STATICALLY DETERMINATE STRUCTURES Loads, Reactions, Stresses, Shears, Bending Moments, Deflections} Equilibrium of Force Systems.. For example, for a space parallel force system acting
Trang 1A13 A14 A15 A18 AI?
A18
A18 A20 A21 A22 A23
A24 A25 A26
TABLE OF CONTENTS
The Work of the Aerospace Structures Engineer
STATICALLY DETERMINATE STRUCTURES
(Loads, Reactions, Stresses, Shears, Bending Moments, Deflections}
Equilibrium of Force Systems Truss Structures Externally Braced Wings Landing Gear
Properties of Sections - Centroids, Moments of Inertia, etc
Generai Loads on Aircraft
Beams - Shear and Moments Beam - Column Moments
Torsion - Stresses and Deflections
Deflections of Structures Castigliano’s Theorem Virtua! Work Matrix Methods
THEORY AND METHODS FOR SOLVING STATICALLY
INDETERMINATE STRUCTURES
Statically indeterminate Structures Theorem of Least Work Virtual Work Matrix Methods
Bending Moments in Frames and Rings by Elastic Center Method
Column Analogy Method
Continuous Structures - Moment Distribution Method
Stope Deflection Method
BEAM BENDING AND SHEAR STRESSES
MEMBRANE STRESSES COLUMN AND PLATE INSTABILITY
Bending Stresses
Bending Shear Stresses - Solid and Open Sections - Shear Center
Shear Flow in Closed Thin-Walled Sections
Membrane Stresses in Pressure Vessels
Bending of Plates
Theory of the instability of Columns and Thin Sheets
INTRODUCTION TO PRACTICAL AIRCRAFT STRESS ANALYSIS
Introduction to Wing Stress Analysis by Modified Beam Theory
Introduction to Fuselage Stress Analysis by Modified Beam Theory
Loads and Stresses on Ribs and Frames
Analysis of Special Wing Problems Cutouts Shear Lag Swept Wing
Analysis by the “Method of Displacements”
THEORY OF ELASTICITY AND THERMOELASTICITY
The 3-Dimensional Equations of Thermoelasticity
The 2-Dimensional Equations of Elasticity and Thermoelasticity
Selected Problems in Elasticity and Thermoelasticity
ee
Trang 2TABLE OF CONTENTS Continued
Chapter No
FLIGHT VEHICLE MATERIALS AND THEIR PROPERTIES
B1 Basic Principles and Definitions
B2 Mechanical and Physical Properties of Metallic Materials for Flight Vehicle Structures
STRENGTH OF STRUCTURAL ELEMENTS AND COMPOSITE STRUCTURES
c1 Combined Stresses Theory of Yield and Ultimate Failure
c2 Strength of Columns with Stable Cross-Sections
œ3 Yield and Ultimate Strength in Bending
C4 Strength and Design of Round, Streamline, Oval and Square Tubing in Tension, Compression, Bending,
Torsion and Combined Loadings
cs Buckling Strength of Flat Sheet in Compression, Shear, Bending and Under Combined Stress Systems
C6 Local Buckling Stress for Composite Shapes
c? Crippling Strength of Composite Shapes and Sheet-Stiffener Panels in Compression, Column Strength
c3 Buckling Strength of Monocoque Cylinders
ca Buckling Strength of Curved Sheet Panels and Spherical Plates Ultimate Strength of
Stiffened Curved Sheet Structures
C10 Design of Metal Beams Web Shear Resistant (Non-Buckling} Type
Part 1 Flat Sheet Web with Vertical Stiffeners Part 2 Other Types of Non-Buckling Webs
C11 Diagonal Semi-Tension Field Design
Part 1 Beams with Flat Webs Part 2 Curved Web Systems
C12 Sandwich Construction and Design
c13 Fatigue
CONNECTIONS AND DESIGN DETAILS
D1 Fittings and Connections Bolted and Riveted
02 Welded Connections
D3 Some Important Details in Structural Design
Appendix A Elementary Arithmetical Rules of Matrices
Trang 3
Accelerated Motion of
Rigid Airplane - A4 8
Aireraft Bolts - 1 ++ DI.2
AircraftNuts oe DI.2
Aircraft Wing Sections -
Types oe ee Alg 1
Aircraft Wing Structure -
Truss Type - 2 eee Al, 14
Air Forces on Wing A4.4
Allowable Stresses (and
to Various Structures + AT.23
Applied Load A4.1
‘Axis of Symmetry A9.4
Beaded Webs - - C10 16
Beam Design - Special Cases D3 10
Beam Fixed End Moments by
Method of Area Moments AT 32
Beam Rivet Design ‹ C10.8
Beam Shear and Bending
Moment .- 2-2-5555 A8.L
Beams - Forces ata Section A5.T
Beams - Moment Diagrams 5.6
Beams with Non-Paralle!
Flanges C11.9
Beams - Shear and Moment
Diagrams A5.2
Beams - Statically Deter~-
minate & Indeterminate 5.1
Bending and Compression
of Columns 2.2 AlBL
Bending Moments Elastic
Center Method ‹ A9.1
Bending Strength - Solid
Round Bar ee eee C3.1
Bending Stresses -‹ À13.1
Bending Stresses - Curved
Beams see eee ,„ A13 l5
Bending Stresses - Elastic
Range - , A18.13
Bending Stresses - Non-
homogeneous Sections + A13 11
Bending Stresses About
Principal Axes 6 0 ee AL3.2
Bending of Thin Plates Al8 10
Bolt Bending Strength « DI.9
Boit & Lug Strength Analysis
Buckling of Stiffened Flat
Sheets under Longitudinal
Compression Buckling under Bending Loads
Buckling under Shear Loads
Buckling under Transverse Shear 2 eee eee eee Carry Over Factor .- Castigliano's Theorem Centroids - Center of Gravity
Cladding Reduction Factors Column Analogy Method
Column Curves - Non- Dimensional .-+- Column Curves - Solution” Column End Restraiat Column Formulas -
Column Strength - Column Strength with Known End Restraining Moment
Combined Axial and Trans- verse Loads - General
Action 4 ee ee eee Combined Bending and
Compression oe Combined Bending and
Combined Bending & Torsion Combined Stress Equations Compatability Equations Complex Bending ~
Single Cell - 2 Flange Beam,
Constant Shear Flow Webs -
Single Cell - 3 Flange Beam
Continuous Structures - Curved Members .- Continuous Structures -
Variable Moment of Inertia Core Shear ne
Correction for Cladding cee
Corrugated Core Sandwich
Curved Web Systems
Cut-Oucs in Webs or Skin Panels
Deflection Limitations in
Plate Analyses .- Deflections by Elastic Weights
C6.4 C5.6
C5.6
C8.14 411.4 ATS A3,1 C5.5
ALO 1
C2.2 C2.13 C2.1 C4.2 Cï.21 C2.16
A5.21 C4, 22
3 10 C4 23
A18 17
CA4.23 1.2
A24.T
c3.9 cat
C8, 22 Al4 10
Al8.3 A15,5 ALL 31 A11 l§
C12.28 CT.4
ALT.4 AT.27
Deflections by Moment Areas
Deflections for Thermal
Deflection Surface %
Discontinuities Distribution of Loads to Sheet Panels
Ductility
Dummy Unit Loads .-
Dynamic Effect of Air Forces
Effect of Axtal Load on
Moment Distribution Effective Sheet Widths
Elastic Buckling Strength of Flat Sheet in Compression
Elastic - Inelastic Action
Elastic Lateral Support
Columns - - - ‹
Elastic Stability of Coiumn
Elastic Strain Energy - Elasticity and Thermo- elasticity - One-Dimensional
Problems
Elasticity and Thermo- elasticity - Two-Dimensional Equations sae Electric Arc Welding eee
End Bay Effects ae
End Moments for Continuous
Frameworks 20:
Equations of Static
Equilibrium ¬
Equilibrium Equations - Failure of Columns by
Fixed End Moments
Fixed End Moments Due to
Support Deflections - Fixity Coefficients
Flange Design « Flange Design Stresses
Flange Discontinuities
Flange Loads
Flange Strength (Crippling) -
Flat Sheet Web with Vertical Stiffeners 2
Flexural Shear Flow Distribution 2 ee Flexural Shear Flow -
Symmetrical Beam Section
Flexurai Shear Stress
AS 12
» ALL, 22 C7, 10 C5.Ỏ BLS C2.17 Al7.2
C1.8
A26.1 A25.L D2,.2 C11 23 All 10 A2.L A24.2 Al8,4 C12.20 BLL c11.4 c1a.8
C10.1
C10,2 C10.7
C1138
Ci0.4 Ci0.¡
A14.5 Al4.Ì
Trang 4
Static Tension Stress-
Strain Diagram BL 2
Statically Determinate
Coplanar Structures and
Loadings 2 eee A2.7
Statically Determinate anc
Indeterminate Structures A2.4
Stiffness & Carry-over
Factors jor Curved Members All 30
Stiffness Factor All.4
Strain - Displacement
Strain Energy ATL
Strain Energy of Plates Due
to Edge Compression and
Strain Energy in Pure Bending
of Plates 2 ee eee A18.12
Streamline Tubing - Strength C4 12
Strength Checking and
Design - Problems C4.5
Strenc*_-! Round Tubes
ander Combined Loadings 4.22
Stress Analysis Formulas €11.15
Stress Analysis of Thin Skin -
Multiple Stringer Cantilever
Wing WaNAMN Al9 10
Stress Concentration Factors C13.10
Stress Distribution & Angle
of Twist for 2-Cell Thin-
Wall Closed Section A6.7
Stress-Strain Curve B17
Stress-Strain Relations A24.6
Stresses around Panel Cutout A22.1
Stresses in Uprights Cll i7
Stringer Systems in Diagonal
Tenion C11.32
Structural Design Philosophy C1.6
Structural Fittings A2.2
Structural Skin Panel Details
Structures with Curved
Members ALL 29
Successive Approximation
Method for Multipie Ceil
Beams - eee eee ALS 24
Symbols for Reacting
Fitting Units A23
Energy eee eee ATS
Theorem of Least Work A8.2 Theorems of Virtual Work and
Minimum Potential Energy A7.$
Thermal Deflections by Matrix Methods A8.39
Thermai §tresses A8.14 Thermal Stresses AB 33 Thermoelasticity - Three-
Dimensional Equations A24,1 Thin Walled Shells Al16,5
Three Cell - Multiple Flange
Beam - Symmetrical about One AxIi8S Al5 lễ
Three Flange - Single Cell
Wing ee eee ee ee Al8.5
Torsion - Circular Sections, A6.1 Torsion - Effect of End
Restraint A6, 16 Torsion ~ Non-circular
Sections 2 2.0 - ABs
Torsion Open Sections Torsion of Thin-Wailed Cylinder having Closed Type
Stffenerg A6 18 Torsion Thin Walled Sections A6.5 Torsional Moments - Beams A5.9
Torsional Modulus of Rupture C4 17
Torsional Shear Flow in Multiple Cell Beams by Method of Successive Corrections A6 10 Torsional Shear Stresses in
Multiple-Cell Thin-Wall
Closed Section - Distribution 6.7
Torsional Strength of Round
Triaxial Stresses
Truss Deflection by Method
of Elastic Weights
Truss Structures Trusses with Double
wee AGE
Two-Dimensional Problems A26.5
Two-Cell Multiple Flange
Beam ~ One Axts of
Symmetry A15 11 Type of Wing Ribs A211 Ultimate Strength in Combined Bending & Flexural Shear C4.25
Ultimate Strength in Combined
Corapression, Bending,
Flexural Shear & Torsion C4 26
Ultimate Strength in Combined
Compression, Bending &
Torsion eee C428
Ultimate Strength in Combined
Tension, Torsion and
Internal Pressure p in pst C4.26
Uniform Stress Condition C1.1
Unit Analysis for Fuselage
Shears and Moments AS, 15, Unsymmetrical Frame A9.2 Unsymmetrical Frames or
Unsymmetrical Frames using
Principal Axes 2 Ag, 13
‘Jasymmetrical Structures Ad 13
vw ; Wy - Load Factor
eT ee =íc ees ALT Wagner Equatlons, C11.4
Web Bending & Shear Stresses C10,5
Web Design C11.18 Web SpHees C10 10
Web Strength Stabie Webs C10,5
Webs with Round Lightening
Holes 0.225500 C10 17
Wing Analysis Problems A13,2 Wing Arrangements A18.1 Wing Effeective Secton A19.12 Wing Internal Stresses A23,14
Wing Shear and Bending
Wing Strength Requirements Ai9.5
Wing Stress Analysis Methods A19.5
Wing - Ultimate Strength A19.12
Work of Structures Group Al.2
Y¥ Stiffened Sheet Panels CT 20
Trang 5
The first controllable human flight in a
heavier than air machine was made by Orville
wright on December 17, 1903, at Kitty Hawk,
North Carolina It covered a distance of 120
feet and the duration of flight was twenty
seconds Today, this initial flight appears
very unimpressive, but it comes into tts true
perspective of importance when we realize that
mankind for centuries has dreamed about doing
or tried to do what the Wright Brothers
accomplished in 1903
The tremendous progress accomplished in the
first 50 years of aviation history, with most
of it occurring in the last 25 years, is almost
unbelievable, but without doubt, the progress
in the second 50 year period will still be more
unbeilevable and fantastic As this its written
in 1964, jet airline transportation at 600 MPH
is well established and several types of
military aircraft nave speeds in the 1200 to
2000 MPH range Preliminary designs of a
supersonic airliner with Mach 3 speed have been
completed and the govermment is on the verge of
sponsoring the development of such a flight
vehicle, thus supersonic air transportation
should become comnon in the early 1970’s The
rapid progress in missile design has ushered
in the Space Age Already many space vehicles
have been flown in search of new knowledge
which is needed before successful exploration
of space such as landings on several planets
can take place Unfortunately, the rapid
development of the missile and rocket power
has given mankind a flight vehicle when combined
with the nuclear bomb, the awesome potential to
quickly destroy vast regions of the earth
While no person at oresent knows where or what
space exploration will lead to, relative to
benefits to mankind, we do know that the next
great aviation expansion besides supersonic
airline transportation will be the full develop-
ment and use of vertical take-off and landing
aircraft Thus persons who will be living
through the second half century of aviation
progress will no doubt witness even more
fantastic progress than oceurred in the first
50 years of aviation history
Al,2 General Organization of an Aircraft Company
Engineering Division,
The modern commercial airliner, military
airplane, missile and space vehicle is a highly
scientific machine and the combined knowledge
and experience of hundreds of engineers and
scientists working in close cooperation is necessary to insure a successful product Thus the engineering division of an aerospace company consists of many groups of specialists whose specialized training covers all ftelds of engineering education such as Physics, Chemical and Metallurgical, Mechanical, Hlectrical and,
of course, Aeronautical Engineering
It so happens that practically all the aerospace companies publish extensive pamphlets
or brochures explaining the organization of the engineering division and the duties and
responsibilities of the many sections and groups
and illustrating the tremendous laboratory and test facilities which the aerospace industry
possesses It is highly recommended that the student read and study these free publications
in order to obtain an early general under- standing on how the modern flight vehicle is
conceived, designed and then produced
In general, the engineering department of
an aerospace company can be broken down into six large rather distinct sections, which in turn are further divided into spectalized groups,
which in turn are further divided into smaller working groups of engineers To illustrate, the
six sections will be listed together with some
of the various groups This {s not a complete
list, but {t should give an idea of the broad engineering set~up that is necessary
1 Preliminary Design Section
Ii Technical Analysis Section
Aerodynamics Group
Structures Group
deignt and Balance Control Group
Power Plant Analysis Group
Materials and Processes Group
Controls Analysis Group
III Component Design Section
(1) Structural Design Group
(ding, Body and Control Surfaces)
(2) Systems Design Group (All mechanical, hydraulic, electrical
and thermal installations)
IV Laboratory Tests Section
ALL
,
Š
Trang 6
Al,2
(1) Wind Tunnel and Fluid Mechanics Test Labs
) Structural Test Labs
) Propulsion Test Labs
) Electronics Test Labs
) Blectro-Mecnanical Test Labs
} Weapons and Controls Test Labs
) Analog and Digital computer Labs
Goauean
V Flignt Test Section
VI Engineering Field Service Section
Since this textbock deals with the subject
of structures, 1t seems appropriate to discuss
in some detail the work of the Structures Group
For the detailed discussion of the other groups,
the student should refer to the various air-
craft company publications
Al.3 The Work of the Structures Group
The structures group, relative to number of engineers, is one of the largest of the many
groups of engineers that make up Section II,
the technical analysis section The structures
group is primarily responsible for the
structural integrity (safety) of the airplane
Safety may depend on sufficient strength or
sufficient rigidity This structural integrity
must be accompanied with lightest possible
weight, because any excess weight has detri-
mental effect upon the performance of aircraft
For example, in a large, long range missile,
one pound of unnecessary structural weight may
add more than 200 lbs to the overall weight or
the missile
The structures group is usually divided
into sub-groups as follows:~
(1) Applied Loads Calculation Group
(2) Stress Analysis and Strength Group
(3) Dynamics Analysis Group
(4) Special Projects and Research Group THE WORK OF THE APPLIED LOADS GROUP
Before any part of the structure can be finally proportioned relative to strength or
rigidity, the true external loads on the air-
craft must be determined Since critical loads
come from many sources, the Loads Group must
analyze loads from aerodynamic forces, as well
as those forces from power plants, aircraft
inertia; control system actuators; launching,
landing and recovery gear; armament, etc The
effects of the aerodynamic forces are initially
calculated on the assumption that the airplane
structure is a rigid body Afters the aircrart
structure is obtained, its true rigidity can
be used to obtain dynamic effects Results of
wind tunnel model tests are usually necessary
in the application of aerodynamic principles to
load and pressure analysis
THE WORK OF THE AEROSPACE STRUCTURES ENGINEER
ne final results of the work o group are formal reports giving comp
load design oriteria, with many
mary tables The final results = plete shear, moment and normal forces ref
to a convenient set of X¥2 axes for major air-
eraft units such as the wing, fuselage, etc
THE WORK OF STRESS ANALYSIS AND STRENGTH G&CUP
Essentially the primary job of the stress
group is to help specify or determine the xind
of material to use and the thickness, size and cross-sectional shape of every structural men-
ber or unit on the airplane or missile, and
also to assist in the design of all joints and comections for such members Safety with light weight are the paramount structural design re- quirements The stress group must constantly
work closely with the Structural Design Section
in order to evolve the best structural over-all arrangement Such factors as power plants,
built in fuel tanks, landing gear retracting wells, and other large cut-outs can dictate the
type of wing structure, as for example, a two
spar single cell wing, or a muitiple svar multiple cell wing
To expedite the initial structural design
studies, the stress group must supply initial structural sizes based on approximate loads
The final results of the work by the stress
group are recorded in elaborate reports which show how the stresses were calculated and how
the required member sizes were obtained to carry Tthase stresses efficiently The final size of
a member may be dictated by one or more factors such as elastic action, inelastic action, ele~ vated temperatures, fatigue, etc To insure
the accuracy of theoretical calculations, the
stress group must have the assistance of the
structures test laboratory in order to obtain
information on which to base allowable design
stresses
THE WORK OF THE DYNAMICS ANALYSIS GROUP
The Dynamics Analysis Group has rapidly
expanded in recent years relative to number of
engineers required because supersonic airplanes
missiles and vertical rising aircraft nave pre-
sented many new and complex problems in the
general field of dynamics In some aircraft
companies the dynamics group is set up as a
separate group outside the Structures Group
The engineers in the dynamics group are
Tesponsible for the investigation of vibration
and shock, aircraft flutter and the establish-
ment of design requirements or changes for its control or correction Aircraft contain dozens
of mechanical installations Vibration of any
part of these installations or systems may be
of such character as to cause faulty operation
or danger of failure and therefore the dynamic
Trang 7ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES characteristics must be changed or modified in
order to insure reliable and safe operation
The major structural units of aircraft such
as the wing and fuselage are not rigid bodies
Thus when a sharp air gust strikes a flexible
wing in high speed flight, we have a dynamic
load situation and the wing will vibrate The
dynamicist must determine whether this vibration
is serious relative to induced stresses on the
wing structure The dynamics group {s also
responsible for the determination of the
stability and performance of missile and flight
vehicle guidance and control systems The
dynamics group must work constantly with che
various test laboratories in order to obtain
reliable values of certain factors that are
necessary in many theoretical calculations
THE WORK OF THE SPSCIAL PROJECTS GROUP
In general, all the various technical
AtiOtLusnC RESEARCH Ì
"AND DeVACPMENT | TT
in the near or distant future as aviation pro- gresses For example, in the “cructures Group, this sub-group might be studying such problems
as: (1) how to calculate the thermal stresses
in the wing structure at super-sonic speeds;
(2) how to stress analyze a new type of wing structure; (3) what type of body structure is best for future space travel and what kind of materials will be needed, etc
Chart 1 tllustrates in general a typical make-up of the Structures Section of a large
aerospace company Chart 2 lists the many
items which the structures engineer must be
concerned with in insuring the structural
integrity of the flight vehicle Both Charts land 2 are from Chance-Vought Structures
Design Manual and are reproduced with their permission
srauctures TEST UM
| sYoRAuc ANO BOWEL Plant TEST una
MACHINE COMPUTATION ROU
Structures Section Organization
Chance-Vought Corp
3
Trang 8THE LINKS TO STRUCTURAL INTEGRITY
++ + ARE NO BETTER THAN THE WEAKEST LINK
MATERIALS OF
CONSTRUCTION FASTENERS
CONTROL SYSTEM STABILITY PANEL FLUTTER-SKIN CONTOURS CONTROL SYSTEM DEFLECTIONS
THERMAL EFFECTS
MECHANICAL VIBRATIONS ROLL POWER+0IVERGENCE AERODYHAMIC CENTER SHIFT DYNAMIC RESPONSE
WELDING
BONDING PLATE AND SAR FORGINGS CASTINGS EXTRUSIONS SHEET METAL SANOWICH PLASTIC LAMINATE BEARINGS
FLIGHT LOAD CRITERIA
GROUND LOAD CRITERIA FLIGHT LOAD DYNAMICS LAUNCHING DYNAMICS, LANDING DYNAMICS DYNAMIC RESPONSE
RECOVERY DYNAMICS
FLIGHT LOAD DISTRIBUTIONS INERTIAL LOAD DISTRIBUTIONS FLEXIBILITY EFFECTS GROUND LOAD DISTRIBUTIONS REPEATED LOAD SPECTRUMS TEMPERATURE DISTRIBUTIONS LOAQS FROM THERMAL DEFORMATIONS
PRESSURES- iMPACT
STRESS ANALYSIS
SKIN PANELS BEAM ANALYSIS STRAIN COMPATIBILITY STRAIN CONCENTRATION JOINT ANALYSIS BEARING ANALYSIS BULKHEAD ANALYSIS FITTING ANALYSIS
THERMAL STRESS
MECHANICAL COMPONENTS EXPERIMENTAL STRESS ANALYSIS
‘CREEP
TAIL ANALYS{S
FUSELAGE SHELL ANALYSIS OEFLECTIONS
" THERMAL EFFECTS THERMAL ANALYSIS
DEFLECTION ANALYSIS STIFFNESS
COMBINED LOADINGS STIFFNESS
SUCKLING
QUALITY CONTROL DUCTILITY STRESS-STRAIN HOMOGENEOUS MATERIAL, RESIOUAL STRESS HEAT TREAT CONTROL STRESS CORROSION STABILITY AT TEMPERATURE SPECIFICATION CONFORMANCE BLUE PRINT CONFORMANCE
Trang 9CHAPTER A2 EQUILIBRIUM OF FORCE SYSTEMS TRUSS STRUCTURES
A2.1 Introduction The equations of static
equilibrium must constantly be used by the
stress analyst and structural designer tn ob-
taining unknown forces and reactions or unknown
internal stresses They are necessary whether
the structure.or machine be simple or complex
The ability to apply these equations is no
doubt best developed by solving many problems
This chapter initiates the application of these
important physical laws to the force and stress
analysis of structures It is assumed that a
student has completed the usual college course
in engineering mechanics called statics
`
A2.2 Equations of Static Equilibrium
To completely define a force, we must Know
its magnitude, direction and point of applica—
tion These facts regarding the force are
generally referred to as the characteristics of
the fore Sometimes the more zeneral term of
line of action or location is used as 2 force
characteristic in place of point of application
designation
A force acting in space is completely
defined if we know its components in three
directions and its moments about 3 axes, as for
example Fy, Fy, Fz and My, My and My For
equilibrium oF a force system there can be no
resultant foree and thus the equations of
equilibrium are obtained by equating the force
and moment components to zero The equations
of static equilibrium for tne various types of
force systems will now De sumnarized
EQUILIBRIUM SCUATIONS FOR GENERAL
SPACE (NON-COPLANAR} FORCE SYSTEM
BFy = 0 mM, = 0
Fy = 0 M20 $ - (2.1)
3Py„ = 0 IM, = 0 3
Thus for 2 general space Zorce system,
there are 6 equations of static equilibrium
available Three of these and no more can be
force equations It is often more convenient
to take the moment axes, 1, 2 and G, as any set
of x, y and z axes All 6 equations could be
moment equaticns about 6 different axes The
force equations are written for 3 mutually
perpendicular axes and need not be tne x, ¥
and 2 axes
SQUILIBRIUM OF SPACE CCNCURRENT
Concurrent means that all
A2
force system pass through a common point The resultant, if any, must therefore be a force and not a moment and thus only 3 equations are necessary to completely define the condition
that the resultant must be zero
A combination of force and moment equations
to make a total of not more than 3 can be used
For the moment equations, axes through the point
of concurrency cannot be used since all forces
of the system pass through this point The
moment axes need not be the same direction as
the directicns used in the force equations but
of course, they could be
NHQUILISRIUM OF SPACE PARALLEL FORCES SYSTEM
In a parallel force system the direction of all forces is known, but the magnitude and
location of each is unknown Thus to determine
magnitude, one equation {ts required and for
location two equations are necessary since the
force is not confined to one plane in general
the 3 equations commonly used to make the re-
sultant zero for this type of force system are one force equation and two moment equations
For example, for a space parallel force system acting in the y direction, the equations of
equilibrium would be:
IFy = 0, If = 0,
EQUILIBRIUM OF GENERAL CO-PLANAR FORCE SYSTEM
In this type of force system all forces lie
in one plane and it es only 3 equations to determine the magnitude, direction and location
or the resultant of such a force system Sither
Force or moment equations can be used, except
that a maximum of 2 force equations can be used
For example, for a force system acting in the
xy plane, the following combination of equili-
(The subscripts 1, 2 and 3 ref
locations for z axes or moment er to different centers.)
Trang 10
A2.2
EQUILIBRIUM OF COPLANAR-CONCURRENT
Since all forces lie in the same plane and
also pass through a common coint, the ™
and direction of the resultant of this
force system is unknown Sut the location {ts
known since the voint of concurrency is on the
line of action cf the resultant Thus only two
equations of equilibrium are necessary to define
she resultant and make It zero The combin-
ations available are,
BF, = 0 5) OFx = 0 4, UFy 20 4, Bg 5 0 } 2.8
3fy =0 =0 Mz 30 Mga =O
(The z axis or moment center locations must be
other than through the point of concurrency)
EQUILIBRIUM OF CO-PLANAR PARALLEL FCRCE SYSTEM
Since the direction of all forces in this
type of force system is known and since the
forces ali lie in the same plane, it only takes
2 equations to define the magnitude and location
of the resultant of such a force system Hencs,
there are only 2 equations of equilibrium avail~
able for this type of force system, namely, a
force and moment equation or two moment
equations For example, for forces parallel to
y axis and located in the xy plane the equili-
brium equations available would be: -
EQUILIBRIUM OF COLINEAR FORCE SYSTEM
A colinear force system is one where all
forces act along the same line or in other
words, the direction and location of the forces
1s known but their magnitudes ere unknown, thus
only magnitude needs to be found to define the
resultant of a colinear force system Thus
only one equation of equilibrium is available,
namely
SF =O or M,=0 +~~+-+ -+
where moment center 1 is not on the line of
action of the force system
A2.3 Structural Fitting Units for Establishing the Force
Characteristics of Direction and Point of Application
To completely define a force in space re- quires 6 equations and 3 equations If the force
is limited to one plane In general a2 structure
is loaded by «mown forces ard these forces are
transferred through the structure in some
manner of internal stress distribution and then
EQUILIBRIUM OF FORCE SYSTEMS
direction of an unknown force cri
application or doth, thus decreast
of unknowns to be determined The
which follow tllustrate tne type of
and Q acting on the bar, the line of such forces must act through the center o°
ball if rotation of the bar is prevented,
a ball and socket Joint can be used to est
or control the direction and line action of
force applied to a structure through cwhis tyr
of fitting Since the joint has no rotationa resistance, mo couples in any plane can oe
For any force such as P and Q acting in the
xy plane, the line of action of such a ?
must pass through the pin center since
fitting unit cannot resist a moment about 2 2 axis through the pin center ‘nersfore, 2or
forces acting in the xy plane, the cirection
and line of action are established Dy the pin joint as illustrated in the figure Since a single pin fitting can resist moments about axes perpendicular to the din axis, the direction and line of action of out of plane forces is there-
If a bar AB has single pin fittings at
each end, then any force P lying in the xy
plane and applied to end B must have a direction
and line of action coinciding with a line join- ing the pin centers at and fitel aA and 3,
since the 7ittings cannot resist oment about
the 2 axis
Trang 11ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES Double Pin - Universal Joint Fittings
Since single pin fitting units can resist
applied moments about axes normal to the pin
axis, a double pin joint as illustrated above
is often used Tnis fitting unit cannot resist
moments about y or 2 axes and thus applied
forces such as P and Q must have a line of
action and direction such as to pass through
the center of the fitting unit as illustrated
in the figure The fitting unit can, however,
resist 4 moment about the x axis or in other
words, a universal type of fitting unit can
resist a torsional moment
Rollers
A
-
t
In order to permit structures to move at
support points, a fitting unit involving the
idea of rollers is often used For example,
the truss in the figure above is supported by
a pin ?itting at (A) which is further attached
to a fitting portion that prevents any nori-
zontal movement of truss at end (A), however,
the other end (B) Ls supported dy a nest of
rollers which provide no horizontal resistance
to a horizontal movement of the truss at end (B)
Tne rollers fix the direction of the reaction
at (B) as perpendicular to the roller bed
Since the fitting unit is joined to the truss
joint by a pin, the point of application of the
reaction {1s also known, hence only one force
characteristic, namely magnitude, 18 unknown
for a roller-pin type of fitting for the
fitting unit at (A), point of application of the
reactton to the truss is knowm because of the
pin, but direction and magnitude are unknown
Lubricated Slot or Double Roller Type of Fitting
Unit
A2.3 Another general fitting type that is used
to establish the direction of a force or reaction
is tllustrated in the figure at the bottom of the
first column Any reacting force at joint (A}
must be horizontal since the support at (A) is
so designed to provide no vertical resistance
Cables - Tie Rods
củ?
P Since a cable or tie rod has negligible
bending resistance, the reaction at Joint B on the crane structure from the cable must be
colinear with the cable axis, hence the cable
establishes the force characteristics of direc-
tion and point of application of the reaction
on the truss at point B
A2.4 Symbols for Reacting Fitting Units as Used in
Problem Solution
In solving a structure for reactions, member stresses, etc., ome must know what force characteristics are unknown and it 1s common
practice to use simple symbols to indicate, what fitting support or attaciment units are to be
used or are assumed to be used in the final
design The following sketch symbols are com-
monly used for coplanar force systems
mn
A small circle at the end of a member or on
a triangle represents a single pin connection and fixes the point of application of forces
acting between this unit and a connecting member
The above graphical symbols represent 2
reaction in which translation of the attach-
ment point (b) is prevented but rotation of the attached structure about (b} can take place
Thus the reaction 1s unknown in direction and
magnitude but the point of application is known, namely through point (b) Instead of using direction as an unknown, {t 1s more convenient
to replace the resultant reaction by two com—
ponents at right angles to each other as indi-
cated in the sketches
Trang 12^A2.4
(b), Knite Edge bapin
The above fitting units using rollers fix
the direction of the reaction as normal to the
roller bed since the fitting unit cannot resist
a horizontal force through point (b) Hence
the direction and point of application of the
reaction are established and only magnitude is
The grapnical symbol above is
represent a rigid support which is
rigidly to a connecting structure The re-
action is completely unknown since ali 3 force
characteristics are unknown, namely, magnitude,
direction and point of application It 1s con-
venient to replace the reaction R by two force
components referred ta some point (bd) plus the
unknown moment M which the resultant reaction R
caused about point (b) as indicated in the
above sketch This discussion applies to a
coplanar structure with all forces in the same
plane For a space structure the reaction
would have 3 further unknowns, namely, Rgs My
and My
A2.5 Statically Determinate and Statically Indeterminate
Structures
A statically determinate structure is one
in which all external reactions and internal
stresses for a given load system can be found
by use of the equations of static equilibrium
and a statically indeterminate structure is
one in which all reactions and internal stresses
cannot be found by using only the equations of
equilibrium
A statically determinate structure is one
that has just enough external reactions, or
just enough tnternal members to make the
structure stable under a load system and if one
reaction or member is removed, the structure is
reduced to 2 linkage or a mechanism and is
therefore not further capable of resisting the
load system If the structure has more ex-
ternal reactions or internal members than is
necessary for stability of the structure under
a given load system it is statically indeter-
EQUILIBRIUM OF FORCE SYSTEMS
minate with respect to external reactions alone
or to internal stresses alene or to doth
The additional equations that are needed
to solve a statically indeterminate structure
are obtained oy considering the distortion of
the structure This means that the size of all
members, the material from which members are
made must be known since distortions must be calculated In 4 statically determinate structure this information on sizes and matertal
is not required but only the configuration of the structure as a whole Thus design analysis for statically determinate structure is straight forward whereas a gensral trial and error pro- cedure is required for design analysis of
statically indeterminate structures
A2.6 Examples of Statically Determinate and Statically
Indeterminate Structures
The first step in analyzing a structure is
to determine whether the structure as presented
is statically determinate If so, the reactions
and internal stresses can de found without xnow¬
ing sizes of members or Kind cf material If not statically determinate, the elastic theory
must be applied to obtain additional equations
The elastic theory is treated in considerable detatl in Chapters A7 to Al2 inclusive
To help the student oecome familiar with the problem of determining whether a structure
is statically determinate, several example problems will be presented,
known forces or loads are the distributed loads shown in Fig, 2.1, the
of 10 1b per inch on member ABD The reactions
at points A and C are unknown The reaction at
C has only one unknown characteristic, namély, magnitude because the point of application of Ro
is through the cin center at C and the directicn
of Ro must be parallel to line CB because thers
is a pin at the other end 3 of member CB At
point A the reaction is unknown in direction
and magnitude but the point of application must
be through the pin center at A Thus there are
2 unknowns at A and one unknown at C or a total