Course book Mech. of Materials content.pptx

Mechanics of Material
Ministry of Higher Education and Scientific research
Salahaddin University – Erbil
College of Engineering
Civil Engineering Department
Dr. Sirwan K. Mala & M. Omer Faisal
Fall Semester 2023/2024

Mechanics of Materials
5 Credits
4 hours per week
Pre request: Engineering Mechanics II
Course overview:
Mechanics of materials deals with the behavior of solid objects
subject to stresses and strains. The study of strength of materials
refers to calculating the stresses and strains in structural
members, such as beams, columns, and shafts. The methods
employed to predict the response of a structure under loading
and its susceptibility to various failure modes takes into account
the properties of the materials such as its yield
strength, ultimate strength, Young's modulus, and Poisson's
ratio; in addition the geometric properties, such as its length,
width, thickness, boundary constraints.

The students will learn the basic concept of stresses and the
corresponding deformations in various structural members.
Considering axial loading, shear, torsion, and bending forces,
considering the related strain (axial strain, shear strain, and
twisting) due to external loads.
Course objective:
1. To provide the basic concepts and principles of mechanics of
materials.
2. To give an ability to calculate stresses and deformations of
objects under external loadings.
3. To give an ability to apply the knowledge of mechanics of
materials on engineering applications and design problems

Student's obligation
• Attendance: Students are required to attend lectures.
The course consists of primarily of theory lectures and
tutorial lectures. Regular attendance is necessary to
maintain pace with the lectures.
Late attendance to class is not allowed.
Maximum allowed absence is 10% (6 hrs.- 3 lectures).
The student will be withdrawn when the absence exceeds
20% (12 hrs.- 6 lectures)
• Students should have Notebook to write the lectures in
the class.

Forms of teaching
Lectures, theoretic and tutorial lectures,
Quizzes are given at the end of second lecture every week.
Examinations: There will be a mid-term examination (01/11/2023), and
final examination will be in (December).
Homework will be given to the students. They should do the homework,
but will not be submitted for evaluation. They are to make the student
ready for exams.
Team work Homework and projects will be given to students monthly.
Exam Weight
Quizzes, Class Activity 15 % + 5%
Mid-term Exam 20 %
Final Exam 60 %
Assessment scheme
The weight percentages are as follows:

Student learning outcome:
Students will have the ability to:
1. Understand the concepts of stress and strain at a point as well
as the stress-strain relationships for homogenous, isotropic
materials.
2. Calculate the stresses and strains in axially-loaded members,
circular torsion members, and members subject to flexural
loadings.
3. Determine the stresses and strains in members subjected to
combined loading and apply the theories of failure for static
loading.
5. Design simple bars, beams, and circular shafts for allowable
stresses and loads

References
1. R.C. Hibbeler `` Mechanics of Material`` Prentice Hall-Pearson,
8th Edition, 2011.
2. F.L. Singer and A. Pytel ``Strength of Materials`` Harper
International Edition, 3rd Edition, 1980.
3. F.L.Bear, E.R. Johnston , and J.T. Dewolf “ Mechanics of Materials “
McGraw Hill Higher Education, 4th Edition, 2006.
4. A.C. Ugural ``Mechanics of Materials `` McGraw Hill. Inc. 1991.
5. R.S. Khurmi`` Strength of Materials( Mechanics of solid) ``SI unit,
S.Ch and company LTD., New Delhi, 2008.
6. J. Case and A.H. Chilver `` Strength of Materials and Structures``
SI Unit, Edward Arnold publisher limited, 1975.
7. Crandall, S. H., N. C. Dahl, and T. J. Lardner.” An Introduction to
the Mechanics of Solids”. 2nd ed. New York, NY: McGraw Hill, 1979.

Syllabus
 Course book, Introduction, Equilibrium of a Deformable Body
 Internal Forces, Stress.
 Average Normal Stress, Average Shear Stress.
 Allowable Stress, design of Simple Connection
 Deformation, Strain
 Hooke’s Law, Elastic deformation
 Axially Loaded Member, Principle of Superposition, Statically Indeterminate
Axially loaded Member.
 Torsion, Angle of Twist.
 Shear Moment Diagram
 Flexure Stress
 Transverse Shear Stress
 Deflection of Beams
 Buckling of Columns

INTRODUCTION
Mechanics of materials is a branch of
mechanics that studies the internal effects of
stress and strain in a solid body that is
subjected to an external loading.
Stress is associated with the strength of the
material from which the body is made, while
strain is a measure of the deformation of the
body.
In addition to this, mechanics of materials
includes the study of the body’s stability
when a body such as a column is subjected to
compressive loading.
A thorough understanding of the
fundamentals of this subject is of vital
importance because many of the formulas
and rules of design cited in engineering
codes are based upon the principles of this
subject.
9

External Loads. A body is subjected to only two
types of external loads; namely, surface forces
or body forces, Fig. 1–1.
Surface forces are caused by the direct contact
of one body with the surface of another. In all
cases these forces are distributed over the area of
contact between the bodies. If this area is small
in comparison with the total surface area of the
body, then the surface force can be idealized as a
single concentrated force, which is applied to a
point on the body.
Equilibrium of a Deformable Body
10

11
Support Reactions

12
Support Reactions

Equations of Equilibrium.. Equilibrium of a body requires both a balance of forces, to
prevent the body from translating or having accelerated motion along a straight or
curved path, and a balance of moments, to prevent the body from rotating.
Often in engineering practice the loading on a body can be represented as a system of
coplanar forces. If this is the case, and the forces lie in the x–y plane, then the conditions
for equilibrium of the body can be specified with only three scalar equilibrium equations;
that is,
13

Here all the moments are summed about point O and so they will be directed along the z axis.
Successful application of the equations of equilibrium requires complete specification of all the
known and unknown forces that act on the body, and so the best way to account for all these
forces is to draw the body’s free-body diagram.
14

Internal Resultant Loadings.
In mechanics of materials, statics is primarily used to determine the resultant loadings
that act within a body.
In order to obtain the internal loadings acting on a specific region within the body, it is
necessary to pass an imaginary section or “cut” through the region where the internal
loadings are to be determined.
The two parts of the body are then separated, and a free-body diagram of one of the
parts is drawn.
Also, if a member is long and slender, as in the case of a rod or beam, the section to be
considered is generally taken perpendicular to the longitudinal axis of the member. This
section is referred to as the cross section.
15

Internal Forces
Normal force, N. This force acts perpendicular to the area. It is developed whenever
the external loads tend to push or pull on the two segments of the body.
ƩFx =N
Shear force, V. The shear force lies in the plane of the area and it is developed when
the external loads tend to cause the two segments of the body to slide over one
another.
ƩFy =V
Bending moment, M. The bending moment is caused by the external loads that tend to
bend the body about an axis lying within the plane of the area.
ƩMo = M
16

1.2 Equilibrium of a Deformable Body
Important Points.
• Mechanics of materials is a study of the relationship between the external
loads applied to a body and the stress and strain caused by the internal
loads within the body.
• External forces can be applied to a body as distributed or concentrated
surface loadings, or as body forces that act throughout the volume of the
body.
• Linear distributed loadings produce a resultant force having a magnitude
equal to the area under the load diagram, and having a location that passes
through the centroid of this area.
• A support produces a force in a particular direction on its attached member
if it prevents translation of the member in that direction, and it produces a
couple moment on the member if it prevents rotation.
17

Important Points.
• The equations of equilibrium
ƩFx = 0
Ʃfy = 0
and ƩM = 0
must be satisfied in order to prevent a body from translating with accelerated
motion and from rotating.
• When applying the equations of equilibrium, it is important to first draw the
free-body diagram for the body in order to account for all the terms in the
equations.
• The method of sections is used to determine the internal resultant loadings
acting on the surface of the sectioned body.
In general, these resultants consist of a:
Normal force (N), shear force (V) , and bending moment (M).
18

The resultant internal loadings at a point located on the section of a body can be obtained using the
method of sections.
Support Reactions.
• First decide which segment of the body is to be considered. If the segment has a support or connection
to another body, then before the body is sectioned, it will be necessary to determine the reactions acting
on the chosen segment. To do this draw the free body diagram of the entire body and then apply the
necessary equations of equilibrium to obtain these reactions.
Free-Body Diagram.
• Keep all external distributed loadings, couple moments, torques, and forces in their exact locations,
before passing an imaginary section through the body at the point where the resultant internal
loadings are to be determined.
• Draw a free-body diagram of one of the “cut” segments and indicate the unknown resultants N, V, M,
and T at the section. These resultants are normally placed at the point representing the geometric
center or centroid of the sectioned area.
• If the member is subjected to a coplanar system of forces, only N, V, and M act at the centroid.
• Establish the x, y, z coordinate axes with origin at the centroid and show the resultant internal
loadings acting along the axes.
Equations of Equilibrium.
• Moments should be summed at the section, about each of the coordinate axes where the resultants
act. Doing this eliminates the unknown forces N and V and allows a direct solution for M (and T).
• If the solution of the equilibrium equations yields a negative value for a resultant, the assumed
directional sense of the resultant is opposite to that shown on the free-body diagram.
19
Procedure for Analysis

20
Example
Determine the internal forces acting on the cross section at C of the cantilevered
beam shown
Solution
Find support reactions
Take section at point C
Use equilibrium equations to find Nc, Vc, Mc.
Nc = 0
Vc = 540 N
Mc = 1080 N.m

21
Example
Determine the internal forces acting on the cross section at C .
Solution
Use equilibrium equations to find
Nc, Vc, Mc.
Nc = 0
Vc = -58.8N
Mc = -5.69 N.m

22
Example
A 500 kg machine is suspended as shown.
Determine the internal forces acting on the cross section at point E .
Solution
Use equilibrium equations to find
Nc, Vc, Mc.

34
Example
For the bar of 35mm width and 10 mm thickness.
Determine the maximum average normal stress when it is subjected to the
loading shown.
Solution
Find internal force for each part
Using section method
Find normal stress in each part
Indicate maximum normal stress in the bar

35
Example
The 80 kg lamp is supported by two rods AB and BC as shown. If AB has a
diameter of 10 mm and BC has a diameter of 8mm.
Determine the average normal stress in each rod.

36
Example
Member AC is subjected to a vertical force of 3 kN.
Determine the position x of this force so that the average normal compressive stress at
the support C is equal to the average tensile stress in the rod AB.
The rod has a cross sectional area of 400 mm2 and the contact area at C is 650 mm2

44
Example
Determine the average shear stress in the 20mm diameter pin at A and the 30
mm diameter pin at B..
Solution

45
Example
The wood joint has a width of 150mm. Determine the average shear stress
along shear planes a-a and b-b.
Solution

50
Example
The shaft shown is supported at C. Determine the largest load P so that the
bearing stress at C does not exceed Ϭallow = 75 Mpa and the normal stress in the
shaft does not exceed Ϭallow = 55 Mpa stress along shear planes a-a and b-b.
Solution

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