THE STATIC AND DYNAMIC STABILITY OF CLAMPED SHALLOW CIRCULAR ARCHES Thesis ...

113 Pages·2012·2.97 MB·English

THE STATIC AND DYNAMIC STABILITY OF
CLAMPED SHALLOW CIRCULAR ARCHES
Thesis by
Man Cheong Cheung
In Partial Fulfillment of the Requirement
For the Degree of
Doctor sf Phils sophy
California Institute of Technology
Pasadena, California
n969
ii
ACKNOWLEDGMENT
The author is deeply indebted to Dr. C. D. Babcock, Jr. for
guidance and encouragement received during the course of this investi-
gation. The valuable suggestions of Dr. E. E. Sechler are gratefully
acknowledged.
Appreciation is expressed for the help received from Mr.
George Carlson of the Aeronautics Shop, Mr. Marvin Yessey of the
Electronics Laboratory. Very much appreciated was the help of Miss
Helen Burrus for the typing and of Mrs. Betty Wood for the graphs and
figures.
This study was supported by the Air Force Office of Scientific.
Research, Office sf Aerospace Research, United States Air Force
under Grant No. AFOSR 68- 1424. The financial aid of a GAECIT
Fellowship is also deeply appreciated.
iii
ABSTRACT
An experimental investigation of the relationship between the
e hen omen on
dynamic buckling and the static buckling was carried out.
The first half of the paper studies the stability of shallow
arches subjected to lateral forces. Concentrated static and step loads
were applied to circular arches with geometric parameters 7 < ?'< 20.
Equilibrium path and the dynamic response at the loading position were
recorded. Dynamic buckling is distinct. Supercritical and subcritical
responses were separated by 0.5 O /o of the total load. The sensitivity
of the arch to loading imperfection was studied. The effect was pro-
nounced in the static tests but moderate in the dynamic responses.
The second half of the paper studies the response of clamped
circular arches under lateral impulse. The motion of the arch was
recorded by a high speed camera and the deflected arch shape was
measured in detail. The response was expressed in terms of the
average displacement and three generalized displacement coordinates
representing the amplitudes of the assumed mode shapes. The critical
impulse is defined as the impulse level at the inflection point of the
curve of maximum response versus impulse.
iv
NOMENCLATURE
2
Cross sectional area of the arch (in )
Arch width (in)
Young's Modulus (psi)
Arch thickness (in)
Central arch rise (in)
2'
Specific impulse (lb- see /in )
4
Moment of inertia (in
41 ' Jk2
Nondimensional impulse =
h 4 p~
Arch length (in)
Distance between load and center of arch (in)
Concentrated load (1bB
PW7r -q '
A l/2
Nondimensional load = (
m
Generalized displacement coordinates
Arch radius (in)
Time (sec)
Arc11 displacement (in)
Nondimensional average displacement
weight of explosive (gm)
Arch coordinate (radians)
Arch half angle (radians)
-
Geometric parameter =
4Rh
-A E
Percentage of eccentricity =
L
3
Mass density (slug/in )
v
NOMENCLATURE (Cont'd)
Subscripts:
CR Experimental critical value
CE Clas sicaP theoretical value
MAX Maximum value
TABLE OF CONTENTS
Page
I. INTRODUCTION 1
11. STABILITY OF CLAMPED SHALLOW CIRCULAR
ARCHES UNDER STATIC CONCENTRATED LOADS
AND STEP CONCENTRATED LOADS
A. Experiment
1. Test Specimens
2. Experimental Setup
3. Test Procedure
B. Test Results
C. Conclusions
119. STABILITY OF CLAMPED SHALLOW CIRCULAR
ARCHES UNDER LJNIFQRaaLY DISTRIBUTED
IMPULSIVE LOAD
A. Experiment
1. Impulsive Load
2. Camera Setup
3. Response Measurement
4. Test Procedure
B. Test Results
C. Conclusion,
REFERENCES
APPENDIX I
APPENDIX I1
TABLES
FIGURES
LIST OF TABLES
TABLE Page
I Geometric Descriptions of the Tested Arches 44
11 Nondimensional Static Critical Load 45
111 Critical Step Load 46
IV Geometric Description of Tested Arches 54
V Surnnnary of Impulse Tests 55
LIST OF FIGURES
Figure Page
I Response of Linear Spring Mass System 56
2 Nonlinear Response of One Degree of Freedom 5 7
System
Nonlinear Response of Two Degrees of Freedom
System
Coordinate System of Circular Arch
Typical Tensile Test of Curved 2024 Aluminum
Specimen
Circular Arch Before and After Installation in
Steel Frame
The Static and Dynamic Test Apparatus
Knife Edge Assembly
Initial Imperfection Measuring Equipment
Measuring of Initial Imperfection of Circuiar Arch
Imperfections of the Tested Arches
Nondimensional Critical Static Load vs. Loading
Imperfection
Comparison of Static Experiments to Theoretical
Static Buckling Load
Dynamic Response of Arch 32, 7 = 18.66, E = 0.00
Dynamic Response of Arch 29, Y = 14.40, = 0.00
Dynamic Response of Arch 28, 7 = 10.96, g: = 0.80
Dynamic Response of Arch 25, = 7. MCBa , = 0.00
Load Deflection Curve for Arch 32, 7 = 18.66
Load Deflection Curve for Arch 29, = 14.40
LIST OF FIGURES' (ContI d)
Figure Page
15c Load Deflection Curve for Arch 28, = 10.96 74
15d Load Deflection Curve for Arch 25, = 7.40 75
.
16 Nondimensional Critical Step Load vs Loading 76
Imperfection
Comparison of Dynamic Experiment to Theoretical 97
Step Buckling Load
Calibration of Explosive (Silver Acetylide-Silver 78
Nitrate)
Camera Setup 79
Film Reader to Measure Deformed Arch Shape 79
Performance and Synchronization of the High Speed 80
Camera
Initial Imperfections of the Tested Arches 8 1
Initial Imperfections of the Tested Arches 82
Impulse vs Maximum Average Displacement for 83
Clamped Arch
Comparison of the Present Results with Humphreys' 84
Experiments
Deformed Shapes of Arch 339 85
Deformed Shapes of Arch 338 86
i
Response of Arch B9, = 8.43 87
Response of Arch B6, .: 10.45 88
Response of Arch B7, += 12. 19 89
Response of Arch B8, ? = 14.82 90
i
Response of Arch C3, = 12.60 9 1
LIST OF FIGURES (Contfd)
Figure Page
26f Response of Arch C2, = 15.08 92
26g Response of Arch C5, -i = 16.66 93
26h Response of Arch C 4 , I = 16.95 94
26i Response of Arch C8, = 20.33 95
i
26j Response of Arch C7, = 21.65 96
T=
26 k Response of Arch C9, 27.75 97
=
27a Trajectories of Arches for Y 10 98
=
27b Trajectories of Arches for 7 20 99
=
28a Arch Antisymmetric Response for 'j' 10 100
28b Arch Antispmetric Response for ? ' = 20 101
29 Coordinate System for Finding Best Fit to the 102
Measured Hnitital Imperfections
Coordinate System for the Clamped Circular
Arches