Lec 3 BeamStress

7/3/2023
AHSANULLAH UNIVERSITY OF SCIENCE

AND TECHNOLOGY (AUST)
ME-2201: MECHANICS OF MATERIALS
BY
ENGR. FAZLAR RAHMAN

Associate Professor
ME Faculty, AUST
(LEC-3)
ME-2201: Mechanics of Materials

BENDING STRESS IN BEAM
Assumptions:
Pure bending (though effects of axial, torsional, and shear
loads are often assumed to have minimal effect on
bending stress of beam).
Material is isotropic and
homogeneous.
Material obeys Hooke’s law.
Beam is initially straight with
constant cross section.
Beam has axis of symmetry in the plane of bending.
Failure of beam is only by bending rather than crushing,
wrinkling, or sidewise buckling.
Plane cross sections remain in plane during bending.
Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3 2
Fazlar Rahman; Asst. Prof.; MPE; AUST; ME-

2201; LEC-3 1
7/3/2023

o Consider a straight portion of a beam subject to positive
bending moment ‘M’. Y-axis is the axis of symmetry and
X-axis is coincident with neutral axis of the section, XZ-
plane is the neutral plane. All beam element coincident
with neutral plane have zero strain.
o Because of curvature, a section AB is originally parallel to
line CD since beam was straight the line AB is rotate
position A’B’ through an angle dφ.
o Assume ρ is the radius of curvature of the neutral axis
and length of differential element of neutral axis is ds and
y is the distance of a elongated fiber from neutral axis.
o Angle between line CD and A’B’ is dφ.
Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3
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BENDING OF BEAM IN TWO PLANE

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PROPERTIES OF CROSS SECTION (SECTION PROPERTIES)
Section Properties:
After determining internal load, each structure or
member should be analyzed for its strength and rigidity.
In stress analysis, fundamental equations relate the
strength & rigidity of the structure with its shape
characteristic.
The shape characteristics which are associated with the
cross section of the member are known as section
properties.
The section properties which are commonly used in
design calculations are the following:

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Section Properties (continue):
o Area (A, in2 or mm2)
o Centroid or neutral axis (N.A ,in or mm)
o Moment of inertia (I, in4 or mm4)
o Polar moment of inertia (J, in4 or mm4 ) and
o Radius of gyration (ρ, in or mm).
Centroid:
o Area:
o Line:

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Moment of Inertia:
The second moment of area about an axis is known
as moment of inertia of plane area or area moment of
inertia or second area moment.
It is a geometrical property of an area and denoted by ‘I’.
It is used to determined the bending stress in a beam and
also design of column.

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Polar Moment of Inertia:
Polar moment of area is a quantity used to predict an
object's ability to resist torsion.
It is denoted by ‘J’.
Relation between J0 and Ix , Iy is possible since r2 =x2+y2.

Radius of gyration:
It is used design of column. Radius of gyration along x and
y axis is given by

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Parallel Axis Theorem:
The parallel axis theorem is used to find moment of inertia
of an area about any axis that is parallel to an axis passing
through the centroid and about which the moment of
inertia is known.

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Example-1:

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Example-2:

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Example-2 (Continue):

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Problem-1(N/A):
Find moment of inertia of the T-section (shown below)
about X-X and Y-Y axes through the centre of gravity of the
section.

Problem-2 (N/A):
Find moment of inertia of the T-section (shown below)
about X-X and Y-Y axes through the centre of gravity of the
section.

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Problem-3 (N/A):
The cross section of a cast iron beam is shown below. Find
moment of inertia of the section about horizontal and
vertical axes passing through the centroid of the cross
section.

Assignmet-5 (N/A):
A built up section is made by two C-channel and strap at the
top & bottom as shown in the figure below. Determine
moment of inertia about X-axis passing through the
centroid of the cross section.

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Problem-4:
A simply supported beam AD is loaded by a concentration force
of 80 KN together with a moment of 30 KN-m as shown in the
figure below. Determine max tension and compression stress in
the beam. Also find maximum shear stress in the beam.
(SEE HAND ANALYSIS)

25

Problem-5: The beam AC is simply supported at both ends
and subject to the uniformly distributed load of 300 N/m
plus a moment of 2700 N m as shown below.
a) Sketch the shear and bending moment diagram.
b) Find the maximum stress (shear) and bending at points
1, 2 and 3. Section of beam is shown below.
(SEE HAND ANALYSIS)

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Problem-6:
Determine maximum tension and compressive stresses in the
beam shown below. Also find max shear stress in the beam.
(SEE HAND ANALYSIS)

SHEAR STRESS IN BEAM
For direct shear (no bending present),
Most beams have both shear forces and bending
moments. Although beam flexure formula is developed
on the assumption of pure bending. We can use same
formula to distribute bending stress in the beam when
shear forces are present in the beam.
Assume segment of beam ‘dx’ is subject to moment ‘M’
on near side and moment ‘M + dM’ on the far side. ‘M’
produce normal stress ‘σ’ and ‘M + dM’ produce ‘σ+dσ ’.
Normal stress on far side is greater than near side which
will cause the section dx to slide along negative X-
direction.
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SHEAR STRESS IN BEAM (Continue)

SHEAR STRESS IN BEAM (Continue)
Q is the first moment of inertia of area about neutral axis.

So, transverse shear stress is always accompanied with
bending stress.
Shear stress at the upper & lower extreme fiber is zero.
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MAX. SHEAR STRESS IN A RECTANGULAR BEAM

BENDING AND SHEAR STRESS IN BEAM
Problem-7:
Determine max bending stresses in the beam. Also find
shear stress distribution in the beam cross section shown in
the picture below.
(SEE HAND ANALYSIS)

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Problem-8 (N/A):
Find (a). Maximum bending stresses and shear stresses in
the beam; and (b). Also find bending and shear stress at
point ‘A’ of the cross section of the beam (cross section is
shown in the next page).
(SEE HAND ANALYSIS)

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Problem-8 (continue):

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7/3/2023

AHSANULLAH UNIVERSITY OF SCIENCE

ME-2201: MECHANICS OF MATERIALS

ENGR. FAZLAR RAHMAN

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; MPE; AUST; ME-

ME-2201: Mechanics of Materials

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3 4

Fazlar Rahman; Asst. Prof.; MPE; AUST; ME-

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3 5

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3 6

Fazlar Rahman; Asst. Prof.; MPE; AUST; ME-

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3 7

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3

Fazlar Rahman; Asst. Prof.; MPE; AUST; ME-

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3 9

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3 10

Fazlar Rahman; Asst. Prof.; MPE; AUST; ME-

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3 12

Fazlar Rahman; Asst. Prof.; MPE; AUST; ME-

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3

ME-2201: Mechanics of Materials

Relation between J0 and Ix , Iy is possible since r2 =x2+y2.

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3

Fazlar Rahman; Asst. Prof.; MPE; AUST; ME-

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3

Fazlar Rahman; Asst. Prof.; MPE; AUST; ME-

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3

Fazlar Rahman; Asst. Prof.; MPE; AUST; ME-

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3 19

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3 20

Fazlar Rahman; Asst. Prof.; MPE; AUST; ME-

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3 21

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3

Fazlar Rahman; Asst. Prof.; MPE; AUST; ME-

ME-2201: Mechanics of Materials

Fazlar Rahman; Asst. Prof.; AUST; ME 2201; LEC-3 23