253 0 -504 0.372 0.372 0.631

TABLE 4e

COLUMN ROTATION UNDER LOADING 1K

LONGITUDINAL DIRECTION

Slab R o t a t i o n at Mirror [degrees x 10'~)

Loaded Load, Ib.

_v

I'm' X Y

z

TABLE

4E

COLUMN

ROTATION

UNDER LOADING IN TORSION

Slab R o t a t i o n a t Mirror [degrees x 1 0 - ~ )

Loaded Load, I b .

_v

TABLE 5

CALCULATED AND MEASURED

FREQUENCIES

OF

VIBRATION

F~equencies from Frequencie

s

from >lea sured

Damning Dynamic T e s t , Hz Computer Analysis, H z Ratio, % of

c r i t i c a l o f D i r e c t ion 1 st Mode 1st Mode 2nd Mode 3rd Mode 1 s t Mode

Transverse 5.77 5 . 7 4 17.14 3 2 . 3 0 1.3

Longitudinal 5 . 8 1 5 . 8 8 17.40 26.16 1 . 4

T o r s i o n 6 . 4 0 6 . 7 3 21.45 44 .69 2.3

TABLE 5a

TABLE 5b

DEFLECTION ASYMMETRICS I N THE LONGITTJDINAL DIRECTION

TABLE 5 c

Figure 3 - M i r r o r s for Measuring Column Rotation and Mounted Strain Gauge.

A

-

Projector

B - M i r r o r mounted on c olumn ( F i g - 3 )

C

-

O n e of 2 reflecting mirrors f o r lengthening

optical arm.

Figure 4

-

O p t i c a l System f o r Measuring C o l m Rotations and G e n e r a l View of Test Operation,

2 x 4 PINE I M P A C T H A M M E R " M 0 " V E L O C I T Y T R A N S D U C E R S W I T H O R I E N T A T I O N I N D I C A T E D BY A R R O W FfGURE 6 D X N A M l C T E S T I N G - T O R S I O N A L D I R E C T I O N

FIGURE 7 V E L O C I T Y T R A C E IF1 T R A N S V E R S E D I R E C t I O N t C - u

2

ILI > M P R E S S E D S C A L E I 1 I I T 1 I 1 I 1 5 C Y C L E S I N 2.6 SEC. I I I I I I 1 I I 0 . 5 1 . 5 2 . Q 2 . 5 3 .O 3 . 5 4 . D 5 . 0 T I M E , S E C . 0 . 5 1 . U 1 . 5 2 . O 2 . 5 3.0 3.5 4 . O 5 .O T I M E , S E C . FIGURE B V E L O C I T Y T R A C E I N L O N G I T U D I N A L D I R E C T I O N

+

r - U

5

LU > I b C Y C L E S 1N 2 . 5 S E C - I F COMPRESSED SCALE r T I M E , S E C . F I G U R E 9 V E L O C I T Y T R A C E 1N T O R S I O N A L M O T I O N