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Strength of
Material (Formula & Short Notes)
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Stress and strain
Stress = Force / Area
( )t
L ChangeinlengthTensionstrain e
L Initial length
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Brinell Hardness Number (BHN)
2 2( )2
P
DD D d
where, P = Standard load, D = Diameter of steel ball, and d = Diameter of the indent.
Elastic constants:
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STRAIN ENERGY Energy Methods:
(i) Formula to calculate the strain energy due to axial loads ( tension):
U = ∫ P ² / ( 2AE ) dx limit 0 to L
Where, P = Applied tensile load, L = Length of the member , A = Area of the member, and
E = Young’s modulus.
(ii) Formula to calculate the strain energy due to bending:
U = ∫ M ² / ( 2EI ) dx limit 0 to L
Where, M = Bending moment due to applied loads, E = Young’s modulus, and I = Moment of
inertia.
(iii) Formula to calculate the strain energy due to torsion:
U = ∫ T ² / ( 2GJ ) dx limit 0 to L
Where, T = Applied Torsion , G = Shear modulus or Modulus of rigidity, and J = Polar
moment of inertia
(iv) Formula to calculate the strain energy due to pure shear:
U =K ∫ V ² / ( 2GA ) dx limit 0 to L
Where, V= Shear load
G = Shear modulus or Modulus of rigidity
A = Area of cross section.
K = Constant depends upon shape of cross section.
(v) Formula to calculate the strain energy due to pure shear, if shear stress is given:
U = τ ² V / ( 2G )
Where, τ = Shear Stress
G = Shear modulus or Modulus of rigidity
V = Volume of the material.
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(vi) Formula to calculate the strain energy , if the moment value is given:
U = M ² L / (2EI)
Where, M = Bending moment
L = Length of the beam
E = Young’s modulus
I = Moment of inertia
(vii) Formula to calculate the strain energy , if the torsion moment value is given:
U = T ²L / ( 2GJ )
Where, T = Applied Torsion
L = Length of the beam
G = Shear modulus or Modulus of rigidity
J = Polar moment of inertia
(viii) Formula to calculate the strain energy, if the applied tension load is given:
U = P²L / ( 2AE )
Where,
P = Applied tensile load.
L = Length of the member
A = Area of the member
E = Young’s modulus.
(ix) Castigliano’s first theorem:
δ = Ә U/ Ә P
Where, δ = Deflection, U= Strain Energy stored, and P = Load
(x) Formula for deflection of a fixed beam with point load at centre:
= - wl3 / 192 EI
This defection is ¼ times the deflection of a simply supported beam.
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(xi) Formula for deflection of a fixed beam with uniformly distributed load:
= - wl4 / 384 EI
This defection is 5 times the deflection of a simply supported beam.
(xii) Formula for deflection of a fixed beam with eccentric point load:
= - wa3b3 / 3 EI l3
Fixed end moments for a fixed beam with the given loading conditions:
Type of loading (A--B) MAB MBA
-wl / 8 wl / 8
-wab2/ l2 wab2/ l2
-wl2 / 12 wl2 / 12
-wa2 (6l2 – 8la + 3a2)/
12 l2
-wa2 (4l-3a)/ 12 l2
-wl2 / 30
-wl2 / 30
-5 wl2/ 96
-5 wl2/ 96
M / 4
M / 4
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Euler’s formula for different end conditions:
1. Both ends fixed:
PE = л 2 EI / ( 0.5L)2
2. Both ends hinged :
PE = л 2 EI / (L)2
3. One end fixed ,other end hinged:
PE = л 2 EI / ( 0.7L)2
4. One end fixed, other end free:
PE = л 2 EI / ( 2L)2 where L = Length of the column
Rakine’s formula:
PR = f C A / (1+ a (l eff / r)2 )
where, PR = Rakine’s critical load
fC = yield stress
A = cross sectional area
a = Rakine’s constant
leff = effective length
r = radius of gyration
Euler’s formula for maximum stress for ‘a’ initially bent column:
σmax = P /A + ( Mmax / Z )= P/ A + P a / ( 1- ( P / PE ))Z
Where, P = axial load
A = cross section area
PE = Euler’s load
a = constant
Z = section modulus
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Euler’s formula for maximum stress for a eccentrically loaded column:
σmax = P /A+( M max /Z) = P/A + ( P e Sec(leff /2 ) √ (P/EI) )/((1- (P / PE )) Z )
Where, P = axial load
A = cross section area
PE = Euler’s load
e = eccentricity
Z = section modulus
EI = flexural rigidity
General expressions for the maximum bending moment, if the deflection curve
equation is given:
BM = - EI ( d 2y / dx 2 )
Maximum Principal Stress Theory ( Rakine’s theory):
σ 1 = f y.
where σ 1 is the maximum Principal Stress, and f y is elastic limit stress.
Maximum Principal Strain Theory ( St. Venant’s theory):
e 1 = f y / E
In 3D, e 1 = 1/E[ σ 1 – (1/m)( σ 2 + σ 3) ] = f y / E → [ σ 1 – (1/m)( σ 2 + σ 3) ] = f y
In 2D, σ 3 = 0 → e 1 = 1/E[ σ 1 – (1/m)( σ 2 ) ] = f y / E → [ σ 1 – (1/m)( σ 2 ) ] = f y
Maximum Shear Stress Theory (Tresca’s theory) :
In 3D, ( σ 1 - σ 3) / 2 = f y /2 → ( σ 1 - σ 3) = f y
In 2D, ( σ 1 - σ 2) / 2 = f y /2 → σ 1 = f y
Maximum Shear Strain Theory (Von –Mises- Hencky theory or Distortion energy
theory):
In 3D, shear strain energy due to distortion:
U = (1/ 12G)[ ( σ 1 - σ 2)2 + ( σ 2 - σ 3) 2 + ( σ 3 - σ 1) 2 ]
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Shear strain energy due to simple tension:
U = f y 2 / 6G
(1/ 12G)[ ( σ 1 - σ 2)2 + ( σ 2 - σ 3) 2 + ( σ 3 - σ 1) 2 ] = f y 2 / 6G
[ ( σ 1 - σ 2)2 + ( σ 2 - σ 3) 2 + ( σ 3 - σ 1) 2 ] = 2 f y 2
In 2D, [ ( σ 1 - σ 2)2 + ( σ 2 - 0) 2 + ( 0 - σ 1) 2 ] = 2 f y 2
Maximum Strain Energy Theory (Beltrami Theory):
In 3D, strain energy due to deformation:
U = (1/ 2E)[ σ 12 + σ 22 + σ 32 -(1/m)( σ 1σ 2 + σ 2σ 2 + σ 2σ 2 )]
Strain energy due to simple tension:
U = f y 2 / 2E
(1/ 2E)[σ 12 + σ 22 + σ 32 -(2/m)( σ 1σ 2 + σ 2σ 2 + σ 2σ 2 )] = f y 2 / 2E
[σ 12 + σ 22 + σ 32 -(2/m)( σ 1σ 2 + σ 2σ 2 + σ 2σ 2 )] = f y 2
In 2D, [ σ 12 + σ 22 - (2/m)( σ 1σ 2 )] = f y 2
Failure theories and its relationship between tension and shear:
1. Maximum Principal Stress Theory ( Rakine’s theory):
ζ y = f y
2. Maximum Principal Strain Theory( St. Venant’s theory):
ζ y = 0.8 f y
3. Maximum Shear Stress Theory ( Tresca’s theory):
ζ y =0.5 f y
4. Maximum Shear Strain Theory ( Von Mises Hencky theory or Distortion energy
theory):
ζ y= 0.577 f y
4. Maximum Strain Energy Theory ( Beltrami Theory):
ζ y= 0.817f y .
Volumetric strain per unit volume:
f y 2 / 2E
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Torque, Power, and Torsion of Circular Bars:
Relation between torque, power and speed of a rotating shaft:
63000
TnH
Where H is power in Hp, T is torque in lb-in, and n is shaft speed in rpm.
In SI units:
TH
Where H is power in Watts, T is torque in N-m, and is shaft speed in rad/s.
The shear stress in a solid or tubular round shaft under a torque:
The shear stress:
J
Tr
J is the area polar moment of inertia and for a solid (di=0) or hollow section,
)(32
44
io ddJ
The angle of rotation of a shaft under torque:
GJ
TL
Axial deflection of a bar due to axial loading
The spring constant is:
L
EAK
Lateral deflection of a beam under bending load:
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3
48
L
EIK
For cantilevered beams of length L:
3
3
L
EIK
Torsional stiffness of a solid or tubular bar is:
L
GJK t
The units are pounds per radians.
Load Distribution between parallel members:
If a load (a force or force couple) is applied to two members in parallel, each member takes
a load that is proportional to its stiffness.
K2 K1
F
T Kt1
Kt2
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The force F is divided between the two members as:
FKK
KFF
KK
KF
21
22
21
11
The torque T is divided between the two bars as:
TKK
KTT
KK
KT
tt
t
tt
t
21
22
21
11
Direct shear stress in pins:
A
F
2
The clevis is also under tear-out shear stress as shown in the following figure (top view):
Tear-out shear stress is:
A
F
4
t
F F
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In this formula A= (Ro-Ri) is approximately and conservatively the area of the dotted
cross-section. Ro and Ri are the outer and inner radii of the clevis hole. Note that there are
4 such areas.
Shear stresses in beams under bending forces:
bI
VQ
Z
11yAQ
Torsion of Thin-walled Tubes:
F
Y
y1
b
A1
y1
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Shear stress:
At
T
2
GtA
TSL24
Where S is the perimeter of the midline, L is the length of the beam, and G is shear modulus.
Stress in Thin-Walled Cylinders
The tangential or hoop stress is:
t
Pdit
2
The axial stress is:
t
Pdia
4
Stresses in Thick-walled Cylinders
The tangential stress:
22
2
2222
io
iooiooii
trr
r
PPrrrPrP
The radial stress is:
22
2
2222
io
iooiooii
rrr
r
PPrrrPrP
When the ends are closed, the external pressure is often zero and the axial stress is:
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22
2
io
iia
rr
rP
Stresses in rotating rings
)3
31)(
8
3( 2
2
22222 r
r
rrrr oioit
))(8
3( 2
2
22222 r
r
rrrr oioir
where is the mass density and is the Poisson’s ratio.
Interface pressure as a result of shrink or press fits
The interface pressure for same material cylinders with interface nominal radius of R and
inner and outer radii of ri and ro:
)(2
))((222
2222
io
ior
rrR
rRRr
R
EP
Impact Forces
For the falling weight:
Wh
F
WW
hkF
st
e
e
211
211
IF h=0, the equivalent load is 2W. For a moving body with a velocity of V before impact, the
equivalent force is:
mkVFe
Failure of columns under compressive load (Buckling)
The critical Euler load for a beam that is long enough is:
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2
2
L
EICPcr
C is the end-condition number.
The following end-condition numbers should be used for given cases:
When both end are free to pivot use C=1. When one end is fixed (prevented from rotation and lateral movement) and the
other is free, use C= 1/4 . When one end is fixed and the other end can pivot, use C=2 when the fixed end is
truly fixed in concrete. If the fixed end is attached to structures that might flex under load, use C=1.2 (recommended).
When both ends are fixed (prevented from rotation and lateral movement), use C=4. Again, a value of C=1.2 is recommended when there is any chance for pivoting.
Slenderness ratio:
An alternate but common form of the Euler formula uses the slenderness ratio which is
defined as follows:
A
Ikwhere
k
LRatiosSlendernes
Where k is the area radius of gyration of the cross-sections.
Range of validity of the Euler formula
Euler formula is a good predictor of column failure when:
yS
EC
k
L 22
If the slenderness ratio is less than the value in the RHS of the formula, then the better
predictor of failure is the Johnson formula:
CEk
LSSAP
y
ycr
1
2
2
Determinate Beams Equations of pure bending:
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Where,
M: Bending Moment [N*m]
σ: normal stress [N/m2]
E: Modulus of elasticity [N/m2]
R: Radius of Curvature [m]
y: Distance from neutral surface [m]
I: Moment of inertia [m4]
Indeterminate Beams
Macaulay’s Method (Singularity functions):
<x-a>ndx=1
n+1<x-a>
n+1
x>a
If positive then the brackets (< >) can be replaced by parentheses. Otherwise the
brackets will be equal to ZERO.
0<x<a<x-a>n= 0
x>a<x-a>n= (x-a)
n
Hooke's Law (Linear elasticity):
Hooke's Law stated that within elastic limit, the linear relationship between simple
stress and strain for a bar is expressed by equations.
M
I=
E
R=σ
y
E Id y
2
dx2 = M
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,
E
P lE
A l
Where, E = Young's modulus of elasticity
P = Applied load across a cross-sectional area
l = Change in length
l = Original length
Poisson’s Ratio:
Volumetric Strain:
V
Changeinvolume Ve
Initial volume V
Relationship between E, G, K and :
Modulus of rigidity:
2(1 )
EG
Bulk modulus:
9
3(1 2 ) 3
E KGK or E
K G
3 2
6 2
K G
K G
Stresses in Thin Cylindrical Shell
Circumferential stress (hoop stress)
2 2
c c
pd pd
t t
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Where, p = Intensity of internal pressure
d = Diameter of the shell
t = Thickness of shell
η = Efficiency of joint
Longitudinal stress
4 4
l l
pd pd
t t