Beams and columns (machine design & industrial drafting )

machine design & industrial drafting

1. Sub:-MDID
Topic:-Beams and Columns
Branch:-
Mechanical Engg.
( 4th SEM)
Prepared by :-
150990119023 -Bhaumik patel
150990119025 - Harsh patel
150990119026 - Heet patel
150990119027 - Jay .P.patel
150990119028 - Jay.R.patel
Guided by:-
Mr. Rudraduttasinh Parmar
Shroff S.R. Rotary Institute of Chemical Technology
Principle Supporter & Sponsor-United Phosphorous Ltd(UPL)/Shroff family
Managed By Ankleshwar Rotary Education Society
Approved by AICTE, New Delhi, Govt. of Gujarat & GTU Affiliated

2. Introduction to beams and
columns
•Beams:- Beam is a structural element, subjected
to a system of loads at right angle to its axis.
Generally the beam is a horizontal member and
the forces acting on it are vertical.
•Columns:- column is a vertical structural element,
subjected to an axial compressive force. Column is
usually long and there is a possibility to buckling of
the column.
•Struct:- It is same as column, but it usually short
and there is no possibility of buckling.

3. Types of supports of beam
1.Simple support
2.Roller support
3.Hinged (Pin) support
4.Fixed support

4. Types of beam
1.Simply supported beam:- A beam with both hinged or both roller or one
hinged and one roller support is known as simply supported beam.
2.Fixed beam:- A beam with both supports fixed is known as fixed beam.
3.Cantilever beam:- A beam with one end fixed and other is free is known
as cantilever beam.

5. 4.Propped cantilever beam:- A beam with one end fixed and other are simply
supported is known as propped cantilever beam.
5.Continuous beam:- A beam supported at more than two locations is known as
continuous beam.

6. Types of loads on beam
1.Point (concentrated) load
2.Uniform distributed load
3.Varying distributed load
4.Moment or couple
1.Point (concentrated) load:-A point load is a load which is considered as applied on a
point. In reality it is applied on a small area.
2.Uniform distributed load:- It is a load which spread on some length of the beam such
that its intensity is constant throughout its length.

7. 3.Varying distributed load:- It is a load which is spread on some length of the
beam such that its intensity in varying linearly throughout its length.
4.Moment or couple:-A beam may also be subjected to a moment or couple at
some point.

8. Types of failures of beam
1.Bending failure
2.Shear failure
3.Failure due to deflection
1.Bending failure:-The beam subjected to different bending moments at
different sections. At one section the bending moment is maximum and that
section is critical from the point of view of bending failure.

9. 2.Shear failure:- The beam is subjected to unbalanced vertical forces at
different sections. This unbalanced vertical force, which is different at different
sections, is known as shear force. At one section, the shear force is maximum
and that section is critical form the point of view of shear failure.
3.Failure due to deflection:- Due to the forces and moments acting on the
beam, the beam gets deflected. The deflection is different at different sections.
The excessive deflection of the beam beyond the permissible limit, at any
section, may be treated as the failure if the beam.

10. Bending stress in beam
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17. Transverse shear stress
•Beams are subjected to different shear forces at different sections.
•Consider a beam is subjected to a shear forces are F at section A-A, as 
shown in fig, 

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21. DEFLECTION OF BEAM BY
CASTIGLIANO’s METHOD
• When an elastic body is subjected to a force or a system of forces, it 
deforms.
• Castigliano’s theorem :-
•The castigliano’s theorem states that when an elastic body is subjected to a 
system of forces, the deflection at any point and in any direction is equal to 
the partial derivative of the total strain energy with respect to a force at that 
point and acting in that direction.

22. •The terms force and displacement in this statement can also be replaced by 
moment and angular displacement respectively.
                                                                 or 
                          
  where U = total strain energy stored in beam      
1
1
P
U
¶
¶
=d
2
2
P
U
¶
¶
=d i
i
P
U
¶
¶
=d

23. Applications of Castigliano’s theorem :-
1.Castigliano’s theorem is used for determining the deflections of simple 
members as well as complex structures.
2.It can be used for finding the reactions of indeterminate structures.
3.Castigliano’s theorem can also be used for finding the deflection at a point 
where no force is acting/ In such an imaginary force ‘Q’ is applied at a desired 
point and in the desired direction.The next step is to calculate the total strain 
energy in terms of an imaginary force ‘Q’. Then the partial derivative ‘δU/δQ’ is 
obtained and finally ‘Q; is equated to zero to obtain the desired deflection.

24. •STRAIN ENERGY FOR DIFFERENT LOADS :-
1.Axial load :-
•.A rod length ‘L’ and c/s ‘A’ , subjected to an axial force’F’. The strain energy 
stored in an elementary length dx of the rod is given by,
                                        but,
Substituting euation b in a,
 
)...(
2
a
Fd
dU
d
=
)...(b
AE
Fdx
d=d
)...(
2
c
AE
FdxF
dU
×
=

25. Total strain energy stored in the rod is given by,
                       or   
(2) Torsional load :- A rod length ‘L’ and c/s are ‘A’., subjected to a torsional 
moment ‘T’. The strain energy stored in an elementary length ‘dx’ of the rod is 
given by,
 For component subjected to torque,
ò
=
L
dUU
0
ò
=
L
AE
dxF
U
0
2
2
)...(
2
d
T
dU
dq
=
L
G
J
Tq
=
GJ
TL
=\q
)...(e
GJ
Tdx
=\dq

26. •Substituting equation (e) in equation (d),
•Total strain energy stored in the rod is given by,
                                    or,
(3) Bending moment :-
Similarly, the strain energy stored in the rod subjected to bending moment M is 
given by,
When a no. is subjected to multiple loads, the total strain energy is sum of strain 
energy due to each individual load.
GJ
dxT
dU
2
2
=
ò
=
L
dUU
0
ò
=
L
EL
dxM
U
0
2
2

27. Buckling of columns
•Compression failure: 
          A short member loaded in pure compression by force acting along the a 
centroid axis will shorten in accordance with Hook’s law until the stress reaches 
the elastic limit of the material. If the load is increased further, the member bulges 
and is squeezed. This is known as compression failure.
•Buckling failure:
          If the member is sufficiently long, as load increases a stage will reach when 
the slightest lateral displacement results eccentric bending moment greater than 
the internal elastic restoring moment and column collapse. This tyupe of failure is 
called buckling failure and the load at which failure takes place is called crippling or 
buckling or critical load.

28. Compression failure Buckling failure
•When a short member is subjected to a compressive force
along the centroidal axis, its length will reduce in accordance
with the Hooke’s law until the compressive stress induced in
the member reaches the elastic limit of material. If the force is
increased further, there is a plastic deformation. The member
gets bulged and is squeezed. This type of failure is called
Compression failure.
•Compression failure is gradual and it takes place with prior
warning or indication. The beginning of the bulging acts as
prior warning.
•The compressive force induced in compression member at
the instant of compression failure is equal to yield strength in
compression.
•The compression failure is due to yielding or plastic
deformation.
•When a sufficient long member is subjected to a compressive
force, as a force increases, a stage will reach when the
slightest lateral displacement results in eccentric bending
moment greater than the internal elastic restoring moment
and the column collapse. This type of failure is called buckling
failure.
•The buckling failure is very dangerous because there is no
prior warning or indication that the critical load has reached>
the column remains straight until the critical load is reached;
after which there is a sudden and total collapse.
•The compressive stress induced in a column at the instant of
buckling failure is much lower than the yield strength in
compression.
•The buckling is due to elastic instability and not because of
yielding.

29. Euler’s Formula:
•Assumptions made in Euler’s theory:
The column is initially perfectly straight and axially loaded.
The column material is perfectly elastic, homogenous and isotropic and obeys
Hooke’s law.
The colunm is uniform in cross-section.
The length of column is very large compared to lateral dimensions.
The direct stress in column is very small.
The column will fail by buckling alone.
The self-weight of column is neglected.

30. •Euler’s formula:
 The critical load :
Where,

2
2
( / )
cr
e
EA
P
L K
p
=
2
2
E=modulus of elasticity of the column material,N/mm
A=cross-section area of the column,mm
K=least radius of gyration of the cross-section,mm
L effective or equivalent length of the column,mm=
L=actua
e
L
C
=
l length of the column,mm
C=end fixing coefficient,depends on end condition of column

31. •Effective length:
•End condition:
•Slenderness ratio: The ratio of effective length of a column to the least radius of
gyration of a column cross-section is called as slenderness ratio.
•Euler’s formula is applicable on long columns only.
Actual length of column
Effective length of column =
And fixity coefficient
L
C
=
End condition End fixity co-efficient ‘c’ Equivalent Length ‘Le’
Both ends fixed 4 0.5L
One end fixed-One end hinged 2 0.707L
Both end hinged 1 L
One end fixed-One end free 0.25 2L
Le
K
l\=

32. J. B. Johnson’s Formula
•It can applicable only on short column. It is given by
•The length of column is determined by slenderness ratio as:
2
2
2
P A 1-
4
where, S Compressive yield strength, N/mm
e
yc
cr yc
yc
L
S
K
S
C Ep
é ù
æ ö
ç ¸ê ú
è ø
ê ú= ´
ê ú
ê ú
ë û
=
2
/ : Column is long (Eular's formula)
/ : Column is short(Johnson's formula)
2
Where X=
e
e
yc
L K x
L K x
E
S
p
>
<

33. Rankine’s Formula:
•The Euler's formula is applicable only for long column while Johnson’s formula is
most suitable for short and moderate columns.
•Rankine’s formula is suitable for any short or long column given by,
2
2
S A
P
1 ( / )
where,
= Renkine constant
1
= mild steel
7500
1
= for aluminium
5000
yc
cr
e
yc
L K
S
E
for
a
a
p
´
=
+
=