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Find Link in the Description.
 COMPLETE SOM FORMULA REVISION
(1) STRESS & STRAIN (4) TRANSFORMATION & (6) SHEAR CENTRE,
1.1 Stress Strain Curve STRESS SPRING & COLUMN
1.2 Stress Tensor Matrix 4.1 Analytical Method
1.3 Properties OF Materials 4.2 Mohr Circle
1.4 Thermal Stress 4.3 Mohr Circle for Strain (7) PRESSURE VESSELS
1.5 Elastic Constant & Deformation 4.4 Strain Rossette 7.1 Thin shell
1.6 Poisson Ratio
7.2 Thick shell
1.7 Impact Loading
(5) TORSION
5.1 Pure Torsion
(2) SFD & BMD 5.2 Power Transmitted by (8) DEFLECTION
2.1 Significance SHAFT 8.1 Various Method
2.2 Numerical 5.3 Torsion Of bars in Series &
Parallel 8.2 Strain Energy Method

(3) BENDING&SHEAR STRESS

3.1 Flexural Formula
3.2 MOR & Beam
3.3 Shear Stress Variation
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REFRENCE TEXT BOOK:
1: Mechanics of material by gare & timoshenko
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IMPORTANT TOPICS / NOT-IMPORTANT TOPICS / BOOKS

SOM
Books:
➢ Gere & Thimshonke - Concepts
➢ BC Punmia - Numericals
Important Not-Important

• Stress & Strain • Column

• Bending Stresses • Shear Center
• Transformation • Pressure Vessels
• of Stress
• Mohr Circle for Strain
• Slope & Deflection
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OA → Linear curve
A → Proportional limit
B → Elastic limit
C → Upper yield point
D → Lower yield point
DE → Plastic region
EF → Strain hardening region
F → Ultimate stress
G → Fracture point
FG → Necking region
COMPLETE SOM FORMULA REVISION

Type of Tension failure in Metal

(a) Ductile metal (Shear failure)

(b) Brittle metal

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Stress-Strain Diagram for Various type of Steel/Material

All grades of steel have same young’s modulus but different yield
stress
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Stress Strain Curve For Brittle Material

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Stress Strain Curve For Different Grade of Steel

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Matrix Representation of Stress and Strain 3-D stress matrix

3-D strain matrix 3-D strain matrix

ϕxy ϕxz
ϵxx
σxx τxy τxz 2 2
τyx σyy τyz ϕyx ϕyz
ϵyy
τzx τzy σzz 2 2
ϕzx ϕzy
ϵzz
2 2
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Relation between E,G,K,μ

• E = 3K(1 − 2μ)

• E = 2G(1 + μ)

9KG
• E = 3K+G

3K−2G
• μ = 6K+2G

Here, E = Young's modulus, G = shear modulus K = Bulk

modulus, μ = Poisson ratio
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Sign Convention for Strain

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Composite Bars

Let α1 > α2 ∆T↑

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Relation Between Elastic Constants

E = 3k (1 – 2μ)
E = 2G (1 + μ)
9kG
E=
3k + G
3k – 2G 1
= =
6k + 2G m

m = modular ratio
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Poisson Ratio

Lateral strain
=–
Longitudnal strain
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μcork = 0

μconcrete = 0.1 to 0.2

μsteel = 0.25 to 0.3

μaluminium = 0.33

μ cast iron = 0.2 to 0.3

μrubber = 0.5

μPOLYMER & μHUMAN TISSUES = –ve

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Tri-axial loading on Rectangular Parallelopipe

  y z l
x x − − =
E E E l
 y x z b
y − − =
E E E b
   y d
z z − x − =
E E E d

Remember:- Sign convention: Tensile is positive, and

Compressive is negative.
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Volumetric Strain of Cylindrical Bar

∈V = Longitudinal Strain + (2 × Diametric stain)

Volumetric Strain of Sphere
∈V = 3 × Diametric strain
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Impact Loading
Loss of potential energy = Gain in strain Energy
2max
w(h + max ) = (AL)
2E

2
   max
w h + max L = AL
 E  2E

2max wL
AL − max − wh = 0
2E E

2w 2wEh
2max − max − =0
A AL
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Axial elongation (Δ) of prismatic bar due to external load

PL
= Here, P = Load applied
AE
L = Length of bar
A = Area of bar
E = Young modulus
PL P
= = K = AE/L = Axial stiffness of bar
EA K
L AE = Axial rigidity
EI/L = Flexural stiffness
EI = Flexural rigidity
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Deflection of (Δ) due to self-weight
Prismatic bar Conical bar
WL x2 x2 WL
= = = =
2AE 2E 6E 2AE
Here,
Here W = Total Self weight
γ = Specific weight
L =Length of bar
E = Young’s modulus

A = Base area of cone = D2
4

Stress diagram
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Deflection (Δ) Tapered Bar
Circular tapering bar Rectangular tapering bar

=
4PL B 
PL log e  2 
ED1 D2  B1 
=
Where, P = Load applied; E  t(B2 − B1 )
Where, t = thickness;
L = Length of bar
P = Load applied
D1 and D2 are Diameter E = Young modulus
COMPLETE SOM FORMULA REVISION

Equivalent Young’s Modulus of Parallel Composite Bar

A1E1 + A2E2
Eequivalent =
A1 + A 2

Where, A1 = Area of first bar; A2 = Area of second bar;

E1 = Young’s modulus of first Young’s modulus of second bar
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Strain Energy
It is the ability of material to absorb energy when it is strained
1 1
U = P   = T   Here, P = Applied load
2 2
δ = Elongation due to applied load
T = Applied torque
θ = Angle of twist due to applied torque
• Resilience : Ability of a material to absorb energy in the
elastic region when it is strained.
1
= Area under P – δ curve = P  
2
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• Proof Resilience : Maximum energy absorbing capacity
of a material in the elastic region is called proof
resilience.
1
= Area under P - δ curve = PEL × δEL
2
Here,
PEL = Load at elatic limit
δEL = Elongation upto elastic limit
2
Proof Resilience EL
Modulus of Resilience = =
Volume 2E
Here,
σEL = Strain at elastic limit
E = Modulus of elasticity
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Types of Beam
a. Cantilevers beam

b. Simply supported beam

c. Proped cantilever beam

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d. Fixed beam

e. Continuous beam
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dm
• = v (shear force)
dx

• dv
= L. I
dx

• ↑ point Load ⇒ S.F. Jumps

↓ Point load ⇒ SF Falls
• Moment ⇒ BMD Jump
⤹ Moment ⇒ BMD Falls
• Upward load ↑↑↑ +ve
↓↓↓ Loading –ve.
 COMPLETE SOM FORMULA REVISION

(1) STRESS & STRAIN (4) TRANSFORMATION & (7) PRESSURE VESSELS
1.1 Stress Strain Curve STRESS 7.1 Thin shell
1.2 Stress Tensor Matrix 7.2 Thick shell
4.1 Analytical Method
1.3 Properties OF Materials 4.2 Mohr Circle
1.4 Thermal Stress 4.3 Mohr Circle for Strain
1.5 Elastic Constant & Deformation 4.4 Strain Rossette
1.6 Poisson Ratio
1.7 Impact Loading (8) DEFLECTION
8.1 Various Method
(2) SFD & BMD (5) TORSION 8.2 Strain Energy Method
2.1 Significance 5.1 Pure Torsion
2.2 Numerical 5.2 Power Transmitted by SHAFT
5.3 Torsion Of bars in Series & Parallel

(3) BENDING&SHEAR (6) SHEAR CENTRE, SPRING & COLUMN

STRESS
3.1 Flexural Formula
3.2 MOR & Beam
3.3 Shear Stress Variation
COMPLETE SOM FORMULA REVISION

σ m E
= =  Bending or Flexural formula
y I R
COMPLETE SOM FORMULA REVISION

Shear Stress In Beams

S (Ay)
τ=
I.b
COMPLETE SOM FORMULA REVISION

S (Ay)
τ=
I.b

S = S.F. at section XX
Ay̅ = Moment of shaded area about N.A
Ay = Area of cross section above EF
y̅ = Dist. of centroid of area A from N.A
I = MOI about N.A.
b = width of EF where shear stress is required.
 COMPLETE SOM FORMULA REVISION

(1) STRESS & STRAIN (4) TRANSFORMATION (7) PRESSURE VESSELS

1.1 Stress Strain Curve & STRESS 7.1 Thin shell
1.2 Stress Tensor Matrix 4.1 Analytical Method 7.2 Thick shell
1.3 Properties OF Materials 4.2 Mohr Circle
1.4 Thermal Stress 4.3 Mohr Circle for Strain
1.5 Elastic Constant & Deformation 4.4 Strain Rossette
1.6 Poisson Ratio
1.7 Impact Loading

(5) TORSION (8) DEFLECTION

(2) SFD & BMD 8.1 Various Method
2.1 Significance 5.1 Pure Torsion
8.2 Strain Energy Method
2.2 Numerical 5.2 Power Transmitted by SHAFT
5.3 Torsion Of bars in Series & Parallel

(3) BENDING&SHEAR (6) SHEAR CENTRE, SPRING & COLUMN

STRESS
3.1 Flexural Formula
3.2 MOR & Beam
3.3 Shear Stress Variation
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σx + σy σx − σy
σx′ = + cos 2 θ + τxy sin 2 θ
2 2
σx + σy σx − σy
σy′ = − cos 2 θ − τxy sin 2 θ
2 2
σx − σy
τx′y′ = − sin 2 θ + τxy cos 2 θ
2
COMPLETE SOM FORMULA REVISION

σx + σy σx − σy 2 σ1 + σ2
2
σ1 /σ2 = ± + τxy a=
2 2 2
Radius of Mohr's circle
σx − σy 2 σ1 − σ2
r= + τ2xy =
2 2
COMPLETE SOM FORMULA REVISION

σx + σy σx − σy
σx′ = + cos 2 θ + τxy sin 2 θ
2 2
σx + σy σx − σy
σy′ = − cos 2 θ − τxy sin 2 θ
2 2
σx − σy
τx′y′ = − sin 2 θ + τxy cos 2 θ
2
COMPLETE SOM FORMULA REVISION

σx + σy σx − σy 2 σ1 + σ2
2
σ1 /σ2 = ± + τxy a=
2 2 2
Radius of Mohr's circle
σx − σy 2 σ1 − σ2
r= + τ2xy =
2 2
 COMPLETE SOM FORMULA REVISION

(1) STRESS & STRAIN (4) TRANSFORMATION & (7) PRESSURE VESSELS
1.1 Stress Strain Curve STRESS 7.1 Thin shell
1.2 Stress Tensor Matrix 7.2 Thick shell
4.1 Analytical Method
1.3 Properties OF Materials 4.2 Mohr Circle
1.4 Thermal Stress 4.3 Mohr Circle for Strain
1.5 Elastic Constant & Deformation 4.4 Strain Rossette
1.6 Poisson Ratio
1.7 Impact Loading (8) DEFLECTION
8.1 Various Method
(2) SFD & BMD (5) TORSION 8.2 Strain Energy Method
2.1 Significance 5.1 Pure Torsion
2.2 Numerical 5.2 Power Transmitted by SHAFT
5.3 Torsion Of bars in Series & Parallel

(3) BENDING&SHEAR (6) SHEAR CENTRE, SPRING & COLUMN

STRESS
3.1 Flexural Formula
3.2 MOR & Beam
3.3 Shear Stress Variation
COMPLETE SOM FORMULA REVISION

Torsion
COMPLETE SOM FORMULA REVISION
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Torsional Formula
τ T Gϕ
= =
r Ip L
COMPLETE SOM FORMULA REVISION

Power transmitted by shaft = P

P=Tω
T = Torque
ω = Angular speed.
 COMPLETE SOM FORMULA REVISION

(1) STRESS & STRAIN (4) TRANSFORMATION & (7) PRESSURE VESSELS
1.1 Stress Strain Curve STRESS 7.1 Thin shell
1.2 Stress Tensor Matrix 7.2 Thick shell
4.1 Analytical Method
1.3 Properties OF Materials 4.2 Mohr Circle
1.4 Thermal Stress 4.3 Mohr Circle for Strain
1.5 Elastic Constant & Deformation 4.4 Strain Rossette
1.6 Poisson Ratio
1.7 Impact Loading (8) DEFLECTION
8.1 Various Method
(2) SFD & BMD (5) TORSION 8.2 Strain Energy Method
2.1 Significance 5.1 Pure Torsion
2.2 Numerical 5.2 Power Transmitted by SHAFT
5.3 Torsion Of bars in Series & Parallel

(3) BENDING&SHEAR (6) SHEAR CENTRE, SPRING & COLUMN

STRESS
3.1 Flexural Formula
3.2 MOR & Beam
3.3 Shear Stress Variation
COMPLETE SOM FORMULA REVISION

6. Shear Centre, Spring & Column

COMPLETE SOM FORMULA REVISION
Deflection of Bream Under Different Loading/Support Condition

Cantilever Beam Slope at B w.r.t A Deflection at B

(θBA) w.r.t. A (ΔBA)

MI MI2
EI 2EI

WI2 WI3
2EI 3EI

WI3 WI4
6EI 8EI
COMPLETE SOM FORMULA REVISION

Cantilever Beam Slope at B w.r.t A Deflection at B

(θBA) w.r.t. A (ΔBA)

WI3 WI4
24EI 30EI

WI3 WI3 WI4 WI4

Slope (BA ) = − Deflection (BA ) = −
6EI 24EI 8EI 30EI
COMPLETE SOM FORMULA REVISION

Simply Supported Slope at C w.r.t Delfection at C

Beam A (θCA) w.r.t. A
(ΔCA = Δmax)

WI2 WI3
16EI 48EI

WI3 5WI4
24EI 384EI
COMPLETE SOM FORMULA REVISION

Fixed Beam Slope Deflection at C

w.r.t A (ΔCA = Δmax)
θA = θ B = 0 WI4
384EI
1
=  max in SS beam
4
θA = θ B = 0 WI4
384EI
1
=  max in SS beam
5
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Analysis of thin cylinder

pd
• Longitudinal Stress L =
4t
pd
• Hoop Stress h =
2t pd
• 
Longitudinal Strain L = (1 − 2)
4t E
pd
• Hoop Strain L = (2 − )
4t E
COMPLETE SOM FORMULA REVISION

Here, p = Pressure of fluid, t = Thickness of cylinder

d = inside diameter, μ = Poision’s ratio

h 2 − 
• Ratio of Hoop Strain to Longitudinal Strain =
L 1 − 2
pd
• Volumetric Strain (∈y) of Cylinder y = (5 − 4)
4tE
COMPLETE SOM FORMULA REVISION

Analysis of thin sphere

pd
• Hoop stress/longitudinal stress; L = h =
4t
pd
• Hoop strain/longitudinal strain; L =h = (1 − )
4t E

3pd
• Volumetric strain of sphere; v = (1 − )
4t E
COMPLETE SOM FORMULA REVISION
Analysis of Thick Cylinder/Lame’s Theorem
• Lame’s Assumption
(i) Material of shell is homeogeneous, isotropic and
line elastic.
(ii) Plane section of cylinder, perpendicular to
longitudinal axis remains. Plane under pressure.
• Lame’s equaitons

B
(i) Hoop stress : x = + A(tensile)
2
x
B
(ii) Radial stress Px = 2
− A(Compressive)
x
Where, B and A are Lame’s constant
COMPLETE SOM FORMULA REVISION

Analysis of Thick spheres

• Lame’s equations:

2B 2B
x = 3
+ A (Tensile) Px = 3
− A (Compressive)
x x
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Euler’s load for different column with different end Condition

End Both end One end Both end One end

condition hinged fixed other fixed fixed and
free other
hinged
Effective L 2L L/2
L
length (le)
2
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Spring
CLOSED COILED HELICAL SPRING (CCHS)
N = no. of turns
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Strain Energy Stored in CCHS [U]

32 P2R 3  n
U=
Gd4
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• Spring stores energy in the form of resilience

Series and parallel arrangement of springs/Equivalent spring

constant (keq)
1 1 1 1
• InSeries : = + + ....
keq k1 k2 kn
• In parallel: k eq = k1 + k2 + .....kn
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Closed coil helical spring under axial pull

16PR
(i) max =
d3
T2L 32P2R3n
(ii) Strain energy stored in spring, U = =
2Gl P Gd4
U 64PR3n
(iii) Axial deflection under load P  ==
P Gd4
P Gd4 1
(iv) Coefficient of stiffness of spring (k) k = = k
 64R3n n
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Remember

actual buckling load.

2
Pe = Buckling load
 EImin
Pe = Imin = Min. Moment of inertia about centroidal axis
l2e
le = Effective lenglh
COMPLETE SOM FORMULA REVISION

Remembers

• It is applicable for long column. Effect of crushing is

neglected.

Column Fails in
1. Short column Crushing
2. Long column Buckling
3. Intermediate column Combined Crushing and
Buckling
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Euler's load for different column with different end Condition

One end
One end
End fixed
Both end fixed Both end
conditio and
hinged other fixed
n other
free
hinged
Effective L L
length L 2L
2 2
𝑙e

• Spring stores energy in the form of resilience.

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Series and parallel arrangement of springs/Equivalent

spring constant (keq)

1 1 1 1
• In Series: k = k + k + ⋯k
eq 1 2 n

• In parallel: k eq = k1 + k 2 + ⋯ k n
 Join my Telegram Group

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Abhinav Sir

• For daily dose to improve concepts

• For daily important questions
• For latest information about exams
• For notification of the class
Civil • For important PDFs & notes
•
By Abhinav Sir For clearing the doubts by experts