SIMPLE STRESS AND STRAIN

STRESS:

STRESS IS THE INTERNAL RESISTANCE OFFERED BY THE BODY TO DEFORMATION.

STRESS IS USUALLY MEASURED BY THE INTENSITY OF FORCE PER UNIT AREA.

STRESS = P/A

WHERE P= FORCE

A= AREA OF CROSS SECTION

UNIT OF STRESS: N/m² ,KN/m²,N/mm²

^{WHY STRESS IS EQUAL TO FORCE/AREA?}^{DID YOU THOUGHT ABOUT IT WHY STRESS IS FORCE/AREA,DON'T WORRY I'LL EXPLAIN,THERE ARE SOME ASSUMPTIONS AND THEY ARE}

THE OBJECT IS HOMOGENEOUS
THE OBJECT IS ISOTROPIC AND
THE OBJECT ACTS AS A SINGLE MEMBER AND NOT AS LAYERS.

WHEN A LOAD IS APPLIED THE WHOLE OBJECT UNDERGOES DEFORMATION AND THE VALUE OF DEFORMATION IS SAME AT EVERY POINT ON THE CROSS SECTION OF THE BEAM PERPENDICULAR TO THE LOAD.THIS HAPPENS DUE TO THE LOAD IS DISTRIBUTED TO THE WHOLE AREA EQUALLY.
NOW, 10X = 10KN ( HERE X= UNIT AREA )
1X = ?
FOR LOAD APPLIED ON UNIT AREA BY CROSS MULTIPLYING WE GET = 1X * 10KN /10X = 1KN
THEREFORE STRESS IS EQUAL TO THE LOAD APPLIED ON UNIT AREA.

TYPE OF STRESSES:

1. DIRECT STRESSES

2. SHEAR STRESS/ TANGENTIAL STRESS

3. BENDING STRESS

4. TORSIONAL STRESS

1) DIRECT STRESS /NORMAL STRESS:

THE STRESS DUE TO AN AXIAL FORCE .IT MAY BE EITHER TENSILE OR COMPRESSIVE.

2) SHEAR STRESS:

WHEN THE TWO PARTS EXERT EQUAL AND OPPOSITE FORCE ON EACH OTHER LATERALLY IN A DIRECTION TANGENTIAL TO THEIR SURFACE IN CONTACT, IT IS CALLED TANGENTIAL FORCE. SHEAR STRESS IS THE TANGENTIAL FORCE PER UNIT AREA.

3) BENDING STRESS(FLEXURAL STRESS):

BENDING STRESS IS DUE TO TRANSVERSE LOADS."IN THE ELEMENT SHOWN, THE LAYERS ABOVE N.A ARE SUBJECTED TO FLEXURAL" COMPRESSIVE STRESS AND THE LAYERS BELOW NA ARE SUBJECTED TO FLEXURAL TENSILE STRESS.

BENDING STRESS = M/Z

WHERE M = BENDING MOMENT

Z = SECTION MODULUS

4) TORSIONAL STRESS:

TWISTING MOMENT RESULTS IN SHEAR STRESS KNOWN AS TORSIONAL STRESS.

TORSIONAL STRESS = T/Z_P

WHERE T = TWISTING MOMENT

Z_P = TORSIONAL SECTIONAL MODULUS

STRAIN:

STRAIN IS THE MEASURE OF DEFORMATION CAUSED DUE TO EXTERNAL LOADING.

STRAIN = ΔL /L

WHERE ΔL = CHANGE IN LENGTH

L = ORIGINAL LENGTH

TYPES OF STRAIN:

1. LONGITUDINAL STRAIN:
THE DEFORMATION CAUSED BY NORMAL/DIRECT FORCE IN ITS DIRECTION .IF STRESS IS TENSILE, IT IS CALLED TENSILE STRAIN AND IF STRESS IS COMPRESSIVE, IT IS CALLED COMPRESSIVE STRAIN.

2. SHEAR STRAIN:
IF THE STRESS IS SHEAR, THE CORRESPONDING STRAIN IS KNOWN AS SHEAR STRAIN. IT IS CHANGE IN ANGLE MEASURED IN RADIANS.

3. VOLUMETRIC STRAIN:
IT IS THE RATIO OF CHANGE IN VOLUME TO THE ORIGINAL VOLUME.

RELATION BETWEEN LOAD,STRESS AND STRAIN :

THE BASIC RELATION IS WHEN A EXTERNAL LOAD IS APPLIED STRESS IS PRODUCED AND DUE TO STRESS, STRAIN OCCURS IN THIS WITHOUT EXTERNAL LOAD THERE IS NO STRAIN RIGHT.
BUT I THINK ITS A MISCONCEPTION BECAUSE WHEN A METAL SHAFT IS SUBJECTED TO EXCESSIVE HEATING THE METAL SHAFT STARTS TO EXTEND(DEFORM) DUE TO THE HEATING BUT HERE WE DIDN'T APPLIED A LOAD SO FROM WHERE THE STRAIN IS PRODUCED.

SO HERE WITHOUT LOAD ALSO THE STRESS CAN BE PRODUCED AND IF THERE IS STRAIN THEN DEFINITELY THERE WILL BE STRESS BUT NOT VICE-VERSA.IN THE ABOVE EXAMPLE WHEN WE RESTRICT THE EXTENSION OF SHAFT THERE WILL BE STRESS BUT NO STRAIN.

HOOK'S LAW :

"HOOK'S LAW STATES THAT WITHIN THE ELASTIC LIMITS, STRESS IS PROPORTIONAL TO STRAIN."

STRESS α STRAIN

STRESS = CONSTANT X STRAIN

"CONSTANT IS CALLED MODULUS OF ELASTICITY ( E )"

E = STRESS/STRAIN

STRESS STRAIN CURVE FOR MILD STEEL :

PROPORTIONALITY LIMIT:

"THE STRAIGHT LINE FROM O TO A REPRESENTS EXTENSION PROPORTIONAL TO THE LOAD . IT IS THE LIMIT, THAT EXTENSIONS CEASE TO BE PROPORTIONAL TO THE CORRESPONDING STRESSES. HOOK 'S LAW IS VALID UPTO LIMIT OF PROPORTIONALITY.

ELASTIC LIMIT:

IT IS THE MAXIMUM STRESS THAT MAY BE DEVELOPED WITH NO PERMANENT / RESIDUAL DEFORMATION AFTER THE REMOVAL OF LOAD .

YIELD POINT:

"IF THE MATERIAL IS LOADED BEYOND THE ELASTIC LIMIT, ELONGATION OR YIELDING OF MATERIAL TAKES PLACE WITHOUT ANY CORRESPONDING INCREASE OF LOAD. IT IS CALLED UPPER YIELD POINT.

ELASTIC RANGE:
IT IS THE REGION OF THE STRESS-STRAIN CURVE BETWEEN THE ORIGIN TO THE ELASTIC LIMIT.

PLASTIC RANGE:

IT IS THE REGION OF THE STRESS-STRAIN CURVE BETWEEN THE ELASTIC LIMIT AND THE POINT OF RUPTURE.

STRAIN HARDENING:

FROM THE POINT ‘D’ TO 'E' IN THE GRAPH ,THE PLASTIC DEFORMATION INCREASES WITHOUT ANY INCREASE IN THE STRESS .DUE TO THE PLASTIC REARRANGEMENT OF PARTICLES IN THE MATERIAL, RESISTANCE TO THE STRESS INCREASES.

ULTIMATE STRENGTH:

IT CORRESPONDS TO THE HIGHEST POINT OF THE STRESS STRAIN CURVE.

ULTIMATE STRESS = MAXIMUM LOAD/ ORIGINAL AREA OF CROSS SECTION

UPTO MAXIMUM LOAD , BARS EXTENDS UNIFORMLY OVER ITS LENGTH ,BUT IF STRESS CONTINUES, A LOCAL REDUCTION IN CROSS SECTION OCCURS LEADING TO THE FORMATION OF NECK ZONE."

RUPTURE STRENGTH:

THE RUPTURE STRENGTH IS THE STRESS CORRESPONDING TO THAT AT FAILURE. THE RUPTURE STRENGTH (REPRESENTED BY F) IS LOWER THAN THE ULTIMATE STRENGTH BECAUSE RUPTURE STRENGTH IS CALCULATED BY THE FAILURE LOAD DIVIDED BY ORIGINAL CROSS SECTION AREA.

ACTUALLY DUE TO NECK FORMATION, CROSS SECTIONAL AREA DECREASES AND THE ACTUAL RUPTURE STRENGTH IS MUCH HIGHER THAN NOMINAL RUPTURE STRENGTH.

NOMINAL STRESS: IT IS THE STRESS CORRESPONDING TO THE ORIGINAL CROSS SECTIONAL AREA .

ACTUAL OR TRUE STRESS: IT IS THE STRESS BASED ON ACTUAL AREA OF THE NECKED PORTION .

STRESS-STRAIN CURVE FOR HIGH YIELD DEFORMED BARS:

FOR TORSTEEL AND ALUMINIUM WITH NO WELL DEFINED YIELD POINT, THE YIELD STRENGTH IS DETERMINED BY OFFSET METHOD. A LINE OFFSET OF 0.2% OF STRAIN IS DRAWN PARALLEL TO THE IN INITIAL LINEAR PART OF THE CURVE. THE POINT OF INTERSECTION OF THE LINE WITH THE CURVE CORRESPONDS TO THE YIELD STRENGTH OF THE MATERIAL.

GAUGE LENGTH AND PERCENTAGE ELONGATION:

GAUGE LENGTH:

IT IS THE LENGTH OF TEST PIECE OVER WHICH EXTENSIONS ARE MEASURED IT IS A FUNCTION OF THE CROSS-SECTIONAL AREA OR THE DIAMETER OF THE TEST BAR.

GAUGE LENGTH = 5.65(a^1/2)

"IT IS INDEPENDENT OF LENGTH, SHAPE OF THE BAR AND RATE OF LOADING."

PERCENTAGE ELONGATION:

IT IS THE PERCENTAGE INCREASE IN LENGTH OF THE GAUGE-LENGTH.

PERCENTAGE ELONGATION = [ FINAL LENGTH- ORIGINAL LENGTH /ORIGINAL LENGTH ]X 100

POISSON’S RATIO:

THE RATIO OF LATERAL STRAIN TO LONGITUDINAL STRAIN IS CALLED POISSON'S RATIO. THE RATIO IS CONSTANT FOR A GIVEN MATERIAL.

POISSON'S RATIO (µ) = LATERAL STRAIN / LONGITUDINAL STRAIN

THE VALUE LIES BETWEEN 0.25 TO 0.34 FOR MOST OF THE METALS .

THE MAXIMUM POSSIBLE VALUE OF POISSON’S RATIO IS 0.5.

FOR AN IDEAL ELASTIC IN-COMPRESSIBLE MATERIAL WHOSE VOLUMETRIC STRAIN IS ZERO

POISSON'S RATIO FOR IMPORTANT MATERIALS :

CORK = 0,                      CAST IRON = 0.2 TO 0.3,   BRASS = 0.34,
CONCRETE = 0.1 TO 0.2,            STEEL = 0.27 TO 0.3, GOLD = 0.44
GLASS = 0.2 TO 0.27,   ALUMINIUM = 0.33

INCOMPRESSIBLE MATERIALS, CLAY, PARAFFIN AND RUBBER = 0.5

ELASTIC CONSTANTS :

1) YOUNG’S MODULUS (OR) MODULUS OF ELASTICITY (E) :

RATIO OF DIRECT STRESS TO DIRECT STRAIN WITHIN LIMIT OF PROPORTIONALITY IS CALLED YOUNG’S MODULUS.

FOR, STEEL - 200 GPA ALUMINIUM - 70 GPA COPPER - 120 GPA
DIAMOND - 1200 GPA BRASS - 100 GPA BRONZE - 80 GPA

2)MODULUS OF RIGIDITY(G):

RATIO OF SHEAR STRESS TO SHEAR STRAIN IS CALLED MODULUS OF RIGIDITY.

3)BULK MODULUS ( K ) :

RATIO OF DIRECT STRESS TO VOLUMETRIC STRAIN IS CALLED BULK MODULUS.

RELATION BETWEEN ELASTIC CONSTANTS :

E =2G( 1 + µ) E = 3K(1 - 2µ)

E = 9KG/(3K + G) µ = (3K + 2G) / (6K + 2G)

STRESSES AND DEFORMATIONS:

l) BARS OF UNIFORM CROSS SECTION:

THE TOTAL EXTENSION 'Δ' OF THE BAR AB WILL BE GIVEN BY

Δ = Δ₁ + Δ₂ + Δ₃
= P(L₁+L₂+L₃)/AE

ll) BARS OF VARYING CROSS SECTION:

Δ = Δ₁ + Δ₂ + Δ₃

= P/E(L₁/A₁+L₂/A₂ +L₃/A₃)

TEMPERATURE STRESSES :

WHEN TEMPERATURE OF A MATERIAL CHANGES, THERE WILL BE CHANGE IN DIMENSIONS. IF THESE CHANGES ARE PREVENTED, NO CHANGE IN LENGTH IS PERMISSIBLE AND STRESS ARE DEVELOPED IN THE MATERIAL. THESE STRESSES ARE KNOWN AS TEMPERATURE STRESS.

1) BAR OF UNIFORM SECTION:

IF THE TEMPERATURE IS INCREASED THROUGH T⁰, THE BAR WILL BE INCREASED IN LENGTH BY AN AMOUNT

Δ = L α T [·: Α = CO-EFFICIENT OF THERMAL EXPANSION]

Δ = PL/E

WHERE P = TEMPERATURE STRESS

P = ΔE/L

= L α T. E/L

P =αTE

WHEN THE ENDS YIELD BY AN AMOUNT ‘a’ , THE TOTAL AMOUNT OF DEFORMATION CHECKED IS (Δ – a)

P =(Δ – a)E/L

2) BAR OF COMPOSITE SECTION :
LET US CONSIDER A BAR CONSIST OF TWO DIFFERENT METAL BARS ‘A’ AND ‘B’:

α_A - COEFFICIENT OF LINEAR EXPANSION IN BAR A

α_B - COEFFICIENT OF LINEAR EXPANSION IN BAR B

IF α_A < α_B

DUE TO TEMPERATURE FALL, BAR A – TENSION

BAR B - COMPRESSION

DUE TO TEMPERATURE RISE, BAR A - COMPRESSION

BAR B - TENSION

STRAIN ENERGY:

IF THE ELASTIC LIMIT IS NOT EXCEEDED, THE WORKDONE IN STRAINING THE MATERIAL IS STORED IN IT IN THE FORM OF STRAIN ENERGY. IT IS ALSO CALLED “RESILIENCE".

ENERGY STORED = (P.A.E.L)/2

U = (1/2) X STRESS X STRAIN X VOLUME

U = (P²/2E) X VOLUME

STRAIN ENERGY IS ALWAYS A POSITIVE SCALAR QUANTITY.

PROOF RESILIENCE (U_P):

THE STRAIN ENERGY STORED AT PROPORTIONALITY LIMIT IS CALLED PROOF RESILIENCE.

U_P= (σ²/2E) X V

MODULUS OF RESILIENCE:

THE PROOF RESILIENCE PER UNIT VOLUME. IT IS THE PROPERTY OF MATERIAL.

MODULUS OF RESILIENCE = σ²/2E

STRAIN ENERGY DUE TO DIFFERENT TYPES OF LOADING:

A) GRADUAL LOADING:

LET THE AXIAL LOAD 'P' BE APPLIED GRADUALLY TO A BAR AND 'Δ' BE THE CORRESPONDING DEFORMATION AND 'R' BE THE TOTAL RESISTANCE SETUP OF THE BODY.

WORKDONE BY THE EXTERNAL LOAD = ½ X PΔ
WORKDONE IN THE BODY = ½ x σAΔ

EQUATING THE WORK STORED TO THE WORKDONE WE GET, σ = P/A

THUS, THE MAXIMUM STRESS SETUP IS EQUAL TO LOAD DIVIDED BY AREA OF SECTION OF THE BAR.

B) SUDDEN LOADING:
IF LOAD 'P' IS APPLIED SUDDENLY, THE DEFORMATION INCREASES FROM ZERO VALUE TO ITS FINAL VALUE 'Δ'.

HENCE, THE WORKDONE ON THE BAR = PΔ

THE WORK STORED = ½ x σAΔ

EQUATING THE WORK STORED TO THE WORK SUPPLIED, σ = 2P/A

THUS, THE MAXIMUM STRESS SETUP IN THE BODY IN THIS CASE IS TWICE THAT IN CASE OF GRADUAL LOADING.

C) IMPACT LOADING:

σ = P/A[1+(1+2Eah/PL)^1/2]

HOWEVER, IF Δ IS NEGLIGIBLE COMPARED TO h, THEN WE HAVE

PH = σ²AL/2E

NOTE : DUE TO STATIC LOADING, STRESS IS INDEPENDENT OF E.

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