This document gives the class notes of Unit 6: Bending and shear Stresses in beams. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Bending Stresses are important in the design of beams from strength point of view. The present source gives an idea on theory and problems in bending stresses.
Bending Stresses are important in the design of beams from strength point of view. The present source gives an idea on theory and problems in bending stresses.
In engineering, deflection is the degree to which a structural element is displaced under a load. It may refer to an angle or a distance.
The deflection distance of a member under a load is directly related to the slope of the deflected shape of the member under that load, and can be calculated by integrating the function that mathematically describes the slope of the member under that load. Deflection can be calculated by standard formula (will only give the deflection of common beam configurations and load cases at discrete locations), or by methods such as virtual work, direct integration, Castigliano's method, Macaulay's method or the direct stiffness method, amongst others. The deflection of beam elements is usually calculated on the basis of the Euler–Bernoulli beam equation while that of a plate or shell element is calculated using plate or shell theory.
This document gives the class notes of Unit 5 shear force and bending moment in beams. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Lecture slides on the calculation of the bending stress in case of unsymmetrical bending. The Mohr's circle is used to determine the principal second moments of area.
Design of steel structure as per is 800(2007)ahsanrabbani
It does not offer resistance against rotation and also termed as a hinged or pinned connections.
It transfers only axial or shear forces and it is not designed for moment
It is generally connected by single bolt/rivet and therefore full rotation is allowed
This document gives the class notes of Unit 3 Compound stresses. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Whenever a body is subjected to an axial tension or compression, a direct stress comes into play at every section of body. We also know that whenever a body is subjected to a bending moment a bending moment a bending stress comes into play.
In engineering, deflection is the degree to which a structural element is displaced under a load. It may refer to an angle or a distance.
The deflection distance of a member under a load is directly related to the slope of the deflected shape of the member under that load, and can be calculated by integrating the function that mathematically describes the slope of the member under that load. Deflection can be calculated by standard formula (will only give the deflection of common beam configurations and load cases at discrete locations), or by methods such as virtual work, direct integration, Castigliano's method, Macaulay's method or the direct stiffness method, amongst others. The deflection of beam elements is usually calculated on the basis of the Euler–Bernoulli beam equation while that of a plate or shell element is calculated using plate or shell theory.
This document gives the class notes of Unit 5 shear force and bending moment in beams. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Lecture slides on the calculation of the bending stress in case of unsymmetrical bending. The Mohr's circle is used to determine the principal second moments of area.
Design of steel structure as per is 800(2007)ahsanrabbani
It does not offer resistance against rotation and also termed as a hinged or pinned connections.
It transfers only axial or shear forces and it is not designed for moment
It is generally connected by single bolt/rivet and therefore full rotation is allowed
This document gives the class notes of Unit 3 Compound stresses. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Whenever a body is subjected to an axial tension or compression, a direct stress comes into play at every section of body. We also know that whenever a body is subjected to a bending moment a bending moment a bending stress comes into play.
This document contains: Mechanics of Materials: Question bank from old VTU Question papers ; Pprepared by Hareesha N G, DSCE, Bengaluru. These questions are picked from last 06 years of old VTU question papers.
This document gives the class notes of Unit 2 stresses in composite sections. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Assembly of Machine vice, Computer Aided Machine Drawing (CAMD) of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
Assembly of Ramsbottom safety valve, Computer Aided Machine Drawing (CAMD) of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
Assembly of Tail stock, Computer Aided Machine Drawing (CAMD) of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
Orthographic projections, Computer Aided Machine Drawing (CAMD) of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
Cotter and knuckle joints, Computer Aided Machine Drawing (CAMD) of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
This document gives the class notes of Unit-8: Torsion of circular shafts and elastic stability of columns. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Assembly of Connecting rod, Computer Aided Machine Drawing (CAMD) of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
This document is an Instruction manual for Computer aided machine drawing
Subject: Computer aided machine drawing (CAMD)
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Assembly of screw jack, Computer Aided Machine Drawing (CAMD) of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
Plummer block, Computer Aided Machine Drawing (CAMD) of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
Section of solids, Computer Aided Machine Drawing (CAMD) of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
This document gives the class notes of Unit-8: Torsion of circular shafts and elastic stability of columns. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
1. In this module we will determine the stress in a
beam caused by bending.
2. How to find the variation of the shear and
moment in these members.
3. Then once the internal moment is determined,
the maximum bending stress can be calculated.
This presentation gives the information about introduction to control systems
Subject: Control Engineering as per VTU Syllabus of Aeronautical Engineering.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Disclaimer:
The contents used in this presentation are taken from the text books mentioned in the references. I do not hold any copyrights for the contents. It has been prepared to use in the class lectures, not for commercial purpose.
This template was created for DSCE, Aeronautical students. You have to replace the institution details.
Create a separate document for each chapter, so that under numbering, you can change the sequence of chapter main heading according to chapter wise. i.e., 2.1, 2.2 etc.
Same procedure is applicable to Figure caption and Table caption.
This template can be used to generate, BE seminar report, M.Tech and Ph.D thesis also.
This template is created to assist UG students in generating their thesis without much hassle.
Contents are taken from VTU website. I don’t hold any copyright for this document.
Hareesha N G
Assistant Professor
DSCE, Bengaluru
This presentation was prepared for a seminar. I have shared this with you. This is not related to curriculam. Please writre your criticisms to: hareeshang@gmail.com.
This presentation gives the information about Screw thread measurements and Gear measurement of the subject: Mechanical measurement and Metrology (10ME32/42) of VTU Syllabus covering unit-4.
This presentation gives the information about Force, Pressure and Torque measurements of the subject: Mechanical measurement and Metrology (10ME32/42) of VTU Syllabus covering unit-6.
This presentation gives the information about mechanical measurements and measurement systems of the subject: Mechanical measurement and Metrology (10ME32/42) of VTU Syllabus covering unit-5.
This CIM and automation laboratory manual covers the G-Codes and M-codes for CNC Turning and Milling operations. Some concepts of Robot programming are also introduced.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
2. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 1
TABLE OF CONTENTS
6.1. INTRODUCTION ................................................................................................................................................ 2
6.2. SIMPLE BENDING ............................................................................................................................................. 2
6.3. ASSUMPTIONS IN SIMPLE BENDING .......................................................................................................... 3
6.4. DERIVATION OF BENDING EQUATION...................................................................................................... 3
6.5. SECTION MODULUS......................................................................................................................................... 5
6.6. MOMENT CARRYING CAPACITY OF A SECTION ................................................................................... 6
6.7. SHEARING STRESSES IN BEAMS.................................................................................................................. 8
6.8 SHEAR STRESSES ACROSS RECTANGULAR SECTIONS ........................................................................ 9
WORKED EXAMPLES ........................................................................................................................................... 11
REFERENCES:......................................................................................................................................................... 15
3. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 2
UNIT-6
BENDING AND SHEAR STRESSES IN BEAMS
Syllabus
Introduction, Theory of simple bending, assumptions in simple bending, Bending stress
equation, relationship between bending stress, radius of curvature, relationship between bending
moment and radius of curvature, Moment carrying capacity of a section. Shearing stresses in
beams, shear stress across rectangular, circular, symmetrical I and T sections. (composite /
notched beams not included).
6.1. INTRODUCTION
When some external load acts on a beam, the shear force and bending moments are set up
at all sections of the beam. Due to the shear force and bending moment, the beam undergoes
certain deformation. The material of the beam will offer resistance or stresses against these
deformations. These stresses with certain assumptions can be calculated. The stresses introduced
by bending moment are known as bending stresses. In this chapter, the theory of pure bending,
expression for bending stresses, bending stress in symmetrical and unsymmetrical sections,
strength of a beam and composite beams will be discussed.
E.g., Consider a piece of rubber, most conveniently of rectangular cross-section, is bent
between one’s fingers it is readily apparent that one surface of the rubber is stretched, i.e. put into
tension, and the opposite surface is compressed.
6.2. SIMPLE BENDING
A theory which deals with finding stresses at a section due to pure moment is called
bending theory. If we now consider a beam initially unstressed and subjected to a constant B.M.
along its length, it will bend to a radius R as shown in Fig. b. As a result of this bending the top
fibres of the beam will be subjected to tension and the bottom to compression. Somewhere
between the two surfaces, there are points at which the stress is zero. The locus of all such points
is termed the neutral axis (N.A). The radius of curvature R is then measured to this axis. For
symmetrical sections the N.A. is the axis of symmetry, but whatever the section the N.A. will
always pass through the centre of area or centroid.
4. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 3
Beam subjected to pure bending (a) before, and (b) after, the moment
M has been applied.
In simple bending the plane of transverse loads and the centroidal plane coincide. The theory of
simple bending was developed by Galelio, Bernoulli and St. Venant. Sometimes this theory is
called Bernoulli's theory of simple bending.
6.3. ASSUMPTIONS IN SIMPLE BENDING
The following assumptions are made in the theory of simple bending:
1 The beam is initially straight and unstressed.
2 The material of the beam is perfectly homogeneous and isotropic, i.e. of the same density
and elastic properties throughout.
3 The elastic limit is nowhere exceeded.
4 Young's modulus for the material is the same in tension and compression.
5 Plane cross-sections remain plane before and after bending.
6 Every cross-section of the beam is symmetrical about the plane of bending, i.e. about an
axis perpendicular to the N.A.
7 There is no resultant force perpendicular to any cross-section.
8 The radius of curvature is large compared to depth of beam.
6.4. DERIVATION OF BENDING EQUATION
Consider a length of beam under the action of a bending moment M as shown in Fig. 6.2a. N-N is
the original length considered of the beam. The neutral surface is a plane through X-X. In the side
view NA indicates the neutral axis. O is the centre of curvature on bending (Fig. 6.2b).
5. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 4
Fig. 6.2
Let R = radius of curvature of the neutral surface
= angle subtended by the beam length at centre O
= longitudinal stress
A filament of original length NN at a distance v from the neutral axis will be elongated to a
length AB
..(i)
Thus stress is proportional to the distance from the neutral axis NA. This suggests that for the
sake of weight reduction and economy, it is always advisable to make the cross-section of beams
such that most of the material is concentrated at the greatest distance from the neutral axis. Thus
there is universal adoption of the I-section for steel beams. Now let A be an element of cross-
sectional area of a transverse plane at a distance v from the neutral axis NA (Fig. 6.2).
For pure bending, Net normal force on the cross-section = 0
6. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 5
This indicates the condition that the neutral axis passes through the centroid of the section. Also,
bending moment = moment of the normal forces about neutral axis
Or
R
E
I
M
(ii)
Where dAyI 2
and is known as the moment of inertia or second moment of area of the
section. From (i) and (ii),
yR
E
I
M
Where,
M = Bending Moment at a section (N-mm).
I = Moment of Inertia of the cross section of the beam about Neutral axis (mm4
).
= Bending stress in a fibre located at distance y from neutral axis (N/mm2
). This stress could be
compressive stress or tensile stress depending on the location of the fibre.
y = Distance of the fibre under consideration from neutral axis (mm).
E = Young's Modulus of the material of the beam (N/mm2
).
R = Radius of curvature of the bent beam (mm).
6.5. SECTION MODULUS
The maximum tensile and compressive stresses in the beam occur at points located farthest from
the neutral axis. Let us denote y1 and y2 as the distances from the neutral axis to the extreme
fibres at the top and the bottom of the beam. Then the maximum bending normal stresses are
t
bc
Z
M
yI
M
I
My
1
1
, bc is bending compressive stress in the topmost layer.
Similarly,
b
bt
Z
M
yI
M
I
My
2
2
, bt is bending compressive stress in the topmost layer.
Here, Zt and Zb are called section moduli of the cross sectional area, and they have dimensions
of length to the third power (ex. mm3
). If the cross section is symmetrical (like rectangular or
square sections), then Zt = Zb = Z, and Z is called as section modulus. Section modulus is defined
7. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 6
as the ratio of rectangular moment of inertia of the section to the distance of the remote layer
from the neutral axis. Thus, general expression for bending stress reduces to
Z
M
It is seen from the above expression that for a given bending moment, it is in the best interests of
the designer of the beam to procure high value for section modulus so as to minimise the bending
stress. More the section modulus designer provides for the beam, less will be the bending stress
generated for a given value of bending moment.
6.6. MOMENT CARRYING CAPACITY OF A SECTION
From bending equation we have
I
My
It shows bending stress is maximum on the extreme fibre where y is maximum. In any design this
extreme fibre stress should not exceed maximum permissible stress. If per is the permissible
stress, then in a design
per max
pery
I
M
Or if M is taken as maximum moment carrying capacity of the section,
pery
I
M
max
Or per
y
I
M
max
The moment of inertia I and extreme fibre distance ymax are the properties of cross-section.
Hence, I/ymax is the property of cross-sectional area and is termed as section modulus and is
denoted by Z. Thus the moment carrying capacity of a section is given by
ZM per
If permissible stresses in tension and compression are different, moment carrying capacity in
tension and compression are found separately by considering respective extreme fibres and the
smallest one is taken as moment carrying capacity of the section.
8. Mechanics of Materials 10ME34
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Expressions for section modulus of various standard cross-sections are derived below.
Rectangular section of width b and depth d:
Hollow rectangular section with symmetrically
placed opening:
Consider the section of size B x
D with
symmetrical opening bx d as shown in Fig..
Circular section of diameter d
Hollow circular section of uniform thickness:
9. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 8
Triangular Section
6.7. SHEARING STRESSES IN BEAMS
we know that beams are usually subjected to varying bending moment and shearing
forces. The relation between bending moment M and shearing force F is dM/dx=F.
Bending stress act longitudinally and its intensity is directly proportional to its distance
from neutral axis. Now we will find the stresses induced by shearing force.
Consider an elemental length of beam between the sections A-A and B- B
separated by a distance dx as shown in Fig. 6.3a. Let the moments acting at A- A and B-B
be M and M+dM.
Fig. 6.3
10. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 9
Let CD be a fibre at a distance y-from neutral axis. Then bending stress at left side of the element
= y
I
M
The force on the element on left side
= ybdy
I
M
Similarly due to bending, force on the right side of the element
= ybdy
I
dMM
Unbalanced force towards right in element
= ybdy
I
dM
ybdy
I
M
ybdy
I
dMM
There are a number of such elements above section CD. Hence unbalance horizontal force above
section CD
=
'
y
y
ybdy
I
dM
This horizontal force is resisted by shearing stresses acting horizontally on plane at CD. Let
intensity of shearing stress be q. Then equating shearing force to unbalanced horizontal force we
get
=
'
y
y
ybdy
I
dM
bdx
Or
'
1
y
y
ya
bIdx
dM
Where a = b dy is area of element.
The term
'
y
y
ya can be looked as
yaay
y
y
'
Where ya is the moment of area above the section under consideration about neutral axis.
From equation, dM/dx=F
ya
bI
F
From the above expression it may be noted that shearing stress on extreme fibre is zero.
6.8 SHEAR STRESSES ACROSS RECTANGULAR SECTIONS
Consider a rectangular section of width b and depth d subjected to shearing force F. Let A-A be
the section at distance y from neutral axis as shown in Fig. 6.4.
We know that shear stress at this section.
ya
bI
F
11. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 10
Fig. 6.4
where ya is the moment of area above this section (shown shaded) about the neutral axis.
i.e., shear stress varies parabolically.
When y=d/2,
y=d/2,
y = 0, is maximum and its value is
4
6 2
3max
d
bd
F
avg
bd
F
5.15.1
Where
bd
F
Area
rceShearingFo
avg
Thus, maximum shear stress is 1.5 times the average shear stress in rectangular section and
occurs at the neutral axis. Shear stress variation is parabolic. Shear stress variation diagram
across the section is shown in Fig.6.4b.
12. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 11
WORKED EXAMPLES
1) A simply supported beam of span 5 m has a cross-section 150 mm * 250 mm. If the
permissible stress is 10 N/mm2
, find (a) maximum intensity of uniformly distributed load it
can carry. (b) maximum concentrated load P applied at 2 m from one end it can carry.
Solution:
Moment carrying capacity M = Z = l0 x 1562500 N - mm
(a) If w is the intensity of load in N/m units, then maximum moment
Equating it to moment carrying capacity, we get maximum intensity of load as
(b) If P is the concentrated load as shown in Fig., then maximum moment occurs under the load
and its value
2) A symmetric I-section has flanges of size 180 mm x 10 mm and its overall depth is 500 mm.
Thickness of web is 8 mm. It is strengthened with a plate of size 240 mm x 12 mm on
compression side. Find the moment of resistance of the section, if permissible stress is 150
N/mm2
. How much uniformly distributed load it can carry if it is used as a cantilever of span
3 m?
Solution
The section of beam is as shown in Fig. Let y be the distance of centroid from the bottom-most
fibre.
13. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 12
Moment of resistance (Moment carrying capacity)
14. Mechanics of Materials 10ME34
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Let the load on cantilever be w/m length as shown in Fig.
3) A T-section is formed by cutting the bottom flange of an I-section. The flange is 100 mm x
20 mm and the web is 150 mm x 20 mm. Draw the bending stress distribution diagrams if
bending moment at a section of the beam is 10 kN-m (hogging).
Solution
M = 10 kN-m = 10 x 106
N mm (hogging)
Maximum bending stresses occur at extreme fibres, i.e. at the top bottom fibres which can be
computed as
I
My
(i)
Moment of inertia is given by
Substituting these values in Eq. (1),
Stress in the top fibre =
Stress in the bottom fibre =
15. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 14
The given bending moment is hogging and hence negative and the tensile stresses occur at top
fibre and compressive stresses in bottom fibres.
4) Fig. shows the cross-section of a beam which is subjected to a shear force of 20 kN. Draw
shear stress distribution across the depth marking values at salient points.
Solution
Let y, be the distance of C.G form top fibre. Then taking moment of area about top
fibre and dividing it by total area, we get
16. Mechanics of Materials 10ME34
Compiled by Hareesha N G, Asst Prof, DSCE Page 15
ya
bI
F
REFERENCES:
1) A Textbook of Strength of Materials By R. K. Bansal
2) Fundamentals Of Strength Of Materials By P. N. Chandramouli
3) Strength of Materials By B K Sarkar
4) Strength of Materials S S Bhavikatti
5) Textbook of Mechanics of Materials by Prakash M. N. Shesha, suresh G. S.
6) Mechanics of Materials: with programs in C by M. A. Jayaram