DIFFERENT STRESS STATE –
DIRECT/UNIAXIAL STRESS & DISTRIBUTION
13/10/2023
CE‐205: Mechanics of Solids‐I
SE‐Civil Engineering (Construction)
FALL 2023
Prof. Dr. Rashid A. Khan
Department of Earthquake Engineering
2
Civil Engineering Department
(OBE) Program Learning Outcomes (PLO)
1. Engineering Knowledge: An ability to apply knowledge of mathematics, science, engineering fundamentals and an
engineering specialization to the solution of complex engineering problems.
2. Problem Analysis: An ability to identify, formulate, research literature, and analyze complex engineering problems reaching
substantiated conclusions using first principles of mathematics, natural sciences and engineering sciences.
3. Design/Development of Solutions: An ability to design solutions for complex engineering problems and design systems,
components or processes that meet specified needs with appropriate consideration for public health and safety, cultural,
societal, and environmental considerations.
4. Investigation: An ability to investigate complex engineering problems in a methodical way including literature survey, design
and conduct of experiments, analysis and interpretation of experimental data, and synthesis of information to derive valid
conclusions.
5. Modern Tool Usage: An ability to create, select and apply appropriate techniques, resources, and modern engineering and IT
tools, including prediction and modeling, to complex engineering activities, with an understanding of the limitations.
6. The Engineer and Society: An ability to apply reasoning informed by contextual knowledge to assess societal, health, safety,
legal and cultural issues and the responsibilities relevant to professional engineering practice and solution to complex
engineering problems.
7. Environment and Sustainability: An ability to understand the impact of professional engineering solutions in societal and
environmental contexts and demonstrate knowledge of and need for sustainable development.
8. Ethics: Apply ethical principles and commit to professional ethics and responsibilities and norms of engineering practice.
9. Individual and Team Work: An ability to work effectively, as an individual or in a team, on multifaceted and /or
multidisciplinary settings.
10. Communication: An ability to communicate effectively, orally as well as in writing, on complex engineering activities with
the engineering community and with society at large, such as being able to comprehend and write effective reports and
design documentation, make effective presentations, and give and receive clear instructions.
11. Project Management: An ability to demonstrate management skills and apply engineering principles to one‟s own work, as a
member and/or leader in a team, to manage projects in a multidisciplinary environment.
12. Lifelong Learning: An ability to recognize importance of, and pursue lifelong learning in the broader context of innovation
and technological developments.
1
DIFFERENT STRESS STATE –
DIRECT/UNIAXIAL STRESS & DISTRIBUTION
3
13/10/2023
CE‐205: Mechanics of Solids‐I
Course Learning Outcomes (CLOs)
Sr.
Taxonomy
CLOs
No.
level
At the end of the course, the student will be able to:
DISCUSS the behavior of members
(bars,
beams)
subjected
to PROGRAMME
different
C2
1 COURSE
LEARNING
OUTCOME AND
ITS MAPPING WITH
LEARNING OUTCOME
sets of loading and states of stresses.
SOLVE problems related to biaxial
C3
2
state of stresses.
PRACTICE experiments to study the
P2
3 material response under different
sets of loadings.
4
Programme learning
outcome (PLO)
Engineering Knowledge
Problem Analysis
Engineering Knowledge
CE‐205: Mechanics of Solids‐I
Text & Reference Books
TEXT BOOKS
MECHANICS OF MATERIALS, R.C. Hibbeler
10th Edition, 2018
MECHANICS OF MATERIALS, James M. Gere & Barry J.
Goodno
9th
Edition, 2018
REFERENCE BOOKS
MECHANICS OF MATERIALS, Ferdinand P. Beer
7th Edition, 2017
MECHANICS OF MATERIALS, Paul S. Steif
2012
2
DIFFERENT STRESS STATE –
DIRECT/UNIAXIAL STRESS & DISTRIBUTION
5
13/10/2023
CE‐205: Mechanics of Solids‐I
Course Profile
LECTURE HOURS:
3 hours per week
CONTACT HOURS:
3 hours per week
ACTIVITIES:
LECTURES, CLASS PARTICIPATION,
CLASS QUIZZES, ASSIGNMENTS,
MID-TERM TEST, LAB WORK
GRADING:
CLASS PARTICIPATION, CLASS
QUIZZES, ASSIGNMENTS, MID-TERM
TEST
ASSESSMENT
Class Activities
Assignments
Test
Mid-semester tests
Final examination.
(Dates to be announced Later)
5%
5%
10%
20%
60%
CE‐205: Mechanics of Solids‐I
6
Different Stress State:
Uniaxial state of stresses and strains, Relationships between elastic
Constants, Response of materials under different sets of monotonic
loading, Normal and shearing stress and strains, Gradually and
suddenly applied loads, Distribution of direct stresses on uniform and
non-uniform members, Thermal stresses and strains
Bending Theory:
Theory of simple bending, position of neutral axis, moment of
resistance and section modulus, Bending and shearing stress
distribution in beams, Relationship between load, shear force and
bending moment, Stresses in composite sections.
Slope and Deflection:
Curvature, slope and deflection of beams using integration methods
Theory of Torsion:
Theory of torsion of solids and hollow circular shafts, shearing
stress
distribution, angle of twist, strength and stiffness of shaft.
Biaxial state of stress:
Biaxial state of stresses, resolution of stresses, Principal
plane, principal stresses and strains, Graphical representation
of stress and strains, Mohr’s circle of stresses and strains.
3
DIFFERENT STRESS STATE –
DIRECT/UNIAXIAL STRESS & DISTRIBUTION
7
13/10/2023
CE‐205: Mechanics of Solids‐I
LECTURES
Introduction
 Introduction
 Relationships between Elastic Constants
Different Stress State
 Uniaxial State of Stresses and Strains
 Distribution of Direct Stresses on Uniform and Non‐uniform
Members
 Shearing Stress and Strains
Bending Theory
 Sectional Properties (position of neutral axis, moment of resistance
and section modulus)
 Theory of simple bending
 Bending and shearing stress distribution in beams
 Relationship between load, shear force and bending moment
 Stresses in composite sections
8
CE‐205: Mechanics of Solids‐I
LECTURES
(After MID-TERM)
Slope and Deflection
 Integration Methods
Biaxial State of Stress
 Biaxial State of Stresses
 Resolution of Stresses
 Principal plane, principal stresses and strains
 Mohr’s circle of stresses and strains
Theory of Torsion
 Theory of Torsion
4
DIFFERENT STRESS STATE –
DIRECT/UNIAXIAL STRESS & DISTRIBUTION
13/10/2023
CE‐205: MECHANICS OF SOLIDS‐I
9
Lecture Etiquette
CE‐205: MECHANICS OF SOLIDS‐I
10
Direct/Normal Stress or Strain
Uniaxial State of Stress & Strain
Stress, like pressure, is a term used to describe the intensity of a force—the
quantity of force that acts on a unit of area.
 Direct Stress (Average Normal Stress) = F/A
 Units: Usually N/m2 (Pa), N/mm2,
GN/m2 or N/cm2
 Note: 1 N/mm2 = 1 MN/m2 = 1 Mpa
MN/m2,
Strain, When a force is transmitted through a body, the body
tends to change its shape or deform. The body is said to be

strained.
Strain   
L
5
DIFFERENT STRESS STATE –
DIRECT/UNIAXIAL STRESS & DISTRIBUTION
11
13/10/2023
CE‐205: MECHANICS OF SOLIDS‐I
Direct/Normal Stress or Strain
Example #1
12
CE‐205: MECHANICS OF SOLIDS‐I
Direct/Normal Stress or Strain
Example #2
6
DIFFERENT STRESS STATE –
DIRECT/UNIAXIAL STRESS & DISTRIBUTION
13/10/2023
UE‐251: MECHANICS OF SOLIDS‐I
13
Direct/Normal Stress or Strain
Example #4
Multiple sections: Find total deformation of end A with respect to D. Area = 20 mm2. Material
is steel w/ E = 200 GPa = 200 x 109 Pa:
100 mm
150 mm
+  A/ D 
()
200 mm

 A/ D 
 A/D 
 A/D 
PL
AE
P L
 CD CD
AE

PAB LAB PBC LBC

AE
AE
(5,000N)(.1m) (3,000N)(.15m) (7,000N)(.2m)


AE
AE
AE
1,350
1,350

 3.38x104 m  0.338mm
9
AE
(.00002)(200x10 )
CE‐205: MECHANICS OF SOLIDS‐I
Different Stress State
Thermal Stresses
• A temperature change results in a change in length or
thermal strain. There is no stress associated with the
thermal strain unless the elongation is restrained by
the supports.
• Treat the additional support as redundant and apply
the principle of superposition.
PL
 T   T L
P 
AE
  thermal expansion coef.
• The thermal deformation and the deformation from
the redundant support must be compatible.
  T   P  0
 T L 
PL
0
AE
  T   P  0
P   AE T 

P
  E   T 
A
7
DIFFERENT STRESS STATE –
DIRECT/UNIAXIAL STRESS & DISTRIBUTION
13/10/2023
CE‐205: MECHANICS OF SOLIDS‐I
15
Distribution of Stress
Axial Stress = P/A
Two variables, if one changes stresses & strain change
Bending Stress = ± M(y)/I = ± Pe/I
NEXT LECTURE
8