SlideShare a Scribd company logo
4
Most read
9
Most read
10
Most read
STRUCTURAL DESIGN I
STRUCTURAL MECHANICS
MOMENT OF INERTIA
LECTURE 10
MOMENT OF INERTIA
• Inertia is the resistance of any physical object to a change in
its state of motion or rest.
• It is represented numerically by an object's mass.
• Moment of Inertia is also known as the second moment of a
force or an area.
• (the first moment is the product of force and the arm)
The figure shows beam sections divided up into a very large number of thin
strips parallel to the neutral axis. If we multiply each strip area by the square
of its distance from the NA and sum up all the quantities obtained, we will
obtain the value of INA for the given beam section.
INA = (a1 x y1²) + (a2 x y2²) + (a3 x y3²) +……
It will be observed that the value of the moment of inertia has nothing to do
with the material of the beam but is only a property of its shape.
.
The value of MoI of certain sections are:
PARALLEL AXIS THEOREM
This theorem states that “ The Moment of Inertia of a lamina about any axis in
the plane of the lamina equals the sum of the following:
i) Moment of Inertia about a centroidal axis parallel to the axis about which
moment of inertia is to be calculated
ii) Product of the area of the laminate and the square of the distance
between the above centroidal axis and the axis about which the moment
of inertia is to be calculated.”
b
dX X’
b
d
X X’’
I about XX’ = bd³/12
I about XX’’ = IXX’ + Ay²
= bd³/12 + (bd×(d/2)²)
POSITION OF NEUTRAL AXIS
We found that the strip load was given by the expression E x ay
R
As long as y is measured downward all the strip loads will represent tension.
If we put y negative i.e. measured the distance from NA , the load would be
compression.
∑E x ay or E x ∑ay (since E and R are constants) will therefore represent a
R R
summation of a large number of positive and negative quantities. But as the total
compressive force is equal to the total tensile force as C and T forma couple
E ∑ay = 0.
R
∑ay = 0 means that the axis , from which y is measured , passes through the
center of gravity of the section.
The neutral axis of a beam section therefore passes through its center of gravity.
The figure shows
the position of
neutral axis of
some sections.
X’X
d
c
b
a
Problem: To find the Moment of Inertia of the lamina shown above about XX’axis .
By parallel axis theorem:
IXX” = (IA + (ab × (b/2)²) + (IB + (dc × (c/2)²) where IA = ab³/12 and IB = dc³/12
A
B
Find IXX’ if a = 3, b = 8, c = 2 and d = 8
While the MoI or I of standard sections are known, I of differently shaped
sections can be computed by the same theory
An I section lamina has top flange 8cm x 2cm, Bottom flange 12 cm x 4 cm and web
3cm x 16 cm. Calculate the centroidal moment of inertia of the lamina.
To solve this question first the CG has to be located.
Then the I of each part around the CG has to be added
Let CG lie at a distance y from bottom of bottom flange
Structure Design-I (Moment of Inertia)
A
C
B
Finding CG
(8×2×21)+(16×3×12)+(12×4×2)=y×(16+48+48)
Y = (336+576+96)÷112=1008÷112=9 cm
Ixx = ((8×2×2×2/12)+(8×2×(21-9)²)) + ((3×16×16×16/12) + (3×16 ×(12-9)²))
+((12×4×4×4/12)+(12×4×(9-2)²))
=(5.33+2304)+(1024+432)+(64+2352)=2309.33+1456+2416=6181.33cm⁴
XX

More Related Content

PPTX
Problems on simply supported beams
PDF
Unit 6: Bending and shear Stresses in beams
PPT
Bending stresses
PPTX
PPT
Shear stresses in beams
PPTX
Bending stresses in beams
PDF
Axial deformation
PPTX
Basic Civil Mechanical Engineering unit-1
Problems on simply supported beams
Unit 6: Bending and shear Stresses in beams
Bending stresses
Shear stresses in beams
Bending stresses in beams
Axial deformation
Basic Civil Mechanical Engineering unit-1

What's hot (20)

PDF
Strength of materials_I
PDF
Reinforced concrete column
PPTX
Theories of columns
PPT
Deflection
PDF
PPTX
PDF
MOMENT OF INERIA
PPTX
Topic1_Castiglianos Theorem.pptx
PDF
Deflection in beams
PPTX
Thermal stesses
PPTX
Axially loaded columns
PDF
single degree of freedom systems forced vibrations
PDF
Parallel axis theorem and their use on Moment Of Inertia
PPTX
Direct and bending stress
DOCX
PPT
Shear Force And Bending Moment In Beams
PPTX
Truss-method of joints
PPTX
Torsion
PPT
Trusses The Method Of Sections
DOCX
Friction full
Strength of materials_I
Reinforced concrete column
Theories of columns
Deflection
MOMENT OF INERIA
Topic1_Castiglianos Theorem.pptx
Deflection in beams
Thermal stesses
Axially loaded columns
single degree of freedom systems forced vibrations
Parallel axis theorem and their use on Moment Of Inertia
Direct and bending stress
Shear Force And Bending Moment In Beams
Truss-method of joints
Torsion
Trusses The Method Of Sections
Friction full
Ad

Similar to Structure Design-I (Moment of Inertia) (20)

PDF
Moment of inertia of non symmetric object
PPTX
Stucture design -I (Centre of Gravity ;Moment of Inertia)
PPT
Lecture material week 6
PDF
inertia notes
PPTX
Properties of surfaces-Centre of gravity and Moment of Inertia
PPTX
Lect9 Moment of Inertia
PDF
Chapter 7-2.pdf. .
DOCX
Centre of Gravity
PPTX
Centroid and Moment of Inertia from mechanics of material by hibbler related ...
DOCX
Assignment no. 5
PDF
Gfffg hedef ghdghftyhjjj_26301324024.pdf
PPTX
Moment of Inertia.pptx
PPTX
momentofinertia-131115093234-phpapp01.pptx
PDF
chapter8.pdf hdjakshdjakshdjkashdjkashdjkashdjaks
PDF
09 review
PPTX
Moment of Inertia by Prof. Malay Badodariya
PPT
Moment of inertia concepts in Rotational Mechanics
PDF
Tutorial 8 slides
PDF
J3010 Unit 2
Moment of inertia of non symmetric object
Stucture design -I (Centre of Gravity ;Moment of Inertia)
Lecture material week 6
inertia notes
Properties of surfaces-Centre of gravity and Moment of Inertia
Lect9 Moment of Inertia
Chapter 7-2.pdf. .
Centre of Gravity
Centroid and Moment of Inertia from mechanics of material by hibbler related ...
Assignment no. 5
Gfffg hedef ghdghftyhjjj_26301324024.pdf
Moment of Inertia.pptx
momentofinertia-131115093234-phpapp01.pptx
chapter8.pdf hdjakshdjakshdjkashdjkashdjkashdjaks
09 review
Moment of Inertia by Prof. Malay Badodariya
Moment of inertia concepts in Rotational Mechanics
Tutorial 8 slides
J3010 Unit 2
Ad

More from Simran Vats (20)

PPTX
CONTEMPORARY PLANNINGS IN INDIA.pptx
PPTX
From Single Function to Integrated: The Evolution of Computer Integrated Buil...
PPTX
The Art of Intelligent Design and Construction
PPTX
Building with the Earth: The Role of Technology in Vernacular Architecture
PPTX
Building Resilience: Vernacular Strategies for Disaster-resistant Structures ...
PDF
Poverty and informal sectors
PPTX
Detailed specifications
PPTX
SPECIFICATION
PPTX
GREEN ARCHITECTURE
PPTX
Forms of human settelments
PDF
Cyclone disaster in Andhra Pradesh
DOCX
Udaipur dustrict disparities
PPT
General theory of Bending
PPT
Structure design -I (Moment of Resistance)
PDF
PRECEPTION (ADT)
PPT
Structure Design -1(Lecture 9 bm and sf solved examples)
PPT
Structure -1(Lecture 8 bm and sf part 2)
PPT
Structure Design-I (Bending moment & Shear force Part II)
PDF
Plannign Theories
PDF
Poverty and Informal sectors
CONTEMPORARY PLANNINGS IN INDIA.pptx
From Single Function to Integrated: The Evolution of Computer Integrated Buil...
The Art of Intelligent Design and Construction
Building with the Earth: The Role of Technology in Vernacular Architecture
Building Resilience: Vernacular Strategies for Disaster-resistant Structures ...
Poverty and informal sectors
Detailed specifications
SPECIFICATION
GREEN ARCHITECTURE
Forms of human settelments
Cyclone disaster in Andhra Pradesh
Udaipur dustrict disparities
General theory of Bending
Structure design -I (Moment of Resistance)
PRECEPTION (ADT)
Structure Design -1(Lecture 9 bm and sf solved examples)
Structure -1(Lecture 8 bm and sf part 2)
Structure Design-I (Bending moment & Shear force Part II)
Plannign Theories
Poverty and Informal sectors

Recently uploaded (20)

PPTX
Week 4 Term 3 Study Techniques revisited.pptx
PPTX
Microbial diseases, their pathogenesis and prophylaxis
PDF
TR - Agricultural Crops Production NC III.pdf
PPTX
Open Quiz Monsoon Mind Game Prelims.pptx
PDF
01-Introduction-to-Information-Management.pdf
PDF
Saundersa Comprehensive Review for the NCLEX-RN Examination.pdf
PDF
BÀI TẬP TEST BỔ TRỢ THEO TỪNG CHỦ ĐỀ CỦA TỪNG UNIT KÈM BÀI TẬP NGHE - TIẾNG A...
PPTX
PPT- ENG7_QUARTER1_LESSON1_WEEK1. IMAGERY -DESCRIPTIONS pptx.pptx
PPTX
school management -TNTEU- B.Ed., Semester II Unit 1.pptx
PPTX
master seminar digital applications in india
PPTX
Cardiovascular Pharmacology for pharmacy students.pptx
PDF
The Lost Whites of Pakistan by Jahanzaib Mughal.pdf
PDF
3rd Neelam Sanjeevareddy Memorial Lecture.pdf
PDF
Basic Mud Logging Guide for educational purpose
PPTX
IMMUNITY IMMUNITY refers to protection against infection, and the immune syst...
PPTX
Pharmacology of Heart Failure /Pharmacotherapy of CHF
PPTX
Renaissance Architecture: A Journey from Faith to Humanism
PDF
Pre independence Education in Inndia.pdf
PPTX
Pharma ospi slides which help in ospi learning
PDF
STATICS OF THE RIGID BODIES Hibbelers.pdf
Week 4 Term 3 Study Techniques revisited.pptx
Microbial diseases, their pathogenesis and prophylaxis
TR - Agricultural Crops Production NC III.pdf
Open Quiz Monsoon Mind Game Prelims.pptx
01-Introduction-to-Information-Management.pdf
Saundersa Comprehensive Review for the NCLEX-RN Examination.pdf
BÀI TẬP TEST BỔ TRỢ THEO TỪNG CHỦ ĐỀ CỦA TỪNG UNIT KÈM BÀI TẬP NGHE - TIẾNG A...
PPT- ENG7_QUARTER1_LESSON1_WEEK1. IMAGERY -DESCRIPTIONS pptx.pptx
school management -TNTEU- B.Ed., Semester II Unit 1.pptx
master seminar digital applications in india
Cardiovascular Pharmacology for pharmacy students.pptx
The Lost Whites of Pakistan by Jahanzaib Mughal.pdf
3rd Neelam Sanjeevareddy Memorial Lecture.pdf
Basic Mud Logging Guide for educational purpose
IMMUNITY IMMUNITY refers to protection against infection, and the immune syst...
Pharmacology of Heart Failure /Pharmacotherapy of CHF
Renaissance Architecture: A Journey from Faith to Humanism
Pre independence Education in Inndia.pdf
Pharma ospi slides which help in ospi learning
STATICS OF THE RIGID BODIES Hibbelers.pdf

Structure Design-I (Moment of Inertia)

  • 1. STRUCTURAL DESIGN I STRUCTURAL MECHANICS MOMENT OF INERTIA LECTURE 10
  • 2. MOMENT OF INERTIA • Inertia is the resistance of any physical object to a change in its state of motion or rest. • It is represented numerically by an object's mass. • Moment of Inertia is also known as the second moment of a force or an area. • (the first moment is the product of force and the arm)
  • 3. The figure shows beam sections divided up into a very large number of thin strips parallel to the neutral axis. If we multiply each strip area by the square of its distance from the NA and sum up all the quantities obtained, we will obtain the value of INA for the given beam section. INA = (a1 x y1²) + (a2 x y2²) + (a3 x y3²) +…… It will be observed that the value of the moment of inertia has nothing to do with the material of the beam but is only a property of its shape. .
  • 4. The value of MoI of certain sections are:
  • 5. PARALLEL AXIS THEOREM This theorem states that “ The Moment of Inertia of a lamina about any axis in the plane of the lamina equals the sum of the following: i) Moment of Inertia about a centroidal axis parallel to the axis about which moment of inertia is to be calculated ii) Product of the area of the laminate and the square of the distance between the above centroidal axis and the axis about which the moment of inertia is to be calculated.”
  • 6. b dX X’ b d X X’’ I about XX’ = bd³/12 I about XX’’ = IXX’ + Ay² = bd³/12 + (bd×(d/2)²)
  • 7. POSITION OF NEUTRAL AXIS We found that the strip load was given by the expression E x ay R As long as y is measured downward all the strip loads will represent tension. If we put y negative i.e. measured the distance from NA , the load would be compression. ∑E x ay or E x ∑ay (since E and R are constants) will therefore represent a R R summation of a large number of positive and negative quantities. But as the total compressive force is equal to the total tensile force as C and T forma couple E ∑ay = 0. R ∑ay = 0 means that the axis , from which y is measured , passes through the center of gravity of the section. The neutral axis of a beam section therefore passes through its center of gravity.
  • 8. The figure shows the position of neutral axis of some sections.
  • 9. X’X d c b a Problem: To find the Moment of Inertia of the lamina shown above about XX’axis . By parallel axis theorem: IXX” = (IA + (ab × (b/2)²) + (IB + (dc × (c/2)²) where IA = ab³/12 and IB = dc³/12 A B Find IXX’ if a = 3, b = 8, c = 2 and d = 8 While the MoI or I of standard sections are known, I of differently shaped sections can be computed by the same theory
  • 10. An I section lamina has top flange 8cm x 2cm, Bottom flange 12 cm x 4 cm and web 3cm x 16 cm. Calculate the centroidal moment of inertia of the lamina. To solve this question first the CG has to be located. Then the I of each part around the CG has to be added Let CG lie at a distance y from bottom of bottom flange
  • 12. A C B Finding CG (8×2×21)+(16×3×12)+(12×4×2)=y×(16+48+48) Y = (336+576+96)÷112=1008÷112=9 cm Ixx = ((8×2×2×2/12)+(8×2×(21-9)²)) + ((3×16×16×16/12) + (3×16 ×(12-9)²)) +((12×4×4×4/12)+(12×4×(9-2)²)) =(5.33+2304)+(1024+432)+(64+2352)=2309.33+1456+2416=6181.33cm⁴ XX