Informacion de vigas y la forma de análisis solución de problemáticas referente a las vigas y su formulación matemática y analítica. Teniendo en cuenta cada una de sus variables como inercia, fuerzas cortantes, flexiones entre muchas más. También se muestran casos de la vida real donde se realiza mal análisis de vigas.
The document discusses torsion and torsional deformation of circular shafts. It defines torsion as a moment that twists a member about its longitudinal axis. For a circular shaft under pure torsion, the angle of twist is linearly proportional to the distance along the shaft. The maximum shear stress occurs at the outer surface of the shaft and is calculated using the torsion formula. Non-uniform torsion is analyzed by dividing the shaft into segments or using differential elements and integrating along the length. The document also provides examples of solving for shear stress and required shaft diameter given applied torques.
Chapter 6: Pure Bending and Bending with Axial ForcesMonark Sutariya
This summary provides the key points about pure bending and bending with axial forces from the document:
1. Pure bending occurs when a beam segment is in equilibrium under bending moments alone, with examples being a cantilever loaded at the end or a beam segment between concentrated forces.
2. For beams with symmetric cross-sections, plane sections remain plane after bending according to the fundamental flexure theory. The elastic flexure formula gives the normal stress as proportional to the bending moment and the distance from the neutral axis.
3. The second moment of area, or moment of inertia, represents the beam's resistance to bending and is used to calculate maximum bending stresses. The elastic section modulus is a ratio of the moment
This document summarizes concepts related to torsion and the torsion of circular elastic bars. It discusses the assumptions made in analyzing torsion, including that shear strain varies linearly from the central axis. It also covers determining shear stress and torque using the polar moment of inertia for circular cross-sections. The relationships between applied torque, shear stress, shear strain, and angle of twist are defined. Stress concentrations and alternative differential equations approaches are also summarized.
The document discusses torsion and torsional loads on circular shafts. It defines torsion as twisting of a structural member when loaded by couples that produce rotation about its longitudinal axis. It describes how torsional loads produce shearing stresses on planes perpendicular to the shaft axis and axial shear stresses. Formulas are provided for shear strain, shear stress, angle of twist, and polar moment of inertia in elastic torsion applications. Examples are included to demonstrate calculating shear stresses and sizing shafts based on allowable stresses. Design considerations for transmission shafts transmitting power at specified speeds are also briefly covered.
Shear Force Diagram is a graphical representation of shear force (V) along the length of a beam subjected to external loads. It is an analytical tool used in structural design to determine member type, size, and material. There are two methods for constructing SFDs - basic and integration. The basic method involves determining support reactions, sectioning the beam, drawing free body diagrams, and calculating shear and moment as a function of position along the beam. SFDs can be used to find maximum shear values, bending moment diagrams, and deflection of beams under different loading conditions such as concentrated, uniform, and variable loads as demonstrated in the examples.
Shear Force Diagram is a graphical representation of shear force (V) along the length of a beam subjected to external loads. It is an analytical tool used in structural design to determine member type, size, and material. There are two methods for constructing SFDs - basic and integration. The basic method involves determining support reactions, sectioning the beam, drawing free body diagrams, and calculating shear and moment as a function of position along the beam. SFDs can be used to find maximum shear values, bending moment diagrams, and deflection of beams under different loading conditions such as concentrated, uniform, and variable loads as demonstrated in the examples.
This document discusses flexural stresses and bending formulas. It defines bending stresses as stresses that occur in a beam under a constant bending moment with zero shear force. It describes the neutral axis and how fibers above and below experience compression and tension. Formulas are derived relating bending stress to radius of curvature and section modulus. Section modulus is defined as the ratio of moment of inertia to the distance of the outermost fiber. Examples are given of calculating section modulus for common shapes and solving bending stress problems for beams.
CE 72.52 Lecture 4 - Ductility of Cross-sectionsFawad Najam
This document provides information on ductility of concrete structures. It discusses how ductility is key to good seismic performance of structures. Ductility is defined and different levels of ductility are described, from the material level to the structural level. Factors that affect ductility include confinement of concrete, reinforcement, cross-section shape, and applied loads. Moment-curvature relationships are used to compute ductility at the cross-section level. Confinement improves concrete ductility by modifying its stress-strain behavior. Spiral reinforcement increases concrete strength under triaxial compression. Moment-curvature curves can indicate yield points and failure mechanisms for different types of sections.
The document discusses torsion and torsional deformation of circular shafts. It defines torsion as a moment that twists a member about its longitudinal axis. For a circular shaft under pure torsion, the angle of twist is linearly proportional to the distance along the shaft. The maximum shear stress occurs at the outer surface of the shaft and is calculated using the torsion formula. Non-uniform torsion is analyzed by dividing the shaft into segments or using differential elements and integrating along the length. The document also provides examples of solving for shear stress and required shaft diameter given applied torques.
Chapter 6: Pure Bending and Bending with Axial ForcesMonark Sutariya
This summary provides the key points about pure bending and bending with axial forces from the document:
1. Pure bending occurs when a beam segment is in equilibrium under bending moments alone, with examples being a cantilever loaded at the end or a beam segment between concentrated forces.
2. For beams with symmetric cross-sections, plane sections remain plane after bending according to the fundamental flexure theory. The elastic flexure formula gives the normal stress as proportional to the bending moment and the distance from the neutral axis.
3. The second moment of area, or moment of inertia, represents the beam's resistance to bending and is used to calculate maximum bending stresses. The elastic section modulus is a ratio of the moment
This document summarizes concepts related to torsion and the torsion of circular elastic bars. It discusses the assumptions made in analyzing torsion, including that shear strain varies linearly from the central axis. It also covers determining shear stress and torque using the polar moment of inertia for circular cross-sections. The relationships between applied torque, shear stress, shear strain, and angle of twist are defined. Stress concentrations and alternative differential equations approaches are also summarized.
The document discusses torsion and torsional loads on circular shafts. It defines torsion as twisting of a structural member when loaded by couples that produce rotation about its longitudinal axis. It describes how torsional loads produce shearing stresses on planes perpendicular to the shaft axis and axial shear stresses. Formulas are provided for shear strain, shear stress, angle of twist, and polar moment of inertia in elastic torsion applications. Examples are included to demonstrate calculating shear stresses and sizing shafts based on allowable stresses. Design considerations for transmission shafts transmitting power at specified speeds are also briefly covered.
Shear Force Diagram is a graphical representation of shear force (V) along the length of a beam subjected to external loads. It is an analytical tool used in structural design to determine member type, size, and material. There are two methods for constructing SFDs - basic and integration. The basic method involves determining support reactions, sectioning the beam, drawing free body diagrams, and calculating shear and moment as a function of position along the beam. SFDs can be used to find maximum shear values, bending moment diagrams, and deflection of beams under different loading conditions such as concentrated, uniform, and variable loads as demonstrated in the examples.
Shear Force Diagram is a graphical representation of shear force (V) along the length of a beam subjected to external loads. It is an analytical tool used in structural design to determine member type, size, and material. There are two methods for constructing SFDs - basic and integration. The basic method involves determining support reactions, sectioning the beam, drawing free body diagrams, and calculating shear and moment as a function of position along the beam. SFDs can be used to find maximum shear values, bending moment diagrams, and deflection of beams under different loading conditions such as concentrated, uniform, and variable loads as demonstrated in the examples.
This document discusses flexural stresses and bending formulas. It defines bending stresses as stresses that occur in a beam under a constant bending moment with zero shear force. It describes the neutral axis and how fibers above and below experience compression and tension. Formulas are derived relating bending stress to radius of curvature and section modulus. Section modulus is defined as the ratio of moment of inertia to the distance of the outermost fiber. Examples are given of calculating section modulus for common shapes and solving bending stress problems for beams.
CE 72.52 Lecture 4 - Ductility of Cross-sectionsFawad Najam
This document provides information on ductility of concrete structures. It discusses how ductility is key to good seismic performance of structures. Ductility is defined and different levels of ductility are described, from the material level to the structural level. Factors that affect ductility include confinement of concrete, reinforcement, cross-section shape, and applied loads. Moment-curvature relationships are used to compute ductility at the cross-section level. Confinement improves concrete ductility by modifying its stress-strain behavior. Spiral reinforcement increases concrete strength under triaxial compression. Moment-curvature curves can indicate yield points and failure mechanisms for different types of sections.
The document discusses methods for determining internal forces like shear, moment and normal force in structural members. It describes:
1) Using a method of sections to determine internal forces at a specified point by analyzing the external loads and support reactions.
2) Shear and moment functions vary across a member depending on the type and location of loads. They can be determined by drawing free body diagrams of small segments.
3) Shear and moment diagrams are created by plotting the variation of shear and moment across a member. Examples show how to construct these diagrams for beams and frames.
This document provides an introduction to bending and shear in simple beams. It discusses the different types of loads, supports, and internal forces that beams experience. Beams can experience bending moments and shear forces from applied loads. Shear and moment diagrams can be constructed to understand the internal forces along the beam. The diagrams are developed using the equilibrium method or semi-graphical method based on the loads, supports, and geometry of the beam. An example problem demonstrates constructing shear and moment diagrams.
CE 72.52 - Lecture 7 - Strut and Tie ModelsFawad Najam
The document discusses the strut-and-tie approach for analyzing concrete structures. It begins with background concepts such as Bernoulli's hypothesis, St. Venant's principle, and the lower bound theorem of plasticity. It then discusses how axial stresses, shear stresses, and the interaction of stresses affect concrete sections. The document outlines the ACI approach to shear-torsion design and provides equations from ACI 318 for calculating the concrete shear capacity. It introduces the concept of modeling concrete as a truss system and compares this to flexural behavior in beams. The strut-and-tie method is presented as a unified approach for considering all load effects. Guidelines are provided for developing an appropriate strut-and-tie model and
Shear Force And Bending Moment In Beams22Amr Hamed
The document discusses concepts related to shear force and bending moment in beams including:
1) Definitions of bending, beams, planar bending, and different types of beams.
2) Methods for simplifying beams, loads, and supports for calculation.
3) Concepts of internal forces including shear force, bending moment, and sign conventions.
4) Relations between shear force, bending moment, and distributed loads.
5) Methods for plotting shear force and bending moment diagrams including using superposition.
6) Application to planar rigid frames.
The document discusses concepts related to shear force and bending moment in beams, including:
- Definitions of bending, beams, planar bending, and types of beams including simple, cantilever, and overhanging beams.
- Calculation sketches simplify beams, loads, and supports for analysis.
- Internal forces in bending include shear force and bending moment. Relations and diagrams relate these to external loads.
- Equations define shear force and bending moment at each beam section. Diagrams illustrate variations along the beam.
Lec03 Flexural Behavior of RC Beams (Reinforced Concrete Design I & Prof. Abd...Hossam Shafiq II
The document discusses the behavior and analysis of reinforced concrete beams. It describes three stages that beams undergo as loading increases: 1) the uncracked concrete stage, 2) the cracked-elastic stage, and 3) the ultimate strength stage. It also discusses assumptions made in flexural theory, stress-strain curves for concrete and steel, and methods for calculating stresses in uncracked and cracked beams using the transformed area method. Key points covered include cracking moment, modular ratio, and the three-step transformed area method for cracked sections.
The document discusses the behavior and analysis of reinforced concrete beams. It describes the three stages a beam undergoes when loaded: uncracked, cracked-elastic, and ultimate strength. The transformed area method is presented for calculating stresses in cracked beams. An example problem demonstrates using this method to find bending stresses in a beam section. The allowable resisting moment is also determined based on specified material stresses.
- The document defines concepts related to forces, stress, strain and elasticity. It also discusses shear force, bending moment, and torsion.
- Examples are provided for drawing shear force and bending moment diagrams for cantilever beams and simply supported beams under different loading conditions like point loads, uniformly distributed loads.
- The key steps involve calculating reactions, determining shear force and bending moment values at different points, and plotting them on diagrams to understand their variations along the beam.
This document discusses torsion and stress distribution in circular torsion bars. It begins with an introduction to torsional loads and deformation of circular bars. It then covers key topics like:
- Shear stress distribution varies linearly with distance from the axis for circular bars.
- Angle of twist is proportional to applied torque and bar length.
- Examples are provided to calculate minimum/maximum shear stresses, required diameters, and angles of twist using elastic torsion formulas.
The document provides an overview of mechanics of materials concepts related to torsion, including:
- Torsion causes shearing stresses that vary linearly from zero at the center to a maximum at the surface for circular shafts.
- Torsion can cause both shearing stresses and normal stresses depending on the orientation of the material element.
- Ductile materials fail in shear while brittle materials fail in tension when subjected to torsion.
- The angle of twist is proportional to the applied torque, material properties, and shaft length based on elastic torsion formulas.
- Stress concentrations can occur due to geometric discontinuities and influence the maximum shearing stress.
This document discusses bending of structural sections and provides examples of calculating stresses due to bending. It begins with an introduction to symmetrical and unsymmetrical bending. Key concepts discussed include the second moment of inertia, stress distribution in symmetrical sections, and the equation for the neutral axis in unsymmetrical bending. Examples are then provided to demonstrate calculating stresses in bent sections, finding the location of the neutral axis, and determining maximum stresses. The document concludes with tutorials for students to practice applying the bending concepts.
This document discusses the design of continuous beams. It notes that continuous beams must be designed to resist hogging moments at supports in addition to sagging moments in spans. An example three-span continuous beam is then designed. The beam has a total factored load of 80.57 kN/m and 6.1m spans. Elastic analysis finds maximum moments of 239.94 kN.m in end spans and -299.80 kN.m at interior supports. The beam is designed with a depth of 530mm and reinforcement is checked for bending, shear, development length, and deflection requirements.
The document discusses centroids and centers of gravity. It defines key terms like centroid of mass, centroid of area, and center of gravity. It describes how to calculate the centroid of common shapes using integrals and the theorems of Pappus-Guldinus. Sample problems show how to determine the centroid of a composite area and the mass and weight of a pulley rim using the centroid concepts. Distributed loads on beams are also discussed.
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.
This document summarizes the moment-area method for calculating deflections in beams. It discusses how the bending moment diagram can be divided into areas that correspond to rotations of the elastic curve. The sum of these areas multiplied by the distance to the centroid gives the tangential deviation, which can be used to determine the deflection. The method is applicable to statically indeterminate beams using superposition. Boundary conditions and how to handle different support types are also covered.
This document provides information about a Mechanics of Materials course, including:
- The instructor's name and credentials.
- An overview of course contents including stresses, strains, torsion, bending, centroids, and beam deflection methods.
- An introduction to mechanics of materials and the objectives to analyze stresses, strains, and displacements in structures.
- Key terms like stress, strain, axial force, normal force, shear force, deformation, prismatic and non-prismatic bars are defined.
- The stress-strain diagram is discussed and key points like the elastic region, proportional limit, yield point, strain hardening, ultimate stress, and necking are explained.
This document summarizes key concepts in strength of materials including:
- Analysis of pure bending and symmetrical sections bending in a plane of symmetry
- Skew loading and bending about axes other than axes of symmetry
- Eccentric loading introducing both direct stress and bending stress
- The middle third rule and middle quarter rule defining safe load application areas to avoid tension
This document discusses structural analysis and beam analysis. It covers topics such as support reactions, shear force and bending moment diagrams, equilibrium of forces in structures, and determinacy. Key points include:
1. To design structures, it is necessary to know bending moments, torsion moments, shear forces, and axial forces in each member.
2. Equilibrium requires that the downward forces from gravity have equal and opposite upward reaction forces.
3. Shear force and bending moment diagrams show the variation of these internal forces along a beam under different loading conditions.
Here are the key points about stresses in a thin-walled pressure vessel:
- Hoop stress acts circumferentially around the vessel and is caused by the internal pressure pushing outward on the wall.
- Hoop stress is calculated as σh = pr/t, where p is the internal pressure, r is the radius, and t is the wall thickness.
- Hoop stress wants to cause the vessel wall to bulge outward. Hoop stresses must be resisted by the vessel material to prevent failure.
- Axial/longitudinal stresses act along the axis of the vessel and are much smaller than hoop stresses. They are caused by the ends of the vessel being pushed together by the internal pressure.
The document discusses methods for determining internal forces like shear, moment and normal force in structural members. It describes:
1) Using a method of sections to determine internal forces at a specified point by analyzing the external loads and support reactions.
2) Shear and moment functions vary across a member depending on the type and location of loads. They can be determined by drawing free body diagrams of small segments.
3) Shear and moment diagrams are created by plotting the variation of shear and moment across a member. Examples show how to construct these diagrams for beams and frames.
This document provides an introduction to bending and shear in simple beams. It discusses the different types of loads, supports, and internal forces that beams experience. Beams can experience bending moments and shear forces from applied loads. Shear and moment diagrams can be constructed to understand the internal forces along the beam. The diagrams are developed using the equilibrium method or semi-graphical method based on the loads, supports, and geometry of the beam. An example problem demonstrates constructing shear and moment diagrams.
CE 72.52 - Lecture 7 - Strut and Tie ModelsFawad Najam
The document discusses the strut-and-tie approach for analyzing concrete structures. It begins with background concepts such as Bernoulli's hypothesis, St. Venant's principle, and the lower bound theorem of plasticity. It then discusses how axial stresses, shear stresses, and the interaction of stresses affect concrete sections. The document outlines the ACI approach to shear-torsion design and provides equations from ACI 318 for calculating the concrete shear capacity. It introduces the concept of modeling concrete as a truss system and compares this to flexural behavior in beams. The strut-and-tie method is presented as a unified approach for considering all load effects. Guidelines are provided for developing an appropriate strut-and-tie model and
Shear Force And Bending Moment In Beams22Amr Hamed
The document discusses concepts related to shear force and bending moment in beams including:
1) Definitions of bending, beams, planar bending, and different types of beams.
2) Methods for simplifying beams, loads, and supports for calculation.
3) Concepts of internal forces including shear force, bending moment, and sign conventions.
4) Relations between shear force, bending moment, and distributed loads.
5) Methods for plotting shear force and bending moment diagrams including using superposition.
6) Application to planar rigid frames.
The document discusses concepts related to shear force and bending moment in beams, including:
- Definitions of bending, beams, planar bending, and types of beams including simple, cantilever, and overhanging beams.
- Calculation sketches simplify beams, loads, and supports for analysis.
- Internal forces in bending include shear force and bending moment. Relations and diagrams relate these to external loads.
- Equations define shear force and bending moment at each beam section. Diagrams illustrate variations along the beam.
Lec03 Flexural Behavior of RC Beams (Reinforced Concrete Design I & Prof. Abd...Hossam Shafiq II
The document discusses the behavior and analysis of reinforced concrete beams. It describes three stages that beams undergo as loading increases: 1) the uncracked concrete stage, 2) the cracked-elastic stage, and 3) the ultimate strength stage. It also discusses assumptions made in flexural theory, stress-strain curves for concrete and steel, and methods for calculating stresses in uncracked and cracked beams using the transformed area method. Key points covered include cracking moment, modular ratio, and the three-step transformed area method for cracked sections.
The document discusses the behavior and analysis of reinforced concrete beams. It describes the three stages a beam undergoes when loaded: uncracked, cracked-elastic, and ultimate strength. The transformed area method is presented for calculating stresses in cracked beams. An example problem demonstrates using this method to find bending stresses in a beam section. The allowable resisting moment is also determined based on specified material stresses.
- The document defines concepts related to forces, stress, strain and elasticity. It also discusses shear force, bending moment, and torsion.
- Examples are provided for drawing shear force and bending moment diagrams for cantilever beams and simply supported beams under different loading conditions like point loads, uniformly distributed loads.
- The key steps involve calculating reactions, determining shear force and bending moment values at different points, and plotting them on diagrams to understand their variations along the beam.
This document discusses torsion and stress distribution in circular torsion bars. It begins with an introduction to torsional loads and deformation of circular bars. It then covers key topics like:
- Shear stress distribution varies linearly with distance from the axis for circular bars.
- Angle of twist is proportional to applied torque and bar length.
- Examples are provided to calculate minimum/maximum shear stresses, required diameters, and angles of twist using elastic torsion formulas.
The document provides an overview of mechanics of materials concepts related to torsion, including:
- Torsion causes shearing stresses that vary linearly from zero at the center to a maximum at the surface for circular shafts.
- Torsion can cause both shearing stresses and normal stresses depending on the orientation of the material element.
- Ductile materials fail in shear while brittle materials fail in tension when subjected to torsion.
- The angle of twist is proportional to the applied torque, material properties, and shaft length based on elastic torsion formulas.
- Stress concentrations can occur due to geometric discontinuities and influence the maximum shearing stress.
This document discusses bending of structural sections and provides examples of calculating stresses due to bending. It begins with an introduction to symmetrical and unsymmetrical bending. Key concepts discussed include the second moment of inertia, stress distribution in symmetrical sections, and the equation for the neutral axis in unsymmetrical bending. Examples are then provided to demonstrate calculating stresses in bent sections, finding the location of the neutral axis, and determining maximum stresses. The document concludes with tutorials for students to practice applying the bending concepts.
This document discusses the design of continuous beams. It notes that continuous beams must be designed to resist hogging moments at supports in addition to sagging moments in spans. An example three-span continuous beam is then designed. The beam has a total factored load of 80.57 kN/m and 6.1m spans. Elastic analysis finds maximum moments of 239.94 kN.m in end spans and -299.80 kN.m at interior supports. The beam is designed with a depth of 530mm and reinforcement is checked for bending, shear, development length, and deflection requirements.
The document discusses centroids and centers of gravity. It defines key terms like centroid of mass, centroid of area, and center of gravity. It describes how to calculate the centroid of common shapes using integrals and the theorems of Pappus-Guldinus. Sample problems show how to determine the centroid of a composite area and the mass and weight of a pulley rim using the centroid concepts. Distributed loads on beams are also discussed.
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.
This document summarizes the moment-area method for calculating deflections in beams. It discusses how the bending moment diagram can be divided into areas that correspond to rotations of the elastic curve. The sum of these areas multiplied by the distance to the centroid gives the tangential deviation, which can be used to determine the deflection. The method is applicable to statically indeterminate beams using superposition. Boundary conditions and how to handle different support types are also covered.
This document provides information about a Mechanics of Materials course, including:
- The instructor's name and credentials.
- An overview of course contents including stresses, strains, torsion, bending, centroids, and beam deflection methods.
- An introduction to mechanics of materials and the objectives to analyze stresses, strains, and displacements in structures.
- Key terms like stress, strain, axial force, normal force, shear force, deformation, prismatic and non-prismatic bars are defined.
- The stress-strain diagram is discussed and key points like the elastic region, proportional limit, yield point, strain hardening, ultimate stress, and necking are explained.
This document summarizes key concepts in strength of materials including:
- Analysis of pure bending and symmetrical sections bending in a plane of symmetry
- Skew loading and bending about axes other than axes of symmetry
- Eccentric loading introducing both direct stress and bending stress
- The middle third rule and middle quarter rule defining safe load application areas to avoid tension
This document discusses structural analysis and beam analysis. It covers topics such as support reactions, shear force and bending moment diagrams, equilibrium of forces in structures, and determinacy. Key points include:
1. To design structures, it is necessary to know bending moments, torsion moments, shear forces, and axial forces in each member.
2. Equilibrium requires that the downward forces from gravity have equal and opposite upward reaction forces.
3. Shear force and bending moment diagrams show the variation of these internal forces along a beam under different loading conditions.
Here are the key points about stresses in a thin-walled pressure vessel:
- Hoop stress acts circumferentially around the vessel and is caused by the internal pressure pushing outward on the wall.
- Hoop stress is calculated as σh = pr/t, where p is the internal pressure, r is the radius, and t is the wall thickness.
- Hoop stress wants to cause the vessel wall to bulge outward. Hoop stresses must be resisted by the vessel material to prevent failure.
- Axial/longitudinal stresses act along the axis of the vessel and are much smaller than hoop stresses. They are caused by the ends of the vessel being pushed together by the internal pressure.
Sri Guru Hargobind Ji - Bandi Chor Guru.pdfBalvir Singh
Sri Guru Hargobind Ji (19 June 1595 - 3 March 1644) is revered as the Sixth Nanak.
• On 25 May 1606 Guru Arjan nominated his son Sri Hargobind Ji as his successor. Shortly
afterwards, Guru Arjan was arrested, tortured and killed by order of the Mogul Emperor
Jahangir.
• Guru Hargobind's succession ceremony took place on 24 June 1606. He was barely
eleven years old when he became 6th Guru.
• As ordered by Guru Arjan Dev Ji, he put on two swords, one indicated his spiritual
authority (PIRI) and the other, his temporal authority (MIRI). He thus for the first time
initiated military tradition in the Sikh faith to resist religious persecution, protect
people’s freedom and independence to practice religion by choice. He transformed
Sikhs to be Saints and Soldier.
• He had a long tenure as Guru, lasting 37 years, 9 months and 3 days
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This is an overview of my current metallic design and engineering knowledge base built up over my professional career and two MSc degrees : - MSc in Advanced Manufacturing Technology University of Portsmouth graduated 1st May 1998, and MSc in Aircraft Engineering Cranfield University graduated 8th June 2007.
This is an overview of my career in Aircraft Design and Structures, which I am still trying to post on LinkedIn. Includes my BAE Systems Structural Test roles/ my BAE Systems key design roles and my current work on academic projects.
6. Symmetric Member in Pure Bending
M
dA
y
M
dA
z
M
dA
F
x
z
x
y
x
x
0
0
• These requirements may be applied to the sums of
the components and moments of the statically
indeterminate elementary internal forces.
• Internal forces in any cross section are equivalent to a
couple. The moment of the couple is the section
bending moment.
• From statics, a couple M consists of two equal and
opposite forces.
• The sum of the components of the forces in any
direction is zero.
• The moment is the same about any axis
perpendicular to the plane of the couple and zero
about any axis contained in the plane.
7. Bending Deformations
Beam with a plane of symmetry in pure bending:
• member remains symmetric
• bends uniformly to form a circular arc
• cross-sectional plane passes through arc
center and remains planar
• length of top decreases and length of bottom
increases
• a neutral surface must exist that is parallel to
the upper and lower surfaces and for which the
length does not change
• stresses and strains are negative
(compressive) above the neutral plane and
positive (tension) below it
8. Strain Due to Bending
Consider a beam segment of length L.
After deformation, the length of the neutral
surface remains L. At other sections,
m
x
m
m
x
c
y
c
ρ
c
y
y
L
y
y
L
L
y
L
or
linearly)
ries
(strain va
9. Stress Due to Bending
• For a linearly elastic material,
linearly)
varies
(stress
m
m
x
x
c
y
E
c
y
E
• For static equilibrium,
dA
y
c
dA
c
y
dA
F
m
m
x
x
0
0
First moment with respect to
neutral plane is zero. Therefore,
the neutral surface must pass
through the section centroid.
• For static equilibrium,
I
My
c
y
S
M
I
Mc
c
I
dA
y
c
M
dA
c
y
y
dA
y
M
x
m
x
m
m
m
m
x
ng
Substituti
2
10. Beam Section Properties
• The maximum normal stress due to
bending,
modulus
section
inertia
of
moment
section
c
I
S
I
S
M
I
Mc
m
A beam section with a larger section
modulus will have a lower maximum
stress
• Consider a rectangular beam cross
section,
Ah
bh
h
bh
c
I
S 6
1
3
6
1
3
12
1
2
Between two beams with the same
cross sectional area, the beam with the
greater depth will be more effective in
resisting bending.
• Structural steel beams are designed to
have a large section modulus.
11. Introduction
• Beams - structural members supporting
loads at various points along the member
• Objective - Analysis and design of beams
• Transverse loadings of beams are classified
as concentrated loads or distributed loads
• Applied loads result in internal forces
consisting of a shear force (from the shear
stress distribution) and a bending couple
(from the normal stress distribution)
• Normal stress is often the critical design
criteria
S
M
I
c
M
I
My
m
x
Requires determination of the location and
magnitude of largest bending moment
13. Shear and Bending Moment Diagrams
• Determination of maximum normal and
shearing stresses requires identification
of maximum internal shear force and
bending couple.
• Shear force and bending couple at a
point are determined by passing a
section through the beam and applying
an equilibrium analysis on the beam
portions on either side of the section.
• Sign conventions for shear forces V and
V’ and bending couples M and M’
14. Sample Problem 5.1
For the timber beam and loading
shown, draw the shear and bend-
moment diagrams and determine
the maximum normal stress due to
bending.
SOLUTION:
• Treating the entire beam as a
rigid body, determine the
reaction forces
• Identify the maximum shear and
bending-moment from plots of
their distributions.
• Apply the elastic flexure formulas
to determine the corresponding
maximum normal stress.
• Section the beam at points near
supports and load application
points. Apply equilibrium
analyses on resulting free-bodies
to determine internal shear
forces and bending couples
15. Sample Problem 5.1
SOLUTION:
• Treating the entire beam as a rigid body,
determine the reaction forces
kN
14
kN
40
:
0
from D
B
B
y R
R
M
F
• Section the beam and apply equilibrium
analyses on resulting free-bodies
0
0
m
0
kN
20
0
kN
20
0
kN
20
0
1
1
1
1
1
M
M
M
V
V
Fy
m
kN
50
0
m
5
.
2
kN
20
0
kN
20
0
kN
20
0
2
2
2
2
2
M
M
M
V
V
Fy
0
kN
14
m
kN
28
kN
14
m
kN
28
kN
26
m
kN
50
kN
26
6
6
5
5
4
4
3
3
M
V
M
V
M
V
M
V
16. Sample Problem 5.1
• Identify the maximum shear and
bending-moment from plots of their
distributions.
m
kN
50
kN
26
B
m
m M
M
V
• Apply the elastic flexure formulas
to determine the corresponding
maximum normal stress.
3
6
3
3
6
2
6
1
2
6
1
m
10
33
.
833
m
N
10
50
m
10
33
.
833
m
250
.
0
m
080
.
0
S
M
h
b
S
B
m
Pa
10
0
.
60 6
m
17. Design of Prismatic Beams for Bending
• Among beam section choices which have an
acceptable section modulus, the one with the smallest
weight per unit length or cross sectional area will be
the least expensive and the best choice.
• The largest normal stress is found at the surface
where the maximum bending moment occurs.
S
M
I
c
M
m
max
max
• A safe design requires that the maximum normal
stress be less than the allowable stress for the
material used. This criteria leads to the determination
of the minimum acceptable section modulus.
all
all
m
M
S
max
min
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