Bruhn - Analysis And Design Of Flight Vehicles Structures Episode 2 Part 10 doc

For example, failure of a structural unit may be due to too high a stress or load causing a complete fracture of the unit while the vehicle is in flight.. Immediately above this point t

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES

In the above derivation the torstonal

stiffness GJ at a representative section is

used The stiffness is obtained from Bredt's

equation for the twist of a single cell thin-

walled tube (Eq 18, Chapter a6}

Here A is the area enclosed within the tube

cross section by the median line of the tube

wall and the integration is carried out around

the tube perimeter (index s gives distance along

the perimeter) For the torque boxes en-

countered in delta wings it ts probably satis-

factory to neglect the ds/t contribution from

the vertical webs, it being small compared with

the corresponding contribution from the cover

sheets

In the example wing three boxes (2-3-5-6,

3-4-6-7 and 6-7-8-9} are to be used These

boxes are each 48 inches square in plan and

have average depths (assumed here to 5e the `

uniform depths of the representative sections)

of 7.26", 5.32" and 4.27", respectively Fig

A23.11 shows the assumed representative cross

section of box 2-3-5-6 and its GJ calculation

A23.5 Complete Wing Stiffness Matrix

The stiffness matrix for the composite wing now may be obtained by forming the sum of the complete stiffness matrices for the beam

elements and the torque boxes For this pur- pose a large matrix table is laid out and en- tries from the individual stiffness matrices

are transferred into the appropriate locations

Wherever multiple entries occur in a box these are summed

Before the large wing matrix is laid out 1t 1S necessary to observe first that the matrix which would be obtained as just indicated would be singular, i.e., its determinant would

be zero, and hence it could not be inverted to yield the flexibility influence coefficients (see Appendix A} This condition artses due to the fact that the equations represented (19 in number in the example problem) are not itnde- pendent; three of these equations can be ob-

tained as linear combinations of the others,

That there are three such interdependencies may be seen from the existence of the three equations of statics which may be applied ta the wing (summation of normal forces and sum-

mation of moments in two vertical planes):

hence three of the reactions expressed by the structural equations in the matrix may be found

7

A23.12

from the others oy the equations of static

To remove the "Singularity" from the stiftness

matrix it is only necessary to drop out three

equations - achieved by removing shree rows

and corresponding columns (so as to retain a

symmetric stiffness matrix)

The act of removing the three equations

Selected 1S also equivalent to assuming the

corresponding deflections to be zero In this

way a reference base for the deflections is

also established The choice of reference base

18 somewhat arbitrary, but, following a Sug~

gestion of Williams (7), a triangular base will

de employed as shown in Fig A23.12

Fig A23.12 Deflection Reference

Base Here the deflections at points 1 and S are zero,

fixing the reference triangle, since the point

corresponding to 5 in the other half of the wing

(say, 5’), will have zero deflection also due to

symmetry The third condition is applied to

point 2: point 2 will be assumed to have zero

rolling rotation (§, = 0) for symmetric wing

loadings and to have zero transverse deflection

(A, = 0) for antisymmetric loadings.”

Hence the following equations (rows and

columns) are to be omitted from the wing stiff-

ness matrix:

Pas Ay

4, for symmetric loadings

for antisymmetric loadings

The

obtained 16 x 16 wing stiffness matrices thus can now be inverted as they are non~

* The antisymmetric loading pattern is one wherein the wing

is loaded equally, but in opposite sense, on corresponding

points of the two wing halves Any general loading may be

resolved into the sum of one symmetric and one anti-

acting alone and supported by

assumed above To account for the presen 1ce

the other nalf of the wing, it is necessary to specify additional geometric conditions along

the airplane centerline This step is accom-

plished by assuming the following deflections

zero (eliminating their corresponding rows and columns from the matrix):

a for symmetric loadings (zero

5 lateral slope or rolling

` rotation along the airpiane

RL centerline)

9, for antisymmetric loadings (zero transverse deflections and piten- ing rotations along the airplane

centerline)

It will be seen that in both cases an

additional 3 equations are eliminated from the

16 x 16 matrices reducing them to 12 x 13's

(In general, there is no reason why the

matrices for the symmetric ane antisymmetric

cases have to be the same siz A retation

%,, for instance, would be saro in one case

only.) Written below is the 13 x 13 wing stiff-

ness matrix for the antisymmetric case (4, 54,7 4,7 4,7 4,59, 50) As explained

earlier, each entry therein is the sum of all corresponding stiffnesses for all elements meeting at the point A typical multiple sentry

occurs for joint 6 - (row P,, column 4,) These

comprise:

0.007178 from spar 3-6~8 0.001832 from rib 5-6-7 0.000310 from d0x 6=7-8-9 0.000898 from box 235-6 0.000482 from box 3-4+6-7 0.01070 TOTAL

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES A23,13

P, |F.003376| 0,a10 | 002604|-.0,<837/-.0,49¢8} 00se471-.0, 441 ~.02228 009947| 04558

P, || 0,510 |¬.00191 |-.0,4537] 001892|~.0.4652 „006810 005306 Pie 0,2Z58 |~.0, 2998 | ~.0, 4642 |~.0,2720 „002561 ¬.001662 002591

My „02760 |~.005066| ,006947 1.585 „1507 „2094 „7168 „1507 [a> bu ~ 0,441 006910} 0O2391] 1507 7211 1507 „3886

THE WING FLEXIBILITY MATRIX:

The wing flexibility influence coefficients are now found by forming the inverse of the above

stiffness matrix (see Eq 2) án automatic digital computing machine is the essential tool for

this step The details of the procedure and techniques employed in forming such inverses are not

of concern here and only the result is presented It 1s assumed that this phase of the work can

be handled with dispatch by experienced computer personnel

[Amn] » Wing Influence Coeffictents, Antisymnetric Case

* In spite of the exercise of great care and much ingenuity, problems of such great size and such a nature will arise occasion-

ally as to defy satisfactory inversion For these problems such clever physical concepts as "block" solutions of portions of

the structure are available The interested reader is referred to a discussion of Ref, 4 and to techniques of block and group

relaxations in the literature on Relaxation Methods (e.g Rei 8)

24S

A23.6 Wing internal Stresses

Following the calculation of the wing

influence coerficients, a relationship is avail-

able between the applied wing loads (assumed

given) and the wing deformations In a symbolic

fashion,

An Py

Earlier in the analysis, deformations of a

structural element (beam or box} were related to

the forces acting on that element by the

{complete} element stiffness matrix:

Now since the deformations of an element

must conform to those of the wing,

Equation (10) provides the desired relation

between external loads applied to the wing and

the forces acting on 2 specific slement These

ces on the element might be locked upon as those necessary to hold the element in the de-

flected shape conforming to that of the wing

Of course, with these element forces known the

finer details of the stress distribution within

the element are readily found by standard techniques

Note that the matrix [XÌm nen in #q (10) wi1l have to be "blown up” to an aparopriate size before it may be premultipliad onto lurve: This enlargement is accomplished oy the in-

serticn of columns of zerces at each column

location corresponding to a wing deformation which does not affect the specific element under

P, |~.O0 š0ø| 0,5Z252|~.G01600|.901458|~.9,4632 P;2|=.G 682| 0,158 | ,0,2z28|~.0,4652| 9,2z29

4.| cày [4a ae dso [8.1% /4/4/4] G lỗ.,|ô,

n, [0] o2303 | 0 |~.ogssos|~.ootse2lo|c| o| o|o| zss+ Jo] o

P, || o [-.co1637] o | oje2s2] o16s |o] 0) 0] 0] o]-.cieos | 0] 0 [*] =5 ỊP, |O| 00Z015| 0 |-.001600! 0,22Z8! 0 | 0] 01010) 0230 010

P,.| 0| 0,2228] 0 [¬.0,4682| 0,22291 0 | 0] 0| 0| 0l-.005421010

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES

Then multiplying out with [Ann] JNING ? er Eq (10),

Fig A23.4-b) in terms of

Bach column of forces in the above matrix

must satisfy the equations of statics on the

element A variety of checks on the accuracy

of such a result are thus available

The student will find it instructive to

Study carefully this last result to observe in

what manner a load applied to the wing appears

on this spar element For instance, one sees

that of the load P, = 1 applied-to the wing,

+3735 goes onto spar 4-7-9-10 xamination of

Fig A23.1 reveals that the remainder, 1.00 ~

+3735 = 6265 must be taken up by rib 5-6-7 and

the torque boxes 3-4-7-6 and 6-7-8-9

REFERENCES 1) Argyris, J H and Kelsey, S., Energy

Theorems and Structural Analysis, Aircraft

ingineering, Oct 1954, et seq

2) Levy, S., Structural Analysis and Influence

Coefitcients for Delta Wings, Journal of the

Aeronautical Sciences, £0, 1953

3) Schuerch, H., Delta Wing Design Analysis,

4) Turner, M J., Clough, R W., Martin, H c.,

and Topp, L J., Stiffness and Deflection Analysis of Complex Structures, Journal of the Aeronautical Structures, 23, September, 1956,

5) Kroll, W., Effect of Rib Flexibility on the

Vibration Modes of a Delta Wing Aircraft, Insti-

tute of Aeronautical Sciences, Preprint No 58,

1956

6) Wooley, Ruth, Check of Method for Computing

Influence Coefficients of Delta and Other Wings, National Bureau of Standards Report 3655, 1954,

{Available as ASTIA No AD46866)

7) Williams, D., Recent Developments in the

structural Apprcach to Aeroelastic Problems, Journal of the Royal Aeronautical Society, 58,

1954 (see also, Aircraft Engineering, 2, 1954)

8) 8outhwell, R.V., "Relaxation Methods In

Engineering Science", Oxford 1940

Society of Automotive Engineers National Aero-

nautic Meeting Preprint No 141, September 1953

AIRCRAFT WITH "DELTA" WING SHAPES

CONVAIR SUPERSONIC F-102 INTERCEPTOR

CONVAIR SUPERSONIC B-58 BOMBER

COMMERCIAL TRANSPORT FOR THE 1970s? This is an artist's conception of a passenger transport of the future cruis-

ing at three to five times the speed of sound at 60, 000 feet or higher This is one of hundreds of configurations considered by

Convair Division of General Dynamics Corporation at San Diego, Calif., in its studies of supersonic airliners

The flight vehicle structures engineer

faces a major design requirement of a high

degree of structural integrity against failure,

but with as light structural weight as possible

Structural failure in flight vehicles can

often prove serious relative to loss of life

and the vehicle However, experience has

shown that if a flight vehicle, whether

military or commercial in type, is to be

satisfactory from a payload and performance

standpoint, major effort must be made to save

structural weight, that is, to eliminate all

structural weight not required to insure

against failure

Since a flight vehicle is sut !ected to

various types of loading such as static,

dynamic and repeated, which may act under a

wide range of temperature conditions, it is

necessary that the structures engineer fave a

broad knowledge of the behavior of matertais

under loading if safe and efficient structures

are to be obtained This Part B provides

information for the structures design engineer

relative to the behavior of the most common

flight vehicle materials under load and the „

various other conditions encountered in flight

environment, such as elevated temperatures

Bl.2 Failure of Structures

A flight vehicle like any other machine,

is designed to do a certain job satisfactorily

If any structural unit of the vehicle suffers

effects which in turn effects in some manner

the satisfactory performance of the vehicle,

the unit can be considered as having failed

Failure of a structural unit 1s therefore 4

rather broad term For example, failure of a

structural unit may be due to too high a stress

or load causing a complete fracture of the unit

while the vehicle is in flight If this unit

Should happen to be one such as a wing beam or

a major wing fitting, the failure is a serious

one as it usually involves loss of the airplane

and loss of life Likewise the collapse of a

trut in landing gear structure during a land-

3g operation of the airplane can be a very

erious failure Failure of a structural unit

cay be due to fatigue and since fatigue failure

748 of the fracture type without warning indi-

‘cations of impending failure, it can also prove

to be a very serious type of failure

BLi

Failure in a different manner can result from a structural unit being too flexible, and this flexibility might influence aerodynamic forces sufficiently as to produce unsatisfac-

tory vehicle flying characteristics In some

cases this flexibility may not be serious Telative to the loss of the vehicle, but it is

still a degree of failure because changes must

be made in the structure to provide a satis- factory operating vehicle In some cases,

excessive distortion such as the torstonal

twist of the wing can be very serious as this

excessive deflection can lead to a build-up

of aerodynamic-dynamic forcas to cause flutter

or violent vibration which can cause failure

involving the loss of the airplane

To illustrate another desgree of failure

of a structural unit, consider a wing built in

fuel tank The stiffened sheet units which

make up the tank are also a part of the wing

structure In general these sheet units are

designed not to wrinkle or buckle under air- plane operating conditions in order to insure

against leakage of fuel around riveted con- nections Therefore if portions of the tank

walls do wrinkle in operation resulting in fuel leakage, which in turn require repair or modification of the structure, we can say failure has occurred since the tank failed to

do its job satisfactory, and involves the item

of extra expense to make satisfactory To illustrate further, the flight vehicle is

equipped with many installations, such as the

control systems for the control surfaces, the

power plant control system etc., which involves many structural units In many cases exces-

sive elastic or inelastic deformation of a unit can cause unsatisfactory operation of the system, hence the unit can be considered

as having failed although it may not be a

serious failure relative to causing the loss

of the vehicle Thus the aero-astro structures engineer is concerned with and responsible for preventing many degrees of

failure of structural units which make up the

flight vehicle and its installations and obviously the greater his knowledge regarding

the behavior of materials, the greater his

chances of avoiding troubles from the many degrees of structural failure

B1,3 General Types of Loading

Failure of a structural member is in- fluenced by the manner in which the load is

Bl.2 BEHAVIOR OF MATERIALS

applied Relative to the length of time in

applying the load to a member, two broad

classifications appear logical, namely,

(1) Static loading and (2) Dynamic loading

For purposes of explanation and general

discussion, these two broad classifications

will be further broken down as follows:-

Static Continuous Loading A continuous load

185 @ load that remains on the member for a long

period of time The most common example ts the

dead weight of the member or the structure it-

self When an airplane becomes airborne, the

weight of the wing and its contents is a con-

tinuous load on the wing A tank subjected to

an internal pressure for a considerable period

of time is a continuous load Since a contin-

uous load is applied for a long time, it is a

type of loading that provides favorable con-

ditions for creep, a term to be explained later,

For airplanes, continuous loadings are usually

associated with other loads acting simultan~

eously

Static Gradually or Slowly Applied Loads A

static gradually applied load is one that

slowly builds up or inersases to its mximm

value without causing appreciable shock or

vibration The time of loading may be a

matter of seconds or even hours The stresses

in the member increases as the load 1s in-

creased and remains constant when the load

becomea constant As an example, an airplane

which is climbing with a pressurized fuselage,

the internal pressure loading on the fuselage

structure is gradually increasing as the

difference in air pressure between the inside

and outside of the fuselage gradually increases

as the airplane climbs to higher altitudes

Static Repeated Gradually Applied Loads If a

gradually applied load is appiifed a large num-

der of times to a member it is referred to as a

repeated load The load may be of such nature

as to repeat a cycle causing the stress in the

member to go to a maximum value and then back

to Zero stress, or from a maximum tensile

stress to a maximm compressive stress, etc

The situation envolving repeated loading is

important because it can cause failure under

a stress in a member which would be perfectly

safe, if the load was applied only once or a

small number of times Repeated loads usually

cause failure by fracturing without warning,

appreciable shock or vibration To produce

such action, the load must be applied far more

rapid than in a static loading This rapid

application of the load causes the stresses

in the member to be momentarily greater than

if the same magnitude of load was applied

statically, that is slowly applied For

example, if a weight of magnitude W ts gradually placed on the end of a cantilever

beam, the beam will bend and gradually reach

a maximum end deflection However if this

same weight of magnitude W is dropped on the end of the beam from even such a small height

as one foot, the maximum end deflection will

be several times that under the same static

lead W The beam will vibrate and finally come to rest with the same end deflection as under the static load W In bringing the dynamic load to rest, the beam must absorb energy equal to the change in potential energy

of the falling load W, and thus dynamic loads

are often referred to as energy loads

Prom the basic laws of Physics, force

equals mass times acceleration (F = Ma) and acceleration equals time rate of change of velocity Thus if the velocity of a body

such as an airplane or missile is changed in

magnitude, or the direction of the velocity

of the vehicle is changed, the vehicle is accelerated which means forces are applied to

the vehicle In severe flight airplane Maneuvers like pulling out of a dive from high speeds or in striking a severe trans—

verse air gust when flying at high speed, or

in landing the airplane on ground or water,

the forces acting externally on the airplane

are applied rather rapidly and are classed

as dynamic loads Chapter A4 discusses the subject of airplane loads relative to whether

they can be classed as static or dynamic and

how they are treated relative to design of aircraft structures

Bl.4 The Static Tension Streas-Strain Diagram

The information for plotting a tension

stress-strain diagram of a material 1s ob~

tained by loading a test specimen in axial

tension and measuring the load with corres—

ponding elongation over a given length, as the specimen is loaded statically (gradually applied) from zero to the failing load To standardize results standard size test specimens are specified by the (ASTM) American Society For Testing Materials The speed of

the testing machine cross-head should not

exceed 1/16 inch per inch of gage length per minute up to the yield point of the material

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES

and it should not exceed 1/2 inch per inch of

gage length per minute from the yield point to

the rupturing point of the material The

instrument for measuring the elongation must

be calibrated to read 0.0002 inches or less

The information given by the tension stress-

strain diagram is needed by the engineer since

it 1s needed in strength design, rigidity

design, energy absorption, quality controi and

Many other uses

Fig Bl.1l shows typical tensile stress-

strain diagrams of materials that fall in three

broad classifications In the study of such

diagrams various facts and relationships have

been noted pelative to behavior of materials

and standard terms and symbols have been pro~

vided for this basic important information

These terms will be explained briefly

Modulus of Elasticity (E) The mechanical

property that defines resistance of a material

in the elastic range is called stiffness and

for ductile materials is measured by the value

termed modulus of Elasticity, and, designated

by the capital letter E Referring to Fig

Bl.1, it ts noticed that the first part of all

three diagrams is a straight line, which indi-

cates a constant ratio between stress and

strain over this range The numerical value of

this ratio is referred to as the modulus of

Eleasticity (EE) B&B 1S; therefore the slope of

the initial “straight portion of the stress~

Strain diagram and its numerical value is

obtained by dividing stress in pounds per

square inch by a strain which is non-dimensionalj

or = = f/E, and thus — has the same units as

stress, namely pounds per square inch

The clad aluminum alloys have two E values

as indicated in the lower diagram of Fig Bl.1l

The initial modulus 1s the same as for other

aluminum alloys, but holds only up to the pro-

portional limit stress of the soft pure

aluminum coating material Immediately above

this point there is a short transition stage

and the material then exhibits a secondary

modulus of Elasticity up to the proportional

limit stress of the stronger core material

This second modulus is the slope of the second

straight line in the diagram Both modulus

values are based on a stress using the gross

area which includes doth core and covering

matertal

Tensile Proportional Limit Stress (Fp) The

proportional limit stress is that stress which

exists when the stress strain curve departs

from the initial straight line portion by a

unit strain of 0.0001 In gensral the pro-

portional limit stress gives a practical

dividing line between the elastic and tnelastic

range of the material The modulus of

elasticity is considered constant up to the

{a) Matertal Having a Definite

Yield Point (such as some Steels)

Strain - Inches Per Inch

Ultimate Tensile Stress

L uea Stress

Proportional Lâmit (bì Matertals not Having a

Definite Yield Point (such as Aluminum Alloys, Magnesium,

and Some Steels)

Strain - Inches Per Inch

Primary Modulus Line

Ultimate Tensile Stresa

Tensile Yield Stress (Fry) In referring

to the upper diagram in Fig Bl.1, we find that some materials show a sharp break at a stress considerably below the ultimate stress and that the material elongates considerably

with little or no increase in load The stress at which this takes place is called

the yield point or yield stress However many materials and most flight vehicle materials do

not show this sharp break, but yield more

gradually as illustrated in the middle diagram

of Fig Bl.1, and thus there is no definite yield point as described above Since

permanent deformations of any appreciable amount are undesirable in most structures or

machines, {t is normal practice to adopt an arbitrary amount of permanent strain that is

considered admissible for design purposes

Test authorities have established this value

of permanent strain or set as 0.002 and the

stress which existed to cause this permanent

strain when released from the material is called the yield stress Fig Bl.1 shows how

bo

it 1s determined graphically by drawing a line

from the 0.002 point parallel to the straight

portion of the stress-strain curve, and where

this line intersects the stress-strain curve

represents the yield strength or yield stress

Ultimate Tensile Stress (Pty) The ultimate

tensile stress 1s that stress under the maxi-

mum load carried by the test specimen It

should be realized that the stresses are based

on the original cross-sectional area of the

test specimen without regard to the lateral

contraction of the specimen during the test,

thus the actual or true stresses are greater

than those plotted in the conventional stress-—

strain curve Fig B1.2 shows the general

relationship between actual and the apparent

stress as plotted in stress-strain curves

The difference 1g not appreciable until the

higher regions of the plastic range are

Figs Bl.3 and Bl.4 compare the shapes of

the tension stress-strain curves for some common aircraft materials

B1,5 The Static Compression Stress-Strain Diagram

Because safety and light structural weight

are so important in flight vehicle structural design, the engineer must consider the entire stress-strain picture through both the tension and compressive stress range This is due to the fact that buckling, both primary and local,

13 a common type of failure in flight vehicle

structures and failure may occur under stresses

in either the elastic or plastic range In general the shape of the stress-strain curve

as it departs away from the initial straight

line portion, is different under compressive

stresses than when under tensile stresses

Furthermore, the various flight vehicle materials have different shapes for the region

of the stress-strain curve adjacent to the

straight portion Since light structural weight is so important, considerable effort is made in design to develop high allowable

compressive stresses, and in many flight

vehicle structural units, these allowable ultimate design compressive stresses fall in the inelastic or plastic zone

Fig B1.5 shows a comparison or the

stress-strain curves in tension and compres-

sion for four Widely used aluminum alloys

Below the proportional limit stress the modulus of elasticity is the same under both tension and compressive stresses The yleld

stress in compression is determined in the Same manner as explained for tension

Compressive Ultimate Stress (F ) Under a

static tension stress, she ultimate tensile stress of a member made from a given material

is not influenced appreciably by the shape of the cross-section or the length of the member,

however under a compressive stress the

ultimate compressive strength of a member is

greatly influenced by both cross~sectional,

shape and length of the member Any nember, unless very short and compact, tends to

buckle laterally as a whole or to buckle

laterally or cripple locally when under

compressive stress If a member is quite Short or restrained against lateral buckling,

then failure for some materials such as stone, wood and a few metals will be by definite fracture, thus giving a definite value for the ultimate compressive stress Most air- craft materials are so-ductile that no fracture

is encountered in compression, but the material

yields and swells out so that the increasing cross-sectional area tends to carry increasing load It 1s therefore practically impossible

to select a value of the ultimate compressive

stress of ductile materials without having

/ [ALCLAD 758-Te SHEET

Vor AND PLATE

THICKNESS 0.016 -0.499-1" 19,

Ì 1 i Ị T5S%-T6 EXTRUSIONS

exiotimsn exit es: % 2a ee Tp

xeCrn mua @ 4107 INJIN E x so-* Psi

Fig B1.5

some arbitrary measure or criteron For

wrought materials it 1s normally assumed that

Foy equals Fyy For brittle materials, that

are relatively weak in tension, an Foy higher

than Fry can be obtained by compressive tests

of short compact specimens and this ultimate

compressive stress is generally referred to

as the block compressive stress

B1.6 Tangent Modulus Secant Modulus

Modern structural theory for calculating

the compressive strength of structural members

as covered in detail in other chapters of this

book, makes use of two additional terms or

values which measure the stiffness of a member

when the compressive stresses in the member

fall in the inelastic range These terms are

tangent modulus of elasticity (Ey) and secant

modulus of elasticity (Eg) These two modi-

fications of the modulus of elasticity (E)

apply in the plastic range and are illustrated

in Fig BL.6 The tangent modulus Et is

determined by drawing a tangent to the stress-

strain diagram at the point under consideration

of change of stress with strain The secant

modulus Eg is determined by drawing a secant

(straight line) from the origin to the point

in question This modulus measures the ratio

between stress and actual strain Curves

which show how the tangent modulus varies with stress are referred to as tangent modulus

curves Fig B1.5 illustrates such curves

for four different aluminum alloys It should

be noted that the tangent modulus is the same

as the modulus of elasticity in the elastic range and gets smaller in magnitude as the

stress gets higher in the plastic range

BL.7 Elastic - Inelastic Action

Tf a member is subjected to a certain stress, the member undergoes a certain strain

If this strain vanishes upon the removal of

the stress, the action is called elastic

Generally speaking, for practical purposes,

a material is considered elastic under stresses

up to the proportional limit stress as previously defined Fig Bl.7 tllustrates alastic action However, if when the stress

is removed, a residual strain remains, the

action is generally referred to as inelastic

or plastic Fig Bl.8 illustrates inelastic action

Elastic Action Inelastic Action

in tension or shear without rupture, as

contrasted with the term brittleness which indicates little capacity for plastic de-

formation without failure From a physical

Standpoint, ductility is a term which measures

the ability of a material to be drawn into a

wire or tube or to be forged or die cast

Ductility 1s usually measured by the percentage

elongation of a tensile test specimen after

failure, for a specified gage length, and is

usually an accurate enough value to compare

matertals

-L,

Percent elongation = (=) 100 = measure of

where L, = original gage length and L, = gage

length after fracture In referring to

ductility in terms of percent alongation, it

is important that the gage length be stated,

since the percent elongation will vary sith

gage length, because a large part of the total

strain occurs in the necked down portion of the

Gage length just before fracture

B1.9 Capacity to Absorb Energy Resilience Toughness

Resilience The capacity of a material to

absorb energy in the elastic range is referred

to as its resilience For measure of

resilience we have the term modulus of

resilience, which 1s defined as the maximum

amount of energy per unit volume which can be

stored in the material by stressing it and

then completely recovered when the stress is

removed Ths maximum stress for elastic

action for computing the modulus of resiltence

is usually taken as the proportional limit

stress Therefore for a unit volume of

material (1 cu in.) the work done in stressing

a material up to its proportional limit stress

would equal the averace stress fp/2 times the

elongation (ep) in one inch If we let U

represent modulus of resilience, then

Under a condition of axlal loading, the modulus

of resilience can be found ag the area under

the stress-strain curve up to the proportional

limit Stress Thus in Fig B1l.9, the area OAB

represents the energy absorbed in stressing

the material from zero to the proportional

limit stress

High resilience is desired in members

subjected to shock, such as springs From

equation (1), a high value of resilience is

obtained when the proportional limit stress is

high and the strain at this stress 1s high,

or from equation (2), when the proportional limit stress is high and modulus of elasticity

18 10W,

In Fig 31.9 if the stress is released from point D in the plastic range, the recovery diagram will be approximately a straight line

DE parallel to AO, and the area CDE represents the energy released, and often referred to as hyper-elastic resilience

Toughness Toughness of a material can be defined as its ability to absorb energy when stressed in the plastic range Since the

tarm energy is involved, another definition

would be the capacity of a material for resisting fracture under a dynamic load

Toughness is usually measured by the term modulus of Toughness which is the amount of

strain energy absorbed per unlt volume when

stressed to the ultimate strength value

In Fig B1.9, let f equal the average

stress over the unit strain distance de from

F to dG Then work done per unit volume in stressing F to G@ is fde which is represented

by the area FGHI The total work done in stressing to the ultimate stress 7, would

1

then equal /°* fde, which is thé area under

the entire stress-strain curve up to the

ultimate stress point, or the area QO AJKO

in Fig Bl.9 and the units are in lb per

eu inch Strictly speaking it should not include the elastic resilience or the energy absorbed in the elastic range, but since this area is small compared to the area under the curve in the plastic range it is usually included in toughness measurements

It should be noted that the capacity of

a member for resisting an axtally applied dynamic load is increased by increasing the length of a member, because the volume is increased directly with length However, the

ultimate strength remains the same since it

is a function of cross-sectional area and not

of volume of the material .