3-Strength of Materials formulaws.pdf

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1. 2
–”‡‰–Š ‘ˆ
Material
‘”—Žƒ¬Š‘”– Notes)

2. 3
–” ‡••ƒ† strain
Stress = Force / A rea
T e n s io n s tr a in ( e )
L C h a n g e in le n g th
L In itial length

3. 4
”‹‡ŽŽ ƒ”† ‡••—„‡”
(BHN)
Žƒ•– ‹……‘•– ƒ–•ã
where, P = Standard load, D = Diameter of steel ball, an d d = Diameter of the indent.

4. 5
š‹ƒŽŽ‘‰ƒ– ‹‘‘ˆ ƒ””‹•ƒ– ‹…ƒ”—‡–‘š –‡”ƒŽ‘ƒ†
Ž‘‰ƒ–‹‘‘ˆ ”‹• ƒ–‹… ƒ” —‡ –‘ ‡Žˆ‡‹‰Š–
Where is specific weight
Ž‘‰ƒ–‹‘‘ˆ ƒ’‡ ”‡ † ƒ”
‹”…—Žƒ” ƒ’‡”‡ †
‡…–ƒ‰—Žƒ”ƒ’‡”‡†
–”‡••†—…‡†„›š ‹ƒŽ– ”‡••ƒ†‹’Ž‡Š‡ƒ”
‘”ƒŽ•– ”‡••
ƒ‰‡ –‹ƒŽ •–”‡ ••
Principal Stresses and Principal Plan es
Major principal stress
Major principal stress

5. 6
” ‹…‹’ƒŽStrain
-

‡”‰›‡–Š‘†•ã
(i) ‘”—Žƒ–‘…ƒŽ…—Žƒ– ‡–Š‡•–”ƒ‹‡‡”‰›†—‡–‘ƒš ‹ƒŽŽ‘ƒ†•– ‡•‹‘ã
limit 0 toL
Where, P = Applied tensile load, L = Length of the member , A = Area of the memb er, and
(ii) ‘”—Žƒ–‘…ƒŽ…—Žƒ– ‡–Š‡•–”ƒ‹‡‡”‰›†—‡–‘„‡†‹‰ã
limit 0 toL
inertia.
(iii) ‘”—Žƒ–‘…ƒŽ…—Žƒ– ‡–Š‡•–”ƒ‹‡‡”‰›†—‡–‘– ‘”•‹‘ ã
limit 0 toL

6. 7
Where, T = Applied Torsion , G = Shear modulus or Modulus of rigidit y, and J = P olar
moment ofinertia
(iv) ‘”—Žƒ–‘…ƒŽ…—Žƒ– ‡–Š‡•–”ƒ‹‡‡”‰›†—‡–‘’—”‡•Š‡ƒ”ã
U =K limit 0 to L
Where, V= Shearload
G = Shear modulus or Modulus of rigidity
A = Area of cross section.
K = Constant depends upon shape of cross section.
(v) ‘”—Žƒ–‘…ƒŽ…—Žƒ– ‡–Š‡•–”ƒ‹‡‡”‰›†—‡–‘’—”‡•Š‡ƒ”ወ •Š‡ƒ”•–”‡••‹•‰‹˜‡ã
Where,
G = Shear modulus or Modulus of rigidity V = Volume of the material.
(vi) ‘”—Žƒ–‘…ƒŽ…—Žƒ– ‡–Š‡•–”ƒ‹‡‡”‰›á‹ˆ –Š‡‘‡– ˜ƒŽ—‡‹•‰‹˜‡ã
U = M ² L / (2EI)
Where, M = Bending moment L = Length of the b eam
I = Moment of inertia
(vii) ‘”—Žƒ–‘…ƒŽ…—Žƒ– ‡–Š‡•–”ƒ‹‡‡”‰›á‹ˆ –Š‡–‘”•‹‘ ‘‡– ˜ƒŽ—‡‹•‰‹˜‡ã
U= T ²L / ( 2GJ)
Where, T = AppliedTorsion
L = Length of the b eam
G = Shear modulus or Modulus of rigidity
J = Polar moment of inertia

7. 8
(viii) ‘”—Žƒ–‘…ƒŽ…—Žƒ– ‡–Š‡•–”ƒ‹‡‡”‰›á‹ˆ –Š‡ƒ’’Ž‹‡†–‡•‹‘Ž‘ƒ†‹•‰‹˜‡ã
U = P² L / ( 2AE )
Where,
P = Applied tensile load.
L = Length of the member
A = Area of the member
(ix) theorem:
(x) ‘”—Žƒˆ‘”†‡ˆŽ‡…–‹‘‘ˆ ƒˆ‹š‡†„‡ƒ™‹–Š’‘‹–Ž‘ƒ†ƒ– centr e:
= - wl
3
/ 192EI
This def ection is ¼ times the deflection of a simply supported beam.
(x i) ‘”—Žƒˆ‘”†‡ˆŽ‡…–‹‘‘ˆ ƒˆ‹š‡†„‡ƒ™‹–Š—‹ˆ ‘”Ž›†‹•– ”‹„—–‡† load:
= - wl
4
/ 384EI
This def ection is 5 times the deflection of a simply supported beam.
(x ii) ‘”—Žƒˆ‘”†‡ˆŽ‡…–‹‘‘ˆ ƒˆ‹š‡†„‡ƒ™‹–Š‡……‡–”‹…’‘‹– load:
= - wa
3
b
3
/ 3 EIl
3
Stresse s due to
Gr adual Loading : -
Sudden Loading: -

8. 9
Impac t Loading : -
Deflection ,
Thermal Stresses: -
‡’‡”ƒ– —”‡ S –” ‡••‡•‹ T ƒ’‡” B ars: -
‡’‡”– ƒ—”‡–”‡••‡•‹‘’‘•‹–‡ƒ”•
‘‘‡5•ƒ™‹‡ƒ”‡Žƒ•–‹…‹–›ã
Hooke's Law stated that within elastic limit, the linear relationship between simple
stress and strain for a b ar is expressed by equations.

9. 10
,
E
Where, E = Young's modulus of elasticity
P = Applied load across a cross - sectional area
Ž = Change in length
Ž= Original length
‘Ž—‡– ”‹…– ”ƒ‹ã

10. 11
‡Žƒ–‹‘ •Š‹’„‡–™‡‡áá K and µ :
‘†—Ž—•‘ˆ rigidit y: -
Bulk modulus: -
Comp ou nd S tresses
“—ƒ– ‹‘‘ˆ —”‡‡†‹‰
‡…–‹‘‘†—Ž—•
Š‡ƒ”‹‰–”‡••
Where,
V = Shearin g forc e
= First moment of a rea
Š‡ƒ”
S –” ‡••
‹
Rectang
—Žƒ”
B eam

11. 12
Š‡ƒ”– ”‡••‹”…—Žƒ”‡ƒ
‘‡–‘ˆ ‡”– ‹ƒ ƒ† ‡… –‹‘ ‘†—Ž— •

12. 13
Direct Stress
where P = axial thrust, A = area of cros s - section
Bend ing Stress
where M = bending moment, y - distan ce of fibre from n eutral axis, I =
moment of inertia.
Torsional Shear Stress
where T = torque, r = ra di us of s haft, J = polar moment of in ertia.
Eq uivalent Tors ional Moment
Eq uivalent Bending Moment

13. 14
Sh ear force and B ending Momen t Relation

14. 15
For both end hin ged ± l
For one en d fixed and other free ±t l
For both end f ixed ± l /2
For one en d fixed and other hi nged ± l /
Slend erness Rati o ( )

15. 16
P
R
=
P
cs

cs
A = Ultimat e crushing l oad for column
‡ˆŽ‡…–‹‘‹†‹ˆ ˆ‡”‡– ‡ƒ•
Torsion
W here, T = Torque,
J = Polar moment of inertia
G = Modulus of ri gidi ty,
= Angle of twis t
L = Length of shaft,

16. 17
‘–ƒ Žƒ ‰Ž‡‘ˆ –™‹•–
GJ = Torsi onal rigidity
= Torsional stiffn ess
= Torsional flexibility
= Axial sti ffn ess
= Axial flexibility
‘ ‡ –‘ˆ ‡”–‹ƒ„‘ —–’‘ Žƒ” š‹•
Moment of Inertia About polar Axis
For hollow ci rcular shaf t
‘ ’‘—†Šƒˆ–
‡”‹‡•…‘ ‡…–‹‘
W here,
1
= Angular deformation of 1
st
shaft
2
= Angular deformation of 2
nd
shaft
ƒ”ƒŽŽ‡Ž‘‡…–‹‘

17. 18
–”ƒ‹ ‡”‰›‹ ‘”•‹‘
For soli d shaf t,
For hollow shaft,
Š‹›Ž‹ †‡”
Circu mferential Stres s /Hoop Stres s
Longitudinal Stres s
Hoop Strain
Longitudinal Strain
Ratio of Hoop S train to Longitu di nal Strain

18. 19
–”‡••‡•‹Š‹ ’Š‡”‹…ƒŽ Š‡ŽŽ
Hoop stress /lon gitud in al s tress
Hoop stres s/longitu din al strain
Vol ume tric s train of s ph ere
Th ickness ratio of Cylind rical Shell w ith Hemis phere E nd s
W here v = Poiss on R atio