MMT

1. Lecture-1
Mechanics of Materials-I

2. Course Instructor:
Waqas Asghar
Lecturer,
Mechanical Department, UET Taxila
PhD. (2017-2021): University of Chinese Academy of Sciences,
China.
MSc. (2013-2016): University of Engineering & Technology, Taxila.
BSc. (2008-2012): University of Engineering & Technology, Taxila.
2
Email: waqas.asghar@uettaxila.edu.pk
Office: Engineering Mechanics Lab

3. Course Contents:
3
• Stresses & Strains
• Mechanical Properties of Materials
• Torsion
• Shearing Force & Bending Moment
• Centroids & Moment of Inertia
• Bending of Beams
• Beam Deflection Methods

4. Course Outcomes:
4

5. Grading Policy:
5
• Quizzes or Assignments: 12%
• Semester Project or CEP: 13%
• Mid-semester Exam: 25%
• Final Exam: 50%

6. Course Books:
6
• Mechanics of Materials by Gere, 6th Edition
• Mechanics of Materials by R. C. Hibbler, 8th Edition
Reference Books:
• Strength of Materials by R. K Rajput
• Mechanics of Materials by Debabrata Nag and Abhijit Chanda
• Mechanics of Materials by Ferdinand P. Beer & Russel Johnston
Jr. McGraw-Hill

7. Introduction
Mechanics of Material: Branch of mechanics that deals with behaviour of materials subjected
to various type of loadings also called Mechanics of solids or Mechanics of deformable bodies.
Main objective of MMT: To find stresses, strains and displacement in structures for the complete
evaluation of mechanical behaviour of materials necessary for safe designing of structures.

8. Introduction
Axial force
Force or Load acting on the axis of member which leads to tension or compression.
Example:
Connecting road of engine, spokes of bicycle, columns of buildings, tow bar to pull aeroplane,
landing gear of plane, etc..
In order to have uniform tension or compression, axial force must act through the centroid of
cross sectional area.

9. Normal Force
• Normal forces act perpendicular to the cross sectional area.
• Angle between the forces and longitudinal axis is 0º
• It tries to changes the axial dimensions of object

10. Shear Force
• Shear forces act parallel to the cross sectional area.
• Angle between the forces and longitudinal axis is 90º
• It tries to changes the shape of object
No rotation

11. Shear Force Examples From Daily Life

12. Shear Force Examples From Daily Life

13. Shear Force Examples From Daily Life

14. Identify the Type of Force??
Elongated spring Crushing failure of Can
Columns of building
Rope in tug of war

15. Types of Bars
Prismatic bar:
A bar having constant cross sectional area throughout its length.
Non-Prismatic bar:
A bar which does not have constant cross sectional area throughout its length.
Prismatic bars Non-Prismatic bars
Cross Section: Section taken perpendicular to the longitudinal axis of the bar, is called a cross section.

16. Stress (𝜎)
In an externally loaded material, stress is the intensity of internal resisting force
which holds the each segment of body in equilibrium against deformation.
OR
Force per unit area is called stress.
Stress may remain constant throughout the body or may change from one point
to another
Direct or Normal Stress:
• Stress produced due to normal force acting on a bar.
• In direct stress, load acts perpendicular to the cross sectional area of body.
• e.g. Axial Stresses, Bending Stresses, Radial Or Hoop Stresses.
Types: Tensile stress (+ive) or compressive ( ̶ ive)
𝝈 =
𝒍𝒐𝒂𝒅 𝒐𝒓 𝑭𝒐𝒓𝒄𝒆
𝒂𝒓𝒆𝒂
=
𝑷
𝑨
𝐔𝐧𝐢𝐭𝐬 = ൗ
𝑁
𝑚2 𝑜𝑟 𝑃𝑎 𝑜𝑟 𝑃𝑠𝑖.
𝝈 =
𝑵𝒐𝒓𝒎𝒂𝒍 𝑭𝒐𝒓𝒄𝒆
𝒂𝒓𝒆𝒂
=
𝑷
𝑨

17. Deformation: Force applied on body changes the shape and size of body, which
is called deformation
Strain: Measure of deformation produced in material due to applied load.
Direct or Normal Strain: Measure of material’s deformation produced due to
normal or direct load.
𝑫𝒊𝒓𝒆𝒄𝒕 𝒐𝒓 𝑵𝒐𝒓𝒎𝒂𝒍 𝑺𝒕𝒓𝒂𝒊𝒏 𝜺 =
∆𝒍
𝒍
Units: Dimensionless
Types & Sign Convention:
• Tensile strain (+ive) or Compressive strain ( ̶ ive)
Mostly, extension of material under load is very small that's why it is expressed as a %age strain, as
𝒔𝒕𝒓𝒂𝒊𝒏 𝜺 =
∆𝒍
𝒍
× 𝟏𝟎𝟎%

18. Example of Tensile & Compressive Load (Force)

19. Uniaxial stress
1D state of stress in which only one normal stress act on body, along any
single direction.
Stress components acting on uniaxial stress system should be equal in
magnitude but opposite in direction.
Biaxial stress
2D state of stress in which two normal stresses act along two different
directions is called biaxial stress
Triaxial stress
3D state of stress in which three normal stresses act along three different
directions is called triaxial stress

20. Shear Stress (𝜏):
• It is defined as the stress applied to the parallel or tangential face of material
𝝉𝒂𝒗𝒈 =
𝑰𝒏𝒕𝒆𝒓𝒏𝒂𝒍 𝒔𝒉𝒆𝒂𝒓 𝒇𝒐𝒓𝒄𝒆
𝑨𝒓𝒆𝒂 𝒓𝒆𝒔𝒊𝒔𝒕𝒊𝒏𝒈 𝒔𝒉𝒆𝒂𝒓
=
𝑽
𝑨
• Shear stress (τ) acts in the same direction as V
• Units: same as normal stress
• Shear stress acts in various types of simple connections, like : bolts, pins, rivets, keys,
welds, and glued joints.
• Normal stress tends to change the size of material and shear stress changes the shape of
material.

21. Examples of Double & Single Shear

22. Shear Strain (γ):
• Measure of change in angle between two lines segments, which were originally perpendicular to
each other.
• It is measured in degrees or radians
• E.g. Bicycle brake block deformed due to application of shear force.
Shear Force
• Sign Convention: γ is positive ----- angle b/w two faces is reduced
γ is negative ---- angle b/w two faces is increased
• Normal strain changes volume/ dimension (elongation or compression) of object.
• Shear strain changes shape of object.

24. Example 1-1 of Gere
Short post constructed from a hollow a Al circular tube supports a compressive load of
26 kips. Inner and outer diameter of tube are di= 4in and do=4.5 in and its length is 16
inch. shortening of post due to load is 0.012 inch. Find compressive stress and strain.
Solution
l=16 in , ∆𝑙= 0.012in,
di=4 in , do=4.5 in
Area of tube= 𝜋𝑟2 =
𝜋𝑑2
4
=
𝜋
4
𝑑𝑜2 − 𝑑𝑖2 =
𝜋
4
4.52 − 42 = 3.38𝑖𝑛2
𝜎𝑐𝑜𝑚𝑝 =
𝑃
𝐴
=
26𝑘𝑖𝑝𝑠
3.38 𝑖𝑛2
=
26000 𝑙𝑏
3.38 𝑖𝑛2
= 7790 𝑝𝑠𝑖
𝜀𝑐𝑜𝑚𝑝 =
∆𝑙
𝑙
=
0.012
16
= 750 × 10−6

25. Curve plotted by using stress and strain data obtained from tensile test of specimen is
called stress strain (σ-ε) diagram or curve.
𝛔 − 𝛆 diagram gives important information about the mechanical properties and
behaviour of material under various stages of loading.
Tensile test on material is performed by gradually applying load (using m/c) and
resulting deformations are measured until failure.
Gauge Length: Part of test specimen, actually being measured for elongation, during
a tensile test.
Extensometer: Device used to measures the elongation produced tensile test.
Stress-Strain Diagram (σ-ε)
Test conducting standards: American Society for Testing and Materials (ASTM), American
Standards Association (ASA) and National Institute of Standards and Technology (NIST).

26. Tensile Testing

27. Stress Strain (σ-ε) Diagram of Ductile Material (Steel)

28. Elastic Region:
• Region in which σ & ε exhibit linear and
proportional relationship.
• Material exhibits exhibit linear elastic behaviour in
this region.
• Slope of Elastic region = Youngs Modulus (E).
• Youngs Modulus (E) = Measure of material’s
stiffness, in its elastic range and has the units of
stress.
𝐒𝐩𝐞𝐜𝐢𝐟𝐢𝐜 𝐬𝐭𝐢𝐟𝐟𝐧𝐞𝐬𝐬 =
𝐘𝐨𝐮𝐧𝐠′𝐬 𝐌𝐨𝐝𝐮𝐥𝐮𝐬
𝐌𝐚𝐭𝐞𝐫𝐢𝐚𝐥′𝐬 𝐃𝐞𝐧𝐬𝐢𝐭𝐲

29. OR
Transition point between linear elastic deformation
region and non-linear elastic deformation.
OR
It determines the greatest stress that is directly
proportional to strain.
• Below proportional limit, material deforms linearly,
elastically & is able regain its original shape on
removal of applied load.
Proportional limit:
Upper stress limit of linear elastic region is called proportional limit (𝜎𝑝𝑙).

30. Yield Point:
• Point on 𝜎 − 𝜀 curve, where elastic behaviour of
material changes into plastic behaviour.
• OR Transition point between elastic & plastic region.
• Below yield point, material deforms non-linearly,
elastically & is able regain its original shape on
removal of applied load.
• In yielding region, material becomes perfectly plastic and continues to elongate without any
noticeable increase of load. This phenomenon is called yielding.
• Low carbon steels exhibit two values of yield point i.e. initial upper yield point followed by certain
decrease in load carrying capacity to a lower yield point.
• Yield strength or Yield stress: Stress corresponding to yield point, after which material begins to
deform permanently or plastically.

31. Strain hardening
• Strengthening the material by plastic deformation.
• It occurs when metal is plastically deformed
beyond yield point.
• Material undergoes change in its crystalline
structure which ultimately increases the strength
of material.
• During strain hardening, elongation of the test
specimen requires an additional amount of tensile
load.
• Finally, load (𝜎 − 𝜀 curve) reaches its maximum value, at which corresponding value of stress
(point D) is called the ultimate stress (𝜎𝑢).
• Strain hardening is also known as work hardening or cold working.

32. Ultimate stress (UTS):
• Maximum stress or load that a stretched material can
withstand, before failure.
• UTS often corresponds to highest point (point D) on
𝜎 − 𝜀 curve
• Beyond UTS, specimen deforms even load is removed.
• Material’s ultimate stress is also called ultimate
strength or ultimate tensile strength of material.
Strength: Capability of structure to resist load.
𝐒𝐩𝐞𝐜𝐢𝐟𝐢𝐜 𝐬𝐭𝐫𝐞𝐧𝐠𝐭𝐡 =
𝐒𝐭𝐫𝐞𝐧𝐠𝐭𝐡 𝐔𝐓𝐒
𝐌𝐚𝐭𝐞𝐫𝐢𝐚𝐥′𝐬 𝐃𝐞𝐧𝐬𝐢𝐭𝐲

33. Necking:
• Type of plastic deformation observed in ductile materials.
• Localized reduction of material’s cross sectional area,
under tensile load.
• Before UTS, entire material undergoes uniform plastic
deformation.
• But just after UTS, cross sectional area of small localized
region decreases more rapidly (by greater proportion), &
eventually adopts neck or V shape.
• Neck exhibits high concentration of local strain.
• Necking behavior is considered in calculating true stress
but disregarded in calculating engineering stress.
Neck

34. Fracture Stress:
• Stress or load at which material failure occurs.
• At fracture stress, necking region ends & strain
reaches its max. value
• Ductile materials: Fracture strength < UTS
• Brittle materials: Fracture strength = UTS

37. 𝛔 − 𝛆 Curves of Various Materials

38. Yield Strength of Materials having No Well-defined
Yield Point
• Aluminum does not show any obvious yield point
• Arbitrary yield stress can be determined by using Offset
method.
• In offset method, 0.2% strain (0.002 in/in ) is chosen as a
reference point on strain axis.
• From this reference point, a line parallel to the initial straight-
line portion of the stress–strain diagram is drawn.
• The point where this line intersects the curve defines the yield
strength.
Yield
Strength